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Changing space and sound: parametric design and variable acoustics
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Changing space and sound: parametric design and variable acoustics
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Content
CHANGING SPACE AND SOUND:
PARAMETRIC DESIGN AND VARIABLE ACOUSTICS
by
Christopher W. Norton
A Thesis Presented to the
FACULTY OF THE USC SCHOOL OF ARCHITECTURE
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF BUILDING SCIENCE
December 2013
Copyright 2013 Christopher W. Norton
2
TABLE OF CONTENTS
LIST OF FIGURES ............................................................................................... 5
LIST OF TABLES ................................................................................................. 7
ABSTRACT .......................................................................................................... 8
1 PARAMETRIC DESIGN IN ARCHITECTURE ......................................... 9
1.1 Opportunities in Acoustics............................................................. 9
1.2 Historic Concert Halls ................................................................. 10
1.3 Challenges with Contemporary Multi-Use Spaces ...................... 12
1.4 Parametric Design in Architectural Applications .......................... 12
1.4.1 Defining Parametric ................................................................... 13
1.4.2 Parametric Uses in Structural Engineering ................................ 15
1.4.3 Parametric Design in Acoustics ................................................. 17
2 FUNDAMENTALS OF SOUND ............................................................. 19
2.1 Sound Generation ....................................................................... 19
2.1.1 Sound as Energy ............................................................. 19
2.1.2 Frequency & Wavelength ................................................ 21
2.1.3 Amplitude ......................................................................... 21
2.1.4 Wave Phenomena ........................................................... 23
2.2 Sound Perception ....................................................................... 26
2.2.1 Human Physiology ........................................................... 26
2.2.2 Perception of Frequency .................................................. 28
2.2.3 Perception of Amplitude ................................................... 29
2.2.4 Perception of Space ........................................................ 30
2.3 Music .......................................................................................... 31
2.3.1 Composition of Music ...................................................... 32
2.3.2 Subjectivity of Music ........................................................ 33
3 ROOM ACOUSTICS.............................................................................. 34
3.1 Behavior of Sound in Enclosed Spaces ...................................... 34
3.1.1 Sound and Material .......................................................... 35
3.1.2 Acoustical Metrics ............................................................ 36
3.1.3 Acoustics and Subjectivity ............................................... 41
3.2 Varying Acoustics ....................................................................... 42
3.2.1 Passive Variable Room Acoustics ................................... 42
3
3.2.2 Active Variable Room Acoustics ...................................... 45
4 CASE STUDY BUILDING AND APPROACH ........................................ 48
4.1 The Harold Lloyd Soundstage ..................................................... 48
4.1.1 Background Information ................................................... 49
4.1.2 Construction ..................................................................... 50
4.2 Ensembles .................................................................................. 53
4.2.1 Orchestra ......................................................................... 53
4.2.2 Percussion ....................................................................... 54
4.2.3 Master Classes ................................................................ 54
4.2.4 Recitals ............................................................................ 54
4.3 Parameters and Constraints ....................................................... 55
4.3.1 Reverberation Time ......................................................... 55
4.3.2 Bass Ratio ....................................................................... 56
4.3.3 Early Energy Ratios ......................................................... 56
4.4 Proposed Solution ....................................................................... 57
4.4.1 General Layout ................................................................ 58
4.4.2 Material ............................................................................ 59
4.4.3 Assumptions and Acknowledgments ............................... 62
5 GEOMETRIC OPTIMIZATION .............................................................. 63
5.1 First Exercise-Galapagos and Reverb Time ............................... 64
5.2 Building the Parametric Definition ............................................... 64
5.2.1 Equations ......................................................................... 65
5.2.2 Base Model ...................................................................... 65
5.2.3 Sequencing/Controlling .................................................... 67
5.3 Resulting Models ........................................................................ 69
5.3.1 Minimum Value ................................................................ 69
5.3.2 Maximum Value ............................................................... 70
5.3.3 Evaluating at Ratio of 3.75 ............................................... 71
5.4 Discussion and Conclusions ....................................................... 72
6 SECOND EXERCISE ............................................................................ 74
6.1 Determining Geometric Relationships ......................................... 74
6.1.1 Equation Overview ........................................................... 75
6.1.2 Early Attempts at Setting Up Data ................................... 75
6.1.3 Families of Curves and Deriving Equations ..................... 76
6.1.4 Sequencing/Controls ....................................................... 82
6.2 Results ........................................................................................ 83
6.3 Discussion and Conclusions ....................................................... 89
4
7 CREATING REFLECTION PATHS ....................................................... 91
7.1 Geometrical Relationships .......................................................... 91
7.1.1 Equations ......................................................................... 91
7.1.2 Sequencing/Controlling .................................................... 94
7.2 Results ........................................................................................ 94
7.3 Discussion and Limitations.......................................................... 96
8 FINAL MODEL ....................................................................................... 97
8.1 Parametric Definition ................................................................... 97
8.1.1 Geometric Set Up ............................................................ 98
8.1.2 Varying Ceiling/Panel Heights ....................................... 102
8.1.3 Varying Absorption Distribution ...................................... 103
8.1.4 Full Grasshopper Definition ........................................... 104
8.2 Results ...................................................................................... 107
8.2.1 Orchestra ....................................................................... 107
8.2.2 Percussion ..................................................................... 109
8.2.3 Master Classes .............................................................. 111
8.2.4 Recitals .......................................................................... 112
9 CONCLUSIONS .................................................................................. 116
10 REFERENCES .................................................................................... 119
11 APPENDIX .......................................................................................... 120
5
LIST OF FIGURES
Figure 2.1 – Constructive Wave Interference 24
Figure 2.2 – Destructive Wave Interference 24
Figure 2.3 – Reflection of Sound Waves 25
Figure 2.4 – The Human Ear 27
Figure 2.5 – Binaural Hearing 31
Figure 3.1 – Masking of Notes in Reverberant Spaces 39
Figure 4.1 – Exterior of Harold Lloyd Soundstage 49
Figure 4.2 – Interior of Soundstage 50
Figure 4.3 – Floor plan of Harold Lloyd Soundstage 51
Figure 4.4 – Interior Wall Construction 52
Figure 4.5 – Ceiling Grid Layout 58
Figure 4.6 – Range of Ceiling Panels 59
Figure 4.7 – Make up of Individual Panel 60
Figure 4.8 – Example Panel 61
Figure 4.9 – Panel Cross Section 61
Figure 5.1 – Model at Largest Ceiling Height Values 66
Figure 5.2 – Model at Smallest Ceiling Height Values 66
Figure 5.3 – Fitness Number Process in Grasshopper 67
Figure 5.4 – Schematic of Grasshopper Sequencing for Exercise 1 68
Figure 5.5 – Full Grasshopper Model for Exercise 1 69
Figure 5.6 – Minimum Ratio Shape 70
Figure 5.7 – Maximum Ratio Shape 71
Figure 5.8 – Models with V/SA Ratio of 3.75 72
Figure 6.1 – Graph of Reverberation Time v Bass Ratio 77
Figure 6.2 – Graph of Bass Ratio as Function of Reverberation Time – 78
Figure 6.3 – Graph of Ceiling Height and Coefficient Values 80
Figure 6.4 – Schematic of Grasshopper Sequencing for Exercise 2 82
Figure 6.5 – Input Values for Orchestra 83
Figure 6.6 – Output Geometry for Orchestra 84
Figure 6.7 – Input Values for Percussion Ensemble 85
Figure 6.8 – Output Geometry for Percussion Ensemble 85
Figure 6.9 – Input Values for Master Classes 86
Figure 6.10 – Output Geometry for Master Classes 87
Figure 6.11 – Input Values for Recitals 88
Figure 6.12 – Output Geometry for Recitals 88
Figure 7.1 – Path of Reflected Ray 92
Figure 7.2 – Construction of Reflection 93
Figure 7.3 – Reflection Points within Room 95
Figure 7.4 – Reflection Points at High Hip in Ceiling 95
Figure 7.5 – Verification of Angles 96
Figure 8.1 – Source/Receiver Locations – Objective Support 99
6
Figure 8.2 – Source/Receiver Locations – Clarity 100
Figure 8.3 – Absorption at Reflection Point 101
Figure 8.4 – Grasshopper Definition Assigning Surface Area Proportions 104
Figure 8.5 – Schematic Diagram of Grasshopper Definition 105
Figure 8.6 – Screenshot of Grasshopper Definition 105
Figure 8.7 – Input Values for Orchestra Configuration 107
Figure 8.8 – Final Configuration of Room Model for Orchestra Configuration 108
Figure 8.9 – Input Values for Percussion Configuration 109
Figure 8.10 – Final Configuration of Room Model for Percussion Configuration
110
Figure 8.11 – Input Values for Master Class Configuration 111
Figure 8.12 – Final Configuration of Room Model for Master Class Configuration
112
Figure 8.13 – Input Values for Recital Configuration 113
Figure 8.14 – Final Configuration of Room Model for Recital Configuration 114
7
LIST OF TABLES
Table 2.1 – Speed of Sound in Various Mediums .............................................. 20
Table 2.2 – Relative Sound Pressure Levels (dB) of Various Sounds ................ 23
Table 2.3 – Dynamics and Associated Sound Pressure Levels ......................... 33
Table 3.1 – Absorption Coefficients of Common Building Materials ................... 36
Table 3.2 – Optimal Reverberation Times .......................................................... 38
Table 4.1 – Acoustic Parameters and Values ..................................................... 57
Table 4.2 – Goal Metrics for Each Condition ...................................................... 62
Table 5.1 – Volume and Surface Areas of Multiple Trials at 3.75 ....................... 72
Table 6.1 – Sample Calculations of RT and BR ................................................. 76
Table 6.2 – Absorption Coefficients of Room Materials ...................................... 78
Table 6.3 – Results of Grasshopper Defined Geometries .................................. 89
Table 8.1 – Final Results .................................................................................. 115
8
ABSTRACT
This thesis examines the potential for parametric design software to create
performance based design using acoustic metrics as the design criteria. A
former soundstage at the University of Southern California used by the Thornton
School of Music is used as a case study for a multiuse space for orchestral,
percussion, master class and recital use. The criteria used for each
programmatic use include reverberation time, bass ratio, and the early energy
ratios of the clarity index and objective support. Using a panelized ceiling as a
design element to vary the parameters of volume, panel orientation and type of
absorptive material, the relationships between these parameters and the design
criteria are explored. These relationships and subsequently derived equations
are applied to Grasshopper parametric modeling software for Rhino 3D (a
NURBS modeling software). Using the target reverberation time and bass ratio
for each programmatic use as input for the parametric model, the genomic
optimization function of Grasshopper – Galapagos – is run to identify the
optimum ceiling geometry and material distribution.
Hypothesis: Parametric modeling software can aid in designing multiuse
performance spaces to accommodate variable acoustic situations.
9
1 PARAMETRIC DESIGN IN ARCHITECTURE
Digital design techniques in architecture have allowed designers to create and
define endless possibilities of shapes that may not have been possible to be
constructed or communicated without the use of computers. The standard image
of contemporary architecture is now one of complexity. The computer allows for a
number of calculations and iterative processes to be done in an instant what
used to take hours or even days by hand. Many building designers use the
power of computers and parametric design technology as a tool to develop their
aesthetic vision, while the incorporation of structural and environmental systems
that may lead to more efficient design and construction is often overlooked; this
is a missed opportunity. There are examples of algorithmic and parametric tools
used in conjunction with quantitative analysis to produce buildings of form as well
as function. The object of this thesis is to utilize parametric design technologies
to develop an effective architectural component that will enhance the acoustic
performance of a multi-use facility.
To achieve this goal, this thesis will look at how to approach the parametric
design process, research the science and art of sound to develop acoustic
criteria for a multiuse facility, and develop a computer model of a case study
building with a ceiling capable of creating a variable acoustic environment.
1.1 Opportunities in Acoustics
The field of architectural acoustics is broad and includes subjects such as the
control of sound isolation between facilities, vibration control of mechanical
10
systems, and room acoustic design in performance halls. The acoustic
performance of concert halls will be the focus of this thesis as they heavily
depend upon the properties of materials and the geometry of the room. Several
different programmatic uses will be proposed for the same room, encompassing
a wide range of values for different acoustic criteria. Solving for these criteria will
involve changing proportions of the materials and the interior shape of the space.
This lends itself to parametric design strategies. While the use of computer
analysis in acoustics has improved the ability to analyze and predict the
performance of concert halls, what many consider to be some of the world’s best
concert halls were built long before the use of computers in acoustics. The
history of concert halls and the thought processes in designing today’s venues
offer some insight into both the success of these older concert halls and the
failings of some of the newer concert halls. The use of parametric software can
aid in providing a solution to the problems associated with multi-use facilities.
1.2 Historic Concert Halls
The concept of an indoor space that had the explicit purpose of exhibiting
musical performances began to take hold in the Classical era of music in Europe
around the mid-18
th
century. Before this, a person’s musical experience existed
mostly in the home or more notably in the Church. The architecture of cathedrals
had a significant impact on the music performed within them. The traditional
chants of services have simple rhythms as the entire congregation acted as
performers. Complex rhythms and the intelligibility of the words would become
11
cluttered in the highly reverberant spaces of cathedrals and keep the
congregation from singing together.
1
In the early days of the Classical period,
music was still performed in ballrooms and salons, which typically had shorter
reverberation times than today’s concert halls, and were performed by smaller
orchestras due to the small size of the room. As public concert halls became
more prevalent in Europe, composers began to compose music for specific
rooms as Haydn did at the Hanover Square Rooms and the Redoutensal at
Hofburg, altering the composition and size of the orchestra to compensate for
acoustic variations including reverberation time and loudness.
2
The Grosser
Musikvereinssaal in Vienna and the Concertgebouw in Amsterdam are regarded
as two of the best concert halls in the world. However, they are each best suited
for a different type of music; the Musikvereinssaal is commonly regarded best for
Classical music while the Concertgebouw is better known for heavy, bombastic
Romantic music, such as Wagner.
3
Resident composers often wrote their music
specifically for the environment it was to be performed in, contributing
significantly to the success of early public concert halls.
1
Marshall Long, Architectural Acoustics (New York, Academic Press, 2006), 10.
2
Long, 21, 23
3
Long, 29
12
1.3 Challenges with Contemporary Multi-Use Spaces
Modern concert hall designers have a significant problem in that they are
expected to provide a space that will provide the greatest value for their clients.
Unfortunately this does not mean that the acoustics will be great and render an
aurally satisfactory experience, but rather that the venue will be able to be
booked year round and host a variety of events including symphonic concerts,
cinema, and stage productions. Most cities have developed performing arts
centers that showcase multiple groups with varying needs, not the least of which
is the concert halls acoustic characteristics. The early multi-use spaces became,
as J. Christopher Jaffe describes, “no-use spaces” due to an acoustic approach
of one-size fits all.
4
In the worst cases, the symphony sounded dry, the theatrical
productions were unintelligible, and no one had a fully satisfying experience.
While a one-size fits all approach to designing performance venues is now
recognized as a major design problem, the current solutions in providing variable
acoustic setting involve either a finite number of possible alternatives or require a
large amount of labor or investment in equipment. Some of these variable
acoustic systems will be presented in Chapter 3.
1.4 Parametric Design in Architectural Applications
Before examining how parametric design can be used for acoustic applications, it
is important to understand what parametric design is and is not, and analyze
4
J. Christopher Jaffe, The Acoustics of Performance Halls: Spaces for Music from Carnegie Hall
to the Hollywood Bowl. (London, WW Norton & Company, 2010), 100
13
other projects in which parametric designs have been used in order to optimize
given criteria. This section will look at the differences between parametric and
algorithmic design methods and look at a few case studies of how parametric
design has been successfully implemented in other projects.
1.4.1 Defining Parametric
Parametric is a common buzzword in architecture; it allows architects to realize
complex geometries that would be otherwise impossible to generate using
standard drafting or modeling. However, not every building with complex
geometries is designed parametrically and similarly not every building designed
parametrically has complex geometries. Parameters in architecture are typically
associated with area and volumetric geometries, which are limiting to achieve
other parameters to achieve criteria set by the designer. Also involved in this
process is the use of algorithms, which are simply a set of processes typically
used to solve a problem. However, most times in architecture algorithms are
utilized as pure form generation instead of in conjunction with parameters such
as height, volume, and weight, or performance based parameters such as cost,
energy efficiency, and constructability.
For instance, Toyo Ito’s design of the 2002 Serpentine Gallery Pavilion used a
geometric algorithm of inscribing a series of squares within the previous square
set at specific ratios between generations. The resulting web of lines was then
used as the structural lattice work for the structure of the pavilion, using the
14
earlier generation lines as the primary load carrying members.
5
While the result
of the algorithm may have an inherent structural hierarchy, the structural
efficiency is not the primary focus of the design.
Parametric design should involve a deeper level of complexity than is apparent at
first glance. The result of the parametric definition should be based upon the
underlying algorithms defining the geometry based on constraints determined by
the project’s program, structure or environmental concerns. The essential pieces
in creating a good parametric design are defined below:
1. Criteria for Success – Having a clear criteria allows the designer to translate
the intent of the design into a quantifiable and therefore measurable result to
judge success or not. The success of a project can be driven by anything
including efficient structure and limited use of materials, ease of construction,
efficient uses of space, or environmental performance.
2. Parameters – The parameters of the project are the conditions that the
designer wishes to achieve and are defined to establish a criteria. For instance,
if a project’s criteria for success is to have a south facing glass façade that will
not have a negative impact on the internal loads of the building, the parameter
may be the amount of direct solar radiation the glazed area receives. This is a
quantifiable variable that can be constrained and is dependent on a number of
55
Sakamoto, Tomoko et. al. From Control to Design: Parametric/Algorithmic Architecture.
(Barcelona:,Actar-D, 2008), 36-43
15
design related issues as well as natural phenomenon including the time of the
day and year.
3. Geometrical and Material Associations – The use of algorithms in
parametric design is found in associating geometry and material properties with
the specified parameters. How geometry is manipulated and evaluated can have
a large impact on the final outcome of the structure. For instance, if the design
intent is to minimize the amount of direct solar radiation on the example south
facing facade, there may be two geometrical associations to control: the size of
the window or the size of the overhang above the window. In both cases, the
amount of direct solar radiation will be minimized. However, in one case the
building will end up with very small windows at the top of the wall, and the other
will end up with a large overhang. There are also material properties associated
with this example that should be considered, including the solar heat gain
coefficient of the glass. There are multiple ways to approach this problem each
with their own positive and negative impacts on different areas of design.
Because of the interrelatedness of material and geometry and their impact on
design, it is crucial to select the appropriate relationships one wishes to
manipulate with the defined parameters.
1.4.2 Parametric Uses in Structural Engineering
Structural engineering is an established field, providing architects and engineers
with hundreds of years of experience, testing, and knowledge that can be
converted into algorithms and design criteria. While the computer age has
16
allowed designers to run calculations at a much faster speed than by hand, the
concept of applying simple parameters to yield complex and functional
geometries is not new. Frei Otto was a pioneer in the field of complex structural
systems utilizing minimal resources and his method involved a parametric
approach even before using computers. His approach to designing buildings
was based on a purely performative aspect, giving little regard to the aesthetic
qualities of the structure. Working from a physical model of hanging chains and
using parameters such as height, span and member size, Otto allowed the
gravity acting on the model to do complex computations in place of today’s
computers and provide him with a structure that performed in pure tension. The
result of turning such a structure upside down is a structure that operates in pure
compression.
6
Computationally, this is a simple example, but the intent to use a
process like Otto’s in order to solve a specific problem (using a minimum amount
of materials) is clear. Today, computer algorithms representing the effects of
gravity and other loads on structure define the geometrical associations that Frei
used hanging chains to determine.
With a shape influenced by banyan trees and with the goal of creating a
structural form with a uniform stress, Matsuro Sasaki “evolved” support structures
for a train station competition in Florence using an iterative parametric analysis of
the structure.
7
In this case, there may have been many unique solutions that
6
Conrad Roland, Frei Otto: Tension Structures. (New York: Praeger Publications. 1970)
7
Sakamoto, Tomoko et. al., 100-105
17
resulted in equally performative structures with different aesthetic qualities.
While defining the structural system of a building may be the most influential use
of parametric software in regards to use of resources and determining the final
form of the building, environmental parameters also significantly influence how a
building will perform. One reason for the lack of precedents specifically in the
field of acoustics is the complex and subjective nature of sound.
1.4.3 Parametric Design in Acoustics
Schroeder diffusers rely upon a series of wells with their width determined by the
frequency the panel is to diffuse. The complex shapes of the diffusers, whose
profiles can be expressed in one or two dimensions, are based on a pattern of
geometrical sequences discovered by Manfred Schroeder. In addition to creating
interesting forms, these panels are able to provide predictable and effective
performance in acoustic diffusion and are based upon the defined parameter of
frequency.
Optimization of the placement of reflecting, diffusing and absorbing panels at a
pavilion in Copenhagen was achieved by use of an algorithm and acoustic
analysis software. The goal of the project was to decrease the sound levels in
the lounge areas and increase the sound levels and even the sound distribution
in the audience areas, by redistributing an equal number of diffusing, reflecting
and absorbing panels. Acoustic analysis was done in the program ODEON and
examined across many points within the audience and lounge areas. Placement
18
of the acoustic reflectors and absorbing panels were analyzed first with the
diffusing panels filling in the remaining spaces.
8
Each of the preceding examples had clear criteria for success and defined
parameters in order to create an object constrained by the criteria of project. The
next few chapters will examine the science of sound, acoustics, and music in
order to understand what the criteria for a successful acoustic environment
should be and what parameters will be the most effective in changing the
acoustics environment of a space.
8
Brady Peters. “Parametric Acoustic Surfaces” ACADIA (2009), 179
19
2 FUNDAMENTALS OF SOUND
Understanding the generation of sound, perception by the listener or audience,
and the relation to music will form the basis for understanding and evaluating the
room acoustic criteria for this thesis. This chapter examines the physical and
quantitative components and the perceptive and qualitative factors of sound and
music.
2.1 Sound Generation
What sound is and how it is generated is fundamental in understanding how it
behaves. This section describes the physical phenomena associated with sound
generation and transmission and the characteristics and physics of wave
behavior.
2.1.1 Sound as Energy
The generation of sound begins with the vibration of an object and the
displacement of a medium. The vibrations of an object, such as a string held in
tension, will displace the medium in contact, usually air, and cause a cycle of
compression and rarefaction through the medium above and below atmospheric
pressure. This sinusoidal cycle of compression and rarefaction creates a wave
with a frequency equal to the number of cycles per second (measured in Hertz,
20
Hz) and an amplitude equal to the maximum displacement of the medium.
9
The
speed of the wave is dependent on the medium. Subsequently, the speed of
sound is variable depending on the medium of travel. Figure 2.1 displays the
speed of sound through common materials.
Substance Temperature (°C) Speed (m/sec) Speed (ft/sec)
Air 0 331.5 1087
Air 20 344 1130
Water 15 1437 4714
Steel - 5000 16,400
Water Vapor 35 402 1320
Table 2.1 – Speed of Sound in Various Mediums
10
As shown in Figure 2.1, sound will travel faster through mediums in which the
molecules are closer together. For air, the speed is very dependent upon
temperature and humidity. The wavelength of the wave, or the distance from
peak to peak displacement of one cycle, is dependent upon both the frequency
and speed of sound and related by the equation:
S = f * λ
Where S is the speed of sound, f is the frequency in Hertz and λ is the
wavelength.
11
9
John Backus, The Acoustical Foundations of Music (London, WW Norton & Company, 1976),
40
10
Backus, 44
11
Backus, 41
21
2.1.2 Frequency & Wavelength
The frequency of a sound wave is typically what is perceived as tone or pitch.
Lower frequencies create “lower” tones and higher frequencies create “higher”
tones. The typical human has a hearing range of approximately 20 Hz to 20,000
Hz, and this range generally decreases as people age.
12
Ultrasonic frequencies
are detectable by other animals such as dogs or bats, but this is not important for
concert hall design.
For analysis and specification purposes, frequencies are broken into octave
bands. Octaves occur at exponential intervals, increasing at a ratio of 2:1, similar
to octaves on a keyboard. The typical octave bands occur at 64, 125, 250, 500,
1000, 2000, 4000, 8000, and 16000 Hz. Harmonic frequencies are important for
the perception of sound and are physical multiples of a base frequency. For
example, a base frequency of 220 Hz will have harmonics at 440 Hz, 660 Hz,
880 Hz, 1100 Hz, etcL
13
The amplitude of a given harmonic is variable and is
dependent on the object vibrating. The relative level of these harmonics is
important in determining the quality of sound in human perception.
2.1.3 Amplitude
The amplitude of a given sound wave is a measure of the physical displacement
of the medium the wave is traveling through. In the typical medium of air, the air
12
Daniel J. Levitin, This Is Your Brain on Music: The Science of a Human Obsession,(New York,
Penguin, 2006) 25
13
Levitin, 42
22
molecules displaced above and below atmospheric pressure have a certain
mass. Due to the internal friction forces of air (or whatever other medium sound
travels through) there is a force resisting the movement caused by a sound
wave. This force multiplied by the displacement, or amplitude, results in the
sound power generated. When sound propagates spherically from the source in
a sphere there is a force of sound pressure impinging on the sphere at any given
time at any given distance. The power acting on this surface area is known as
the sound pressure and is used in calculating the relative loudness of a sound.
14
The standard measure of loudness is the decibel (dB) and is calculated as
dB =20log
P
spl
P
ref
Where P
spl
is the measure sound pressure level and P
ref
is the reference sound
pressure level. The reference sound pressure that is used as a basis for
measurement is about 20 uPa which equates to threshold of hearing. Because
of this logarithmic reference, doubling of the sound level equals a gain of about 6
dB. For example:
20log(2) =6.02
Table 2.2 displays the sound level in dB for everyday sounds as a reference.
0 dB Mosquito flying in a quiet room, ten feet away from your ears
20 dB A recording studio or a very quiet executive office
35 dB A typical quiet office with the door closed and computers off
14
Backus, 53
23
50 dB Typical conversation in a room
75 dB Typical, comfortable music listening level in headphones
100-105 dB Classical music or opera concert during loud passages; some
portable music players go to 105 dB
110 dB A jackhammer three feet away
120 dB A jet engine heard on the runway from three hundred feet away; a
typical rock concert
126-130 dB Threshold of pain and damage; a rock concert by The Who
180 dB Space shuttle launch
Table 2.2 – Relative Sound Pressure Levels (dB) of Various Sounds
15
2.1.4 Wave Phenomena
Just like ocean waves, sound waves are capable of interacting with one another.
The results of these interactions are important in the discussion of room
acoustics, which will be discussed in Chapter 3. Interference, reflection, and
absorption are three wave behaviors that are the basis for most acoustic studies
and analogous to the study of light waves.
Waves traveling in the same space have the ability to create interference. This is
a bit misleading as the waves do not interfere with each other but rather affect
the medium they pass through. Two waves with identical frequencies in the
same location will result in some of the amplitude of the waves, either
constructively or destructively interfering (Figures 2.1 and 2.2).
16
Noise canceling
technologies are based upon destructive interference.
15
Levitin, 71
16
Backus, 48
24
Figure 2.1 – Constructive Wave Interference
Figure 2.2 – Destructive Wave Interference
25
Sound waves approaching an obstacle will reflect a portion of their energy, while
a portion will be absorbed by the material and another portion is transmitted
through the material.
17
Although transmission is another important part of the
overall study of sounds and acoustics, this thesis will focus on the reflection and
absorptive properties of the room boundaries. The material finishes and
geometries in an enclosed space are the key components in controlling sound
wave behavior and the level of sound energy. Like light, the angle of reflectance
is dependent upon the angle of incidence. Figure 2.3 demonstrates a typical
reflection assuming a sound wave as a one-dimensional line.
Figure 2.3 – Reflection of Sound Waves
17
Backus, 46
26
Sound absorption at certain frequencies is dependent on the type and
construction of the material the sound wave strikes. Different materials will
absorb different quantities of sound energy across the range of frequencies. This
will be discussed more in depth in the next chapter.
2.2 Sound Perception
While the physics of sound generation is a simple theory and relatively well
understood, what creates the challenges in acoustics is human perception of
sound. This section will explain the physiological and psychological phenomena
linked to the human understanding of sound.
2.2.1 Human Physiology
How humans understand sound is based on the interpretation of the vibrations
generated by anything and everything in the physical world. Our interpretation of
these vibrations begins with the ear. The ear consists of three distinct areas. The
outer ear, the pinna, acts as an auditory funnel to direct sound through the
auditory canal to the middle ear. At the interface between the outer and middle
ear is the ear drum or tympanic membrane. As the name alludes to, the ear
drum, the tympanic membrane, consists of a membrane stretched across the
auditory canal that vibrates when subjected to sound pressure. The rest of the
middle ear consists of the hammer, anvil, and stirrup, which are interconnected to
transfer the vibrational energy from the ear drum to the inner ear through the oval
window to the liquid filled cochlea. The cochlea is a rolled organ containing a fluid
27
and lined with hair cells (the basilar membrane) and as the vibrations at the oval
window create a wave of the fluid in the cochlea, these hair cells transfer the
mechanical energy of the wave into neural information (electrical signals)
processed by the brain as sound.
18
Figure 2.4 – The Human Ear
18
Yoichi Ando, Concert Hall Acoustics (New York, Springer, 1985), 27
28
2.2.2 Perception of Frequency
The frequency that is determined by the brain is a result of where on the basilar
membrane the wave is at its peak amplitude. The human ear is capable of
hearing from 20 Hz to 20 kHz, although this range decreases with age. The
typical human adult cannot hear beyond 12 kHz.
19
This loss of hearing range is
normal, but hearing can also be damaged by over exposure to loud sounds. The
ability to recognize specific frequencies varies widely between people. While
people with healthy hearing are able to determine frequencies by absolute or
“perfect” pitch, most people identify frequencies relative to one another, always
requiring a base tone on which to reference the second tone.
Our perception of frequencies is extremely important to our recognition of
sounds, especially the combination of different frequencies. No sound, except
for a sine wave, is heard as a pure tone, generated at a single frequency.
Instead, sounds are produced with a variety of harmonic frequencies as
previously discussed. These harmonic or overtones series in a sound source are
what give different qualities or timbres to a specific source. This is important in
speech as everyone’s vocal cords generate a unique timbre, which allows others
to recognize individuals’ voices. These harmonics are a product of the natural
resonance frequencies of the material the sound is generated from. The key of A
above middle C played on a Stradivarius will sound different than the key of A on
a piano because the construction and material composition of the instruments
19
Levitin, 25
29
influence the perceived timbre. The timbre or tone quality of specific instruments
is important in music as is discussed further in Section 2.3.
2.2.3 Perception of Amplitude
Our perception of loudness aids us in determining what is important and what is
noise. The level of ambient or background noise for a given space is called the
noise floor. Other sounds, if they are to be heard, must be louder than the noise
floor or will blend in. This is also referred to as masking.
20
Some acoustic
situations, such as recording studios, require bringing the noise floor as low as
possible. Emergency sirens are always louder than the ambient street or
freeway traffic in order to be heard. There are of course exceptions to this,
especially if there is a large difference between the frequencies of the ambient
and source sounds, but generally we perceive the louder sounds as more
important. However, how we perceive loudness is not a constant across all
frequencies.
Humans require significantly more sound energy at the low end of the frequency
spectrum in order to create the same perceived level of loudness at 500-1000
Hz. Less sound energy is required to create the same perceived level of
loudness between 2-5kHz. Physically this makes sense as the average auditory
canal has a resonance frequency in this region.
21
20
Backus, 101
21
Backus, 96
30
2.2.4 Perception of Space
Our sense of hearing affects our perception of space in two different ways:
identifying the scale of our surroundings and locating the source of sounds.
Through the reverberance of an enclosed space (or lack thereof in an open
area), we can identify the scale of the space we are in. Although the time
difference between reflections from surfaces 40 feet away is on the scale of
microseconds, the brain is able to process these differences and give us a sense
of where we are.
22
Other processes aid us in identifying where sounds come
from. Just as having two eyes enables us to perceive depth of space, having two
ears allows us to perceive the location of sound source. Though the time
difference between a sound reaching one ear versus the other is short, it is large
enough for our brain to register and triangulate the placement of the source in
space.
23
22
Levitin, 108
23
Backus, 90
31
Figure 2.5 – Binaural Hearing
Stereophonic hearing allows us to place which direction an emergency vehicle is
coming from or where a dangerous animal is lurking in the jungle. While we are
rarely faced with dodging predators in the jungle, binaural hearing plays a large
part in music, allowing us to associate the instruments we see with the sounds
we hear. Failing acoustic situations may have issues with reflections and
skewed aural images, creating a disconnection between what we see and what
we hear.
2.3 Music
This section will identify the composition of music, define musical terms for the
non-musician, and identify the difficulties of analyzing music in scientific terms.
32
2.3.1 Composition of Music
Music at its simplest is the organization of sound in the form of pitch, rhythm, and
dynamics. Frequencies determine the tone or pitch of the melody and harmonies
and define the key in which the piece is set. The key signature is a general rule
for structuring which notes are important and “allowed” within the framework of
the piece. Traditional Western music is comprised of 12 notes.
Rhythm is not related to a physical quality that has been discussed, but is rather
a derivative of the amount of time notes are held, relative to one another. Like
pitch, rhythm is also structured within the piece of music through the time
signature. Time signature dictates what length of note is considered the beat
and how many beats are in a measure. Another time related quality of music is
tempo, or how fast a piece is played. Tempo is measured in beats per minute.
Tempo is an important consideration in the acoustics of performance spaces
when considering the clarity of individual notes. Music with faster tempos
requires a higher level of clarity in order to distinguish and interpret notes and
rhythms.
Dynamics, or the relative loudness of both different instruments and different
passages within a piece, help to shape the flow and emotional qualities of the
music. Dynamic markings are noted as variations of piano (quiet) and forte
(loud). Table 2.3 displays the common dynamic markings and the typical sound
pressure level of an orchestra.
33
Dynamic Marking Sound Pressure Level (dB)
Threshold of feeling 120
fff 100
f 80
p 60
ppp 40
Threshold of hearing 0
Table 2.3 – Dynamics and Associated Sound Pressure Levels
24
2.3.2 Subjectivity of Music
Music varies worldwide. For example, traditional Asian music uses a different set
of frequencies that take advantage of the quarter-tones between the traditional
Western notes. As with any other art, the music one likes is personal and
subjective and based upon previous history and experiences with music.
Subjective preference makes music and related fields of study an inexact
science.
The science of sound is very much linked to physical quantities such as
distances, areas, and orientations of reflecting planes. These are important
when examining the geometrical and material relationships of a space. However,
simply knowing how a sound wave will move in a room and how much the
amount of sound drops over time is not enough information to define a
successfully designed space. The next chapter will examine the principals of
acoustic design and some of the general criteria acousticians review when
designing a performance facility.
24
Backus, 92
34
3 ROOM ACOUSTICS
This chapter will focus on how sound behaves in enclosed spaces, examining the
reaction of sound to materials and geometry, what metrics have been established
to quantify acoustic parameters, and the issues of subjective preference.
Acoustics as a science is young, having only been formally developed within the
last 100 years. The equipment necessary to measure and record sound data
has only been available since the invention of electronic amplification and
microphones in the early part of the 20
th
century.
25
While there has been
significant progress in defining how sound behaves, achieving specific metrics for
judging architectural acoustics have no guarantee of producing a space with a
satisfying listening experience. This chapter will also discuss the current
technologies and methods utilized to vary acoustics in both passive and active
form.
3.1 Behavior of Sound in Enclosed Spaces
Sound in a free field with no obstructions is heard directly from the source.
Outdoor venues are common but are rarely considered good performance
spaces without the addition of some architectural elements or technological
system. The acoustics are usually enhanced with an orchestral shell or the use of
electronic reinforcement to project the sound to the audience. The difficult part of
analyzing sound is its interactions with the boundary of an enclosed space. This
25
Leo Beranek, Music, Acoustics, and Architecture.(New York, John Wiley & Sons, Inc. 1962),4-5
35
section discusses the effects that material and geometry have on the behavior of
sound in relation to music, the standard metrics of acoustic analysis, and the
difficulties in establishing an “ideal” set of criteria for all acoustic situations.
3.1.1 Sound and Material
As with structures, material plays an important part in sound, either weakening or
reinforcing the sound within a space. As mentioned in Chapter 2, one of the
attributes of sound wave is that it will reflect off of a material. However, this is
only one of the interactions a sound wave will have when encountering a material
at the boundary of a space. In addition to reflecting sound energy, a surface may
absorb or transmit sound energy. As sound is simply the vibration of molecules,
the absorption, transmission and reflection characteristics of a material are
dependent on the density and composition of the material. The reflective and
absorptive qualities of materials, not transmission, are most important for
acoustics in a space. Sound absorption occurs when a material converts sound
energy into another form of energy, usually heat. The absorption coefficient
(alpha) of a material is a measure of the percentage of sound energy absorbed
by the material and will vary with respect to frequency. For example, heavy
carpet on concrete will absorb 14% of the sound energy in a 500 Hz sound wave;
therefore the alpha is 0.14. A table of sample materials and their absorption
coefficients is displayed in Table 3.1.
36
Frequency (Hz)
Material 125 250 500 1k 2k 4k
Marble or glazed tile .01 .01 .01 .01 .02 .02
Concrete, unpainted .01 .01 .01 .02 .02 .03
Asphalt tile on concrete .02 .03 .03 .03 .03 .02
Heavy carpets on
concrete
.02 .06 .14 .37 .60 .65
Heavy carpets on felt .08 .27 .39 .34 .48 .63
Plate glass .18 .06 .04 .03 .02 .02
Plaster on lath on studs .30 .15 .10 .05 .04 .05
Table 3.1 – Absorption Coefficients of Common Building Materials
26
3.1.2 Acoustical Metrics
A series of metrics have been devised to describe acoustical performance
characteristics. One of the most important measures already mentioned is how
reverberant a space is or how long a sound remains in a space after the source
of sound has ceased. This is measured by an acoustic parameter called
Reverberation time (RT
60
) or the length of time (in seconds) it takes for a sound
source to drop 60 dB after it has ceased. It is a function of the volume and the
absorption properties of the surface materials in the room. Reverberation time
can be expressed by
RT
60
=
(.049)V
s
n
a
n
1
n
∑
where V is equal to the total volume of the room (in cubic feet), s is the exposed
surface area of a material (in square feet), and a is the absorption coefficient of
26
Backus, 172
37
that material. Each surface multiplied by its absorption coefficient are summed
up.
Reverberation time will influence the clarity of notes and different styles of music
will require different reverberation times. Spoken word will require a much
shorter reverberation time in order to provide the speech intelligibility needed to
understand each word. The optimal reverberation time for any given musical
setting may vary based upon the volume of the room. Figure 2.2 displays
common performance types and corresponding reverberation times (at 500 Hz)
for spaces of a volume similar to the case study developed in Chapter 3,
approximately 70,000 ft
3
(2,500 m
3
) .
Sound Source Optimal Reverberation Time (s)
Spoken Word 0.7
Jazz/Pop 0.8
Orchestra
Chamber 1.4 - 1.7
Opera 1.3 - 1.8
Romantic/Classical
1.8 - 2.2
38
Table 3.2 – Optimal Reverberation Times
27
A derivative of reverberation time, which aids in describing the presence of lower
frequencies, is the bass ratio. Bass ratio is defined as
1000 500
500 125
RT RT
RT RT
BR
+
+
=
where RT
n
is the reverberation time (in seconds) at n Hz. While no studies are
available to describe the optimal bass ratio for given musical settings, a general
rule of thumb is to provide a bass ratio of at least 1.2 seconds in order to provide
a desirable “warmth” of sound.
Early Decay Time (EDT) is also similar to the reverberation time but is defined as
the time for sound to decay 10 dB and is associated with the perceived
reverberation time of space. Beranek argues that EDT is a better gauge of the
subjective preference of concert halls, especially those in which detailed rhythms
need to be heard as EDT is more on scale with the time between successive
notes.
28
Figure 3.1 demonstrates the principle of masking of successive notes as
a product of reverberation.
27
S Ellison, R Schwenke. “The Case for Widely Variable Acoustics” Proceedings of the International
Symposium on Room Acoustics, (2010), 2
28
Beranek. 24
39
Figure 3.1 – Masking of Notes in Reverberant Spaces
The Clarity or Definition Indices are acoustic metrics that also play an important
role in determining how easy it is to distinguish separate notes. This is a
measured as a ratio (in decibels) of early sound energy (earlier than 80
milliseconds) to the reverberant sound energy (after 80 milliseconds).
29
In most
modern situations, it is preferable to have a ratio favoring more early sound
energy as this is better to distinguish individual instruments and rhythms. A hall
29
Beranek. 23
40
for medieval choir music, originally performed in Gothic cathedrals with large
spaces and a long reverberation time, would be better designed with less clarity.
The Intimacy Index, or the feeling of being close to the sound source, is a metric
based on the specific geometry of the room. This is measured as the interval of
time between the direct sound and the first reflection. The standard time interval
of the initial time delay for creating an intimate space is considered to be 20 ms.
30
Determining intimacy requires more information about the geometry of the room
and will change based upon where the sound source and receiver are placed.
The first reflections in most concert halls come from the sidewalls as opposed to
the ceiling, which is further away.
There are other considerations, particularly regarding the musician as a listener,
in acoustic design that lack clear metrics but are crucial to understand in order to
create an optimal acoustic setting. The Objective Support, or ability for
performers to hear one another is extremely important in enabling performers to
play as a cohesive group, or ensemble. Good ensemble can be achieved by
ensuring that reflectors on stage reflect sound from one part of the orchestra to
the other side of the orchestra.
31
Objective Support is defined as the energy
from reflections arriving within 100 ms of the direct source sound from a receiver
30
Backus, 179.
31
Beranek. 32
41
placed 3 meters from the source and measured in decibels. A desirable
measure for this metric is between -13 and -11 dB.
32
A musician should also be able to sense the acoustics of the room and how it
responds to what he or she is playing. The immediacy of response, or attack, of
a venue is similar to the intimacy felt by the audience in that it is dependent on
the first reflections of the source sound arriving back at the musician’s ears.
33
3.1.3 Acoustics and Subjectivity
Part of the problem with defining a “perfect” set of acoustical parameters is that
different sounds are perceived as better in different contexts. A drum set solo in
a large Gothic cathedral would not offer the same experience as listening to the
same solo in a recording studio. Yoichi Ando argues that the preferred qualities
of the space are dependent on the function and qualities of the music (or other
sounds) within the space.
34
To fully enjoy the rhythmic intricacies of a rock drum
solo, the listener must be able to distinguish each attack of the drum, a quality
not afforded in the highly reverberant spaces of a stone Gothic cathedral. This is
not to say that all music will sound terrible in this setting. In fact, Gregorian
chants have the best effect when performed in a highly reverberant space.
35
32
Michael Barron, Auditorium Acoustics and Architectural Design, (New York, Routledge, 1993),
61
33
Beranek. 33
34
Ellison, Schwenke, 2
35
Beranek, 44
42
Ando’s argument for matching performance spaces with the qualities of music
performed there is true, but historically composers have done the inverse and
designed their music for the spaces in which they were to be performed. In
either case, the goal is the same – the properties of the space enhance the
performance.
3.2 Varying Acoustics
Not all music calls for the same acoustic environment. As demonstrated by the
reverberation time, different styles of music can require widely different
situations. Historically, composers in residence would compose for the setting in
which their works were to be performed. Now, with a wealth of historic music
that was composed for specific settings, modern concert venues attempt to
provide multiple acoustic settings in order to provide the optimal concert going
experience for the audience. This section focuses on the ways in which
acousticians use variable and active means of manipulating the actual, or
perceived, acoustics of a concert hall.
3.2.1 Passive Variable Room Acoustics
Passive variable systems manipulate the acoustical qualities of the room by
changing the material and geometrical properties of the room. The acoustical
parameter with the greatest impact on the quality of listening is the reverberation
time. Changing the reverberation time of a space requires changing the volume
of the room and the sound absorbing and sound reflecting surfaces.
43
RT
60
=
(.049)V
s
n
a
n
1
n
∑
Other factors in changing the acoustic properties of the space, which influence
other important metrics such as clarity and definition, include relocating and
moving ceiling and wall panels.
Changing the volume of a room can be accomplished in multiple ways. The first
involves moving an entire partition of the room, such as a wall, or in most cases
raising or lowering the ceiling. Many modern concert halls make use of the fly
loft above the stage (used to “fly” set pieces in and out for theatrical productions)
in order to add volume to the hall, raising and lowering a drop ceiling as
necessary. Another practical system involves shutters that open up to include a
larger volume of space.
36
The fly loft at the Bass Performance Hall in Fort Worth, Texas allows the hall to
have a reverberation time of 1.6 seconds for opera and a reverberation time 1.9
seconds for concerts by deploying a concert hall shaper, sealing off the fly loft.
37
While there is no fly loft at the KKL Concert Hall in Lucerne, Switzerland, there is
36
Barron, 340
37
Beranek, 550
44
a large amount of volume on the sides of the hall accessed by moveable doors.
The reverberation time can be varied by accessing the volume through opening
or closing to the doors to any degree.
38
This, in combination with varying the
absorbing surfaces, allows the reverberation time of the hall to range from 1.60 to
2.15 seconds.
39
Varying the sound absorbing characteristics of the room is a much easier task
and is a more common occurrence than changing the volume of a space. This
usually involves movable panels or more commonly installing a sound absorbing
banner across sound reflecting walls or in the attic spaces above the ceiling
reflectors. Movable curtains, for example, are aesthetic and practical methods of
creating more sound absorbing surfaces. Reflectors are traditionally used to
bounce sound from the orchestra up to balcony areas and are relocated
generally to enhance early reflections associated with speech. Just using the
deployable curtains at the KKL Concert Hall with the doors closed allows the
reverberation of the hall to range from 1.60 to 1.95 seconds.
40
38
Beranek, 550
39
Beranek, 613
40
Beranek, 660
45
The movable reflector at the Queen Elizabeth Hall in London is used primarily to
provide more early reflections from overhead during speech and rotated upward
for orchestral performances.
41
Many spaces combine these methods of varying the reverberation time. The
Paris Espace de Projection is a space in which the entirety of the room is
variable. The ceiling consists of three separate panels capable of raising and
lowering in order to change the room volume. The largest of these configurations
offers a volume four times larger than the smallest. In addition to changing the
volume of the space, the surface characteristics of the ceiling and walls are
variable through use of rotating prisms, each with an absorbing surface and two
reflective surfaces, one specular and the other diffuse. This design allows for
reverberation times ranging between 0.5 to 2.0 seconds.
42
3.2.2 Active Variable Room Acoustics
Active systems use electronics to manipulate the sound and re-project it through
speakers. The challenge in using electronic methods in acoustic variation is
integrating the electronic sources (speakers) with the natural environment of the
room. A good active system is one that is not noticed.
43
41
Barron, 343
42
Barron, 343, 344
43
Barron, 348
46
In order to lengthen the low frequency reverberation times at the Royal Festival
Hall in London, an active system termed Assisted Resonance was used to boost
the sound energy of the lower frequencies. Multiple microphones across the
room engage 72 different frequencies, all below 1kHz. The microphone
placements throughout the room are based upon the natural peaks of that
frequency due to the room. As lower frequencies have a longer wavelength, the
peak position is much less susceptible to changes in temperature, humidity etc.
For this reason, reinforcing of higher frequencies, which have shorter
wavelengths and are much more influenced by environmental factors, is done
across broadband channels.
44
For venues in which voice is to be reinforced, it is important that the audience
feel connected to the person and not the microphone projecting his or her voice.
For this reason the Palast der Republik in Berlin, a 5000 seat hall, utilized the
Delta Stereophony System or DSS to reinforce the speaker’s voice but also
provide a sense of directionality to the speaker’s voice. This is accomplished by
creating a delay in the signal from the microphone to the speaker. As long as the
audience hears the first sound from the natural source first, the sounds arriving
later from the electronic speakers will appear to be attributed to the natural
source.
45
44
Barron, 349, 350
45
Barron 348
47
New active room acoustic systems use computer simulations and auralization
techniques to pair the actual space with a digital model and reinforce the sound
in order to achieve a number of desired effects.
In the next chapter, a specific multi-use space will be evaluated, first to determine
the ideal acoustic characteristics for the different types of programs it is used for
and then looking for ways to improve it. These will help to determine the
constraints of the parameters and formulate a parametric definition.
48
4 CASE STUDY BUILDING AND APPROACH
Creating a parametric model based on structural requirements is fairly
straightforward (although computationally complex); success is defined by a
structure that will resist the specified loads. As discussed, designing a
successful acoustic environment depends on a number of criteria. While there
are various acoustic criteria to assess room acoustics, the physical parameters
affecting these criteria are based on the room size, geometry, surface finishes,
and the location of the sound source and listener. Assessing the room acoustics
of an existing space will provide a practical approach by examining how various
programmatic uses can be used in the existing building geometry. This chapter
discusses a case study building, its programmatic uses, and the acoustic
parameters that will be used to define an acoustic element within the venue.
4.1 The Harold Lloyd Soundstage
The Harold Lloyd Soundstage at the University of Southern California was
originally built for the School of Cinematic Arts as a soundstage for film
production. The Thornton School of Music acquired the building in 2010 for use
as a rehearsal space for multiple ensembles. This section will give an overview of
the building and space, its history, and the physical attributes of the space as it
exists, construction assembly, and existing acoustic environment.
49
4.1.1 Background Information
The Harold Lloyd Soundstage was originally built as a soundstage for the
production of student film projects at the University of Southern California. It is
located on the University Park Campus, at 3450A W 35
th
St in Los Angeles, CA.
Figure 4.1 – Exterior of Harold Lloyd Soundstage
50
Figure 4.2 – Interior of Soundstage
After the completion of a new Cinematic Arts facility, including the construction of
brand new soundstages immediately to the west of the existing soundstage, the
Harold Lloyd Soundstage was transferred to the Thornton School of Music at the
beginning of the 2010 fall semester.
4.1.2 Construction
The Harold Lloyd Soundstage is constructed of load bearing 8 inch concrete
masonry unit (CMU) walls, with a total floor area of approximately 3,072 sq ft.
The north and south walls are approximately 64 feet long, while the east and
west walls measure approximately 48 feet. The interior height is 30 ft at the
51
corners and approaches 35 ft at the center of the hip. An interior ceiling is
framed out of dimensional lumber at 30 feet above the floor. A floor plan from
the original construction is shown in Figure 4.3.
Figure 4.3 – Floor plan of Harold Lloyd Soundstage
For purposes of sound recording, a soundstage requires good sound isolation
from the exterior environment. A secondary interior wall is built leaving a 6 inch
52
space between it, and the CMU wall filled with fiberglass insulation. The interior
wall is constructed of one layer of 3/8-inch gypsum drywall behind two layers of
½-inch fiber-board insulation. Figure 4.4 illustrates the wall construction.
Figure 4.4 – Interior Wall Construction
The soundstage is considered acoustically dead, meaning the reverberation time
is extremely low. Although good for a soundstage, an extremely low
reverberation time is not considered good for most music performances. Since
acquiring the building, Thornton has attempted to temporarily mitigate the
“deadness” of the soundstage by installing a series of 4 x 8 foot wood veneer
panels at 4 foot intervals around the perimeter of the facility at approximately 3 ft
off the floor to provide more reflective surfaces. Since these panels are
53
temporary installations, they will not be considered in the calculations for the
existing conditions.
4.2 Ensembles
This section describes four types of musical ensembles that use the soundstage
and require specific acoustic environments for each. Each ensemble using the
space will require a different set of acoustic parameters. This section will focus
on four varied acoustic environments to achieve an acceptable condition that
meets the acoustic criteria.
4.2.1 Orchestra
As the Thornton School of Music’s flagship ensemble, it is important that the
space be best tuned for the symphony orchestra. A symphony orchestra
instrumentation typically consists of strings (including violins, violas, cellos and
string basses), woodwinds (flutes, clarinets, oboe and bassoon), brass
(trumpets, horns, trombones and tubas), and percussion including drums mallets
and auxiliary. This is a rough guide as instrumentations change based on the
piece played; a performance of The Rite of Spring includes more than the
average number of musicians, and some performances require specific
instruments such as a harp, grand piano, or even large wood box that are not
typical. Depending on the selection of music, a symphony orchestra will require
an appropriate reverberation time to realize an ideal acoustic setting. Ideal
acoustic settings are also subjective to the conductor and opinions can vary
54
considerably from conductor to conductor. Composer and USC faculty member
Frank Tichelli prefers a less reverberant room for his works, which tend to have
more rhythmic intricacy.
46
Based on the style of music and the director’s own
preferences, the reverberation time will range from 1.8-2.2.
4.2.2 Percussion
The Percussion Ensemble consists of a variety of percussive instruments
including drums, mallets, and auxiliary equipment. Typically percussion focuses
on the rhythmic intricacies of music. This necessitates a space in which rhythms
can be clearly understood and perceived. Similar to jazz, this will necessitate a
lower reverberation time of near 0.8 seconds.
4.2.3 Master Classes
Master Classes are instructional periods with a “master” musician aimed at
improving technical performance in a group setting. Master Classes exist for the
range of instruments, but the essential criteria is that the instructor is intelligible
to the students. This will necessitate a reverberation time of not more than 0.7
seconds.
4.2.4 Recitals
The School also expressed interest in being able to utilize the soundstage as a
location for recitals. Recitals exist for every instrument and typically involve an
46
Ellison, Schwenke, 2
55
ensemble of instruments, featuring a soloist. Unlike the other uses of the facility,
the acoustics in a recital format would involve an audience in addition to the
musicians. As recitals occur for every instrument, the reverberation time can
vary from 0.8 for percussion to 2.2 for more Classical pieces.
4.3 Parameters and Constraints
After examining the potential uses of the space and having knowledge of the
ways in which acoustics are measured and assessed, this section will explain the
metrics that will serve as the principal criteria for achieving acoustic success and
act as the drivers for altering the space. The metrics were chosen based upon
their ability to account for the range of physical quantities of sound including
amplitude, frequency, and timing. Reverberation time, bass ratio, and early
energy ratios are critical metrics.
4.3.1 Reverberation Time
The single most common metric in changing to assess the acoustic quality in a
space is reverberation time, RT. This will act as the principal parameter to
control the physical changes room acoustic quality of the space. The RTs
designed in the space will be based on use and will range from 0.7 – 2.2
seconds. In order to change the reverberation time, both total room volume and
the amount of sound absorbing material will need to be variable.
56
4.3.2 Bass Ratio
Varying the reverberation time of the room will have the greatest effect on the
performance of the room, but is not the only acoustic parameter that should be
considered. The Avery Fisher Hall in New York had a similar reverberation time
to other concert halls when it opened in 1962, but was perceived as being a
substandard performance venue for its lack of “warmth,” technically a measure of
the balance of low frequencies and bass ratio (despite several attempts to
remedy the acoustical problems with the hall, many exist to this day).
47
While
there will not be as much acoustic variability with Bass Ratio based on program
use, it will be important to constrain the bass ratio to a range of possible values.
This range of possible values will be considered 1.2-3. Constraining bass ratio
will depend on the total room volume as well as the sound absorbing properties
at the first 4 octave bands, 125, 250, 500 and 1000 Hertz.
4.3.3 Early Energy Ratios
The final metric taken into consideration will be Objective Support for rehearsal
settings and Clarity for recital settings. These metrics will have more influence
over the individual panel height and orientation and distribution of sound
absorptive material. The range for constraining Objective Support will be
between -13 and -11 dB. Because the Clarity index is in part based upon the late
arriving sounds energy, which will not be calculated, the constraint will be to
47
Backus, 180-181
57
maximize the amount of sound energy arriving at each location prior to 80
milliseconds.
Table 4.1 summarizes the constraints that each parameter will be held to as a
criteria for judging success in each situation.
Acoustic
Parameters
Space Use RT60 Bass Ratio Objective
Support
Clarity
Orchestra 1.8-2.2 s 1.2+ -13 – -11dB
Percussion .8 s 1.2+ -13 – -11dB
Master Class .7 s 1.2+ -13 – -11dB
Recital .7-2.2 s 1.2+ - Max. E
80
Table 4.1 – Acoustic Parameters and Values
4.4 Proposed Solution
The proposed solution to varying reverberation time, bass ratio, and the amount
of early energy is to design a ceiling canopy system that is able to change the
room’s volume, shape, and percentage of materials able to add more or less
absorption to the room.
Changing the ceiling’s height provides variable acoustic volumes.
Manipulating the ceiling shape will vary the sound reflecting surfaces.
Exposing different materials through a panel system will allow the room to
have varied amounts and type of sound absorption.
The geometry of the canopy system will be modeled in Rhino 3D and the
parametric design will be done using a third party plugin called Grasshopper.
58
These programs will be further explained in the following chapter. This section
will detail the geometry and construction of the proposed ceiling canopy system
and the ability of each component to impact varying the established metrics.
4.4.1 General Layout
The ceiling canopy will be divided into 32 triangular panels measuring 16 x 12
feet. The use of approximate measurements at the nearest whole number is
intended to simplify the calculations involved. The system will use 25 different
nodes in order to control the varying heights of the triangles’ corners.
Figure 4.5 – Ceiling Grid Layout
59
The range of heights will be from 15-30 feet and flexible material will be placed
between each panel to allow for the differential movement between the panel
edges. For the purposes of this study, it will be assumed that the space between
the panels is negligible and will not contribute to increased absorption.
Figure 4.6 – Range of Ceiling Panels
This configuration allows for an easy algorithmic definition and a manageable
scale for this project to identify and manipulate the movement of each panel.
4.4.2 Material
Similar to the prisms at the Espace de Projection in Paris, the system will utilize
different materials to provide more or less sound absorption. Each triangular
60
panel will have a series of smaller triangular panels capable of rotating 360
degrees about one axis in order to expose one of two materials.
Figure 4.7 – Make up of Individual Panel
This will operate as the method to change the sound absorption characteristics of
the ceiling. Specific materials will be chosen during the study based upon
effectiveness and ability to change the acoustic environment.
61
Figure 4.8 – Example Panel
Figure 4.9 – Panel Cross Section
62
4.4.3 Assumptions and Acknowledgments
There are many issues that need to be examined when doing a proper acoustic
study. The following case studies will operate on a number of assumptions; there
will be gaps in the process that may not lead to the completely ideal acoustic
situation if actually built. The case studies illustrate a method of studying
optimization of a room for multiple uses, but do not take into account all the
known constraints for designing. Weight is given to reverberation time, bass
ratio, and clarity. For the purpose of this thesis, the floor treatment will be
assumed as a wood floor over the existing concrete, while the wall treatments will
be determined in conjunction with the materials for the ceiling panels. It is also
assumed that the direction the sound arrives from does not matter, and hence
only the first reflections from surfaces in the room will be calculated.
The main goal by the end of all the case studies is to determine what the ceiling’s
geometric configuration needs to be and what materials need to be applied to
achieve the following metrics:
Acoustic
Parameters
Space Use RT60 Bass Ratio Objective
Support
Clarity
Orchestra 1.8-2.2 s 1.2+ -13 – -11dB
Percussion .8 s 1.2+ -13 – -11dB
Master Class .7 s 1.2+ -13 – -11dB
Recital .7-2.2 s 1.2+ - Max. E
80
Table 4.2 – Goal Metrics for Each Condition
63
5 GEOMETRIC OPTIMIZATION
In order to understand the different effects that distribution of absorption, ceiling
height and the geometry of reflective surfaces have on reverberation time, bass
ratio and objective support, a series of exercises were completed that focused on
how to control the geometry of a room based upon these criteria. This chapter
will cover the first exercise, the parametric definition and its results, the
limitations and problems with the exercise and the lessons learned moving
forward. These case studies use Rhino, Grasshopper, and Galapagos.
Rhino is a 3D computer modeling program that uses mathematical
representations to define curves and lines that form surfaces and solids in 3D
space.
Grasshopper is a plug-in for Rhino that utilizes a graphic interface to create
parametric modeling streams through existing Rhino tools. Numbered “sliders”
are used to vary input numbers.
Galapagos is a genomic algorithm within Grasshopper that generates a set of
potential options (the base population) based upon the range of constraints
designated as variable with Galapagos (the genome). Each proceeding
generation of models is created from mating and eliminating previous options
based upon their fitness level, as Galapagos attempts to optimize the value set
by the user. The optimization is limited to a single fitness value but it could
represent any combination of variables that can be evaluated within the
Grasshopper definition. Galapagos allows the user to control the size of the
64
initial population, the maximum number of generations before terminating, the
number of “fruitless” generations before terminating, and a tolerance for
approaching the fitness number.
5.1 First Exercise-Galapagos and Reverb Time
Since varying Reverberation Time will have the biggest impact on the quality of
the acoustic performance, it is important to understand what effects moving the
ceiling around will have on this criteria. The goal of this exercise was to begin to
understand the effects of changing ceiling height on the ratio between volume
and surface area. This relationship is necessary to understand how exposing
more of the material on the walls will impact changing the volume of the space.
This exercise covers varying the total room volume as a function of the height
and surface area of a hypothetical ceiling panel system and also experiments in
using the Galapagos genomic optimization feature in Grasshopper.
5.2 Building the Parametric Definition
Building the definition of the model is the real challenge of this thesis. The basis
of the definition will lie in the relation of the room geometry to the desired
performance characteristics. This exercise is the first step to identify the
relations between the most important characteristic, reverberation time, and the
room geometry as well as how to control this relation through Grasshopper.
65
5.2.1 Equations
Recalling that reverberation time is a function of both volume and surface area
and that changing the ceiling height will have an impact on both total volume and
surface area, this exercise will simplify the equation for Reverberation Time from
Chapter 3 from
RT
60
=
(.049)V
s
n
a
n
1
n
∑
to
RT
60
=
V
S
Where V is the room volume and S is the total surface area. While this is an
oversimplification of the equation, it negates the effects of different materials and
allows one to study just the impact of room geometry.
5.2.2 Base Model
The base model is a 30 by 30 foot room with a square grid ceiling composed of
nine 10 by 10 foot panels. The ceiling height at each corner or node of the
ceiling is independently variable from 15-30 feet. This also necessitates that the
walls are variable as well from 15-30 feet as well. With these constraints, the
largest possible dimensions, 30x30x30 ft, give a volume of 27000 cubic feet and
a surface area of 5400 square feet.
66
Figure 5.1 – Model at Largest Ceiling Height Values
This results in a volume to surface area ratio of 5. When set at the smallest
possible dimensions, 30x30x15 ft, the model room has a volume of 13500 cubic
feet and a surface area of 3600 square feet with a ratio of 3.75.
Figure 5.2 – Model at Smallest Ceiling Height Values
67
5.2.3 Sequencing/Controlling
The first challenge in setting up the definition is getting Galapagos to work in this
context. Because Galapagos is set to optimize the fitness number, it is
necessary to translate the target value into a value that Grasshopper will take
and attempt to optimize. The way this was handled was taking the negative of
the absolute value of the difference between the target and calculated values.
This makes the fitness number optimize to 0, equal to no difference between the
target value and the calculated value.
Figure 5.3 – Fitness Number Process in Grasshopper
The Genome or variable part of Galapagos is associated with the height of each
ceiling panel node. The height of each node of the surface is variable from 15-30
ft, while the X and Y coordinates remain fixed. Surface area and room volume
are determined by evaluation functions within Grasshopper and are fed back to
the fitness number for the Galapagos function as previously discussed.
68
Figure 5.4 – Schematic of Grasshopper Sequencing for Exercise 1
69
Figure 5.5 – Full Grasshopper Model for Exercise 1
5.3 Resulting Models
In order to evaluate this definition and gauge if Galapagos is a suitable tool for
this application, the ratio for the minimum ratio, the maximum ratio, and multiple
simulations at one target value were evaluated. Within Galapagos, the specific
values were set as follows:
Maximum Generations 80. This number defines the maximum number of
generations the application will run for before terminating.
Maximum Fruitless Generations 20. This number defines how many generations
the application will run for with no improvement before terminating.
Initial Population 50. This number defines the initial number of solutions the
application will evaluate.
Inbreeding Factor 50. This number defines how likely the breeding populations
will combine with similar solutions (value of 100) or dissimilar solutions (value of -
100).
Maintain High Fitness 10. This number defines the percentage of solutions
retained from previous generations that outperform the new generation.
5.3.1 Minimum Value
The target slider was set to zero to obtain the smallest possible volume to
surface area ratio, calculated as around 3.37. This required extending the total
number of generations up to 240. The shape produced makes sense as a low
70
ratio is produced by a high surface area and a small volume. The inverted dome
shape of the roof provides this sort of geometry, maximizing the amount of
surface area for amount of volume gained.
Figure 5.6 – Minimum Ratio Shape
5.3.2 Maximum Value
The target slider was set to 50 to obtain the largest possible volume to surface
area ratio, which resulted in a ratio of 5.09. The shape of this model is essentially
an inversion of the previous model using a domed shape to maximize the volume
to surface area ratio.
71
Figure 5.7 – Maximum Ratio Shape
5.3.3 Evaluating at Ratio of 3.75
The slider was set to 3.75, the ratio calculated using the lowest height for each
node, to see what would be produced by Galapagos.
72
Figure 5.8 – Models with V/SA Ratio of 3.75
Each model produced had a volume to surface area ratio of 3.75, but each had
different configurations and total volume and surface area.
Volume Surface Area Ratio
Model 1 18958.43 5055.51 3.7500
Model 2 17120.43 4565.32 3.7501
Model 3 17967.00 4791.20 3.7500
Model 4 17794.45 4745.06 3.7501
Table 5.1 – Volume and Surface Areas of Multiple Trials at 3.75
5.4 Discussion and Conclusions
If this technique will be applied to finding the shape of the ceiling panel system, it
will be necessary to restrict the height of the nodes relative to each other. While
the panels can be expected to accommodate some flexibility and movement from
73
panel to panel, there needs to be a tolerance limit in order to maintain the
assumption that the space between the panels is negligible.
Using Galapagos also takes care in realizing if the optimized number has
reached its limit and in setting up the fitness number so that it is solving for a
specific value rather than for the largest or smallest value. This requires that the
genome actually has an effect on the fitness number, and that enough
generations are programmed into Galapagos, especially when the genome
consists of a large amount of sliders.
The result of using Galapagos to perform this function is that it produces an array
of variations with very different geometries. Limiting the variation in height from
panel to panel will limit these variations, but there is the potential that some of
these models will result in unfavorable geometries.
74
6 SECOND EXERCISE
As demonstrated by the failures of Avery Fisher Hall, the balance of frequencies
is an important factor in the quality of the acoustics of a space. Varying both
ceiling height and the different type of absorptive materials should be able to
achieve the control of both reverberation time and bass ratio. The goal of this
exercise was to examine the interaction of absorption and room volume with the
effects on reverberation time and bass ratio and determine an equation for
material ratios and room volume as a function of reverberation time and bass
ratio. A secondary goal in this exercise was to establish the best materials to use
on the walls, floor, and ceiling that would allow the widest range of possible
values.
6.1 Determining Geometric Relationships
The equations for reverberation time and bass ratio are written such that they are
functions of (a) room volume, (b) the properties of the finishes, and (c) the
surface area of the finish materials. In order to vary the acoustic parameters of
the room, it will be necessary to rewrite these equations so that the room volume
(or at least the ceiling height) and the percentage of each material is a function of
the prescribed reverberation time and bass ratio. This required some
assumption of the materials within the space, deriving equations for the general
form of the equation as well as calculating coefficients for the use of specific
materials.
75
6.1.1 Equation Overview
This exercise will assume four materials present in the room. The ceiling will
consist of the two materials under investigation, the floor as carpet over the
existing concrete floor, and the walls as plaster with 25% of the wall space
covered by 2” fiberglass absorbers. A flat ceiling will be used in this case study in
order to keep variables to a minimum. Under these conditions the equation for
reverberation time will look like:
RT
60
=
(.049)*3072*h
s
C1
a
C1
+s
C2
a
C2
+s
F
a
F
+s
W
a
W
∑
Where 3072 is the square footage of the floor, h is the height of the ceiling, C1
and C2 are the first and second ceiling materials, F is the floor, and W is walls.
Reverberation time will be assumed to be at 500 Hz, but will also have to be
calculated at 125, 250, 500 and 1000 Hz to figure out bass ratio:
BR =
RT
125
+RT
250
RT
500
+RT
1000
6.1.2 Early Attempts at Setting Up Data
This exercise began as a trial and error attempt to try to confirm the possibility of
achieving a targeted reverberation time of 2.0 seconds and a bass ratio of 1.2.
Using the assumptions previously mentioned, the height and ceiling material
percentages were varied within a Microsoft Excel spreadsheet (Table 6.1) to try
76
to achieve the specified parameters. This would help later in verifying the results
in Galapagos.
Variables Reverberation Times Results
Ceiling
Height
Material 1
(%)
Material 2
(%)
125
Hz
250
Hz
500
Hz
1
kHz
2
kHz
4
kHz
RT @
500 Bass Ratio
15
1 0 1.59 2.66 2.06 1.85 1.87 1.79 2.06 1.09
Table 6.1 – Sample Calculations of RT and BR
This resulted in two positive outcomes: an understanding of what types of
materials would be most useful and a form for generating different sets of curves
based upon two materials. The trial and error portion became a very tedious task
as it involved a very limited view of the effect of using different materials. This
led to the creation of full spreadsheets of reverberation times and bass ratios at
the full range of material percentages (0-100%) at 5% intervals and the full range
of ceiling heights (15-30 feet) at 5 foot intervals. A full set of these spreadsheets
can be found in the Appendix. These spreadsheets were also useful in
transforming the numerical data into a visual display of curves and to fit
equations through Excel’s curve fitting functions.
6.1.3 Families of Curves and Deriving Equations
The graphs of the 15 and 30 foot curves were used to determine which sets of
materials would be the most useful in looking at how the curves compared with
the target values of reverberation time and bass ratio. Figure 6.1 illustrates an
example of the material curves used in this process.
77
Figure 6.1 – Graph of Reverberation Time v Bass Ratio
After examining a number of common building materials, pageboard over 1” of
fiberboard and ½” plasterboard appeared to be the most viable as ceiling
materials. Plasterboard, or drywall, is 1/2 in. gypsum sandwiched between fiber
or paper facers. The pageboard material is a perforated wood particleboard over
a 1 in. thick fiber board and has a unique characteristic in absorbing a
percentage of the mid-range frequencies. The absorption coefficients for all of
the materials in the room are as follows:
78
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
Plasterboard 0.29 0.1 0.06 0.05 0.04 0.04
Pageboard 0.08 0.32 0.99 0.76 0.34 0.12
Walls 0.12 0.13 0.21 0.26 0.27 0.28
Floor 0.04 0.04 0.07 0.06 0.06 0.07
Table 6.2 – Absorption Coefficients of Room Materials
Figure 6.2 displays the resulting curves of bass ratio as a function of
reverberation time for pageboard and plasterboard as materials, with a range of
each material from 0-100% at the 15, 20, 25 and 30 foot ceiling heights.
Figure 6.2 – Graph of Bass Ratio as Function of Reverberation Time –
Plasterboard and Pageboard
79
Fitting a power trend line (black line in the above graphs) for each of the ceiling
heights yields the following equations:
15 Feet
917 . 0
60
) ( 9833 . 1
- = RT BR 9903 . 0
2
= R
20 Feet
907 . 0
60
) ( 341 . 2
- = RT BR 9922 . 0
2
= R
25 Feet
899 . 0
60
) ( 6237 . 2
- = RT BR 9936 . 0
2
= R
30 Feet
893 . 0
60
) ( 8529 . 2
- = RT BR 9947 . 0
2
= R
The high R squared numbers (least squares regression) indicate a strong fit for a
power trend line yielding an equation relating Reverberation Time and Bass Ratio
in the general form of
BR = f(h)* RT
60
k
Where the coefficient f(h) is a function dependent upon the height of the ceiling
and k is based on the height of the ceiling and the absorption coefficients of both
materials. As there is a negligible variation in k from 15-30 feet, this value will
remain constant across each height. For these materials k will be equal to -0.90.
Taking the coefficient f(h) and plotting it as a function of ceiling height yields the
following curve:
80
Figure 6.3 – Graph of Ceiling Height and Coefficient Values
Creating a logarithmic trend line yields the equation of form:
€
f(h) =a*ln(h) +b
Where a and b are derivatives of the specific ceiling materials. For plasterboard
and pageboard a is 1.1814 and b is -1.2616.
Substituting this into the first equation gives
€
BR = (1.1814 *ln(h) - 1.2616)*RT
60
- 0.90
Rearranged in terms of BR and RT gives an equation for the height of the room:
81
h =e
BR
RT
60
- 0.90
+1.2616
1.1814
Being able to calculate the height of the ceiling in terms of the reverberation time
and bass ratio removes another variable from the original reverberation time
equation. Recall:
RT
60
=
(.049)*3072*h
s
C1
a
C1
+s
C2
a
C2
+s
F
a
F
+s
W
a
W
∑
The only unsolved variables are now the total surface areas of C1 and C2.
However, there is another equation relating the two considering the total surface
area is constrained to 3072 square feet, therefore:
s
C1
+s
C2
=3072
Substituting for S
C1
yields
s
C1
=
(.049)(3072)h
RT
60
- (s
F
a
F
+s
W
a
W
+(3072)a
C2
)
a
C1
- a
C2
Now that one can determine the total room volume and area of both materials
based upon Bass Ratio and Reverberation Time, this sequence can be brought
into Grasshopper.
82
6.1.4 Sequencing/Controls
While the underlying equations on this sequence are fairly complex to derive, the
processing for incorporating this into a Grasshopper model is straightforward.
The inputs consist of the target reverberation time and bass ratio.
Figure 6.4 – Schematic of Grasshopper Sequencing for Exercise 2
The outputs are the ceiling height and the percentage of the ceiling area covered
by each material (recall that there are two materials).
Within Grasshopper, the algorithm begins with inputting the reverberation time
and bass ratios into the sliders. Based upon the reverberation time, a general
guideline is given for the lower and upper limits of the bass ratio.
The reverberation time and bass ratio are run through the functions and the
ceiling height is applied as the magnitude of a vector in the z direction to move a
rectangle representing the ceiling. As a visual representation of the percentage
83
of materials, the ceiling is divided into 3072 different squares representing
individual panels that have an area of one square foot each.
6.2 Results
The results in this section will survey the reverberation times of each scenario
and confirm that the output height and material proportions will create the desired
reverberation time and bass ratio.
Orchestra
At a reverberation time of 1.8 and a bass ratio of 1.4 (Figure 6.5), the resulting
geometry is shown in Figure 6.6. Note that for this case study, the location of
the two ceiling materials (red and yellow) do not matter; it is the amount of them
that is critical. Also, recall that the ceiling is constrained to being flat.
Figure 6.5 – Input Values for Orchestra
84
Figure 6.6 – Output Geometry for Orchestra
The height of this ceiling is 21.7 ft and the material breakdown is 2633 square
feet of plasterboard and 439 square feet of pageboard. To validate the result,
these values were substituted back into the original equations; this yields a
reverberation time of 1.80s and a bass ratio of 1.45, what was input as the goal.
Percussion Ensemble
At a reverberation time of 0.8s and a bass ratio of 2.5 (Figure 6.7), the resulting
geometry is shown in Figure 6.8.
85
Figure 6.7 – Input Values for Percussion Ensemble
Figure 6.8 – Output Geometry for Percussion Ensemble
The height of this ceiling is 16.4 ft and the material breakdown is 999 sq ft of
plasterboard and 2073 sqft of pageboard. Substituting these back into the
86
original equations, this yields a reverberation time of .80s and a bass ratio of
2.52; this checks with the original values.
Master Classes
At a reverberation time of .7 s and a bass ratio of 1.4 (Figure 6.9), the resulting
geometry is shown in Figure 6.10.
Figure 6.9 – Input Values for Master Classes
87
Figure 6.10 – Output Geometry for Master Classes
The height of this ceiling is 15.3 ft and the material breakdown is 734 sq ft of
plasterboard and 2338 sq ft of pageboard. Substituting these back into the
original equations, this yields a reverberation time of 0.70s and a bass ratio of
2.77.
Recitals
At a reverberation time of 2.2 and a bass ratio of 1.3 (Figure 6.11), the resulting
geometry is shown in Figure 6.12.
88
Figure 6.11 – Input Values for Recitals
Figure 6.12 – Output Geometry for Recitals
89
The height of this ceiling is 27.25 ft and the material breakdown is 2858 sq ft of
plasterboard and 213 sq ft of pageboard. Substituting these back into the original
equations, this yields a reverberation time of 2.2 and a bass ratio of 1.37.
Table 6.3 summarizes the input values for each case, the heights, and material
portions and the recalculated reverberation times and bass ratios. The “results
column” shows that the hand calculations agree with the Galapagos calculated
results.
Input Variables Grasshopper Results Results (% dif)
Setting RT60
(s)
BR Height
(ft)
Plasterboard/
Pageboard
(sq ft)
RT60
(s)
BR
Orchestra 1.8 1.4 21.7 2633/439 1.8
(0%)
1.45
(3.4%)
Percussion 0.8 2.4 16.4 999/2073 0.8
(0%)
2.52
(0.8%)
Master Class 0.7 2.7 15.3 734/2338 0.7
(0%)
2.77
(2.5%)
Recital 2.2 1.3 27.25 2858/214 2.2
(0%)
1.37
(5.1%)
Table 6.3 – Results of Grasshopper Defined Geometries
6.3 Discussion and Conclusions
Recalculating the reverberation times and bass ratios was important to ensure
that the assumptions made in forming the equations did not cause a substantial
deviation in the actual results. Overall, the reverberation times matched hand
calculations to a 1/100
th
of a second and the bass ratio times varied up to 5.1%
from the input bass ratio.
90
Several limitations of this method were made apparent. There is a general
limitation based upon the materials -- it is the inability to find a consistent range
of bass ratios across the entire range of reverberation times. The general lower
limit for bass ratios was established at 1.2, but the lower limit, based upon
materials, is well above this for the lower reverberation times. This may be a
general limitation of the materials and could be eliminated by introducing a third
material variable. Another one of the limitations of this set of equations is that
the height of the ceiling influences two different other variables: the amount of
exposed wall area and the room volume. As seen in the first exercise there can
be a number of different geometries that yield the same physical measurements,
so it will be safe to assume that for any given total volume and wall surface area
values, there will be a variety of ceiling shapes that fit this criteria. Deciding
which of these variations is the most ideal will require the input of another criteria,
clarity/support, which is explored in the next exercise.
91
7 CREATING REFLECTION PATHS
While the previous two exercises have concentrated on the general geometry of
the room and reverberation time, this exercise will focus on the specific geometry
of the space and the ceiling’s effect on reflections. The object of this specific
exercise is to use Grasshopper to provide a simple ray tracing analysis limited to
one reflection between a source and a receiver. The length of each ray and the
specific point of reflection will be used later to calculate early sound energy
measurements.
7.1 Geometrical Relationships
This exercise, more than the previous two, will depend on the actual geometry of
the room and the shape of the ceiling surface. For simplicity’s sake, this exercise
will only deal with creating the reflections and measuring results, not attempting
to optimize or target a specific value.
7.1.1 Equations
Both types of early energy measurements, Clarity and Support, will depend on
the amount of sound energy arriving at a location relative to a particular source.
The equation for the intensity at a location based upon a singular specular
reflection is
I =
(1- α)
4π(x + y)
2
92
where I is the intensity, α is the absorption coefficient, and x and y total the entire
path of the incoming and reflected wave, as illustrated in Figure 7.1.
Figure 7.1 – Path of Reflected Ray
Because the measurements used will be relative and not absolute, the sound
power will be assumed as 1. The numerator is the total intensity after absorption
lost due to the reflection off the surface while the denominator is associated to
the intensity lost due to distance from the source (the sound pressure at a point
of an equivalent sphere).
When calculating by hand, a reflection path in two dimensions can be found by
first reflecting the receiver point, R, over the reflection plane, creating R’, and
drawing a line between the source, S, and R’. The intersection of this line with
the reflection plane is the point of reflection. A line between this point and the
receiver completes the path of the sound ray.
93
Figure 7.2 – Construction of Reflection
This geometry can be checked by measuring the angle between each of the ray
paths and the normal vector of the plane and verifying that the angles are equal.
This process can be duplicated in Grasshopper by using the transformation tools.
The other measurement needed in this process is the amount of time it takes for
the sound to travel the full distance from the source to the receiver. This can be
calculated by dividing the distance by the speed of sound (C), resulting in time.
t =
(x + y)
C
94
7.1.2 Sequencing/Controlling
To start the construction of the reflection paths within Grasshopper, two points
are defined within the space of a 48 foot by 64 foot rectangular room. The source
point (S) and receiver point (R) are each located 3 feet off of the floor and 12 feet
from the center of the room on opposite sides in the center the room. The roof of
the room is split into two halves (each 32 foot by 48 foot) allowing for variation at
the middle inflection point (Figure 7.4). The receiver point is reflected on opposite
sides of the planes defined by the walls, floor and ceiling panels, creating a
series of R’s, one for each reflecting plane. A line is created between the source
and each R’ and using the Plane/Line Intersection tool, a point is established at
the intersection of each of each plane and line. If the intersection of the line and
plane does not exist with the specified surface, the point is determined as “null”.
Lines are then constructed between the intersection point and the source and
receiver. The measurement tool is able to determine the length of the lines and
feed this into the intensity function. That length is also used in determining the
time it takes for the sound to travel from the source to the receiver. For
measuring support, the time for each ray is used to separate the respective rays
into groups, from 0-10 ms and 20-80 ms. These groups can then be summed
and used to create a ratio of early to late energy values.
7.2 Results
95
Figure 7.3 – Reflection Points within Room
Figure 7.4 – Reflection Points at High Hip in Ceiling
13.86 ft
12.05 ft
96
To verify the reflected angles at the ceiling are correct and the algorithm
functions as intended, the angle at the plane of each line was calculated and
compared within Grasshopper, as shown in Figure 7.5.
Figure 7.5 – Verification of Angles
7.3 Discussion and Limitations
The virtual image method of defining in this context works as the program is able
to calculate the information needed to process calculations related to early
energies. This method of ray tracing only creates a very small portion of the
reflections happening within the room. To create a series of higher order
reflections and rays is beyond the scope of this exercise when associated with
this genomic optimization tool, but further study could improve upon this method
to allow for more reflections.
97
8 FINAL MODEL
The final model incorporates elements of each of the previous exercises: the use
of Galapagos from Exercise 1, the calculation of the total room volume and wall
surface area described in Exercise 2, and ray tracing from Exercise 3. In
addition, there is a combination of ray tracing from Exercise 3 and the use of
Galapagos from Exercise 1 in order to determine individual ceiling panel
orientation and sound absorption material distribution. This chapter details the
algorithm used to determine the ceiling configuration and the final results for
each programmatic use. For this model, the ceiling will be defined by 25
separate nodes capable of moving independently with each of the 32 triangular
panels made up of 100 panels capable of rotating to expose either of two
materials, as described in Chapter 4.
8.1 Parametric Definition
The added complexity of controlling the amount of early energy (either the
Objective Support or Clarity) indices will be dependent on the specific geometry
and material of the ceiling panels. Because these indices are generally averaged
over a number of areas, the model will have to incorporate and average the
targets from a number of specific locations. This section details the parametric
definition used to create the ceiling panel system including the heights of each
node and the determination of the location of each type of material.
98
8.1.1 Geometric Set Up
The room volume and wall surface area are determined by the height equation
produced in Exercise 2:
h =e
BR
RT
60
- 0.90
+1.2616
1.1814
These variables are a function of the reverberation and bass ratio as well as the
overall proportion of material between pageboard and plasterboard is determined
by the equation:
s
C1
=
(.049)(3072)h
RT
60
- (s
F
a
F
+s
W
a
W
+(3072)a
C2
)
a
C1
- a
C2
As illustrated in previous chapters, these two equations will ensure that the first
two parameters, reverberation and bass ratio, are satisfied. The goal of the final
exercise is to use either the Support or Clarity indices (dependent on the
application) to influence the orientation and absorptive qualities of the individual
ceiling panels while still maintaining the desired reverberation time and bass
ratio. The Support and Clarity indices are averaged from four locations within the
room.
For Support, the source locations are based on the extents of where musicians
would be positioned for a rehearsal at 12 feet from the longer walls and 16 feet
from the shorter walls. The receiver locations are placed at a distance of 3 feet
99
away from the source location, directed towards the center of the room as
detailed in Figure 8.1.
Figure 8.1 – Source/Receiver Locations – Objective Support
S
2
S
1
R
1
R
2
S
3
R
3
S
4
R
4
3 ft
12 ft
16 ft
100
For Clarity, the source location is placed midway between the longer walls at a
distance of 8 from the wall as a performer would be situated for a recital. The
receiver locations are located in 4 different areas where an audience would be
seated as detailed in Figure 8.2.
Figure 8.2 – Source/Receiver Locations – Clarity
The ray tracing aspects of Exercise 3 are used to create reflection paths from the
ceiling panels in order to calculate the distance traveled, the time it took, and the
12 ft
16 ft
8 ft
101
relative sound energy lost. In constructing the reflection paths, if the line does not
intersect the plane where the surface of the panel exists, the line is defined as
null and is removed from the list. Not only is this model more complex than
Exercise 3 because of the number of panels, but it must also account for the
absorption coefficient of each panel. This equation assumes the sound power is
1 where x and y are the distances from the source to the reflecting plane and the
plane to the receiver position, respectively.
I =
(1- α)
4π(x + y)
2
x
y
α
Figure 8.3 – Absorption at Reflection Point
102
8.1.2 Varying Ceiling/Panel Heights
In Exercise 2, the target reverberation times and bass ratios defined a height for
the ceiling. Because the room in this instance was a box, the height defined both
a room volume and an amount of wall surface area. The challenge in using 32
panels instead of one large ceiling panel comes in ensuring that with each
different geometric possibility, the volume and wall surface area will remain
consistent. Once the height of the room is determined, this is used as a basis to
vary the ceiling panels form. To produce the heights by which the panels will
vary, a group of eight number sliders is used to produce a list of numbers from 0
to 2.5. The negatives of these numbers are then taken and added to the list.
The full range of numbers (-2.5 to 2.5) ensures that no two panels are more than
5 feet different in height. The outer ring of nodes, which affect the amount of wall
area exposed, are grouped together to ensure the wall surface area stays as
close as possible to the one given by a flat ceiling shape. A similar set of four
sliders is used to control the second ring of eight nodes and the center node
operates off of its own slider from -2.5 to 2.5. Each set of numbers is “jittered” to
scramble their order using a pseudorandom number generator application within
Grasshopper. The number generator creates repeatable lists based off of seed
number that is defined by another number slider. The seed number and other
number sliders are all then defined as genome terms by Galapagos. To ensure
that no nodes go lower than 15 feet or higher than 30 feet, a series of “if-then”
103
statements are applied to each number to constrict the possible range of
numbers.
8.1.3 Varying Absorption Distribution
The target reverberation time and bass ratio also produce a square footage
number for the entire ceiling for both materials, but the distribution of this material
can have an effect on the early energy levels. This is different from Exercise 2
where only the relative amount of each of the two materials was important, not
their locations. Determining how the material is spread around the room is
determined by first assigning each panel a number 0-100 by a number slider and
summing these numbers. Dividing this number by the individual panel number
will result in a percentage of the total area of pageboard each panel will
contribute. Each panel has a maximum limited to the area of the panel of 96 sq
ft. The number sliders are also a part of the genome of Galapagos.
104
Figure 8.4 – Grasshopper Definition Assigning Surface Area Proportions
8.1.4 Full Grasshopper Definition
Inputs to this definition are the reverberation time and bass ratio. There are two
separate halves of the definition, the first of which involves these inputs and the
second half involving changing the target value for Galapagos to either a
rehearsal or recital setting and running Galapagos. A schematic diagram of the
definition and the full Grasshopper definition are pictured in Figure 8.5 and Figure
8.6 respectively..
105
Figure 8.5 – Schematic Diagram of Grasshopper Definition
Figure 8.6 – Screenshot of Grasshopper Definition
106
Each node is separately defined in a graphical layout to keep each point
organized especially while defining the planes of the walls. While the x and y
coordinates are static, the z coordinates of each node is determined by the
method discussed previously. The first half of the definition is exactly the same
as Exercise 2, determining the height and surface area mathematically. The
second half of the definition is similar to Exercise 3 as it measures, separates
and sums relative sound energy according to either a rehearsal or recital
program usage. Running Galapagos is the final step, specifying the target
number as -12 for dB for rehearsal settings or maximizing the target number for a
recital setting.
In summary, the basic room geometry is defined by set points in Grasshopper
and remains static. The average ceiling height and total proportions of materials
are defined by the equations derived in Exercise 2. The distribution of materials
on the ceiling surface and orientation of ceiling panels is defined by Galapagos,
based on early energy calculations for either Clarity or Objective Support
(depending on the program use). For Objective Support, there are four receiver
and source nodes placed in each quadrant of the room, and located 3 feet apart.
For Clarity there is a single source at the front of the room with four receivers in
the audience area. In each case the user inputs are the Reverberation Time and
Bass Ratio. The Grasshopper results will define the surface geometry and the
material distribution.
107
8.2 Results
This section details the results of the parametric definition based on inputs
specific to the programmatic uses described in Chapter 4. To verify the program,
the reverberation times and bass ratios are calculated from the actual room
volume and surface areas defined by the model.
8.2.1 Orchestra
A reverberation time of 1.8 and a bass ratio of 1.4 (Figure 8.7) were used as the
input data.
Figure 8.7 – Input Values for Orchestra Configuration
108
Figure 8.8 – Final Configuration of Room Model for Orchestra Configuration
The resultant volume of the space was 66863 with a wall surface area of 4870
square feet and the ceiling divided into 2633 square feet of plasterboard (86%)
and 439 square feet of pageboard (14%). The corresponding reverberation time
and bass ratio for these conditions are 1.79 seconds and 1.45 respectively. This
is a variation of 0.01 seconds (0.6%) from the input reverberation time and a
variation of 0.05 (3.6%) from the input bass ratio. The resulting average
Objective Support value was -13.399 dB.
109
8.2.2 Percussion
A reverberation time of 0.8 and a bass ratio of 2.5 (Figure 8.9) were used as the
input data.
Figure 8.9 – Input Values for Percussion Configuration
110
Figure 8.10 – Final Configuration of Room Model for Percussion Configuration
The resultant volume of the space was equal to 50,596 cubic feet with a wall
surface area of 3676 square feet, and the surface area of the ceiling divided in
1188 square feet of plasterboard (39%) and 1884 square feet of pageboard
(61%). The corresponding reverberation time and bass ratio for these conditions
are 0.85 seconds and 2.37 respectively. This is a variation of 0.05 seconds
(6.3%) from the input reverberation time and a variation of 0.13 (5.2%) from the
input bass ratio. The resulting average Objective Support value was -12.824 dB.
111
8.2.3 Master Classes
A reverberation time of 0.7 and a bass ratio of 2.7 (Figure 8.11) were used as the
input data.
Figure 8.11 – Input Values for Master Class Configuration
112
Figure 8.12 – Final Configuration of Room Model for Master Class Configuration
The resultant volume of the space was equal to 47,066 cubic feet with a wall
surface area of 3413 square feet, and the surface area of the ceiling divided in
729 square feet of plasterboard (24%) and 2333 square feet of pageboard (76%).
The corresponding reverberation time and bass ratio for these conditions are 0.7
and 2.78 respectively. This is a variation of 0.0 seconds (0%) from the input
reverberation time and a variation of 0.08 (2.9%) from the input bass ratio. The
resulting average Objective Support value was -12.399 dB.
8.2.4 Recitals
A reverberation time of 2.2 and a bass ratio of 1.3 (Figure 8.12) were used as the
input data.
113
Figure 8.13 – Input Values for Recital Configuration
114
Figure 8.14 – Final Configuration of Room Model for Recital Configuration
The resultant volume of the space was equal to 83,654 cubic feet with a wall
surface area of 6,102 square feet, and the surface area of the ceiling divided into
2850 square feet of plasterboard (93%) and 222 square feet of pageboard (7%).
The corresponding reverberation time and bass ratio for these conditions are
2.17 seconds and 1.38 respectively. This is a variation of 0.03 (1.3%) from the
input reverberation time and a variation of 0.08 (6.1%) from the input bass ratio.
For the Clarity Index calculation, the average amount of sound energy arriving
prior to 80 milliseconds is 2.9319 x10
-4
of the original source sound energy.
115
Table 8.1 summarizes the input values for each case, the volumes and material
portions and the recalculated reverberation times and bass ratios and the related
energy metrics.
Input
Variables
Calculated Results Results (% dif) Early Energy
Measurement
Setting RT60
(s)
BR Avg Height
(ft)
Plasterboard/
Pageboard (sq ft)
RT60 (s) BR
Orchestra 1.8 1.4 21.7 2633/439 1.79(0.6%) 1.45(3.8%) -13.399 dB
Percussion .8 2.5 16.43 1188/1884 .85(6.3%) 2.37(5.2) -12.824 dB
Master Class .7 2.7 15.27 729/2333 .7(0%) 2.78(2.9%) -12.399 dB
Recital 2.2 1.3 27.25 2850/222 2.17 (1.3%) 1.38(6.1%) 2.9319x10
-4
Table 8.1 – Final Results
116
9 CONCLUSIONS
From the results of the final model, it is clear that it is possible to create forms
and provide a solution to variable acoustic settings using a parametric definition
for an architectural element. However, there are several shortcomings with this
model, both from a design standpoint and from a practicality standpoint.
Due to the optimization tool used within the algorithm, there are many different
versions of each ceiling profile that may result from running the model. Each of
these versions may have a different effect on other parameters not designated as
part of the algorithm. The effect of multiple reflections, acoustical metrics such
as loudness, and the overall impulse response of the room are not taken into
account in the model but are factors that should be analyzed to create an overall
successful acoustic environment.
The metrics used may be improved upon, namely the objective support. As a
metric that looks at a measurement in close proximity to the source, this may not
give an accurate feel for the entire room. The number of source –receiver pairs
and locations have a large effect on the final shape of the ceiling.
The design of a large ceiling panel works within the context of this case study as
it is a significant portion of the overall surface area. There may not be a direct
correlation of such a system having success in a larger, more audience centric
environment. For instance, in a concert theater the reflections off the walls have
a much more significant impact on early reflections and would make more sense
as the architectural element to manipulate. However, a wall system would not
117
necessarily have the same flexibility to move in and out as the ceiling panel in
this example. Another method of altering the space would have to be explored.
The practicality of such a system would have to be analyzed as well. The
structural requirements for the ceiling system to be installed would have to be
looked at as well as the cost effectiveness of such a system as compared to the
installation of an electronic system, capable of simulating variable acoustic
settings. These current limitations lead to potential areas of future work.
Further Research Topics
There are many areas where work in acoustic parameter optimization can be
improved including researching methods of variability, effects based on different
metrics, and deeper analysis.
Further exploration into the research of parametric design and variable acoustics
should look at other settings and venues that might deploy a variable acoustic
program such as larger concert halls or performing arts centers. Within these
venues there is more opportunity to explore other ways of altering the acoustics
including different architectural elements such as the walls or concert shell as
well as different ways to alter their properties. Absorption panels that are reliant
on the depth of their assemblies and Schroeder diffusers both naturally offer
opportunities for parametric design.
There are many other metrics within the field of acoustics that were not a part of
this study that have an impact on the overall performance that should be looked
118
at including loudness and the overall impulse response of the room. The
distribution and density of sound reflections across the audience listening plane
would also be an effective tool to shape the ceiling panels. Some of these
metrics are extremely complex and rely on many different parameters. Analysis
for larger more complex venues with multiple reflections would require more
computing time to handle multiple reflections or coordination with a
comprehensive acoustic analysis program to provide a thorough analysis of
multiple audience points and with multiple reflections and the ability to calculate
the late sound energy used to fully define Clarity. Additionally, comprehensive
analysis software would allow for qualitative analysis with auralization software.
To fill the gap between design and actual performance, real time feedback may
be included with microphones at certain positions to analyze and confirm the
performance or make adjustments as needed. The real life feasibility of such a
system should be examined as well as many assumptions are made in this study
to make an idealized condition. The assembly and mechanics of such a system
will have its own effect on the acoustics and physical mockups should be made
to evaluate the actual acoustic conditions.
119
10 REFERENCES
Ando, Yoichi, Concert Hall Acoustics (New York, Springer, 1985)
Backus, John, The Acoustical Foundations of Music (London, WW Norton &
Company, 1976)
Barron, Michael, Auditorium Acoustics and Architectural Design, (New York,
Routledge, 1993)
Beranek, Leo, Music, Acoustics, and Architecture.(New York, John Wiley & Sons,
Inc. 1962)
Ellison, S, & Schwenke, R. “The Case for Widely Variable Acoustics”
Proceedings of the International Symposium on Room Acoustics, (2010)
Jaffe, J. Christopher, The Acoustics of Performance Halls: Spaces for Music from
Carnegie Hall to the Hollywood Bowl. (London, WW Norton & Company, 2010)
Levitin, Daniel J., This Is Your Brain on Music: The Science of a Human
Obsession,(New York, Penguin, 2006)
Long, Marshall, Architectural Acoustics (New York, Academic Press, 2006)
Peters, Brady, “Parametric Acoustic Surfaces” ACADIA (2009)
Roland, Conrad, Frei Otto: Tension Structures. (New York: Praeger Publications.
1970)
Sakamoto, Tomoko et. al. From Control to Design: Parametric/Algorithmic
Architecture. (Barcelona:,Actar-D, 2008)
120
11 APPENDIX
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 1.59 2.66 2.06 1.85 1.87 1.79 2.06 1.09
15 0.95 0.05 1.63 2.55 1.82 1.70 1.80 1.77 1.82 1.19
15 0.9 0.1 1.67 2.46 1.63 1.57 1.74 1.76 1.63 1.29
15 0.85 0.15 1.71 2.37 1.48 1.46 1.68 1.74 1.48 1.39
15 0.8 0.2 1.75 2.29 1.35 1.36 1.62 1.72 1.35 1.49
15 0.75 0.25 1.80 2.22 1.25 1.28 1.57 1.71 1.25 1.59
15 0.7 0.3 1.85 2.14 1.16 1.20 1.52 1.69 1.16 1.69
15 0.65 0.35 1.90 2.08 1.08 1.14 1.48 1.68 1.08 1.79
15 0.6 0.4 1.95 2.02 1.01 1.08 1.43 1.66 1.01 1.90
15 0.55 0.45 2.00 1.96 0.95 1.03 1.39 1.65 0.95 2.01
15 0.5 0.5 2.06 1.90 0.89 0.98 1.35 1.63 0.89 2.12
15 0.45 0.55 2.13 1.85 0.85 0.93 1.32 1.62 0.85 2.23
15 0.4 0.6 2.19 1.80 0.80 0.89 1.28 1.60 0.80 2.35
15 0.35 0.65 2.26 1.75 0.76 0.86 1.25 1.59 0.76 2.48
15 0.3 0.7 2.34 1.71 0.73 0.82 1.22 1.57 0.73 2.61
15 0.25 0.75 2.42 1.66 0.70 0.79 1.19 1.56 0.70 2.74
15 0.2 0.8 2.51 1.62 0.67 0.76 1.16 1.55 0.67 2.89
15 0.15 0.85 2.60 1.59 0.64 0.73 1.13 1.54 0.64 3.04
15 0.1 0.9 2.70 1.55 0.62 0.71 1.11 1.52 0.62 3.21
15 0.05 0.95 2.81 1.51 0.59 0.69 1.08 1.51 0.59 3.38
15 0 1 2.93 1.48 0.57 0.66 1.06 1.50 0.57 3.57
20 1 0 1.94 3.04 2.27 1.99 2.00 1.92 2.27 1.17
20 0.95 0.05 1.98 2.94 2.05 1.85 1.94 1.90 2.05 1.26
20 0.9 0.1 2.02 2.85 1.86 1.74 1.88 1.89 1.86 1.35
20 0.85 0.15 2.07 2.76 1.71 1.64 1.83 1.87 1.71 1.44
20 0.8 0.2 2.12 2.68 1.58 1.54 1.78 1.86 1.58 1.53
20 0.75 0.25 2.17 2.60 1.47 1.46 1.73 1.85 1.47 1.62
20 0.7 0.3 2.22 2.52 1.38 1.39 1.69 1.83 1.38 1.71
20 0.65 0.35 2.27 2.45 1.29 1.32 1.65 1.82 1.29 1.81
20 0.6 0.4 2.33 2.39 1.22 1.26 1.61 1.80 1.22 1.90
20 0.55 0.45 2.39 2.33 1.15 1.21 1.57 1.79 1.15 2.00
20 0.5 0.5 2.45 2.27 1.09 1.16 1.53 1.78 1.09 2.10
20 0.45 0.55 2.52 2.21 1.04 1.11 1.50 1.77 1.04 2.20
20 0.4 0.6 2.59 2.16 0.99 1.07 1.46 1.75 0.99 2.31
20 0.35 0.65 2.66 2.11 0.94 1.03 1.43 1.74 0.94 2.42
20 0.3 0.7 2.74 2.06 0.90 0.99 1.40 1.73 0.90 2.53
20 0.25 0.75 2.82 2.01 0.87 0.96 1.37 1.72 0.87 2.65
20 0.2 0.8 2.91 1.97 0.83 0.92 1.34 1.70 0.83 2.77
20 0.15 0.85 3.00 1.92 0.80 0.89 1.32 1.69 0.80 2.91
20 0.1 0.9 3.10 1.88 0.77 0.87 1.29 1.68 0.77 3.04
20 0.05 0.95 3.21 1.84 0.74 0.84 1.26 1.67 0.74 3.19
20 0 1 3.32 1.81 0.72 0.81 1.24 1.66 0.72 3.34
25 1 0 2.23 3.33 2.41 2.08 2.08 2.00 2.41 1.24
25 0.95 0.05 2.28 3.23 2.21 1.96 2.03 1.99 2.21 1.32
25 0.9 0.1 2.32 3.14 2.04 1.86 1.98 1.98 2.04 1.40
25 0.85 0.15 2.37 3.06 1.89 1.76 1.94 1.97 1.89 1.48
25 0.8 0.2 2.42 2.97 1.76 1.68 1.89 1.95 1.76 1.57
25 0.75 0.25 2.47 2.90 1.65 1.60 1.85 1.94 1.65 1.65
25 0.7 0.3 2.52 2.82 1.56 1.53 1.81 1.93 1.56 1.73
25 0.65 0.35 2.58 2.75 1.47 1.46 1.77 1.92 1.47 1.82
25 0.6 0.4 2.64 2.69 1.39 1.40 1.73 1.90 1.39 1.90
25 0.55 0.45 2.70 2.62 1.32 1.35 1.70 1.89 1.32 1.99
25 0.5 0.5 2.76 2.56 1.26 1.30 1.66 1.88 1.26 2.08
25 0.45 0.55 2.83 2.51 1.20 1.25 1.63 1.87 1.20 2.17
25 0.4 0.6 2.90 2.45 1.15 1.21 1.60 1.86 1.15 2.27
25 0.35 0.65 2.97 2.40 1.10 1.17 1.57 1.85 1.10 2.37
25 0.3 0.7 3.05 2.35 1.06 1.13 1.54 1.84 1.06 2.47
25 0.25 0.75 3.13 2.30 1.02 1.09 1.51 1.82 1.02 2.57
25 0.2 0.8 3.22 2.25 0.98 1.06 1.48 1.81 0.98 2.68
25 0.15 0.85 3.31 2.21 0.94 1.03 1.45 1.80 0.94 2.80
25 0.1 0.9 3.41 2.16 0.91 1.00 1.43 1.79 0.91 2.92
25 0.05 0.95 3.51 2.12 0.88 0.97 1.40 1.78 0.88 3.04
25 0 1 3.62 2.08 0.85 0.94 1.38 1.77 0.85 3.18
30 1 0 2.48 3.56 2.52 2.15 2.15 2.07 2.52 1.29
30 0.95 0.05 2.53 3.46 2.33 2.04 2.10 2.05 2.33 1.37
30 0.9 0.1 2.57 3.38 2.17 1.95 2.06 2.04 2.17 1.44
30 0.85 0.15 2.62 3.29 2.03 1.86 2.01 2.03 2.03 1.52
30 0.8 0.2 2.67 3.21 1.91 1.78 1.97 2.02 1.91 1.60
30 0.75 0.25 2.72 3.14 1.80 1.71 1.93 2.01 1.80 1.67
30 0.7 0.3 2.78 3.07 1.70 1.64 1.90 2.00 1.70 1.75
30 0.65 0.35 2.83 3.00 1.62 1.58 1.86 1.99 1.62 1.83
30 0.6 0.4 2.89 2.93 1.54 1.52 1.83 1.98 1.54 1.91
30 0.55 0.45 2.95 2.87 1.47 1.46 1.79 1.97 1.47 1.99
30 0.5 0.5 3.02 2.81 1.40 1.41 1.76 1.96 1.40 2.07
30 0.45 0.55 3.08 2.75 1.34 1.37 1.73 1.95 1.34 2.15
30 0.4 0.6 3.15 2.70 1.29 1.32 1.70 1.94 1.29 2.24
30 0.35 0.65 3.22 2.64 1.24 1.28 1.67 1.93 1.24 2.33
30 0.3 0.7 3.30 2.59 1.19 1.24 1.64 1.92 1.19 2.42
30 0.25 0.75 3.38 2.54 1.15 1.21 1.62 1.91 1.15 2.51
30 0.2 0.8 3.46 2.49 1.11 1.17 1.59 1.90 1.11 2.61
30 0.15 0.85 3.55 2.45 1.07 1.14 1.56 1.89 1.07 2.71
30 0.1 0.9 3.64 2.40 1.03 1.11 1.54 1.88 1.03 2.82
30 0.05 0.95 3.74 2.36 1.00 1.08 1.52 1.87 1.00 2.93
30 0 1 3.84 2.32 0.97 1.05 1.49 1.86 0.97 3.04
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.29 0.1 0.06 0.05 0.04 0.04
0.08 0.32 0.99 0.76 0.34 0.12
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Plasterboard and Pageboard
Plasterboard
Pageboard
Walls
Floor
Reverberation Times Variables
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Plasterboard and Pageboard
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 1.59 2.66 2.06 1.85 1.87 1.79 2.06 1.09
15 0.95 0.05 1.60 2.60 2.03 1.84 1.86 1.77 2.03 1.08
15 0.9 0.1 1.60 2.55 2.00 1.83 1.84 1.76 2.00 1.08
15 0.85 0.15 1.60 2.49 1.97 1.82 1.83 1.74 1.97 1.08
15 0.8 0.2 1.60 2.44 1.94 1.81 1.82 1.73 1.94 1.08
15 0.75 0.25 1.60 2.40 1.91 1.81 1.80 1.72 1.91 1.08
15 0.7 0.3 1.60 2.35 1.88 1.80 1.79 1.70 1.88 1.07
15 0.65 0.35 1.61 2.31 1.86 1.79 1.78 1.69 1.86 1.07
15 0.6 0.4 1.61 2.26 1.83 1.78 1.76 1.68 1.83 1.07
15 0.55 0.45 1.61 2.22 1.81 1.77 1.75 1.66 1.81 1.07
15 0.5 0.5 1.61 2.18 1.78 1.76 1.74 1.65 1.78 1.07
15 0.45 0.55 1.61 2.14 1.76 1.75 1.73 1.64 1.76 1.07
15 0.4 0.6 1.61 2.11 1.74 1.75 1.71 1.62 1.74 1.07
15 0.35 0.65 1.62 2.07 1.72 1.74 1.70 1.61 1.72 1.07
15 0.3 0.7 1.62 2.04 1.69 1.73 1.69 1.60 1.69 1.07
15 0.25 0.75 1.62 2.00 1.67 1.72 1.68 1.59 1.67 1.07
15 0.2 0.8 1.62 1.97 1.65 1.71 1.67 1.57 1.65 1.07
15 0.15 0.85 1.62 1.94 1.63 1.70 1.66 1.56 1.63 1.07
15 0.1 0.9 1.63 1.91 1.61 1.70 1.65 1.55 1.61 1.07
15 0.05 0.95 1.63 1.88 1.59 1.69 1.63 1.54 1.59 1.07
15 0 1 1.63 1.85 1.57 1.68 1.62 1.53 1.57 1.07
20 1 0 1.94 3.04 2.27 1.99 2.00 1.92 2.27 1.17
20 0.95 0.05 1.94 2.99 2.24 1.98 1.99 1.90 2.24 1.17
20 0.9 0.1 1.94 2.93 2.21 1.97 1.98 1.89 2.21 1.17
20 0.85 0.15 1.95 2.88 2.18 1.96 1.96 1.88 2.18 1.16
20 0.8 0.2 1.95 2.83 2.16 1.96 1.95 1.87 2.16 1.16
20 0.75 0.25 1.95 2.78 2.13 1.95 1.94 1.85 2.13 1.16
20 0.7 0.3 1.95 2.74 2.10 1.94 1.93 1.84 2.10 1.16
20 0.65 0.35 1.95 2.69 2.08 1.93 1.92 1.83 2.08 1.16
20 0.6 0.4 1.96 2.65 2.06 1.93 1.91 1.82 2.06 1.16
20 0.55 0.45 1.96 2.60 2.03 1.92 1.90 1.81 2.03 1.15
20 0.5 0.5 1.96 2.56 2.01 1.91 1.88 1.79 2.01 1.15
20 0.45 0.55 1.96 2.52 1.99 1.90 1.87 1.78 1.99 1.15
20 0.4 0.6 1.96 2.49 1.97 1.90 1.86 1.77 1.97 1.15
20 0.35 0.65 1.97 2.45 1.94 1.89 1.85 1.76 1.94 1.15
20 0.3 0.7 1.97 2.41 1.92 1.88 1.84 1.75 1.92 1.15
20 0.25 0.75 1.97 2.38 1.90 1.87 1.83 1.74 1.90 1.15
20 0.2 0.8 1.97 2.34 1.88 1.87 1.82 1.73 1.88 1.15
20 0.15 0.85 1.97 2.31 1.86 1.86 1.81 1.72 1.86 1.15
20 0.1 0.9 1.98 2.28 1.84 1.85 1.80 1.71 1.84 1.15
20 0.05 0.95 1.98 2.25 1.82 1.85 1.79 1.70 1.82 1.15
20 0 1 1.98 2.22 1.81 1.84 1.78 1.69 1.81 1.15
25 1 0 2.23 3.33 2.41 2.08 2.08 2.00 2.41 1.24
25 0.95 0.05 2.23 3.28 2.38 2.07 2.07 1.99 2.38 1.24
25 0.9 0.1 2.24 3.22 2.36 2.07 2.06 1.98 2.36 1.23
25 0.85 0.15 2.24 3.17 2.33 2.06 2.05 1.97 2.33 1.23
25 0.8 0.2 2.24 3.13 2.31 2.05 2.04 1.96 2.31 1.23
25 0.75 0.25 2.24 3.08 2.29 2.05 2.03 1.95 2.29 1.23
25 0.7 0.3 2.24 3.03 2.26 2.04 2.02 1.94 2.26 1.23
25 0.65 0.35 2.25 2.99 2.24 2.03 2.01 1.93 2.24 1.23
25 0.6 0.4 2.25 2.95 2.22 2.03 2.00 1.92 2.22 1.22
25 0.55 0.45 2.25 2.90 2.20 2.02 1.99 1.91 2.20 1.22
25 0.5 0.5 2.25 2.86 2.17 2.01 1.98 1.90 2.17 1.22
25 0.45 0.55 2.25 2.82 2.15 2.01 1.97 1.89 2.15 1.22
25 0.4 0.6 2.26 2.78 2.13 2.00 1.96 1.88 2.13 1.22
25 0.35 0.65 2.26 2.75 2.11 1.99 1.95 1.87 2.11 1.22
25 0.3 0.7 2.26 2.71 2.09 1.99 1.95 1.86 2.09 1.22
25 0.25 0.75 2.26 2.68 2.07 1.98 1.94 1.85 2.07 1.22
25 0.2 0.8 2.27 2.64 2.05 1.97 1.93 1.84 2.05 1.22
25 0.15 0.85 2.27 2.61 2.04 1.97 1.92 1.83 2.04 1.22
25 0.1 0.9 2.27 2.57 2.02 1.96 1.91 1.82 2.02 1.22
25 0.05 0.95 2.27 2.54 2.00 1.96 1.90 1.81 2.00 1.22
25 0 1 2.27 2.51 1.98 1.95 1.89 1.80 1.98 1.22
30 1 0 2.48 3.56 2.52 2.15 2.15 2.07 2.52 1.29
30 0.95 0.05 2.48 3.50 2.49 2.14 2.14 2.06 2.49 1.29
30 0.9 0.1 2.49 3.46 2.47 2.14 2.13 2.05 2.47 1.29
30 0.85 0.15 2.49 3.41 2.45 2.13 2.12 2.04 2.45 1.29
30 0.8 0.2 2.49 3.36 2.43 2.12 2.11 2.03 2.43 1.29
30 0.75 0.25 2.49 3.32 2.40 2.12 2.10 2.02 2.40 1.28
30 0.7 0.3 2.49 3.27 2.38 2.11 2.09 2.01 2.38 1.28
30 0.65 0.35 2.50 3.23 2.36 2.11 2.08 2.00 2.36 1.28
30 0.6 0.4 2.50 3.19 2.34 2.10 2.07 1.99 2.34 1.28
30 0.55 0.45 2.50 3.14 2.32 2.09 2.06 1.98 2.32 1.28
30 0.5 0.5 2.50 3.10 2.30 2.09 2.06 1.97 2.30 1.28
30 0.45 0.55 2.50 3.07 2.28 2.08 2.05 1.96 2.28 1.28
30 0.4 0.6 2.51 3.03 2.26 2.08 2.04 1.95 2.26 1.28
30 0.35 0.65 2.51 2.99 2.24 2.07 2.03 1.94 2.24 1.28
30 0.3 0.7 2.51 2.96 2.22 2.06 2.02 1.93 2.22 1.27
30 0.25 0.75 2.51 2.92 2.21 2.06 2.01 1.92 2.21 1.27
30 0.2 0.8 2.51 2.89 2.19 2.05 2.01 1.92 2.19 1.27
30 0.15 0.85 2.52 2.85 2.17 2.05 2.00 1.91 2.17 1.27
30 0.1 0.9 2.52 2.82 2.15 2.04 1.99 1.90 2.15 1.27
30 0.05 0.95 2.52 2.79 2.14 2.04 1.98 1.89 2.14 1.27
30 0 1 2.52 2.76 2.12 2.03 1.97 1.88 2.12 1.27
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.29 0.1 0.06 0.05 0.04 0.04
0.28 0.22 0.17 0.09 0.1 0.11
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Plasterboard and Plywood
Variables Reverberation Times
Plasterboard
3/8 in. Plywood
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Plasterboard and Plywood
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.93 1.48 0.57 0.66 1.06 1.50 0.57 3.57
15 0.95 0.05 2.81 1.49 0.59 0.68 1.08 1.50 0.59 3.38
15 0.9 0.1 2.71 1.51 0.61 0.71 1.10 1.50 0.61 3.20
15 0.85 0.15 2.61 1.53 0.63 0.73 1.12 1.50 0.63 3.04
15 0.8 0.2 2.52 1.54 0.65 0.76 1.14 1.50 0.65 2.88
15 0.75 0.25 2.44 1.56 0.68 0.78 1.16 1.51 0.68 2.74
15 0.7 0.3 2.36 1.57 0.71 0.81 1.18 1.51 0.71 2.59
15 0.65 0.35 2.29 1.59 0.74 0.84 1.21 1.51 0.74 2.46
15 0.6 0.4 2.22 1.61 0.77 0.88 1.23 1.51 0.77 2.33
15 0.55 0.45 2.15 1.63 0.80 0.91 1.26 1.51 0.80 2.21
15 0.5 0.5 2.09 1.65 0.84 0.95 1.28 1.51 0.84 2.09
15 0.45 0.55 2.03 1.66 0.88 1.00 1.31 1.51 0.88 1.97
15 0.4 0.6 1.98 1.68 0.92 1.04 1.34 1.52 0.92 1.86
15 0.35 0.65 1.93 1.70 0.97 1.09 1.37 1.52 0.97 1.75
15 0.3 0.7 1.88 1.72 1.03 1.15 1.40 1.52 1.03 1.65
15 0.25 0.75 1.83 1.74 1.09 1.22 1.43 1.52 1.09 1.55
15 0.2 0.8 1.79 1.76 1.16 1.29 1.47 1.52 1.16 1.45
15 0.15 0.85 1.74 1.79 1.25 1.37 1.50 1.52 1.25 1.35
15 0.1 0.9 1.70 1.81 1.34 1.46 1.54 1.53 1.34 1.26
15 0.05 0.95 1.67 1.83 1.45 1.56 1.58 1.53 1.45 1.16
15 0 1 1.63 1.85 1.57 1.68 1.62 1.53 1.57 1.07
20 1 0 3.32 1.81 0.72 0.81 1.24 1.66 0.72 3.34
20 0.95 0.05 3.21 1.82 0.74 0.84 1.26 1.66 0.74 3.19
20 0.9 0.1 3.11 1.84 0.77 0.86 1.28 1.66 0.77 3.04
20 0.85 0.15 3.02 1.86 0.79 0.89 1.30 1.66 0.79 2.90
20 0.8 0.2 2.93 1.88 0.82 0.92 1.32 1.66 0.82 2.77
20 0.75 0.25 2.84 1.89 0.85 0.95 1.34 1.67 0.85 2.64
20 0.7 0.3 2.76 1.91 0.88 0.98 1.36 1.67 0.88 2.52
20 0.65 0.35 2.68 1.93 0.91 1.01 1.39 1.67 0.91 2.40
20 0.6 0.4 2.61 1.95 0.95 1.05 1.41 1.67 0.95 2.29
20 0.55 0.45 2.55 1.97 0.99 1.09 1.44 1.67 0.99 2.18
20 0.5 0.5 2.48 1.99 1.03 1.13 1.46 1.67 1.03 2.07
20 0.45 0.55 2.42 2.01 1.08 1.17 1.49 1.67 1.08 1.97
20 0.4 0.6 2.36 2.03 1.13 1.22 1.52 1.68 1.13 1.87
20 0.35 0.65 2.31 2.05 1.18 1.28 1.55 1.68 1.18 1.77
20 0.3 0.7 2.25 2.07 1.24 1.34 1.58 1.68 1.24 1.68
20 0.25 0.75 2.20 2.10 1.31 1.40 1.61 1.68 1.31 1.59
20 0.2 0.8 2.15 2.12 1.39 1.47 1.64 1.68 1.39 1.50
20 0.15 0.85 2.11 2.14 1.47 1.55 1.67 1.68 1.47 1.41
20 0.1 0.9 2.06 2.17 1.57 1.63 1.71 1.68 1.57 1.32
20 0.05 0.95 2.02 2.19 1.68 1.73 1.74 1.69 1.68 1.24
20 0 1 1.98 2.22 1.81 1.84 1.78 1.69 1.81 1.15
25 1 0 3.62 2.08 0.85 0.94 1.38 1.77 0.85 3.18
25 0.95 0.05 3.51 2.10 0.88 0.97 1.40 1.77 0.88 3.04
25 0.9 0.1 3.41 2.12 0.90 0.99 1.42 1.77 0.90 2.92
25 0.85 0.15 3.32 2.14 0.93 1.02 1.44 1.78 0.93 2.79
25 0.8 0.2 3.23 2.16 0.96 1.05 1.46 1.78 0.96 2.68
25 0.75 0.25 3.15 2.18 0.99 1.08 1.48 1.78 0.99 2.57
25 0.7 0.3 3.07 2.20 1.03 1.12 1.50 1.78 1.03 2.46
25 0.65 0.35 3.00 2.22 1.06 1.15 1.52 1.78 1.06 2.35
25 0.6 0.4 2.93 2.24 1.10 1.19 1.55 1.78 1.10 2.25
25 0.55 0.45 2.86 2.26 1.15 1.23 1.57 1.78 1.15 2.15
25 0.5 0.5 2.79 2.28 1.19 1.27 1.60 1.78 1.19 2.06
25 0.45 0.55 2.73 2.30 1.24 1.32 1.62 1.79 1.24 1.97
25 0.4 0.6 2.67 2.32 1.29 1.37 1.65 1.79 1.29 1.88
25 0.35 0.65 2.61 2.34 1.35 1.42 1.67 1.79 1.35 1.79
25 0.3 0.7 2.56 2.37 1.42 1.48 1.70 1.79 1.42 1.70
25 0.25 0.75 2.51 2.39 1.49 1.54 1.73 1.79 1.49 1.62
25 0.2 0.8 2.46 2.41 1.57 1.61 1.76 1.79 1.57 1.53
25 0.15 0.85 2.41 2.44 1.65 1.68 1.79 1.79 1.65 1.45
25 0.1 0.9 2.36 2.46 1.75 1.76 1.82 1.80 1.75 1.37
25 0.05 0.95 2.32 2.49 1.86 1.85 1.86 1.80 1.86 1.29
25 0 1 2.27 2.51 1.98 1.95 1.89 1.80 1.98 1.22
30 1 0 3.84 2.32 0.97 1.05 1.49 1.86 0.97 3.04
30 0.95 0.05 3.75 2.34 1.00 1.08 1.51 1.86 1.00 2.93
30 0.9 0.1 3.65 2.36 1.03 1.11 1.53 1.86 1.03 2.82
30 0.85 0.15 3.56 2.38 1.06 1.14 1.55 1.86 1.06 2.71
30 0.8 0.2 3.48 2.40 1.09 1.17 1.57 1.86 1.09 2.61
30 0.75 0.25 3.40 2.42 1.12 1.20 1.59 1.86 1.12 2.50
30 0.7 0.3 3.32 2.44 1.16 1.23 1.61 1.86 1.16 2.41
30 0.65 0.35 3.25 2.46 1.20 1.27 1.63 1.87 1.20 2.31
30 0.6 0.4 3.18 2.48 1.24 1.31 1.65 1.87 1.24 2.22
30 0.55 0.45 3.11 2.50 1.28 1.35 1.68 1.87 1.28 2.13
30 0.5 0.5 3.05 2.52 1.33 1.39 1.70 1.87 1.33 2.05
30 0.45 0.55 2.98 2.54 1.38 1.43 1.72 1.87 1.38 1.96
30 0.4 0.6 2.93 2.56 1.44 1.48 1.75 1.87 1.44 1.88
30 0.35 0.65 2.87 2.59 1.50 1.53 1.77 1.87 1.50 1.80
30 0.3 0.7 2.81 2.61 1.56 1.59 1.80 1.87 1.56 1.72
30 0.25 0.75 2.76 2.63 1.64 1.65 1.83 1.87 1.64 1.64
30 0.2 0.8 2.71 2.66 1.71 1.71 1.85 1.88 1.71 1.57
30 0.15 0.85 2.66 2.68 1.80 1.78 1.88 1.88 1.80 1.49
30 0.1 0.9 2.61 2.71 1.89 1.86 1.91 1.88 1.89 1.42
30 0.05 0.95 2.57 2.73 2.00 1.94 1.94 1.88 2.00 1.34
30 0 1 2.52 2.76 2.12 2.03 1.97 1.88 2.12 1.27
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.08 0.32 0.99 0.76 0.34 0.12
0.28 0.22 0.17 0.09 0.1 0.11
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Pageboard and Plywood
Variables Reverberation Times
Pageboard
3/8 in. Plywood
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Bass Ratio
RT60 (sec)
Pageboard and Plywood
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.09 0.78 0.57 0.55 0.55 0.54 0.57 2.57
15 0.95 0.05 2.06 0.81 0.59 0.57 0.57 0.56 0.59 2.48
15 0.9 0.1 2.03 0.83 0.61 0.59 0.59 0.58 0.61 2.39
15 0.85 0.15 2.01 0.86 0.63 0.61 0.61 0.60 0.63 2.31
15 0.8 0.2 1.98 0.89 0.65 0.64 0.63 0.62 0.65 2.22
15 0.75 0.25 1.95 0.92 0.68 0.66 0.66 0.64 0.68 2.14
15 0.7 0.3 1.93 0.95 0.71 0.69 0.68 0.67 0.71 2.06
15 0.65 0.35 1.90 0.98 0.74 0.72 0.71 0.70 0.74 1.98
15 0.6 0.4 1.88 1.02 0.77 0.75 0.74 0.73 0.77 1.91
15 0.55 0.45 1.85 1.06 0.80 0.79 0.78 0.76 0.80 1.83
15 0.5 0.5 1.83 1.10 0.84 0.83 0.82 0.80 0.84 1.76
15 0.45 0.55 1.81 1.15 0.88 0.87 0.86 0.84 0.88 1.69
15 0.4 0.6 1.79 1.20 0.92 0.92 0.91 0.88 0.92 1.62
15 0.35 0.65 1.77 1.25 0.97 0.98 0.96 0.93 0.97 1.55
15 0.3 0.7 1.74 1.32 1.03 1.04 1.02 0.99 1.03 1.48
15 0.25 0.75 1.72 1.38 1.09 1.11 1.09 1.05 1.09 1.41
15 0.2 0.8 1.70 1.46 1.16 1.19 1.17 1.12 1.16 1.34
15 0.15 0.85 1.68 1.54 1.25 1.28 1.25 1.20 1.25 1.27
15 0.1 0.9 1.67 1.63 1.34 1.39 1.36 1.29 1.34 1.21
15 0.05 0.95 1.65 1.73 1.45 1.52 1.48 1.40 1.45 1.14
15 0 1 1.63 1.85 1.57 1.68 1.62 1.53 1.57 1.07
20 1 0 2.48 1.00 0.72 0.68 0.68 0.67 0.72 2.48
20 0.95 0.05 2.45 1.03 0.74 0.71 0.70 0.69 0.74 2.40
20 0.9 0.1 2.42 1.06 0.77 0.73 0.73 0.71 0.77 2.32
20 0.85 0.15 2.39 1.09 0.79 0.76 0.75 0.74 0.79 2.25
20 0.8 0.2 2.36 1.12 0.82 0.78 0.78 0.76 0.82 2.18
20 0.75 0.25 2.33 1.16 0.85 0.81 0.80 0.79 0.85 2.10
20 0.7 0.3 2.31 1.19 0.88 0.84 0.84 0.82 0.88 2.03
20 0.65 0.35 2.28 1.24 0.91 0.88 0.87 0.85 0.91 1.97
20 0.6 0.4 2.25 1.28 0.95 0.91 0.90 0.88 0.95 1.90
20 0.55 0.45 2.23 1.33 0.99 0.95 0.94 0.92 0.99 1.83
20 0.5 0.5 2.20 1.38 1.03 1.00 0.98 0.96 1.03 1.77
20 0.45 0.55 2.18 1.43 1.08 1.04 1.03 1.00 1.08 1.70
20 0.4 0.6 2.15 1.49 1.13 1.10 1.08 1.05 1.13 1.64
20 0.35 0.65 2.13 1.55 1.18 1.16 1.14 1.10 1.18 1.58
20 0.3 0.7 2.11 1.62 1.24 1.22 1.20 1.16 1.24 1.51
20 0.25 0.75 2.09 1.70 1.31 1.29 1.27 1.22 1.31 1.45
20 0.2 0.8 2.06 1.78 1.39 1.37 1.35 1.29 1.39 1.39
20 0.15 0.85 2.04 1.87 1.47 1.47 1.43 1.37 1.47 1.33
20 0.1 0.9 2.02 1.97 1.57 1.57 1.53 1.46 1.57 1.27
20 0.05 0.95 2.00 2.09 1.68 1.70 1.65 1.57 1.68 1.21
20 0 1 1.98 2.22 1.81 1.84 1.78 1.69 1.81 1.15
25 1 0 2.79 1.19 0.85 0.80 0.80 0.78 0.85 2.41
25 0.95 0.05 2.76 1.22 0.88 0.83 0.82 0.81 0.88 2.34
25 0.9 0.1 2.73 1.26 0.90 0.85 0.85 0.83 0.90 2.27
25 0.85 0.15 2.70 1.29 0.93 0.88 0.87 0.86 0.93 2.21
25 0.8 0.2 2.67 1.33 0.96 0.91 0.90 0.88 0.96 2.14
25 0.75 0.25 2.64 1.37 0.99 0.94 0.93 0.91 0.99 2.08
25 0.7 0.3 2.61 1.41 1.03 0.97 0.96 0.94 1.03 2.01
25 0.65 0.35 2.59 1.46 1.06 1.01 1.00 0.98 1.06 1.95
25 0.6 0.4 2.56 1.51 1.10 1.05 1.04 1.01 1.10 1.89
25 0.55 0.45 2.53 1.56 1.15 1.09 1.08 1.05 1.15 1.83
25 0.5 0.5 2.51 1.62 1.19 1.14 1.12 1.09 1.19 1.77
25 0.45 0.55 2.48 1.68 1.24 1.19 1.17 1.14 1.24 1.71
25 0.4 0.6 2.46 1.74 1.29 1.24 1.22 1.19 1.29 1.66
25 0.35 0.65 2.43 1.81 1.35 1.30 1.28 1.24 1.35 1.60
25 0.3 0.7 2.41 1.89 1.42 1.36 1.34 1.30 1.42 1.54
25 0.25 0.75 2.38 1.97 1.49 1.44 1.41 1.36 1.49 1.49
25 0.2 0.8 2.36 2.06 1.57 1.52 1.48 1.43 1.57 1.43
25 0.15 0.85 2.34 2.15 1.65 1.60 1.57 1.51 1.65 1.38
25 0.1 0.9 2.32 2.26 1.75 1.70 1.66 1.59 1.75 1.33
25 0.05 0.95 2.30 2.38 1.86 1.82 1.77 1.69 1.86 1.27
25 0 1 2.27 2.51 1.98 1.95 1.89 1.80 1.98 1.22
30 1 0 3.05 1.37 0.97 0.91 0.90 0.88 0.97 2.35
30 0.95 0.05 3.02 1.40 1.00 0.93 0.92 0.91 1.00 2.29
30 0.9 0.1 2.98 1.44 1.03 0.96 0.95 0.93 1.03 2.23
30 0.85 0.15 2.95 1.48 1.06 0.99 0.98 0.96 1.06 2.17
30 0.8 0.2 2.93 1.52 1.09 1.02 1.01 0.99 1.09 2.11
30 0.75 0.25 2.90 1.57 1.12 1.05 1.04 1.02 1.12 2.05
30 0.7 0.3 2.87 1.61 1.16 1.09 1.07 1.05 1.16 2.00
30 0.65 0.35 2.84 1.66 1.20 1.12 1.11 1.09 1.20 1.94
30 0.6 0.4 2.81 1.71 1.24 1.16 1.15 1.12 1.24 1.88
30 0.55 0.45 2.79 1.77 1.28 1.21 1.19 1.16 1.28 1.83
30 0.5 0.5 2.76 1.83 1.33 1.25 1.24 1.20 1.33 1.78
30 0.45 0.55 2.73 1.89 1.38 1.30 1.28 1.25 1.38 1.72
30 0.4 0.6 2.71 1.96 1.44 1.36 1.33 1.30 1.44 1.67
30 0.35 0.65 2.68 2.03 1.50 1.41 1.39 1.35 1.50 1.62
30 0.3 0.7 2.66 2.11 1.56 1.48 1.45 1.41 1.56 1.57
30 0.25 0.75 2.64 2.20 1.64 1.55 1.52 1.47 1.64 1.52
30 0.2 0.8 2.61 2.29 1.71 1.63 1.59 1.54 1.71 1.47
30 0.15 0.85 2.59 2.39 1.80 1.71 1.67 1.61 1.80 1.42
30 0.1 0.9 2.57 2.50 1.89 1.81 1.76 1.69 1.89 1.37
30 0.05 0.95 2.55 2.62 2.00 1.91 1.86 1.78 2.00 1.32
30 0 1 2.52 2.76 2.12 2.03 1.97 1.88 2.12 1.27
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.18 0.76 0.99 0.99 0.99 0.99
0.28 0.22 0.17 0.09 0.1 0.11
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Fiberglass and Plywood
Variables Reverberation Times
2" Fiberglass Board
3/8 in. Plywood
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Fiberglass and Plywood
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.09 0.78 0.57 0.55 0.55 0.54 0.57 2.57
15 0.95 0.05 2.06 0.81 0.59 0.57 0.57 0.56 0.59 2.47
15 0.9 0.1 2.03 0.84 0.62 0.59 0.59 0.58 0.62 2.38
15 0.85 0.15 2.00 0.88 0.64 0.61 0.61 0.60 0.64 2.29
15 0.8 0.2 1.97 0.91 0.67 0.64 0.64 0.63 0.67 2.21
15 0.75 0.25 1.94 0.95 0.70 0.67 0.67 0.65 0.70 2.12
15 0.7 0.3 1.91 0.99 0.73 0.70 0.69 0.68 0.73 2.04
15 0.65 0.35 1.89 1.04 0.76 0.73 0.73 0.71 0.76 1.96
15 0.6 0.4 1.86 1.09 0.80 0.76 0.76 0.75 0.80 1.88
15 0.55 0.45 1.83 1.15 0.85 0.80 0.80 0.79 0.85 1.81
15 0.5 0.5 1.81 1.21 0.89 0.85 0.85 0.83 0.89 1.73
15 0.45 0.55 1.79 1.28 0.95 0.90 0.90 0.88 0.95 1.66
15 0.4 0.6 1.76 1.36 1.01 0.95 0.95 0.93 1.01 1.59
15 0.35 0.65 1.74 1.45 1.08 1.01 1.01 0.99 1.08 1.53
15 0.3 0.7 1.72 1.55 1.16 1.08 1.08 1.06 1.16 1.46
15 0.25 0.75 1.69 1.66 1.25 1.16 1.17 1.13 1.25 1.39
15 0.2 0.8 1.67 1.80 1.35 1.26 1.26 1.22 1.35 1.33
15 0.15 0.85 1.65 1.96 1.48 1.37 1.37 1.33 1.48 1.27
15 0.1 0.9 1.63 2.14 1.63 1.50 1.51 1.45 1.63 1.21
15 0.05 0.95 1.61 2.37 1.82 1.65 1.67 1.60 1.82 1.15
15 0 1 1.59 2.66 2.06 1.85 1.87 1.79 2.06 1.09
20 1 0 2.48 1.00 0.72 0.68 0.68 0.67 0.72 2.48
20 0.95 0.05 2.45 1.03 0.74 0.71 0.70 0.69 0.74 2.40
20 0.9 0.1 2.41 1.07 0.77 0.73 0.73 0.72 0.77 2.32
20 0.85 0.15 2.38 1.11 0.80 0.76 0.76 0.74 0.80 2.24
20 0.8 0.2 2.35 1.15 0.83 0.79 0.78 0.77 0.83 2.16
20 0.75 0.25 2.32 1.20 0.87 0.82 0.81 0.80 0.87 2.09
20 0.7 0.3 2.29 1.25 0.90 0.85 0.85 0.83 0.90 2.02
20 0.65 0.35 2.26 1.30 0.94 0.89 0.88 0.87 0.94 1.95
20 0.6 0.4 2.23 1.36 0.99 0.93 0.92 0.91 0.99 1.88
20 0.55 0.45 2.20 1.43 1.04 0.97 0.97 0.95 1.04 1.81
20 0.5 0.5 2.18 1.50 1.09 1.02 1.02 0.99 1.09 1.74
20 0.45 0.55 2.15 1.58 1.15 1.07 1.07 1.04 1.15 1.68
20 0.4 0.6 2.13 1.67 1.22 1.13 1.13 1.10 1.22 1.62
20 0.35 0.65 2.10 1.77 1.29 1.19 1.19 1.16 1.29 1.56
20 0.3 0.7 2.08 1.88 1.38 1.26 1.26 1.23 1.38 1.50
20 0.25 0.75 2.05 2.01 1.47 1.35 1.35 1.31 1.47 1.44
20 0.2 0.8 2.03 2.16 1.58 1.44 1.44 1.40 1.58 1.38
20 0.15 0.85 2.01 2.33 1.71 1.55 1.55 1.50 1.71 1.33
20 0.1 0.9 1.98 2.52 1.86 1.67 1.67 1.62 1.86 1.28
20 0.05 0.95 1.96 2.76 2.05 1.82 1.82 1.75 2.05 1.22
20 0 1 1.94 3.04 2.27 1.99 2.00 1.92 2.27 1.17
25 1 0 2.79 1.19 0.85 0.80 0.80 0.78 0.85 2.41
25 0.95 0.05 2.76 1.23 0.88 0.83 0.82 0.81 0.88 2.34
25 0.9 0.1 2.72 1.27 0.91 0.85 0.85 0.84 0.91 2.27
25 0.85 0.15 2.69 1.32 0.94 0.88 0.88 0.86 0.94 2.20
25 0.8 0.2 2.66 1.37 0.98 0.91 0.91 0.89 0.98 2.13
25 0.75 0.25 2.63 1.42 1.02 0.95 0.94 0.93 1.02 2.06
25 0.7 0.3 2.60 1.48 1.06 0.98 0.98 0.96 1.06 2.00
25 0.65 0.35 2.57 1.54 1.10 1.02 1.02 1.00 1.10 1.93
25 0.6 0.4 2.54 1.60 1.15 1.06 1.06 1.04 1.15 1.87
25 0.55 0.45 2.51 1.68 1.20 1.11 1.10 1.08 1.20 1.81
25 0.5 0.5 2.48 1.76 1.26 1.16 1.15 1.13 1.26 1.75
25 0.45 0.55 2.45 1.84 1.32 1.21 1.21 1.18 1.32 1.70
25 0.4 0.6 2.43 1.94 1.39 1.27 1.27 1.24 1.39 1.64
25 0.35 0.65 2.40 2.05 1.47 1.34 1.33 1.30 1.47 1.59
25 0.3 0.7 2.38 2.16 1.56 1.41 1.40 1.37 1.56 1.53
25 0.25 0.75 2.35 2.30 1.65 1.49 1.48 1.44 1.65 1.48
25 0.2 0.8 2.33 2.45 1.76 1.58 1.58 1.53 1.76 1.43
25 0.15 0.85 2.30 2.62 1.89 1.68 1.68 1.63 1.89 1.38
25 0.1 0.9 2.28 2.82 2.04 1.79 1.79 1.73 2.04 1.33
25 0.05 0.95 2.25 3.06 2.21 1.93 1.93 1.86 2.21 1.28
25 0 1 2.23 3.33 2.41 2.08 2.08 2.00 2.41 1.24
30 1 0 3.05 1.37 0.97 0.91 0.90 0.88 0.97 2.35
30 0.95 0.05 3.01 1.41 1.00 0.93 0.93 0.91 1.00 2.29
30 0.9 0.1 2.98 1.46 1.03 0.96 0.95 0.94 1.03 2.22
30 0.85 0.15 2.95 1.51 1.07 0.99 0.98 0.97 1.07 2.16
30 0.8 0.2 2.91 1.56 1.11 1.02 1.02 1.00 1.11 2.10
30 0.75 0.25 2.88 1.62 1.15 1.06 1.05 1.03 1.15 2.04
30 0.7 0.3 2.85 1.68 1.19 1.10 1.09 1.07 1.19 1.98
30 0.65 0.35 2.82 1.74 1.24 1.13 1.13 1.11 1.24 1.93
30 0.6 0.4 2.79 1.82 1.29 1.18 1.17 1.15 1.29 1.87
30 0.55 0.45 2.76 1.89 1.34 1.22 1.22 1.19 1.34 1.81
30 0.5 0.5 2.73 1.98 1.40 1.27 1.27 1.24 1.40 1.76
30 0.45 0.55 2.71 2.07 1.47 1.33 1.32 1.29 1.47 1.71
30 0.4 0.6 2.68 2.17 1.54 1.39 1.38 1.35 1.54 1.66
30 0.35 0.65 2.65 2.28 1.62 1.45 1.44 1.41 1.62 1.61
30 0.3 0.7 2.63 2.40 1.70 1.52 1.52 1.48 1.70 1.56
30 0.25 0.75 2.60 2.54 1.80 1.60 1.59 1.55 1.80 1.51
30 0.2 0.8 2.58 2.70 1.91 1.69 1.68 1.63 1.91 1.47
30 0.15 0.85 2.55 2.87 2.03 1.78 1.78 1.72 2.03 1.42
30 0.1 0.9 2.53 3.07 2.17 1.89 1.88 1.82 2.17 1.38
30 0.05 0.95 2.50 3.29 2.33 2.01 2.01 1.94 2.33 1.34
30 0 1 2.48 3.56 2.52 2.15 2.15 2.07 2.52 1.29
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.18 0.76 0.99 0.99 0.99 0.99
0.29 0.1 0.06 0.05 0.04 0.04
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Fiberglass and Plasterboard
Variables Reverberation Times
2" Fiberglass Board
Plasterboard
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Fiberglass and Plasterboard
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.09 0.78 0.57 0.55 0.55 0.54 0.57 2.57
15 0.95 0.05 2.12 0.80 0.57 0.55 0.56 0.56 0.57 2.60
15 0.9 0.1 2.15 0.82 0.57 0.56 0.58 0.58 0.57 2.63
15 0.85 0.15 2.19 0.84 0.57 0.56 0.59 0.60 0.57 2.67
15 0.8 0.2 2.22 0.87 0.57 0.57 0.61 0.62 0.57 2.71
15 0.75 0.25 2.25 0.89 0.57 0.57 0.62 0.64 0.57 2.74
15 0.7 0.3 2.29 0.91 0.57 0.58 0.64 0.67 0.57 2.78
15 0.65 0.35 2.32 0.94 0.57 0.58 0.66 0.70 0.57 2.82
15 0.6 0.4 2.36 0.97 0.57 0.59 0.68 0.73 0.57 2.87
15 0.55 0.45 2.40 0.99 0.57 0.60 0.70 0.76 0.57 2.91
15 0.5 0.5 2.44 1.03 0.57 0.60 0.72 0.79 0.57 2.96
15 0.45 0.55 2.48 1.06 0.57 0.61 0.75 0.83 0.57 3.00
15 0.4 0.6 2.52 1.09 0.57 0.61 0.77 0.88 0.57 3.05
15 0.35 0.65 2.57 1.13 0.57 0.62 0.80 0.92 0.57 3.11
15 0.3 0.7 2.61 1.17 0.57 0.62 0.83 0.98 0.57 3.16
15 0.25 0.75 2.66 1.21 0.57 0.63 0.86 1.04 0.57 3.22
15 0.2 0.8 2.71 1.26 0.57 0.64 0.89 1.11 0.57 3.28
15 0.15 0.85 2.76 1.31 0.57 0.64 0.93 1.18 0.57 3.35
15 0.1 0.9 2.81 1.36 0.57 0.65 0.97 1.27 0.57 3.42
15 0.05 0.95 2.87 1.42 0.57 0.66 1.01 1.38 0.57 3.49
15 0 1 2.93 1.48 0.57 0.66 1.06 1.50 0.57 3.57
20 1 0 2.48 1.00 0.72 0.68 0.68 0.67 0.72 2.48
20 0.95 0.05 2.51 1.02 0.72 0.69 0.70 0.69 0.72 2.51
20 0.9 0.1 2.55 1.04 0.72 0.70 0.71 0.71 0.72 2.54
20 0.85 0.15 2.58 1.07 0.72 0.70 0.73 0.74 0.72 2.57
20 0.8 0.2 2.61 1.10 0.72 0.71 0.75 0.76 0.72 2.60
20 0.75 0.25 2.65 1.12 0.72 0.71 0.77 0.79 0.72 2.63
20 0.7 0.3 2.68 1.15 0.72 0.72 0.79 0.82 0.72 2.67
20 0.65 0.35 2.72 1.18 0.72 0.72 0.81 0.85 0.72 2.70
20 0.6 0.4 2.76 1.22 0.72 0.73 0.83 0.88 0.72 2.74
20 0.55 0.45 2.80 1.25 0.72 0.74 0.85 0.92 0.72 2.78
20 0.5 0.5 2.84 1.29 0.72 0.74 0.88 0.96 0.72 2.82
20 0.45 0.55 2.88 1.32 0.72 0.75 0.91 1.00 0.72 2.86
20 0.4 0.6 2.93 1.36 0.72 0.76 0.93 1.04 0.72 2.91
20 0.35 0.65 2.97 1.41 0.72 0.76 0.96 1.09 0.72 2.95
20 0.3 0.7 3.02 1.45 0.72 0.77 0.99 1.15 0.72 3.00
20 0.25 0.75 3.06 1.50 0.72 0.78 1.03 1.21 0.72 3.05
20 0.2 0.8 3.11 1.55 0.72 0.78 1.07 1.28 0.72 3.10
20 0.15 0.85 3.16 1.61 0.72 0.79 1.10 1.36 0.72 3.16
20 0.1 0.9 3.21 1.67 0.72 0.80 1.15 1.45 0.72 3.22
20 0.05 0.95 3.27 1.74 0.72 0.81 1.19 1.54 0.72 3.28
20 0 1 3.32 1.81 0.72 0.81 1.24 1.66 0.72 3.34
25 1 0 2.79 1.19 0.85 0.80 0.80 0.78 0.85 2.41
25 0.95 0.05 2.82 1.22 0.85 0.81 0.81 0.81 0.85 2.44
25 0.9 0.1 2.86 1.25 0.85 0.81 0.83 0.83 0.85 2.46
25 0.85 0.15 2.89 1.27 0.85 0.82 0.85 0.86 0.85 2.49
25 0.8 0.2 2.93 1.30 0.85 0.83 0.87 0.88 0.85 2.52
25 0.75 0.25 2.96 1.33 0.85 0.83 0.89 0.91 0.85 2.55
25 0.7 0.3 3.00 1.37 0.85 0.84 0.91 0.94 0.85 2.58
25 0.65 0.35 3.03 1.40 0.85 0.85 0.94 0.97 0.85 2.61
25 0.6 0.4 3.07 1.44 0.85 0.85 0.96 1.01 0.85 2.65
25 0.55 0.45 3.11 1.48 0.85 0.86 0.98 1.05 0.85 2.68
25 0.5 0.5 3.15 1.52 0.85 0.87 1.01 1.09 0.85 2.72
25 0.45 0.55 3.19 1.56 0.85 0.87 1.04 1.13 0.85 2.75
25 0.4 0.6 3.23 1.60 0.85 0.88 1.07 1.18 0.85 2.79
25 0.35 0.65 3.28 1.65 0.85 0.89 1.10 1.23 0.85 2.83
25 0.3 0.7 3.32 1.70 0.85 0.90 1.13 1.29 0.85 2.87
25 0.25 0.75 3.37 1.76 0.85 0.90 1.17 1.35 0.85 2.92
25 0.2 0.8 3.41 1.81 0.85 0.91 1.20 1.42 0.85 2.97
25 0.15 0.85 3.46 1.87 0.85 0.92 1.24 1.49 0.85 3.01
25 0.1 0.9 3.51 1.94 0.85 0.93 1.29 1.57 0.85 3.06
25 0.05 0.95 3.56 2.01 0.85 0.94 1.33 1.67 0.85 3.12
25 0 1 3.62 2.08 0.85 0.94 1.38 1.77 0.85 3.18
30 1 0 3.05 1.37 0.97 0.91 0.90 0.88 0.97 2.35
30 0.95 0.05 3.08 1.40 0.97 0.91 0.92 0.91 0.97 2.38
30 0.9 0.1 3.11 1.43 0.97 0.92 0.94 0.93 0.97 2.40
30 0.85 0.15 3.14 1.46 0.97 0.92 0.96 0.96 0.97 2.43
30 0.8 0.2 3.18 1.49 0.97 0.93 0.98 0.99 0.97 2.45
30 0.75 0.25 3.21 1.53 0.97 0.94 1.00 1.02 0.97 2.48
30 0.7 0.3 3.25 1.56 0.97 0.95 1.02 1.05 0.97 2.51
30 0.65 0.35 3.28 1.60 0.97 0.95 1.04 1.08 0.97 2.54
30 0.6 0.4 3.32 1.64 0.97 0.96 1.07 1.12 0.97 2.57
30 0.55 0.45 3.36 1.68 0.97 0.97 1.09 1.16 0.97 2.60
30 0.5 0.5 3.40 1.72 0.97 0.97 1.12 1.20 0.97 2.63
30 0.45 0.55 3.44 1.77 0.97 0.98 1.15 1.24 0.97 2.67
30 0.4 0.6 3.48 1.82 0.97 0.99 1.18 1.29 0.97 2.70
30 0.35 0.65 3.52 1.87 0.97 1.00 1.21 1.34 0.97 2.74
30 0.3 0.7 3.56 1.92 0.97 1.00 1.25 1.40 0.97 2.78
30 0.25 0.75 3.61 1.98 0.97 1.01 1.28 1.46 0.97 2.82
30 0.2 0.8 3.65 2.04 0.97 1.02 1.32 1.52 0.97 2.86
30 0.15 0.85 3.70 2.10 0.97 1.03 1.36 1.59 0.97 2.90
30 0.1 0.9 3.75 2.17 0.97 1.04 1.40 1.67 0.97 2.95
30 0.05 0.95 3.79 2.24 0.97 1.05 1.44 1.76 0.97 2.99
30 0 1 3.84 2.32 0.97 1.05 1.49 1.86 0.97 3.04
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.18 0.76 0.99 0.99 0.99 0.99
0.08 0.32 0.99 0.76 0.34 0.12
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Fiberglass and Pageboard
Variables Reverberation Times
2" Fiberglass Board
Pageboard
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Fiberglass and Pageboard
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.09 0.78 0.57 0.55 0.55 0.54 0.57 2.57
15 0.95 0.05 2.14 0.82 0.59 0.57 0.57 0.56 0.59 2.54
15 0.9 0.1 2.18 0.85 0.62 0.59 0.59 0.58 0.62 2.50
15 0.85 0.15 2.23 0.89 0.64 0.62 0.61 0.60 0.64 2.47
15 0.8 0.2 2.27 0.93 0.67 0.64 0.64 0.63 0.67 2.44
15 0.75 0.25 2.32 0.97 0.70 0.67 0.67 0.66 0.70 2.40
15 0.7 0.3 2.38 1.02 0.74 0.70 0.70 0.69 0.74 2.36
15 0.65 0.35 2.43 1.07 0.77 0.73 0.73 0.72 0.77 2.33
15 0.6 0.4 2.49 1.13 0.81 0.77 0.77 0.76 0.81 2.29
15 0.55 0.45 2.55 1.20 0.86 0.81 0.81 0.80 0.86 2.24
15 0.5 0.5 2.61 1.27 0.91 0.86 0.86 0.84 0.91 2.20
15 0.45 0.55 2.68 1.36 0.97 0.91 0.91 0.89 0.97 2.15
15 0.4 0.6 2.75 1.46 1.03 0.97 0.97 0.94 1.03 2.10
15 0.35 0.65 2.82 1.57 1.11 1.03 1.03 1.01 1.11 2.05
15 0.3 0.7 2.90 1.70 1.20 1.11 1.11 1.08 1.20 2.00
15 0.25 0.75 2.98 1.85 1.30 1.19 1.19 1.16 1.30 1.94
15 0.2 0.8 3.07 2.04 1.42 1.29 1.30 1.26 1.42 1.89
15 0.15 0.85 3.16 2.26 1.56 1.41 1.42 1.37 1.56 1.83
15 0.1 0.9 3.26 2.55 1.74 1.55 1.57 1.51 1.74 1.76
15 0.05 0.95 3.37 2.91 1.96 1.73 1.75 1.67 1.96 1.70
15 0 1 3.48 3.39 2.25 1.95 1.97 1.88 2.25 1.64
20 1 0 2.48 1.00 0.72 0.68 0.68 0.67 0.72 2.48
20 0.95 0.05 2.53 1.04 0.75 0.71 0.70 0.69 0.75 2.45
20 0.9 0.1 2.57 1.08 0.77 0.73 0.73 0.72 0.77 2.42
20 0.85 0.15 2.62 1.12 0.80 0.76 0.76 0.74 0.80 2.39
20 0.8 0.2 2.67 1.17 0.84 0.79 0.79 0.77 0.84 2.36
20 0.75 0.25 2.72 1.22 0.87 0.82 0.82 0.80 0.87 2.33
20 0.7 0.3 2.78 1.28 0.91 0.86 0.85 0.84 0.91 2.29
20 0.65 0.35 2.83 1.34 0.95 0.89 0.89 0.87 0.95 2.26
20 0.6 0.4 2.89 1.41 1.00 0.93 0.93 0.91 1.00 2.22
20 0.55 0.45 2.95 1.49 1.05 0.98 0.98 0.96 1.05 2.18
20 0.5 0.5 3.02 1.57 1.11 1.03 1.03 1.00 1.11 2.15
20 0.45 0.55 3.08 1.67 1.17 1.08 1.08 1.06 1.17 2.11
20 0.4 0.6 3.15 1.78 1.25 1.14 1.14 1.11 1.25 2.06
20 0.35 0.65 3.22 1.91 1.33 1.21 1.21 1.18 1.33 2.02
20 0.3 0.7 3.30 2.05 1.42 1.29 1.29 1.25 1.42 1.98
20 0.25 0.75 3.38 2.22 1.53 1.37 1.38 1.34 1.53 1.93
20 0.2 0.8 3.46 2.41 1.65 1.47 1.48 1.43 1.65 1.88
20 0.15 0.85 3.55 2.65 1.79 1.59 1.59 1.54 1.79 1.83
20 0.1 0.9 3.64 2.93 1.97 1.72 1.73 1.67 1.97 1.78
20 0.05 0.95 3.74 3.29 2.17 1.88 1.89 1.82 2.17 1.73
20 0 1 3.84 3.74 2.43 2.07 2.08 2.00 2.43 1.68
25 1 0 2.79 1.19 0.85 0.80 0.80 0.78 0.85 2.41
25 0.95 0.05 2.84 1.24 0.88 0.83 0.82 0.81 0.88 2.38
25 0.9 0.1 2.88 1.28 0.91 0.86 0.85 0.84 0.91 2.36
25 0.85 0.15 2.93 1.33 0.95 0.88 0.88 0.87 0.95 2.33
25 0.8 0.2 2.98 1.39 0.98 0.92 0.91 0.90 0.98 2.30
25 0.75 0.25 3.03 1.44 1.02 0.95 0.95 0.93 1.02 2.27
25 0.7 0.3 3.09 1.51 1.06 0.99 0.98 0.96 1.06 2.24
25 0.65 0.35 3.14 1.58 1.11 1.03 1.02 1.00 1.11 2.21
25 0.6 0.4 3.20 1.66 1.16 1.07 1.07 1.04 1.16 2.18
25 0.55 0.45 3.26 1.74 1.22 1.12 1.11 1.09 1.22 2.14
25 0.5 0.5 3.32 1.83 1.28 1.17 1.16 1.14 1.28 2.11
25 0.45 0.55 3.39 1.94 1.35 1.22 1.22 1.19 1.35 2.07
25 0.4 0.6 3.45 2.06 1.42 1.29 1.28 1.25 1.42 2.04
25 0.35 0.65 3.52 2.19 1.50 1.35 1.35 1.32 1.50 2.00
25 0.3 0.7 3.60 2.34 1.60 1.43 1.43 1.39 1.60 1.96
25 0.25 0.75 3.67 2.51 1.71 1.52 1.51 1.47 1.71 1.92
25 0.2 0.8 3.75 2.71 1.83 1.61 1.61 1.56 1.83 1.88
25 0.15 0.85 3.83 2.95 1.97 1.72 1.72 1.66 1.97 1.84
25 0.1 0.9 3.92 3.22 2.13 1.84 1.84 1.78 2.13 1.80
25 0.05 0.95 4.01 3.56 2.33 1.99 1.99 1.91 2.33 1.75
25 0 1 4.10 3.98 2.56 2.15 2.16 2.07 2.56 1.71
30 1 0 3.05 1.37 0.97 0.91 0.90 0.88 0.97 2.35
30 0.95 0.05 3.09 1.42 1.00 0.93 0.93 0.91 1.00 2.33
30 0.9 0.1 3.14 1.47 1.04 0.96 0.96 0.94 1.04 2.30
30 0.85 0.15 3.19 1.52 1.07 0.99 0.99 0.97 1.07 2.28
30 0.8 0.2 3.23 1.58 1.11 1.03 1.02 1.00 1.11 2.25
30 0.75 0.25 3.28 1.65 1.15 1.06 1.06 1.04 1.15 2.22
30 0.7 0.3 3.34 1.71 1.20 1.10 1.09 1.07 1.20 2.20
30 0.65 0.35 3.39 1.79 1.25 1.14 1.13 1.11 1.25 2.17
30 0.6 0.4 3.45 1.87 1.30 1.19 1.18 1.15 1.30 2.14
30 0.55 0.45 3.50 1.96 1.36 1.23 1.23 1.20 1.36 2.11
30 0.5 0.5 3.56 2.06 1.42 1.28 1.28 1.25 1.42 2.08
30 0.45 0.55 3.63 2.17 1.49 1.34 1.33 1.30 1.49 2.05
30 0.4 0.6 3.69 2.29 1.57 1.40 1.40 1.36 1.57 2.01
30 0.35 0.65 3.75 2.43 1.65 1.47 1.46 1.43 1.65 1.98
30 0.3 0.7 3.82 2.58 1.75 1.54 1.54 1.50 1.75 1.95
30 0.25 0.75 3.89 2.76 1.85 1.63 1.62 1.57 1.85 1.91
30 0.2 0.8 3.97 2.96 1.97 1.72 1.71 1.66 1.97 1.88
30 0.15 0.85 4.04 3.19 2.11 1.82 1.81 1.76 2.11 1.84
30 0.1 0.9 4.12 3.46 2.26 1.93 1.93 1.86 2.26 1.81
30 0.05 0.95 4.21 3.77 2.44 2.06 2.06 1.99 2.44 1.77
30 0 1 4.29 4.16 2.65 2.21 2.21 2.13 2.65 1.74
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.18 0.76 0.99 0.99 0.99 0.99
0.04 0.04 0.03 0.03 0.02 0.02
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Fiberglass and Glass
Variables Reverberation Times
2" Fiberglass Board
Glass
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Fiberglass and Glass
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 1.59 2.66 2.06 1.85 1.87 1.79 2.06 1.09
15 0.95 0.05 1.64 2.69 2.07 1.86 1.88 1.79 2.07 1.10
15 0.9 0.1 1.68 2.71 2.08 1.86 1.88 1.80 2.08 1.12
15 0.85 0.15 1.73 2.75 2.09 1.86 1.89 1.80 2.09 1.13
15 0.8 0.2 1.79 2.78 2.09 1.87 1.89 1.81 2.09 1.15
15 0.75 0.25 1.84 2.81 2.10 1.87 1.90 1.81 2.10 1.17
15 0.7 0.3 1.90 2.84 2.11 1.88 1.90 1.82 2.11 1.19
15 0.65 0.35 1.97 2.87 2.12 1.88 1.91 1.82 2.12 1.21
15 0.6 0.4 2.03 2.91 2.13 1.89 1.91 1.82 2.13 1.23
15 0.55 0.45 2.11 2.94 2.14 1.89 1.92 1.83 2.14 1.25
15 0.5 0.5 2.19 2.98 2.15 1.90 1.92 1.83 2.15 1.28
15 0.45 0.55 2.27 3.02 2.16 1.90 1.93 1.84 2.16 1.30
15 0.4 0.6 2.36 3.05 2.17 1.91 1.93 1.84 2.17 1.33
15 0.35 0.65 2.46 3.09 2.18 1.91 1.94 1.85 2.18 1.36
15 0.3 0.7 2.57 3.13 2.19 1.92 1.94 1.85 2.19 1.39
15 0.25 0.75 2.68 3.17 2.20 1.92 1.95 1.86 2.20 1.42
15 0.2 0.8 2.81 3.21 2.21 1.93 1.95 1.86 2.21 1.46
15 0.15 0.85 2.95 3.26 2.22 1.93 1.96 1.87 2.22 1.50
15 0.1 0.9 3.11 3.30 2.23 1.94 1.96 1.87 2.23 1.54
15 0.05 0.95 3.28 3.35 2.24 1.94 1.97 1.88 2.24 1.59
15 0 1 3.48 3.39 2.25 1.95 1.97 1.88 2.25 1.64
20 1 0 1.94 3.04 2.27 1.99 2.00 1.92 2.27 1.17
20 0.95 0.05 1.99 3.07 2.27 1.99 2.00 1.92 2.27 1.19
20 0.9 0.1 2.04 3.10 2.28 2.00 2.01 1.93 2.28 1.20
20 0.85 0.15 2.10 3.13 2.29 2.00 2.01 1.93 2.29 1.22
20 0.8 0.2 2.15 3.16 2.30 2.00 2.02 1.93 2.30 1.23
20 0.75 0.25 2.21 3.19 2.31 2.01 2.02 1.94 2.31 1.25
20 0.7 0.3 2.28 3.22 2.31 2.01 2.02 1.94 2.31 1.27
20 0.65 0.35 2.35 3.25 2.32 2.02 2.03 1.94 2.32 1.29
20 0.6 0.4 2.42 3.29 2.33 2.02 2.03 1.95 2.33 1.31
20 0.55 0.45 2.50 3.32 2.34 2.03 2.04 1.95 2.34 1.33
20 0.5 0.5 2.58 3.35 2.35 2.03 2.04 1.96 2.35 1.36
20 0.45 0.55 2.67 3.39 2.36 2.03 2.05 1.96 2.36 1.38
20 0.4 0.6 2.76 3.42 2.36 2.04 2.05 1.96 2.36 1.40
20 0.35 0.65 2.86 3.46 2.37 2.04 2.05 1.97 2.37 1.43
20 0.3 0.7 2.97 3.50 2.38 2.05 2.06 1.97 2.38 1.46
20 0.25 0.75 3.09 3.53 2.39 2.05 2.06 1.98 2.39 1.49
20 0.2 0.8 3.21 3.57 2.40 2.06 2.07 1.98 2.40 1.52
20 0.15 0.85 3.35 3.61 2.41 2.06 2.07 1.98 2.41 1.56
20 0.1 0.9 3.50 3.65 2.42 2.06 2.08 1.99 2.42 1.60
20 0.05 0.95 3.66 3.69 2.43 2.07 2.08 1.99 2.43 1.64
20 0 1 3.84 3.74 2.43 2.07 2.08 2.00 2.43 1.68
25 1 0 2.23 3.33 2.41 2.08 2.08 2.00 2.41 1.24
25 0.95 0.05 2.28 3.36 2.42 2.09 2.09 2.01 2.42 1.25
25 0.9 0.1 2.34 3.39 2.42 2.09 2.09 2.01 2.42 1.27
25 0.85 0.15 2.40 3.41 2.43 2.09 2.10 2.01 2.43 1.28
25 0.8 0.2 2.46 3.44 2.44 2.10 2.10 2.02 2.44 1.30
25 0.75 0.25 2.52 3.47 2.45 2.10 2.10 2.02 2.45 1.32
25 0.7 0.3 2.59 3.50 2.45 2.10 2.11 2.02 2.45 1.34
25 0.65 0.35 2.66 3.53 2.46 2.11 2.11 2.03 2.46 1.35
25 0.6 0.4 2.73 3.56 2.47 2.11 2.11 2.03 2.47 1.37
25 0.55 0.45 2.81 3.59 2.48 2.11 2.12 2.03 2.48 1.39
25 0.5 0.5 2.89 3.63 2.48 2.12 2.12 2.04 2.48 1.42
25 0.45 0.55 2.98 3.66 2.49 2.12 2.12 2.04 2.49 1.44
25 0.4 0.6 3.07 3.69 2.50 2.12 2.13 2.04 2.50 1.46
25 0.35 0.65 3.17 3.72 2.51 2.13 2.13 2.05 2.51 1.49
25 0.3 0.7 3.28 3.76 2.51 2.13 2.14 2.05 2.51 1.51
25 0.25 0.75 3.39 3.79 2.52 2.14 2.14 2.05 2.52 1.54
25 0.2 0.8 3.51 3.83 2.53 2.14 2.14 2.06 2.53 1.57
25 0.15 0.85 3.64 3.87 2.54 2.14 2.15 2.06 2.54 1.60
25 0.1 0.9 3.78 3.90 2.55 2.15 2.15 2.06 2.55 1.64
25 0.05 0.95 3.94 3.94 2.55 2.15 2.15 2.07 2.55 1.67
25 0 1 4.10 3.98 2.56 2.15 2.16 2.07 2.56 1.71
30 1 0 2.48 3.56 2.52 2.15 2.15 2.07 2.52 1.29
30 0.95 0.05 2.53 3.58 2.52 2.15 2.15 2.07 2.52 1.31
30 0.9 0.1 2.59 3.61 2.53 2.15 2.15 2.07 2.53 1.32
30 0.85 0.15 2.65 3.63 2.54 2.16 2.15 2.07 2.54 1.34
30 0.8 0.2 2.71 3.66 2.54 2.16 2.16 2.08 2.54 1.35
30 0.75 0.25 2.77 3.69 2.55 2.16 2.16 2.08 2.55 1.37
30 0.7 0.3 2.84 3.72 2.56 2.17 2.16 2.08 2.56 1.39
30 0.65 0.35 2.91 3.75 2.56 2.17 2.17 2.09 2.56 1.41
30 0.6 0.4 2.98 3.77 2.57 2.17 2.17 2.09 2.57 1.42
30 0.55 0.45 3.06 3.80 2.58 2.18 2.17 2.09 2.58 1.44
30 0.5 0.5 3.14 3.83 2.58 2.18 2.18 2.10 2.58 1.46
30 0.45 0.55 3.23 3.86 2.59 2.18 2.18 2.10 2.59 1.49
30 0.4 0.6 3.32 3.89 2.60 2.19 2.18 2.10 2.60 1.51
30 0.35 0.65 3.42 3.93 2.60 2.19 2.19 2.10 2.60 1.53
30 0.3 0.7 3.52 3.96 2.61 2.19 2.19 2.11 2.61 1.56
30 0.25 0.75 3.63 3.99 2.62 2.20 2.19 2.11 2.62 1.58
30 0.2 0.8 3.75 4.02 2.63 2.20 2.20 2.11 2.63 1.61
30 0.15 0.85 3.87 4.06 2.63 2.20 2.20 2.12 2.63 1.64
30 0.1 0.9 4.00 4.09 2.64 2.21 2.20 2.12 2.64 1.67
30 0.05 0.95 4.14 4.12 2.65 2.21 2.21 2.12 2.65 1.70
30 0 1 4.29 4.16 2.65 2.21 2.21 2.13 2.65 1.74
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.29 0.1 0.06 0.05 0.04 0.04
0.04 0.04 0.03 0.03 0.02 0.02
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Plasterboard and Glass
Variables Reverberation Times
Plasterboard
Glass
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Plasterboard and Glass
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 2.93 1.48 0.57 0.66 1.06 1.50 0.57 3.57
15 0.95 0.05 2.95 1.52 0.59 0.69 1.09 1.51 0.59 3.49
15 0.9 0.1 2.97 1.57 0.62 0.71 1.11 1.53 0.62 3.42
15 0.85 0.15 3.00 1.62 0.64 0.74 1.14 1.54 0.64 3.34
15 0.8 0.2 3.02 1.67 0.67 0.76 1.17 1.56 0.67 3.27
15 0.75 0.25 3.05 1.72 0.70 0.79 1.20 1.58 0.70 3.19
15 0.7 0.3 3.07 1.78 0.74 0.83 1.23 1.60 0.74 3.10
15 0.65 0.35 3.10 1.84 0.77 0.86 1.27 1.61 0.77 3.02
15 0.6 0.4 3.12 1.91 0.81 0.90 1.30 1.63 0.81 2.93
15 0.55 0.45 3.15 1.98 0.86 0.94 1.34 1.65 0.86 2.85
15 0.5 0.5 3.18 2.06 0.91 0.99 1.38 1.67 0.91 2.76
15 0.45 0.55 3.21 2.14 0.97 1.04 1.42 1.69 0.97 2.66
15 0.4 0.6 3.23 2.24 1.03 1.10 1.47 1.71 1.03 2.57
15 0.35 0.65 3.26 2.34 1.11 1.16 1.52 1.73 1.11 2.47
15 0.3 0.7 3.29 2.44 1.20 1.23 1.57 1.75 1.20 2.36
15 0.25 0.75 3.32 2.56 1.30 1.31 1.62 1.77 1.30 2.26
15 0.2 0.8 3.35 2.70 1.42 1.41 1.68 1.79 1.42 2.14
15 0.15 0.85 3.38 2.84 1.56 1.51 1.75 1.81 1.56 2.03
15 0.1 0.9 3.41 3.00 1.74 1.63 1.82 1.83 1.74 1.90
15 0.05 0.95 3.45 3.19 1.96 1.78 1.89 1.86 1.96 1.77
15 0 1 3.48 3.39 2.25 1.95 1.97 1.88 2.25 1.64
20 1 0 3.32 1.81 0.72 0.81 1.24 1.66 0.72 3.34
20 0.95 0.05 3.34 1.86 0.75 0.84 1.27 1.67 0.75 3.28
20 0.9 0.1 3.37 1.91 0.77 0.87 1.29 1.69 0.77 3.21
20 0.85 0.15 3.39 1.96 0.80 0.90 1.32 1.70 0.80 3.15
20 0.8 0.2 3.41 2.02 0.84 0.93 1.35 1.72 0.84 3.08
20 0.75 0.25 3.44 2.07 0.87 0.96 1.38 1.73 0.87 3.01
20 0.7 0.3 3.46 2.14 0.91 1.00 1.41 1.75 0.91 2.94
20 0.65 0.35 3.49 2.21 0.95 1.03 1.45 1.76 0.95 2.86
20 0.6 0.4 3.51 2.28 1.00 1.08 1.48 1.78 1.00 2.79
20 0.55 0.45 3.54 2.35 1.05 1.12 1.52 1.79 1.05 2.71
20 0.5 0.5 3.56 2.44 1.11 1.17 1.56 1.81 1.11 2.63
20 0.45 0.55 3.59 2.52 1.17 1.22 1.60 1.83 1.17 2.55
20 0.4 0.6 3.62 2.62 1.25 1.28 1.64 1.85 1.25 2.47
20 0.35 0.65 3.64 2.72 1.33 1.35 1.68 1.86 1.33 2.38
20 0.3 0.7 3.67 2.83 1.42 1.42 1.73 1.88 1.42 2.29
20 0.25 0.75 3.70 2.95 1.53 1.50 1.78 1.90 1.53 2.20
20 0.2 0.8 3.73 3.08 1.65 1.58 1.83 1.92 1.65 2.11
20 0.15 0.85 3.75 3.22 1.79 1.68 1.89 1.94 1.79 2.01
20 0.1 0.9 3.78 3.38 1.97 1.80 1.95 1.96 1.97 1.90
20 0.05 0.95 3.81 3.55 2.17 1.92 2.02 1.98 2.17 1.80
20 0 1 3.84 3.74 2.43 2.07 2.08 2.00 2.43 1.68
25 1 0 3.62 2.08 0.85 0.94 1.38 1.77 0.85 3.18
25 0.95 0.05 3.64 2.13 0.88 0.97 1.41 1.78 0.88 3.12
25 0.9 0.1 3.66 2.19 0.91 1.00 1.43 1.80 0.91 3.06
25 0.85 0.15 3.68 2.24 0.95 1.03 1.46 1.81 0.95 3.00
25 0.8 0.2 3.70 2.30 0.98 1.06 1.49 1.82 0.98 2.94
25 0.75 0.25 3.73 2.37 1.02 1.10 1.52 1.84 1.02 2.87
25 0.7 0.3 3.75 2.43 1.06 1.13 1.55 1.85 1.06 2.81
25 0.65 0.35 3.77 2.50 1.11 1.17 1.58 1.87 1.11 2.74
25 0.6 0.4 3.80 2.57 1.16 1.22 1.61 1.88 1.16 2.68
25 0.55 0.45 3.82 2.65 1.22 1.26 1.65 1.90 1.22 2.61
25 0.5 0.5 3.84 2.74 1.28 1.31 1.68 1.91 1.28 2.54
25 0.45 0.55 3.87 2.82 1.35 1.37 1.72 1.93 1.35 2.47
25 0.4 0.6 3.89 2.92 1.42 1.42 1.76 1.94 1.42 2.39
25 0.35 0.65 3.92 3.02 1.50 1.49 1.80 1.96 1.50 2.32
25 0.3 0.7 3.94 3.13 1.60 1.56 1.85 1.97 1.60 2.24
25 0.25 0.75 3.97 3.24 1.71 1.63 1.89 1.99 1.71 2.16
25 0.2 0.8 3.99 3.37 1.83 1.71 1.94 2.00 1.83 2.08
25 0.15 0.85 4.02 3.50 1.97 1.81 1.99 2.02 1.97 1.99
25 0.1 0.9 4.05 3.65 2.13 1.91 2.04 2.04 2.13 1.90
25 0.05 0.95 4.07 3.81 2.33 2.02 2.10 2.05 2.33 1.81
25 0 1 4.10 3.98 2.56 2.15 2.16 2.07 2.56 1.71
30 1 0 3.84 2.32 0.97 1.05 1.49 1.86 0.97 3.04
30 0.95 0.05 3.86 2.37 1.00 1.08 1.52 1.87 1.00 2.99
30 0.9 0.1 3.88 2.43 1.04 1.11 1.54 1.88 1.04 2.94
30 0.85 0.15 3.90 2.49 1.07 1.14 1.57 1.89 1.07 2.88
30 0.8 0.2 3.93 2.55 1.11 1.18 1.60 1.91 1.11 2.83
30 0.75 0.25 3.95 2.61 1.15 1.21 1.62 1.92 1.15 2.77
30 0.7 0.3 3.97 2.68 1.20 1.25 1.65 1.93 1.20 2.71
30 0.65 0.35 3.99 2.75 1.25 1.29 1.68 1.94 1.25 2.65
30 0.6 0.4 4.01 2.82 1.30 1.33 1.71 1.96 1.30 2.59
30 0.55 0.45 4.03 2.90 1.36 1.38 1.75 1.97 1.36 2.53
30 0.5 0.5 4.06 2.98 1.42 1.43 1.78 1.98 1.42 2.47
30 0.45 0.55 4.08 3.07 1.49 1.48 1.82 2.00 1.49 2.40
30 0.4 0.6 4.10 3.16 1.57 1.54 1.85 2.01 1.57 2.34
30 0.35 0.65 4.12 3.26 1.65 1.60 1.89 2.02 1.65 2.27
30 0.3 0.7 4.15 3.36 1.75 1.66 1.93 2.04 1.75 2.20
30 0.25 0.75 4.17 3.47 1.85 1.74 1.97 2.05 1.85 2.13
30 0.2 0.8 4.19 3.59 1.97 1.81 2.02 2.07 1.97 2.06
30 0.15 0.85 4.22 3.72 2.11 1.90 2.06 2.08 2.11 1.98
30 0.1 0.9 4.24 3.85 2.26 1.99 2.11 2.10 2.26 1.90
30 0.05 0.95 4.27 4.00 2.44 2.10 2.16 2.11 2.44 1.82
30 0 1 4.29 4.16 2.65 2.21 2.21 2.13 2.65 1.74
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.08 0.32 0.99 0.76 0.34 0.12
0.04 0.04 0.03 0.03 0.02 0.02
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Pageboard and Glass
Variables Reverberation Times
Pageboard
Glass
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Pageboard and Glass
15 Feet
20 Feet
25 Feet
30 Feet
Ceiling Height Material 1 (%) Material 2 (%) 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz RT @ 500 Bass Ratio
15 1 0 1.63 1.85 1.57 1.68 1.62 1.53 1.57 1.07
15 0.95 0.05 1.67 1.90 1.60 1.69 1.64 1.54 1.60 1.08
15 0.9 0.1 1.72 1.94 1.62 1.70 1.65 1.56 1.62 1.10
15 0.85 0.15 1.77 1.99 1.65 1.72 1.67 1.57 1.65 1.12
15 0.8 0.2 1.82 2.04 1.67 1.73 1.68 1.59 1.67 1.13
15 0.75 0.25 1.88 2.09 1.70 1.74 1.70 1.60 1.70 1.15
15 0.7 0.3 1.94 2.14 1.73 1.75 1.71 1.62 1.73 1.17
15 0.65 0.35 2.00 2.20 1.76 1.77 1.73 1.64 1.76 1.19
15 0.6 0.4 2.07 2.26 1.79 1.78 1.75 1.65 1.79 1.21
15 0.55 0.45 2.14 2.33 1.82 1.79 1.76 1.67 1.82 1.24
15 0.5 0.5 2.22 2.40 1.85 1.81 1.78 1.69 1.85 1.26
15 0.45 0.55 2.30 2.47 1.88 1.82 1.80 1.70 1.88 1.29
15 0.4 0.6 2.39 2.55 1.92 1.83 1.82 1.72 1.92 1.32
15 0.35 0.65 2.49 2.63 1.96 1.85 1.83 1.74 1.96 1.35
15 0.3 0.7 2.59 2.71 1.99 1.86 1.85 1.76 1.99 1.38
15 0.25 0.75 2.71 2.81 2.03 1.87 1.87 1.78 2.03 1.41
15 0.2 0.8 2.84 2.91 2.07 1.89 1.89 1.80 2.07 1.45
15 0.15 0.85 2.97 3.02 2.11 1.90 1.91 1.82 2.11 1.49
15 0.1 0.9 3.12 3.13 2.16 1.92 1.93 1.84 2.16 1.54
15 0.05 0.95 3.29 3.26 2.20 1.93 1.95 1.86 2.20 1.58
15 0 1 3.48 3.39 2.25 1.95 1.97 1.88 2.25 1.64
20 1 0 1.98 2.22 1.81 1.84 1.78 1.69 1.81 1.15
20 0.95 0.05 2.03 2.26 1.83 1.85 1.79 1.70 1.83 1.17
20 0.9 0.1 2.08 2.31 1.85 1.86 1.81 1.71 1.85 1.18
20 0.85 0.15 2.14 2.36 1.88 1.87 1.82 1.73 1.88 1.20
20 0.8 0.2 2.19 2.41 1.90 1.88 1.83 1.74 1.90 1.22
20 0.75 0.25 2.25 2.47 1.93 1.89 1.85 1.75 1.93 1.23
20 0.7 0.3 2.32 2.52 1.96 1.90 1.86 1.77 1.96 1.25
20 0.65 0.35 2.38 2.58 1.99 1.91 1.88 1.78 1.99 1.27
20 0.6 0.4 2.46 2.65 2.01 1.93 1.89 1.80 2.01 1.30
20 0.55 0.45 2.53 2.71 2.04 1.94 1.91 1.81 2.04 1.32
20 0.5 0.5 2.61 2.78 2.07 1.95 1.92 1.83 2.07 1.34
20 0.45 0.55 2.70 2.85 2.10 1.96 1.94 1.84 2.10 1.37
20 0.4 0.6 2.79 2.93 2.14 1.97 1.95 1.86 2.14 1.39
20 0.35 0.65 2.89 3.01 2.17 1.98 1.97 1.88 2.17 1.42
20 0.3 0.7 3.00 3.10 2.20 2.00 1.98 1.89 2.20 1.45
20 0.25 0.75 3.11 3.19 2.24 2.01 2.00 1.91 2.24 1.48
20 0.2 0.8 3.23 3.29 2.28 2.02 2.02 1.93 2.28 1.52
20 0.15 0.85 3.37 3.39 2.31 2.03 2.03 1.94 2.31 1.55
20 0.1 0.9 3.51 3.50 2.35 2.05 2.05 1.96 2.35 1.59
20 0.05 0.95 3.67 3.61 2.39 2.06 2.07 1.98 2.39 1.64
20 0 1 3.84 3.74 2.43 2.07 2.08 2.00 2.43 1.68
25 1 0 2.27 2.51 1.98 1.95 1.89 1.80 1.98 1.22
25 0.95 0.05 2.33 2.56 2.00 1.96 1.90 1.81 2.00 1.23
25 0.9 0.1 2.38 2.61 2.03 1.97 1.92 1.82 2.03 1.25
25 0.85 0.15 2.44 2.66 2.05 1.98 1.93 1.83 2.05 1.26
25 0.8 0.2 2.50 2.71 2.08 1.99 1.94 1.85 2.08 1.28
25 0.75 0.25 2.56 2.77 2.10 2.00 1.95 1.86 2.10 1.30
25 0.7 0.3 2.62 2.82 2.13 2.01 1.96 1.87 2.13 1.32
25 0.65 0.35 2.69 2.88 2.15 2.02 1.98 1.89 2.15 1.34
25 0.6 0.4 2.77 2.95 2.18 2.03 1.99 1.90 2.18 1.36
25 0.55 0.45 2.84 3.01 2.21 2.04 2.00 1.91 2.21 1.38
25 0.5 0.5 2.93 3.08 2.23 2.05 2.02 1.93 2.23 1.40
25 0.45 0.55 3.01 3.15 2.26 2.06 2.03 1.94 2.26 1.43
25 0.4 0.6 3.10 3.22 2.29 2.07 2.04 1.95 2.29 1.45
25 0.35 0.65 3.20 3.30 2.32 2.08 2.06 1.97 2.32 1.48
25 0.3 0.7 3.30 3.39 2.35 2.09 2.07 1.98 2.35 1.51
25 0.25 0.75 3.41 3.47 2.39 2.10 2.08 2.00 2.39 1.54
25 0.2 0.8 3.53 3.56 2.42 2.11 2.10 2.01 2.42 1.57
25 0.15 0.85 3.66 3.66 2.45 2.12 2.11 2.03 2.45 1.60
25 0.1 0.9 3.80 3.76 2.49 2.13 2.13 2.04 2.49 1.63
25 0.05 0.95 3.94 3.87 2.52 2.14 2.14 2.06 2.52 1.67
25 0 1 4.10 3.98 2.56 2.15 2.16 2.07 2.56 1.71
30 1 0 2.52 2.76 2.12 2.03 1.97 1.88 2.12 1.27
30 0.95 0.05 2.58 2.80 2.14 2.04 1.98 1.89 2.14 1.29
30 0.9 0.1 2.63 2.85 2.16 2.05 1.99 1.90 2.16 1.30
30 0.85 0.15 2.69 2.90 2.18 2.06 2.01 1.91 2.18 1.32
30 0.8 0.2 2.75 2.96 2.21 2.06 2.02 1.93 2.21 1.34
30 0.75 0.25 2.81 3.01 2.23 2.07 2.03 1.94 2.23 1.35
30 0.7 0.3 2.88 3.07 2.25 2.08 2.04 1.95 2.25 1.37
30 0.65 0.35 2.95 3.12 2.28 2.09 2.05 1.96 2.28 1.39
30 0.6 0.4 3.02 3.19 2.30 2.10 2.06 1.97 2.30 1.41
30 0.55 0.45 3.10 3.25 2.33 2.11 2.07 1.98 2.33 1.43
30 0.5 0.5 3.18 3.32 2.36 2.12 2.08 2.00 2.36 1.45
30 0.45 0.55 3.26 3.38 2.38 2.13 2.10 2.01 2.38 1.47
30 0.4 0.6 3.35 3.46 2.41 2.14 2.11 2.02 2.41 1.50
30 0.35 0.65 3.45 3.53 2.44 2.15 2.12 2.03 2.44 1.52
30 0.3 0.7 3.55 3.61 2.47 2.15 2.13 2.05 2.47 1.55
30 0.25 0.75 3.65 3.69 2.50 2.16 2.15 2.06 2.50 1.58
30 0.2 0.8 3.76 3.77 2.53 2.17 2.16 2.07 2.53 1.60
30 0.15 0.85 3.88 3.86 2.56 2.18 2.17 2.08 2.56 1.63
30 0.1 0.9 4.01 3.96 2.59 2.19 2.18 2.10 2.59 1.67
30 0.05 0.95 4.15 4.06 2.62 2.20 2.20 2.11 2.62 1.70
30 0 1 4.29 4.16 2.65 2.21 2.21 2.13 2.65 1.74
125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz
0.28 0.22 0.17 0.09 0.1 0.11
0.04 0.04 0.03 0.03 0.02 0.02
0.12 0.13 0.21 0.26 0.27 0.28
0.04 0.04 0.07 0.06 0.06 0.07
Walls
Floor
Plywood and Glass
Variables Reverberation Times
Plywood
Glass
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
RT60 (sec)
Plywood and Glass
15 Feet
20 Feet
25 Feet
30 Feet
Abstract (if available)
Abstract
This thesis examines the potential for parametric design software to create performance based design using acoustic metrics as the design criteria. A former soundstage at the University of Southern California used by the Thornton School of Music is used as a case study for a multiuse space for orchestral, percussion, master class and recital use. The criteria used for each programmatic use include reverberation time, bass ratio, and the early energy ratios of the clarity index and objective support. Using a panelized ceiling as a design element to vary the parameters of volume, panel orientation and type of absorptive material, the relationships between these parameters and the design criteria are explored. These relationships and subsequently derived equations are applied to Grasshopper parametric modeling software for Rhino 3D (a NURBS modeling software). Using the target reverberation time and bass ratio for each programmatic use as input for the parametric model, the genomic optimization function of Grasshopper – Galapagos – is run to identify the optimum ceiling geometry and material distribution.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Norton, Christopher W.
(author)
Core Title
Changing space and sound: parametric design and variable acoustics
School
School of Architecture
Degree
Master of Building Science
Degree Program
Building Science
Publication Date
11/26/2013
Defense Date
09/10/2013
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Acoustics,Architecture,OAI-PMH Harvest,parametric design
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kensek, Karen M. (
committee chair
), Carlson, Anders (
committee member
), Noble, Douglas (
committee member
), Valmont, Elizabeth (
committee member
)
Creator Email
cwnort@gmail.com,cwnorton@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-352379
Unique identifier
UC11288099
Identifier
etd-NortonChri-2189.pdf (filename),usctheses-c3-352379 (legacy record id)
Legacy Identifier
etd-NortonChri-2189.pdf
Dmrecord
352379
Document Type
Thesis
Rights
Norton, Christopher W.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
parametric design