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Raga structure: Geometric and generative models

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Raga structure: Geometric and generative models

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Content R A G A STRUCTURE - GEO M ETRIC A N D G EN ER ATIVE M ODELS
by
Shivani Yardi
A Thesis Presented to the
FA C U LTY OF TH E G R AD U ATE SCHOOL
U N IV E R S ITY OF SOUTHERN C A LIFO R N IA
In P artial FnlG Ihnent o f the
Requirem ents fo r the Degree
M ASTER OF SCIENCE
(E LE C TR IC A L EN G IN EER IN G )
M ay 2004
C opyright 2004 Shivani Y a rd i
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Acknowledgements
I w ould lik e to express m y gratitude fo r Professor Elaine Chew fo r her invaluable
guidance and support in directing the research. I w ould also lik e to thank her fo r her
assistance in the preparation o f th is m anuscript.
A special note o f thanks to m y Mends who offered me valuable insights on N orth
Indian C lassical M usic.
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Il l
Table of Contents
Acknowledgem ents ii
L is t o fT a b le and Figures v i
A bstract v iii
CHAPTER 1 N orth Indian C lassical M usic and its Elements
1.1 Introduction to N orth Indian C lassical M usic 1
1.2 Raga and its Elements
1.2.1 DeSning Raga 2
1.2.2 Raga: E m otion and Ornam entations 4
1.3 Representation o f P itch Classes in N IC M 6
1.4 Im provisation in N IC M 7
1.5 Tonal H ierarchies in N IC M 8
1.6 Perform ance Sections in N IC M
1.6.1 Alaap 12
1.6.2 K hayal 13
1.6.3 D rut 14
CHAPTER 2 D escriptive M odels
2.1 Introduction and O verview 15
2.2 The H arm onic N etw ork 16
2.3 M apping tA aatf in N IC M to the H arm onic N etw ork 17
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IV
2.4 Structures on the H arm onic N etw ork
2.4.1 Introduction 18
2.4.2 Structures on the H arm onic N etw ork 19
2.4.3 Sym m etry Axes and N etw ork Coverage 22
2.4.4 Prim e A xis, P itch D istrib u tio n and E m otion 23
2.4.5 Raga and Tim e o f D ay 26
2.5 Raga and E m otion 26
2.6 Im plications 29
CHAPTER 3 G enerative M odels
3.1 Introd u ctio n and O verview 31
3.2 Raga Yaman - Structure and C haracteristics
3.2.1 Pakad 34
3.2.2 Ornam entations in Yaman 34
3.2.3 T ypical P itch Transitions in Yaman 36
3.2.4 H eld Notes in Raga Yaman 39
3.2.5 Exam ple o f a M elodic T ransition in Yaman 41
3.3 General Rules o f Generation 43
3.4 Special cases: Sequential Patterns 48
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3.5 Im plem entation
3.5.1 Representation using P itch T ransition N etw ork (PTN ) 53
3.5.2 Flow chart fo r the Generation A lg o rith m 56
3.6 G eneration A lg orith m : Sample Results and A nalysis
3.6.1 Stepwise Generation: O utput Sequence I 58
3.6.2 Samples o f Generated Patterns and th e ir A nalysis 60
3.7 Beyond the Alaap: Generation w ith R hythm 66
3.8 P ractical A pplications o f the Generated A lg o rith m 67
References 68
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VI
L is t o fT a b le and Figures
Table 1. L in kin g perform ance tim e to structure 28
Figure 1. M elodic movement in Raga M arw a 5
Figure 2 Representation o f Sharp and F lat P itch Classes 7
Figure 3 .Tonal H ierarchies in a Jüugu in N IC M . 9
Figure 4. M elodic movement in Raga B hoopali. 11
Figure 5. Pitches represented on the H arm onic N etw ork 16
Figure 6. Scales o f the ten parent classes in N IC M 17
Figure 7. P itch distances between pairs o f thaats. 18
Figure 8. Thaats on harm onic netw ork 19-22
Figure 9. Recognized emotions in m usic 28
Figure 10. Ascending and descending scales o f Raga Yaman. 32
Figure 11. Pakad (catch phrase) o f Raga Yaman 34
Figure 12. Ornam entations associated w ith the pakad. 35
Figure 13. Glissandos in Raga Yaman 35
Figure 14 . M elodic movement in Yaman. 36
Figure 15. W ide p itch leaps in Yaman. 37
Figure 16. W ide p itch leaps in Yaman. 38
Figure 17. H eld Notes in Yaman 39
F igure 18. H eld notes in Y aman. 40
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Figure 19. H eld note in Yaman - special caae. 41
F igure 20. M elodic phrase in Raga Yaman. 42
F igure 21. R esting nodes in Raga Y aman 44
Figure 22. M om entary holding on a p itch in Raga Yaman. 45
Figure 23. Exam ple o f a linear sequential pattern. 49
Figure 24. Exam ple o f a linear sequential pattern. 49
Figure 25. Exam ple o f a linear sequential pattern. 50
Figure 26. Exam ple o f a non-hnear sequential pattern. 51
Figure 27 Exam ple o f a non-hnear sequential pattern. 51
Figure 28. Exam ple o f a non-liuear sequential pattern. 52
Figure 29. Random m elodic sequence in Yaman. 53
Figure 30. P itch T ransition N etw ork o f Raga Yaman. 54
Figure 31.Flow chart descrihing the generation algorithm fo r Raga Yaman. 56
Figure 32. Stepwise procedure o f m elodic generation based on rules. 58
Figure 33. M elodic pattern generated based on algorithm 59
Figure 34. H ierarchy o f generation rules 60
Figure 35.Generated phrase not conform ing to the aesthetics o f Raga Yaman. 61
Figure 36. Sample generated sequence in Raga Yaman 63
Figure 37. Sample generated sequence in Raga Yaman 64
Figure 38. Sample generated sequence in Yaman. 65
Figure 39.O verlapping tonal and rhythm hierarchies. 66
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Vlll
A bstract
The H arm onic N etw ork used in W estern C lassical m usic to represent cords
can be a useful to o l in representing the structures o f ragas in N orth Indian C lassical
M usic (N IC M ). Such a representation leads to conclusive inferences re la tin g the
m ood conveyed by the raga to its geom etric structure on the H arm onic N etw ork. We
also 6nd a hnk between the w eight d istrib u tio n o f the pitches around a prim e axis to
the perform ance tim e o f the raga. The symm etries o f the structures obtained &om the
m apping fu rth e r ju s tify the choice o f the ten parent classes adopted in N IC M from
w hich a ll other ragas are derived.
M elodic phrases can be generated in a raga from its p itch tra n sitio n netw ork.
G eneration o f m elodic sequences has been presented in Raga Yam an after capturing
its structural and m usical info rm a tio n in the algorithm . This m ethod can be extended
to create m elodic patterns in any raga in N IC M .
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Chapter 1
North Indian Classical Music and its Elements
1.1 Introduction to North Indian Classical Music
There are tw o genres o f Indian Classical M usic, N orth Indian Classical M usic
(N IC M ) popularly known as /fiw A trfaM i m usic, and South Indian Classical M usic
(SIC M ) known as Cam a/ic M usic. As the name suggests, N IC M is indigenous to the
northern parts o f India, w hile SICM is more often practiced in the south India. Both
schools o f music d iffe r in style and approach, but are based on the system,
where a Æaga is the most fundamental element. Musicians elaborate on a single
JZ^a in detail in N IC M , largely through hrgirovisation but also based on Garmal
conqx)sitions. Individual m elodic forms are shorter in SICM and performances
consist o f selective pieces in contrasting i(agas.
In both and Camafzc m usic, compositions are often preceded by
an im provised non-metered prelude known as .^/aqp, w hich is sometimes extensive.
This is follow ed by the com position section in w hich a speciGc rhythm ic cycle (7b/)
is provided by a percussion accompaniment. Although it is usually based upon a pre
existing com position, there are specific higrrovisational features to this section as
w ell. This aspect earns Indian classical music coirgiarisons w ith Western Jazz, w ith
w hich it shares some demands.
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The techniques and styles adopted by follow ers o f these tw o schools o f music
d iffe r more on a cultural Aont, rather than horn a music study point o f view . A
may be recognized by a d ifk re n t name in the N IC M and SICM and may even sound
d ifferent due to the d if& re n t techniques used to portray the structure o f the
The ly ric a l content o f melodies in and C anw tic music may d iffe r as the
latter is m ainly geared towards or die 'devotional type o f m usic', w hile
the form er usually captures fo lk stories and poems. music evolved as a
devotional medium to eiqiress gratitude to the Creator in temples in ancient India.
N IC M on the other hand developed thousands o f years ago during the reigns o f kings
and was adopted by court musicians as a means o f entertainment. Both schools o f
thought are high ly disciplined in th e ir study and students o f N IC M and SICM
undergo rigorous training under a teacher to attain a level o f perfection.
It must also be bome in m ind that Indian Classical music is considered to be
spiritual in nature and even in today's tim es, is considered to be m editative and to an
extent, sacred, perhaps ju sti^d n g the importance o f em otional content.
1.2 Raga and its Elements
1.2.1 DeSnlng Raga
A collection o f pitch classes w ith d istin ct inter-pitch relations form s the
underlying stm cture o f a Each collection consists o f Gve, six or seven pitches
selected h"om among a to ta l o f tw elve p itch classes. Carnatic music makes use o f 22
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3
pitch classes as opposed to the 12 p itch classes used in N IC M . However, a raga is
not sh^)ed by a mere collection o f pitch classes, but extends to the interplay o f its
pitches w ith one another that follow s a hierarchy o f pitch importance and thus lends
its e lf to the creation o f unique m elodic gestures. According to the m usicologist
Vishnu Bhatkhande, the diHerent characteristics o f a raga can be combined as
fo llo w s -
# Each raga has a Gxed set o f pitches ranging &om 5 to 7 in number. The
concept is sim ilar to that o f a W estern solfege. The presence o f two pitches is
im perative; one o f w hich is the tonic 0 and the other is the Fourth or the F ifth .
# There exists a modal structure fo r a raga, where the mode is the tAaat or the
parent class o f the raga from w hich it is derived.
# A raga has its own set o f ascending and descending scales w hich may or not
may not be linear.
# Tonal hierarchies in a raga result 6om varying levels o f signiGcance
associated w ith certain pitches. The vadi is the most im portant pitch in the
raga and the samvadi is the second most im portant pitch.
# The pakad or the catch phrase o f a raga is its most defining & ctor and forms
the typical m elodic transitions in the raga.
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1^.2 Raga: Emotion and Ornamentations
Each Aaga acquires a d istinct character depending on the pitches it makes use
o f and more im portantly, the manner in w hich the pitches are used. The form o f the
Jgqga is determined by the patterns o f ascent and descent o f the pitches, w hich may
or may not be linear in nature. Although it m ight appear to the ear that a musician
makes use o f the 12 pitch classes in a performance, in re a lity, he touches iqxm the
hundreds o f microtones existing between these tones by means o f a smooth,
continuous glissando 6om one p itch to the other.
A n in trin sic part o f any Rqgo is the mood or emotion it seeks to evoke. A
popular legend associated w ith Raga ZXpak is its a b ility to lig h t up lan çs when
elaborated upon by a musician w ith sufGcient s k ill and dedication. Raga M z/Aar is
believed to have the capability to bring down the rains. Although the scientific
authenticity o f such a statement is questionable, it is not entirely unim aginative that
some Ragas can be closely associated w ith a particular feeling. Ragas have been
associated w ith seasons and times o f the day as w ell. Such stipulations o f tim e and
season are influenced by the pitches present in the Raga and th e ir usage w ith respect
to emphasis, frequency o f use and the ornamentation technique. The semantics
governing a Rcga are intricate and a mere h in t o f a wrong note shifted in fequency
can invoke the theme o f a d ifferent Raga.
Ragas are identiGed by a set o f catch phrases that guide its m elodic
movement. In addition to the use o f pitches, N IC M makes use o f a number o f pitch
and melodic embellishments that enhance the aesthetic appeal o f a Raga.
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Embellishments o r ornamentations occur as glissando, vibrato, oscillation, je rk on a
pitch, intensity m odulation o f pitches and m elodic phrases that are based on
arithm etic sequences. Pitches are rarely used as discrete tones in N IC M and are often
merged into one another using various techniques o f ornamentations 6)r seamless
transitions. These embellishments reinforce the expression o f an individual m elodic
movement. It is at the discretion o f the m usician to choose the ^p ro p rié té dynamics,
articulations and nuances o f a pitch in addition to the tempo to effective ly convey the
intended expression. The use o f ornamentations is not entirely libe ra l and it is
expected to use a speciGc kind o f ornamentation w ith a pitch in a given
Generating anticipation fo r a p itch in melodies is another kind o f ornamentation that
involves tempo m odiEcation, change in intensity o f a pitch or a com bination o f both.
Consider Raga Marwa where the use o f the tonic 0 is hugal and the wide
pitch leap &om 1 to 7, a common phenomenon. This is elaborated in the fo llo w in g
example in F ig u re l.
Rag# Marwa -Melodic Movemeut
1 0
3
6
I
4
I
I
0
> 1
0 1 7""6 7 8 71°"“ 4 1 § » - 0 -.
Ea !Re P a -- M 2# Pa IDha Fa IRe Ga IRe IDha --Sa-
Figure 1. Melodic movement in Raga Marwa
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There is a strong expectation generated in the minds o f the listeners fo r die
stable tone 0. This summarizes one technique o f using ornamentation to bring to
surface the underlying expression in a raga.
U Representation of Pitch Ciasses in N IC M
Since N IC M and SICM d iffe r more in th eir cultural origios, rather than the
study and theory associated w ith them, one can claim that tw elve pitch classes is
suGScient to represent a ll tones present in Indian classical music. Krishnaswamy
(2003) showed using pitch-tracking methods ^ lie d to South Indian Classical
(C am aiic) M usic per&rmances that tw elve distinct intervals are sufGcient to
rqiresent a ll intervals present in C om oiic music. According to Krishnaswamy, even
though pitch inflexions are a commcm feature in Indian classical music, some can be
classiGed as ornamentations and odiers as d ifferent versions o f a fundamental tone.
These twelve pitch intervals are represented as shown below.
0 1 2 3 4 5 6 7 8 9 10 1 1
In the above pitch class representation, each succeeding pitch is a semitone higher
than the previous one. Pitch classes 0 and 7 are the natural tones or those that do not
have fla t or sharp counterparts. Figure 2 shows the pitch classes and th e ir
counterparts in N IC M . This representation o f pitches w ill be used throughout our
discussion.
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0 1 2 3 4 5 6 7 8 9 10 1 1 12
/Ga G o Mz A6z# fa /DAa DAa /M M &z*
Notations
! -fla tto n e
* - pitch class represented in the next higher octave
# - sharp tone
^ - pitch class represented in the low er octave
Figure 2. Representation o f Sharp and Flat Pitch Classes
1.4 Improvisation in N IC M
hrgnovisatioa plays a key role in N IC M . It is an essential feature o f Indian
music that depends upon the im agination and creativity o f an artist in painting a
mood in the given f aga that the listener can relate to. Im provisation on the set o f
notes or scale o f a JRog a occurs in the form o f m elodic and rhythm ic inventions. The
extent o f &eedom allow ed in im provisation is large, although it lies w ith in certain
lim itations. Some o f these lim itations are:
1. R estrictions on distance o f pitch leaps.
2. Im provisations that adhere to the importance and interplay o f the
and the fa/Mvorfi pitches fo r that Raga.
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3. Im provisations must be able to capture the catch phrase o f that particular
Thzga.
Im provisation triggers patterns o f thought around the Gxed set o f pitches in a
given raga and the a b ility o f a musician to e ffectively emote through his creations is
a measure o f his true success. N IC M is largely im provisational and in addition to
mastering the musical skills, it is essential to understand the technique to develop a
raga e fkctive ly.
1 j Tonal Hierarchies in N IC M
Tonality is an irrqx)rtant structural property o f any raga and has been
described by psychologist, Carol Krum hansl (1990) as a hierarchical ordering o f the
pitches o f he chromatic scale where each pitch is perceived in relation to a stable
pitch known as the tonic. Such a hierarchy o f pitches is evident in the listeners'
perceptions o f the pitch sta b ility in tonal contexts.
It is essential to study the tonal hierarchies in a IZaga to understand its
structure and the appropriate usage o f certain pitches. The generation algorithm
discussed in Chapter 3 makes use o f inform ation about the tonal hierarchy in l(aga
fomam to generate m elodic sequences. N ot a ll pitches in a JZaga are assigned equal
importance because o f w hich certain pitches occiqiy strategic positions such as being
the ending pitch starting pitch or a resting node in a melody. We w ill now study the
tonal hierarchies pertaining to a ll Kagas in N IC M .
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Figure 3 shows the pitch or tonal hierarchies in a in N IC M . The importance o f
pitches decreases as we move down the pyram id in the given TZngn. It should not be
misunderstood that pitches in the low er rungs o f the pyram id do not play a
significant role in the development o f the They are as involved as the other
pitches in lending the form and structure o f the JZogo, but they usually do not occupy
strategic locations in melodies such as resting points and starting and ending points.
0.7
Remaining pitches in
the Raga.
itches omitted in the Raga.
Ftgore 3. Tonal Hierarchies in a in NICM.
The tw o most stable tones or the natural tones 0 and 7 occupy the topmost
rung in the pitch hierarchy. These tw o pitches establish the key or the constant drone
used to establish the locations o f aU other pitches. Pitches are dedned relative to
these tw o pitches and any s h ift in these tones is reflected in a universal sh ift in the
tonality o f the Tüogn.
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1 0
Some make rare use o f the pitch class 7 and some even p itch class 0.
However, even in those Tkgas, these tw o pitches play the most im portant role in
defin in g the spatial positions o f the rem aining pitches in the scale o f the
The second rung in the pitch hierarchy is occupied the vodf (sonant) and
jom vwA (consonant) pitches. These pitches occupy tw o diSerent registers in the
scale o f the For example, i f the vacR lies in the pitch range 6om 0 to 7, the
fom votA w d l lie in the 7 to 1 1 range and vice versa. They are separated by at least a
Fourth. Azgu famoM has pitch 4 as its vod; and pitch 1 1 as the fumvodli. The
hequency o f use the v a fi and jomvaÆ pitches w ill be noticeable higher than the
others. They oAen serve as starting, ending and resting nodes in melodies are often
emphasized by the musician.
The th ird rung in the pitch hierarchy is occrçied by the rem aining pitches in
the realm o f the Azga. These tones m ainly bridge the gap between the and
samvofA and are rarely used as resting points in a m elody. Their presence aids
smooth transition 6om one p itch to the other and play an im portant role in
im plem enting techniques such as glissando, where these pitches are only touched
upon during a transition between tw o boundary pitches. Figure 4 illustrates this
concept in (Ka/yw : . The vadf and .y aT M va df o f Bhupali are
4 and 9 respectively.
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Rmg* %mMÜl - Mdodic Movement
I
1 2
10
8
6
4
2
0
7 9 7 4 2 4 2 4 0
P itdiC lasgee
4 7 9 7 (4) 2 4 (2) 0
Ga f a DAa f a f e Ga Sa
FIgnre 4. Melodic movement in Raga Bhoopali.
The wide p itch leap 6om 7 to 2 is not discrete, but a smooth glissando
touching upon pitch class 4 in its transition. In the latter part o f the m elodic phrase,
pitch class 2 is not e ^g lic itly used in the m elodic phrase, but used as a muted pitch in
the glissando from 4 to 0.
The Snal and the fourth rung o f the pyram id is occupied by those pitches w hich
are om itted in the scales o f the known as swor. M usicians sometimes
make use o f a pitch in the to bring in an element o f surprise, w ithout
digressing bom the essence o f the Azgu. Such a technique requires expertise w ithout
w hich the use o f voiyrt pitch sounds discordant in the bam ework o f the These
pitches are rarely found in any part o f the Tfaga, but may occasionally be found in
semi-classical o r 'lig h t' melodies.
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1.6 Performance Sections in N IC M
A perfbnnance in N IC M consists o f the m etrically bound and the unm etrically
bound sections. The concept o f meter and tenqx) in relation to the m elodic content is
very intricate. There are over 30 rhythm ic patterns in N IC M , each w ith varying
beats/measure, measures/cycle and the number o f cycles. There are three basic
components o f a N IC M performance pertaining to m elody and rhythm .
1. yf/oqp (%ee o f meter)
2. Khayal (m etrically bound)
3. D rut (m etrically bound)
1.6.1 Alaap
This is an introductory part o f a m usical performance based on a rhythm less,
&ee m elodic elaboration on a Kago. The can vary 6om a period o f a few
minutes to an hour. The A rgo is gradually revealed to the listener in a hm n o f
im provisation w hich is a m editative e]q)loration o f the J&rgo. It is considered to be
the ideal (and d ifB cu lt) manner o f presenting the The a rtist is expected to
adhere stric tly to the rules o f the Kago in this phase and the m elodic phrases
im provised by the m usician are expected to be the idealistic note combinations in
that Kogo. A n in depth rendition o f a Æago requires a sound understanding o f the
structure o f the l(oga w hich is p a rtia lly captured in its ascending and descending
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1 3
scales, vodf (sonant pitch) and (consonant pitch ), catch phrases, resting
pitches, and starting and Gnal pitches.
F irst the tonic is established and the detailed introduction o f the TZog o begins.
The m elodic features o f the Aago are exposed in an unmetered harm, elaborately and
system atically, gently unveiling the beauty o f the j&zgo's scale, its deSnitive phrases,
as w e ll as notes and th e ir transition. Id ea lly, the scale is traversed slow ly starting at
the tonic (note Sa), follow ed by exploration o f the low er octave ranges, one note at a
tim e. When the embellishment o f a note o r phrase has been completed, the artist
returns to the tonic and enphasizes it w ith a characteristic pattern called the mohra.
This phrase helps provide a sense o f tem poral variation in an otherwise 6ee and un
metered melody. The uAzqp extends hrom the low er register to the higher register and
touches upon the tonic o f the next octave. The descent back to the tonic o f the low er
register follow s the same rules o f the .Raga.
In Chapter 3 on Generation o f m elodic phrases in j&zga fom on, I have
highlighted generation in the phase. S im ilar rules can be applied to other parts
o f a performance such as the Khayal or the D rut, but an additional variable that
w ould need to be considered is the metered rhythm accompanying the melodies.
1.6.2 Khayal
The w ord Khayal, o f Persian o rig in translates into 'thought' or 'conception'.
It is a slow rendition o f the I&zga w ith a rhythm ic accompaniment. The m ain element
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1 4
o f the Khayal is its efBcacy in expressing a mood or a ras. The form is 6ee-flow ing
and decorative, unlike the w hich attempts to dehne the JCaga and its m elodic
nature. The Khayal is also entirely im provisational and can last fo r iq ito 3 hours
during a m usical performance. The tempo gradually increases w ith tim e and it
provides the musician an ocean o f opportunities to explore the Azga and ac%)t it to
the A ythm .
1.6J Drut
The D rut A llo w s the Khayal or A e vfZaqp i f the Khayal is skipped and has a
faster tempo than both. Since Ae Aundations o f Ae have already been
established by Ae vf/uqp and Ae Khayal, this phase allow s more Aeedom to explore
a ll possible m elodic comhmations, boA typ ica l and atypical A A e raga. The
interplay o f A ythm w iA m elody reaches its peak m Ae D rut section and Ae tenqx)
gradually mcreases Awards A e enA It is often Ae m usician's technical expeAse m
keepmg w iA A e rhythm at a fast pace that is brought A surface m th is part o f Ae
perArmance. The A ythm cycle is doubled and trebled A r Ae same m elody and Ae
musician attempts A 6 t A e m elody m Ae moAGed beat measures.
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Chapter 2
Descriptive Models
2.1 Introduction and Overview
This section explores the use o f Harmonic N etw ork in representing pitch
relations in N IC M . Numerous theorists have used the Harm onic N etwork, also
known as the fonnetz, to study and model roMo/iry, the system o f pitch relations
underlying western tonal music (see Lew in 1982 and Cohn 1998). In this discussion,
we w ill see how the Harmonic N etw ork can be an effective model 5)r pitch relations
in N IC M as w ell.
The scales o f the ten fAoo/s o r parent classes wiU be mapped on the harmonic
network and we w ill study the geom etric structures that result from the mapping.
Further, we w ill study a direct correspondence that exists between the mood or
emotion conveyed by the pitch patterns o f a Aago and its geometric conGguration on
the Harmonic N etwork.
N IC M is monophonic in nature and the concept o f harmony is absent.
However, the use o f the Harmonic N etw ork in this study is lim ite d to p lo ttin g the
spatial coordinates o f the pitches o f a raga w ith respect to a prim e axis. The
Harmonic N etw ork has proved to be a useful to o l in mapping ragas and
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1 6
understanding th e ir structure and em otional content purely based on th e ir geometric
shapes obtained &om the mapping.
2 J 2 The Harmonic Network
—
F C -G — .D A -
A A / \ / \ / \
Figure 5. Pitches rqiresented on the Harmonic Network
The harmonic network sown in Figure 5 represents harmonic relationships
where each lin k represents a prim e harmonic interval, and a ll lin ks representing a
particular prim e interval are parallel. Pitch classes on the horizontal axis are related
by hequency ratios o f 2:3, and pitch classes along the diagonal are related by
hequency ratios o f 3:4. The harmonic netw ork is particularly h e lp fu l in id e n ti^ in g
cords w hich are essentially triads form ed by jo in in g any three at^acent pitches.
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1 7
2J M o p in g in N IC M to the Harmonic Network
T T t w o r e a u R : 1 ( > jnsccyrnizxxl fA cK z/s c f jparent (classes izi IflC IN f licM ii v/bicdi otlwar
^bg o sa re d e n ve i, 7(agos derived ûom a fAaof w ill contain at least Gve o f the pitches
p % e % K X Q t iui idle A b o w # ' , aiai w ill iikw) display ty]ikxdtCHial]patk%nis clwu%w ck %istic to üs
parent class. The ten fAaots and th e ir pitches are shown in Figure 6
0 2
4
6 7 9 1 1
5a fg Ga Mz# f a DAa
0 1 4
5 7 8 1 1
5a ./fg Ga A/a fa ./DAa M
7 9 1 1
f a DAa iVz
2 3 5 7 9 10
fg /Ga A A z fa DAa /iV i
0 1 3 5 7 8 10
5 1 a /fg /Ga Mz fa /DAa
0 1 3 6 7 8 1 1
5a /fg /Ga JW h # fa /DAa M
0 1 4 6 7 8 1 1
5a ./fg Ga Mz# fa .^D A a M
0 2 3 5 7 8 1 1
5a /fg / Ga A & z fa /DAa M
0 2 4 5 7 9 10
5a fg Ga A A z fa DAa /M
1 3 5 7 8 1 1
/ f g /Ga Mz fa /DAa M
B ilaw al
Ka5
Kalyan
Bhairav
Bhairavi
P urvi
Marwa
Todi
Kham ^
Asavari
figu re 6. Scales o f the ten parait classes in NICM
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1 8
Kaû Todi Khamai Bilawal Kalyan Bhairavi Asavari Purvi Bhairav Marwa
0 0 0 0 0 0 0 0 0 0
2 2 2 2 2 1 1 1 1 1
3 3 4 4 4 3 3 3 4 4
5 5 5 5 6 5 5 6 5 6
7 7 7 7 7 7 7 7 7 7
9 8 9 9 9 8 8 8 8 8
10 11 10 1 1 11 10 1 1 11 1 1 11
Figure 7. Pitch distance between pairs of thaats.
The thaats in Figure 7 have been re-arranged such that tw o adjacent thaats d iffe r in
not more than 2 pitches. Such an arrangement could explain the choice o f the ten
thaats.
2.4 Structures on the Harmonic Network
2.4.1 Introduction
We now map the 10 thaats on the harmonic netw ork to analyze the various
geometric structures that we obtain. Based on the ten structures obtained 6om the
m op in g , we can analyze the symmetry and w eight distribution o f the structures.
The symmetry structures help to explain the choice o f the ten tAaofs, w hile the
distribution can be linked d ire ctly to the em otional content o f the 7(agas. The
conclusions derived horn these observations can be extended to other ragas. The
details o f the analyses are described in greater detail in the fo llo w in g sections.
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2.4.2 Structures on the Harmonic Network
Figure 8 (a) Bhairav
Figure 8 (b) B hairavi
Figure 8 (c ) P urvi
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Figure 8 (d) Marwa
Figure 8 (e) B ilaw al
Figure 8 (f) Kalyan
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21
Figure 8(g) KaS
Figure 8 (i) Kham^ÿ
Figure 8 (h) Asavari
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22
Figure 8 (j) Todi
2.4J Symmetry Aies and Network Coverage
I posit that the choice o f the ten tAaafs m axim ize the coverage on the
Harmonic N etw ork using compact structures. M axim al coverage is achieved by
mirror images reflected across a stable axis. Such m irro r image pairs m axim ize the
coverage o f different pitch patterns around the region o f greatest s ta b ility and can be
used to group a large number o f iZagos having a sim ilar structure. Consider the axis
containing the pitches 5 — 0 - 7 to be the prim e axis since it contains the tw o most
stable tones, 0 and 7. It is seen that among these ten parent classes, many are m irror
images o f each other. Some m irro r image pairs are:
1. (shown in Figure 8 (g) ) and (Figure 8 (a));
2. .BW roW (Figure 8 (b)) and (Figure 8 (e));
3. .BW ruW (Figure 8 (b)) and (Figure 8 (f)); and,
4. visavari (Figure 8 (h) and TbrA (Figure 8 (j)).
The parent classes .BW ravr (Figure 8 (b)) and .KAama/ (Figure 8 (i)) are translations
one o f another. S im ilarly, the parent classes vfja va n (8 (h )), jKafya» (8 (f)) and
^i/a w a / (8 (e)) are translations on the plane.
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The m irro r image pairs give m axim al coverage o f the network. For example,
BAarrmv (8 (b)) and^f/m va/ (8 (e)) cover a ll pitch classes except "6 ".
(8 (b)) and its derived j(qgos w ould a ll extend below die 5-0-7 axis
covering the pitch space in that direction. (8 (e)), and its derived J&rgos
w hich extend in the opposite direction w ould cover the p itch space above the prime
axis. In addition, jPAairav (8 (a)) and Kad (8 (g)) cover a ll pitch classes except "6 ".
(8 (b)) and Kalyan (8 (f)) together cover a ll pitch classes.
(8 (b)) and Xxavan (8 (h)) have identical spatial arrangements, a tw o-der
pattern w ith the 5 - 0 - 7 axis on the top. It is interesting to note that both the
BW rovz and A ro vw i cover a range o f sim ilar emotions, a sim ila rity that is paralleled
in th e ir pitch patterns.
2.4.4 Prime Axis, Pitch Distribution & Emotion
Since the 5 - 0 - 7 axis is c ritic a l in the structure o f each tAaat, another way
o f classifying the structures is by what we ca ll "top heavy" o r "bottom heavy"
natures. This is determined by the location o f the m ^o rity o f pitches above or below
die prim e axis. The pitches on the harmonic netw ork are arranged such that the tones
diat appear fla t relative to the prim e axis occur below and die tones d ia t appear sharp
or natural relative to the prim e axis occur above. In N IC M , fla t tones are
characterized by a poignant mood, and combinadons o f fla t and natural/sharp tones
occur in most JZagas. Depending on the distribudon o f fla t and natural/sharp tones in
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a its corresponding structure on Ihe harmonic netw ork w ill be top-heavy or
bottom -heavy.
The /(agos associated w ith a grave or solemn em otion tend to be bottom -
heavy. For example, Todi (8 (j)), Ayavan (8 (h)) and ^Aarravz (8 (b)) have a
m ^o rity o f th e ir pitch w eight below the prim e axis. This structure indicates a
predominance o f flattened notes in the Aaga, giving rise to a more grave or solemn
emotion.
Top-heavy shuctures are associated w ith jo y fu l jZagas. Aafyan is one such
parent class having a top-heavy structure. Consider Tfaga ^A aapa/i, belonging to the
Kalyan class, w ith the hallowing ascending and descending scales:
0 2 4 7 9 12 / 12 9 7 4 2 0
It is an established 6 c t that Aago jBAoqpo/z, an early evening Jüogo, is light-hearted
and in ça rts a sense o f tra n q u ility to its listeners w ith its hugal and pure notes.
Other top-heavy /(ogos, consisting p rim a rily o f "pure" or "natu ra l" tones, are the
KTwMq/, B ilaw al and fam an. These T&zgos often lighten iq) the mood o f the listener
by evoking a lig h te r or a happier emotion. Æ W nq/ is often used in sem i-classical
musical compositions such '^ u m ris " (melodies belonging to the lig h t classical
genre) and devotional fo lk songs as it has a liltin g sp irit.
In contrast, consider J&rgo (6om the 7 b < a K parent class) that has
the follow ing ascending and descending scales:
0 1 4 7 8 12 / 12 8 7 4 1 0
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The p itch set is essentially lik e that o f the Æo/yon except fo r tw o diSerences. B y
flattening tw o pitches, the sound and feel o f the Tktgo BAoqpaZ; is changed horn
jo y fu l to grave. In addition, this Azgo is typ ica lly perform ed in the late night or
early m orning before dawn. As eqiected, the parent class exhibits a bottom -
heavy structure.
In general, the top-heavy and bottom-heavy classification is an appropriate
indication o f the em otional content o f the JZagu. We now address the exceptions:
the M zrw a (8 (d )) parent class and the fw rw i (8 (c )) parent class. These /Aoats
appear as top-heavy structures and one m ight eiq)ect a lig h te r mood to be projected
by these j&zgas. However, the M zrw a and fw rv i express renunciation and pathos.
The structural explanation is that: the Grst tie r above the prim e axis contains pitches
that are sharp/natural w ith respect to the prim e axis. The second tie r contains pitches
that have equivalent representations in the Grst tie r below the prim e axis. Hence,
pitches in the second tie r (except fo r "6 ") should be classiGed as below the prim e
axis. The pitch class "6 " is unusual, occurrmg only in the parent classes: Marwa,
fw rw and Xa/yon. "6 " w ith respect to is the classical frim ne, an interval that
u n til recent times is avoided in western classical music. In N IC M , enharmonic
equivalence can be assumed in general. However, the use o f "6 " (w ith respect to
"0 ") is always considered a sharp version o f "5 ". Another reason fo r th is treatm ent
o f "6 " is that 0 - 7 are the most stable tones and are never sharpened o r flattened.
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2.4.5 jZagaandTim eofday
In N IC M , there is a customary tim e o f day associated w ith the performance
o f m ost TZqgos. These performance guidelines ensure that each jZngo achieves its
desired e% ct. We posit that it is possible to in fe r the tim e o f day associated w ith a
Æzga by observing its structure on the Harmonic N etwork. I(ogos that are perform ed
during the day (late m orning, afternoon and early evening) have a top-heavy
structure, and A%^os that are performed between late night and early m orning have a
bottom -heavy structure.
2.5 jfggg and Emotion
The success o f a musician is determined not only his technical expertise and
s k ill, but also by his a b ility to convey a particular mood or an em otion resulting &om
the Kogn to the listeners. Each Thaga w ith a set o f pitches and characteristic m elodic
patterns is capable o f painting a mood in the minds o f the listeners. One o f the most
inqx)rtant concepts in music is the conveying o f a particular kin d o f
em otion through m elodic im provisation on a Gxed set o f p itch classes. M usic has
been the medium adopted to narrate stories o f heroism, devotion and defeat fa r
thousands o f years across India, w hich is w hy communicating an em otion plays a
v ita l role in this 6)rm o f music. However, pitches by themselves do not have the
power to emote. It is only when they are used in conjunction w ith others do they
have a capability to bring about a certain mood in the m ind o f the listener. Sets o f
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27
pitches consistÎDg o f distinct seven tones form parent classes. M athem atically, there
can be a to ta l o f 792 (Ihe to ta l number o f ways to choose seven elements ûom
tw elve) possible parent classes. In practice, only ten such classes have been adopted
to serve as the foundation o f N IC M 6om w hich hundreds o f Azgos have been
derived.
As described in the previous section, each Azga is uniquely deGned by its
pitch collection and characteristic phrase. Tkga is also a Sanskrit w ord meaning
'c o lo r' or 'passion. Each Tfoga has some mood associated w ith it that can be related
to its pitches and th eir relations one w ith another. Certain pitch classes commonly
occur in Argos conveying a particular kind o f emotion. For example, in relation to
pitch class 0, the pitch class 1 is a flattened counterpart o f pitch class 2 and hence
brings about a more pensive and grave mood. Although ind ividu a l pitches are not
solely responsible fo r a particular em otion and cannot always be characterized as
such, it can be inferred &om sufficient observations that certain pitches in
com bination w ith others do bring about a mood that is conveyed w ith greater
enq)hasis than are the others.
The p itch set o f a and its characteristic phrase establish the fla vo r or
mood o f the It is a well-accepted notion that there are 11 basic moods (based
on "A lg a , the soul o f classical m usic") in N IC M that can be depicted through a
combination o f music, dance and poetry. M usic alone cannot convey a ll 1 1
sentiments and its scope is lim ited to the Grst 7 moods as shown in Figure 9. In
contrast, literature is considered the most pow erful medium o f expression, a medium
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lhat can express a ll 11 emotions. In the realm o f m usic, individual notes by
themselves are not capable o f generating e^gressive effects. It is the interplay o f
notes one w ith another, w ith the use o f proper stressing on pitches, resting places,
ornamentation that goes a long way in creating any sentiment. The mapping o f any
j& jga to the Harmonic Network considers only the p itch set and not the typical note
progressions.
1. Aanm : evoking pathos
2. 5%nngor: rom antic
3. peaceful
4. Fieer courage/victory
5. anger
6. varragya: ascetic
7. .BAatrf: devotion
M usic
8. Fearful
9. Hauya: comic
--------- L ite ra tu re
FIgare 9. Recognized emodons in music
P arent Class Tim e o f day S tru ctu re
Bhairav Pre-Dawn Bottom -heavy
Bhairavi Dawn Bottom -heavy
Asavari E arly M orning Bottom -heavy
Todi E arly M orning / Late
N ight
Bottom -heavy
KaS Afternoon Top-heavy
Marwa Late N ight Bottom -heavy
Purvi Dusk Bottom -heavy
Khamzy A ll Day (except Late
N ight & E arly M orning
Top-heavy
Kalyan Evening Top-heavy
B ilaw al Late M orning Top-heavy
Table 1: Linking performance time to structure.
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Vishnu Bhatkhande, the nineteenth century Indian m usicologist, suggested a theory
lin kin g the and pitches o f a JZaga to its associated tim e in the day.
According to him , j&zgos w ith (heir vocA pitch in the upper register (between pitch
classes 8 -1 1 ) were to be per&rm ed between 12 am-12pm and those w ith the in
the low er register (0-7) were to be performed 6om 12 pm-12 am. Furthermore,
Æzgas were assigned a tim e Aame depending on th e ir use o f certain pitch classes:
{2,4,9} are linked to 7-10 am/pm, {1,4,11} to 4-7 am/pm and {3,10} to 10-4 am/pm.
The m orning (am) or evening (pm ) tim e is further decided by the location o f the
in the low er or upper registers. For example, A rg o Foman, a derivative o f the parent
class Kalyan, has (he assigned tim e between 7-10 am/pm since it uses the pitches 2,4
and 9. Since its is 4, lyin g in the low er register, its tim e 6ame is narrowed
down to 12pm-12 am. Based on this reasoning, i&rga Faman is considered an
evening .Ruga. One can prcpose a sim pler way to determine the tim e o f day. Table 1
shows the performance tim e fo r Ragas ûom each parent class, and the associated
structure. We conclude (hat top-heavy Ragas are perform ed in the day, and bottom -
heavy Ragas in (he night.
2.6 Implications
It has been shown that (he Harm onic N etw ork is a suitable model fo r Ragas
and their parent classes in N IC M . The physical location o f pitches and th e ir spatial
relation one to another on (he harmonic netw ork can help erqrlain the choice o f (he
ten parent classes, and can be used to in fe r the mood and performance tim e fo r a
given Raga. This could be o f tremendous use in understanding the N IC M traditiorL
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30
L ike Longuet-Higgm s & Steedman's algorithm fo r key-Gndmg, one can use the
Harm onic N etw ork and the rAoof templates to determine the parent class o f a
B y extension, one can also determine the mood conveyed by that parent class and the
typ ica l performance tim e fa r that T & zga .
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CHAPTER 3
GENERATIVE MODELS
3.1 Introduction and Overview
I chose Jkga l'am an to demonstrate m elodic generation in the vi/aop phase o f
the performance. The Alaap phase is when the m usician inq)rovises w ith in the
structure o f the raga to create m elodic patterns representative o f that raga. This
choice was m otivated by the heptatonic scale o f the TZaga l'am an leading to creation
innumerable pitch combinations. Besides, Raga Yaman is a common favorite
amongst connoisseurs o f N IC M .
The to o l used fo r im plem enting the generation scheme was a C++ program
that is fed w ith details about die structure and m elodic movements in Raga Yaman.
The m elodic sequences generated w ill be more accurate as one includes more details
about the raga. The algorithm starts on the tonic 0 and continues to traverse the paths
along the Transition N etw ork t ill it completes its iterations. The length o f the Alaap
and the length o f individual output string can be selected by the user.
The generation algorithm was run several times and it was found that close to
90% o f the m elodic sequences generated were true to the form and structure o f Raga
Yaman.
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3.2 J R a y g flgfmw - Structure and Characteristics
Æaga yoTwm is one o f the most popular TZ u gu s in N IC M w ith the audience
because o f its light-hearted, jo y fu l expression, and w ith musicians as it lends its e lf to
detailed elaborations. One o f the most fundamental 6cets o f N IC M is the
characteristic m elodic gestures o f a jZuga, known as Kagu-vf (h t. the body o f the
raga). These m elodic gestures are inherited from its thaat (parent class) Kalyan. It is
essential to study and understand these m elodic clusters o f a Kugu in order to
im provise m elodic patterns. The ascending and descending scales o f Raga Yaman
are represented in Figure 10.
Ragm Yaman Scales
16 n
14 -
/'ll 2 4 6 9 11 0* 0* 11 9 7 6 4 2 0
Pitch Classes
/ 'l l 2 4 6 9 11 0 * / 0* 11 9 7 6 4 2 0
Re Go Mz# Dha M J a */' M Dha fa M z# Ga Re Ra
Figure 10. Ascending and descending scales o f Raga Yaman.
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There are tw o pitches deGned in Raga Yaman, one o f w hich is the sonant (vadi) and
the other is the consonant (samvadi). The sonant pitch denotes the most im portant
pitch in the raga w hile the consonant pitch indicates the second most inqw rtant pitch.
These tw o pitches are oAen sounded more frequently than the others. There also
exists a catch phrase, also known as pakad in each raga that is equivalent to a m o tif
in W estern Classical M usic. The pakad captures the most typ ica l p itch transitions
akin to a raga.
FocK (sonant) 4
Samvadi (consonant) 1 1
Pakad (catch phrase) '^11 2 4 6 7 6 4 2 '^ ll 2 0
Performance tim e - Evening
Thaat (parent class or mode) - Kalyan
In addition to the characteristic phrases, there are some other aspects to the
d e finitio n o f a Rago that must be home in m ind, some o f w hich are :
1. H eld notes in a JRago known as Nyasa. These notes are elongated in a
succession o f tones and held steady fa r a b rie f period o f tim e. In the tonal
hierarchy o f a raga, these Nyasa notes occupy the top tw o rungs o f the
hierarchical structure.
2. Characteristic elaborations fa r transitioning between d istinct pitch
pairs in a raga.
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fAoof makes use o f a ll pure or natural tones, except fo r the Fourth, w hich is
sharpened.
3J1.1 Pakad
The pakad o f Yaman is shown in Figure 11. The deliberate skipping o f the
tonic 0 in the ascending mode is an emblematic feature o f this its parent class
Kalyan.
Raga Yanma- Pakad
O
§
7
6
5
4
3
2
1
0
2 0 2 4
Pitch Classes
^11 2 4 2 0
'^N i Re Ga Re Sa
figure 11. Pakad (catch phrase) o f Raga Yaman
3.2.2 O rnam entations in Yam an
It has been emphasized that pitch transition '^ ll 2 4 is the most prom inent
pointer to Yaman. However, these pitches alm ost never occur discrete in ûequency.
A more accurate representation o f this pitch transition is shown in Figure 12.
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Yaman- Pakad ornamentation
7
2 6
i:
g 3
lî
0
'11
Pitch Classes
(0) " " Il (4) 2 4
(Sa) ''N i (Ga) Re Ga
Figure 12. Ornamentations associated with the pakad.
The smooth curves indicate a glide from the preceding p itch to the next.
Yamam - CBssaudos
14
8 1 2
S 10
5 8
S 6 -
a
0
1 1 9 1 1
Pitch Chssea
4 6 (11) 9 11 (9) 7
Ga Ma# (Ni) Dha Ni (Dha) Pa
Figure 13. Glissandos in Raga Yaman
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There exists a seamless transition &om pitch 11 to 9 and 6om 9 to 7, where
only pitches 9 and 7 are e xp licitly pronounced. The adorning pitches 11 and 9 shown
in braces are muted and touched upon lig h tly . In im plem enting a smooth glide 6om
one pitch to the other, the musician in re a lity touches upon a number o f intermediate
m icrotonal &equencies between the boundary pitches.
3.2.3 Typical Pitch Transitions in Yaman
A typ ica l tonal phrase in fbm on is that w hich involves the transition ûom
pitch 7 to 2. This is elaborated in Figure 14.
Ragm Ymmmm - between Mtche* 7 m m d 2
10
s
6
4
2
0
2 4 6 6 7 4 2 0
Pitch Clames
^11
/'N i
2 4 6 7 ( 6 4) 2 0
Re Ga Ma# Pa (M a# Ga) Re Sa
Mgarc 14. Melodic movement in Yaman.
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The leap hrom p itch 7 to 2 is not discrete, but a continuons and a smooth transition
touching upon the m icrotonal ùequencies o f the intermediate pitch classes 6 and 4.
Pitches 4 and 6 are shown in braces to indicate that they are not pronounced
e x p lic itly , but used to bridge the wide leap 6om 7 to 2. We w ill see that sim ilar
pitch transitions are prevalent in Æaga Toman. The intentional om ission o f the stable
tones 0 and 7 in the ascending movements lends a d istinct fla vo r to Raga Yaman.
Another striking feature o f Raga Taman is the w ide pitch leap between 4 and
11 and between 6 and 11 w hich are shown in Figure 15.
wide Pitch Le»pe In Rag* Y m n m m
14
12
10
8
6
4
2
0
6 2 0 4 6 11 9 7 4
Pitch Chews
4 6 11 (9) 7 (6 4 ) 2 0
Ga M z# M (D hql fa Re Ra
PTgore 15. Wide pitch leaps in Yaman.
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Wide PKch kmp* In Rngm Ynm m n
10
2 11 4 9 7
Pitch Classes
^11 2 4 11 (9) 7
'Wf Go M (DAq) fa
Figure 16. Wide pitch leaps in Yaman.
The wide pitch leaps &om 11 to 7, 7 to 2, 4 to 1 1 do not stand by themselves, but
make use o f the invisible intermediate pitches (shown in brackets) to make a
seamless transition as shown in Figure 16.
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3.2.4 H eld Notes in Raga Yam an
foToaM, sim ilar to other derived Tfqgas from Kalyan /Aaof makes use o f
certain pitches as held notes in melodies. These prim e held nodes are pitches 0 , 4 ,7
and 11. You w ill also Snd that pitch 2 also serves as a resting p oint in some strategic
m elodic locations in j(ogas derived from Kalyan. The only pitch excluded from the
category o f held notes is the pitch class 9. Resting on pitch 9 potentially im pairs the
flo w o f the Raga.
These concepts are better explained using the examples in Figure 17 and Figure 18.
Held N ote: In Yanain
16 1
14
1 2
1 0
8
6
4
2 -I
0
0»
Pkch Classes
0* 0* O'
4 6 9 0 * - - -
Ga Ma# Dha Sa* —
figu re 17. Held Notes in Yaman
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1 0
9
8 y
7
6
5
4
3
2
1
0
Held N ote: in Raga Yaman
Pilch Oaasea
2 7 6 4 - - -
Re Pa Ma# Ga —
Figure 18. Held notes in Yaman.
Pitch class 6 can also be used a held note on rare occasions, but it m ust be follow ed
by a m elodic phrase that ends on one o f the stable tones. Consider the example
shown in Figure 19.
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41
S
I
Held Notes in Ymmna
9
8
7
6
5
4
3
2
1
0
4 6 6 2 6 4 4 4 2 4 6
Pkch Oasses
^11 2 4 6 - - 2 6 4 - - -
^N i Re Ga Ma# - - Re Ma# Ga —
figure 19. Held note in Yaman— special case.
3.2.5 Example of a Melodic Transition In Yaman
Putting together the various features o f Raga Yaman described in the
previous sections, one can create a m elodic pattern as is shown in Figure 20.
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42
M elodk Phrm se la Raga Yam m n
10 -
S
I
0 ^11 2 4 - - - , 2 0, 4 6 9 11 7
7(g Ga— ,7k Go ATo# DAa M fo
Figure 20. Melodic phrase in Raga Yaman.
Thus, one caw iniakeuaeoflliese characteristic m elodic gestures fcM m dinthe]xirent
class o f a 7(ogo to generate m elodic patterns in that Tkga. The generation algorithm
makes use o f a ll such pieces o f inform ation about Tkgo fonzan.
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43
3 J General Rules of Generation
This section involves abstracting o f rules &om traditional knowledge o f
N IC M . I t is possible to summarize certain rules o f a raga and use them towards
computer generation o f m elodic phrases. The m elodic sequences generated in Koga
fom on must conform to the fundamental structure o f the JZago. Therefore, it is
essential to lis t both in ç lic it and e xp licit rules that are true to f&zgo fam on. These
are explained below.
(a) Starting pitches
The tonic 0 often is the opening pitch o f m elodic phrases in the firs t h a lf o f the /oqp
section. This inform ation was fed to the algorithm , and it was made to select 0 as the
starting pitch fo r the Grst h a lf o f the yf/oqp section.
(b) Ending pitches
The most common end pitch is the tonic 0. However, in the m iddle section o f the
Alaap, it is not necessary to always come back to the tonic to end a phrase. The
resting nodes often act as the ending pitches in such cases.
(c) Held Notes
The held notes are usually the tonic 0, vat/; 4 , and the F ifth 7. These pitches occupy
the top tw o rungs o f the tonal hierarchy ofYam an. See Figure 21.
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44
14 -
Held Note: In Ymmn n
12 -
10 -
1
I
1 1 9 7 6 7 4 6 9 7 7 7 2 2 4 2 0 0 0 0
Pitch Chsses
11 9 7 6 7 4 6 9 7 - - - 2- ^11 2 0-
Ni Dha Pa Ma# Pa Ga Ma# Dha Pa — Re - ^Ni Re Sa
Figure 21. Resting nodes in Raga Yaman
Note the longer duration o f holding on pitch 7 as compared to that on 2. The duration
o f holding on a particular pitch has a strong correspondence w ith th e ir relative
locations on the tonal hierarchy.
M elodic phrases can also rest on the th ird tie r pitches in the tonal hierarchy o f
the raga to create a sense o f anticipation fo r the im pending stable tone or the varf;
pitch in the minds o f the listener. Consider such a momentary resting on the p itch 6
in Raga fd/na» as shown in Figure 22.
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45
Held Notes In Ymnan
9
8
7
8 6
I:
2
0
6 4 4 4 4 2 6 2 2 0 0 0
Pitch Classes
"^11 ^11 2 6# - - - 4
M g # — Go
figure 22. Momentaty holding on a pilch in Raga Yaman.
The m usical rest on p itch 6 prolongs the landing on the stable tone 4 creating an
expectation in the minds o f the listeners. M elodic phrases cannot end on pitch 6 in
Yaman because it does not oBer a stable landing point in the tonal context o f the
raga.
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46
(d) Characteristic phrases
The characteristic phrases could be the pakad or subsets o f the pakad. This is the
most inqxxrtant to o l used in the generation algorithm . Some o f the characteristic
phrases in J&zgo f o T M a M are listed below.
^1 1 2 4
^ N i Re Ga
6 9 1 1
Ma# Dha N i
7 6 4 2
Pa M a# Ga Re
These short pitch combinations are signatures o f Raga Yaman and are to be found in
several locations in a m usical com position.
(e) Use of lower, middle and higher registers/octaves.
The o/oqp always starts in the m iddle and low er registers and gradually progresses to
the higher register w ith tim e. The algorithm must be made to start by establishing the
tonic 0 and then traverse the m elodic contours in the low er registers before
proceeding to the higher one.
(f) Frequency of use of pitches
Depending on the pitch hierarchy o f a raga, the ûequency o f usage o f some o f its
pitches varies w ith respect to the others. In raga Yaman, the pitches most commonly
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47
returned to are 0 and 4. M elodic phrases often begin and end on 0, but may also end
on the pitch 4. The pitches in the top tw o rungs o f the tonal hierarchy are heard
more d istin ctly than the others and also more hequently. S im ila rly, by setting these
pitches to occupy the resting, start and end nodes in the algorithm , the ir relative
inqw rtance w ould be made distinct.
(g) Repetitive melodic phrases
These phrases are emphasized throughout a com position and are equivalent to or are
subsets o f the characteristic phrases themselves.
The p itch com bination '^ l 1 - 2 - 4 is the most popular phrase in Yaman and is an
instant identiSer o f the raga. Other such phrases include the transition 6 -9 -1 1 and
1 1 - 9 - 7 - 6 .
(h) Length of melodic phrases
The length o f m elodic phrases varies a ll through the alaap phase, but can be
categorized to be short, medium and long w ith tim e. M elodic sequences become
more intricate and complex as the Alaap progresses in tim e, and w ith that, th e ir
length increases proportionately.
In the generation algorithm discussed, the length o f m elodic phrases has been kept
constant through each cycle.
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48
3.4 Special Cases: Sequential Patterns
M elodies in N IC M make Sequent use o f sequences w hich are mathematical
patterns o f pitches, known as to create melodic phrases in a given raga.
Let us consider a raga w ith 5 pitches a,b,c,d,e.
B y selecting tw o, three or fo u r pitches at a tim e, we can expand the m elodic
sequence to obtain linear patterns (Figures 23-25) and non-linear patterns
(Figures 26-28).
Linear Patterns
Pattern 1: a b, b c, c d, d e, e a
Pattern 2: abc, bed, cde, d e a
Pattern 3: a b c d , b c d e , c d e a
N on-Linear Patterns
Pattern 4: a c b a ,b d c b, c e d c
Pattern 5: c b a b ,d c b c, e d c d
Pattern 6: c a b c ,d b c d ,e c d e
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49
linear SeqaenÜai PaW erm 1
c d
Pitch Classes
Figure 23. Exançle o f a linear sequential pattern.
Llm em r Sequentlm] Pattern 2
b e d
P h d i C h a s e s
Figure 24. Example o f a linear sequential pattern.
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50
lin e a r S e q u e n tia l P a tte rn 3
7
6
5
4
3
2
1
0
d a b d d b c
Pitch Q e s s c s
Figure 25. Example o f a lin ar sequential pattern.
There exist innumerable such mathematical progressions that can be tgypUed
to a pitch set in a raga. These patterns are independent o f the actual pitches o f a raga
and the Bequency distance separating tw o successive tones. Im provisation centers on
the creation o f patterns o f melodies in a given raga.
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51
§
Nom -Um em r Seqaesdal Pm tterm 4
5
3
2
1
a c b b d c b c e d c
Pitch Clamea
figu re 26. Example o f a non-linear sequential pattern.
Nou-Umear Sequential Pattern 5
5
4
3
2
1
c b b d c b c e d c d
Pitch Classe:
Figure 27 Example o f a non-linear sequential pattern.
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52
N@n-Une*rSequemd*l Pmttem 6
6
5
1
0
d b d b d c a c c e c o
P k d i Classes
Figure 28. Exanqile o f a non-linear sequential pattern.
The patterns described in Sgures hallow arithm etic progression. They diS er 6om
other random ly generated patterns in their slopes when plotted on Cartesian axes.
A random ly generated sequence can also be a perfect Gt in the raga, but the
progression o f pitches may or may not be linear w ith tim e. A n example o f a
random ly generated m elodic pattern in Raga Yaman is shown in Figure 29.
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53
Random Melodic Pattern in Yaman
1 6 1
10 -I
0
1
Pitch Classes
Figure 29. Random melodic sequence in Yaman.
3.5 Implementation
The rules o f generation described in section 3.3 were used to im plem ent the
generation algorithm . The method o f inq)lem entation is discussed in detail in this
section.
3.5.1 Representation using Pitch Transition Network (PTN)
The pitch transition netw ork (PTN) shown in Figure 30. embodies a ll
possible pitch transitions in Yaman. . These transitions have been summarized 6om
listening to recordings in jZqga by masters o f N IC M such as K ishori
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54
Am onkar, Padma Talw aikar and Pandit Jasr^. The links connecting tw o pitches can
be either one directional or bi-directional depending on the perm issible transitions.
The transition &om 0 to '^ l 1 in an exanq)le o f a uni-directional pitch transition. I f one
were to allow '^ l 1 to transition to 0 in the opposite direction, it w ould be against the
grain o f the raga. Pitch ^11 always transitions to 2 in the ascending mode.
Figure 30. Pitch Transition Network o f Raga Yaman.
The PTN acts as a guideline to attain next p itch states during the generation process.
This inform ation coigiled w ith arithm etic progressions o f m usical sequences gives
rise to numerous possible m elodic combinations.
Although such a representation is h e lp fu l in assigning the 'next pitch states'
in the generation algorithm , it does not hold inform ation about the fin e r aspects o f
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55
j&zga faTMo» such as omameutatÎQiis specific to a pitch, intensity o f volume on
various pitches and several other details. I f one were to include these details in the
generation algorithm , one w ould require additional variables that w ould represent
these details.
In the generation algorithm discussed, the transition netw ork helps to create
m elodic sequences characteristic to Tkga Yaman w ith respect to p itch transitions.
The tim e variable introduced in the algorithm keeps track o f the ^p ro p ria te pauses
on certain pitches and works in the Aamework o f the Æzga to generate m usically
accurate m elodic phrases. The length o f the vfZoqp can be selected by assigning an
integer value to the number o f m elodic phrases desired at the output.
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3.52 Flowchart for the Generation Algorithm
56
Select num ber o f strings
Select string leogA.
Select generation
Select register
Select starting pitch.
P ick direction o f p itch travel
M inim ize distance o f pitch leap
Choose the path o f the characteristic
P itch rests
Endm tches
Lormmg / repetition o f phrases
End program when iterations = # o f strings.
Figure 31. Flowchart describing the generation algorithm fo r Raga Yaman.
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57
3.6 Generation Algorithm: Sample results and analysis
This section presents fo r analysis, some o f the o n ^n t m elodic sequences
generated by the algorithm . It was observed that close to 90% o f the generated
patterns were representative o f Raga Yaman when the ouq)ut string length was kept
between 8-15 pitches. As the output string lengths were increased, there were an
increased number o f patterns that did not closely Gt the structure ofYam an.
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3.6.1 Stepwise Generation: Output Sequence 1
I T " " " "
S taiting on 0
Nearest pitch
Nearest pitch
Charactensbc
phrase
Octave
D irection o f Resting pitch
travel
Nearest pitch Octave
N o re p e titio Nearest pitch
Nearest p itch Resting pitch
Nearest pitch D irection o f
travel
D ire c tirm o f
trave
C haractaistic
phrase
Characteristic
phrase
Octave
End p itch 0
58
©
/-ii
0
0
©
0
0
©
( 0
0
Figure 32. Stepwise procedure o f melodic generation based on rules.
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59
The m elodic pattern obtained by fo llo w in g the flow chart described in Figure 32 is
shown in Figure 33.
O utput Sequence 1
0
o
c n
7
6
5
4
3
2
1
0
0 4 0 0 0 0 2 4 4
Pitch Classes
0 '^ ll 2 4 -- 2 6 4 -- 2 /"II 2 0 --
'W ; Go - - M z# Go - - - -
Figure 33. Melodic pattern generated based on the algorithm.
A t each stage in the algorithm , there is a choice o f more than one possible
next state pitch, w hich is governed by the rules o f generation described in section
3.3.1. A decision needs to be made to elim inate a ll except one pitch since a u n it in
tim e can be occupied by only one p itch in N IC M . We can introduce a hierarchy o f
generation rules to influence the decision.
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60
Characteristic Phrase
M inim ize pitch leap
Continue direction o f travel
R epetition o f patterns
Decreasing order
o f p rio rity
V
Figure 34. Hierarchy o f generation rules.
Thus, by selecting die boundary pitches or the octave, and applying the basic rules o f
Raga Taman to the algorithm , we can obtain numerous such m elodic patterns.
3.6.2 Samples of generated patterns and their analysis
Although a ll such generated patterns fo llo w the rules o f the Raga Taman, not a ll are
m usically and aesthetically appealing 6om Raga Taman's perspective. This is
because the algorithm is not enable o f generating jGner details such as smooth glides
between two distant pitches creating a d isjo in t effect. Some transitions appearing on
the transition network are always used in a specific context and th e ir use outside o f it
can sound inharmonious.
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61
The m elodic pattern in Figure 35. is one such exançle.
O utput Sequence 2
10 -
8
I
< 5
a
g
4 6 7 9 11 11 9 7 6 4 2 Ml 2 0 0 0
Pitch Classes
4 6 7 9 11- 9 7 6 4 2 ^11 2 0 - -
Ga M g# fu D A a JVg-D Aafa M z# Ga 7 k 'W i 7(e &%-
F%ure 35. Generated phrase not conforming to the aesthetics o f Raga Yaman.
Although, the m elodic phrase in Table 18 is in tune w ith the rules o f TZ a ga
fam a», the linear pitch transition 6 7 9 1 1 is rarely found in this 7(aga. The
algorithm generates such transitions in order to avoid excessive repetitions in one
generation cycle. The transition 6 7 9 11 was found to occur several times towards
the end o f an iteration when use o f the characteristic transitions was exhausted.
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62
Pitch transitions 6om 6 9 11-9 7isa more popular phrase. Such m inute details
can be captured in die characteristic phrases o f this JZaga and made available to the
algorithm .
The F ifth (7) is used commonly in the descending scale o f this whereas
the ascending pitch transition 6om 6 to 9 is a straight leap. Thus, according the rule
o f m inim iTing pitch leaps, 6 wiU transition to 9 in the ascending scale.
We w ill further discuss and analyze some o f the results obtained &om the
generation algorithm and comment on th e ir m usical appeal and accuracy. The output
string length was set to 20 and the number o f iterations was 25. We now discuss
three more output patterns (Figures 36-38) generated by the algorithm .
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63
S
0
1
Output Sequence 3
8
6
4
2
0
-2
-4
-6
0__ A ll A Ç Aq ^11 2 0 - ^11 2 4 - - 2 ^11 2 0
& % - - "W z ^ A a 'Wz &z- Ga - - 7(e &z
Figure 36. Sample generated sequence in Raga Yaman
In sequence 3 in Figqre 36., there are appropriate rests on the tonic 0 and the vazA 4.
The characteristic phrase 1 - 2 — 4 is also clearly dehned and the transition horn
'^ l 1 to 2 is prevalent.
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64
I
B
§
Ou^nt Sequence 4
18
16
14
12
1 0
8
6
4
2
0
4 6 9 1 1 7 7 7 4 6 9 11 2* 0*0* 11 9 7 6 4 4 4
Pitch Chases
4 6 9 11 7- 4 6 9 11 2* 0* -- 11 9 7 6 4
Ga Ma# Dha Ni Pa-Ga Ma# Dha Ni Re* Sa*--Ni Dha PaMa#Ga
Figure 37. Sang)le generated sequence in Raga Yaman
Sequence 4 shown in Figure 37. has been picked up hrom the m iddle portion o f the
alaap and involves pitches in the m iddle and higher octaves. This m elodic phrase is
representative o f the raga due to its rests on pitches 7, 4 and 0 and the typical
transition &om 11-2.
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65
Ootpnt Sequoice 5
1 4 1
10 -
Pitch Classes
0 2 4 6 9 7 - 2 4 2 ^11 ^ M l 2 0 - 6 9 11 9 7
Sa Re Ga M a # Dha Pa - Re Ga Re ^N i '^Dha ^N i Re Sa - M a# Dha N i Dha Pa
Figure 38. Sample generated sequmce in Yaman.
This phrase was picked up &om towards the end o f the generation cycle and is not in
keeping w ith the form ofYam an, Pitch transitions 6om 0 - 2 - 4 is never to be found
in this raga and the algorithm made use o f such a phrase when other combinations
were exhausted.
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66
3.7 Beyond the Alaap: Generation with Rhythm
The alaap phase o f a N IC M performance is free o f rhythm and can take any
shape and farm . The metered parts o f a performance such as the Khayal and other
medium to j&st tempo melodies require adhering to a hxed rhythm ical cycle. There
are over 20 diSerent rhythm cycles known as 7b/ in N IC M . Each o f these rhythm ic
patterns have a unique com bination o f measures/cycle and beats/measure. M oreover,
there exists a hierarchy o f significance o f beats in a rhythm cycle parallel to the tonal
hierarchies in a raga. It is a common tendency fo r a m usician to o v e rly the tonal and
the rhythm hierarchies. The result o f such an overlap w ould be the simultaneous
occurrence in tim e o f the vadi and the most im portant beat in the rhythm cycle. A n
illu stra tio n supporting the discussion is presented in Figure 39.
1 2 3 4 5 6 7 8 9 10
X X X X
VadI
__
PoeilionB oco^iied by pitdies in the tq i two
nn%aofthe tonal h icm d iy o f the laga.
Ftgore 39.0verlappmg tonal and ihythm hiérarchies.
A complete rhythm cycle in Ih ^ ta l is divided into 4 measures w ith
alternating tw o and three beats per measure. The Erst beat indicated by X in Figure?
is the most im portant beat o f the rhythm cycle and is emphasized in intensity by the
percussionisL The Erst beats o f the rem aining measures indicated by 'x ' occupy the
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67
Icnanariiing <)f lüie hierarclgf o f inipoitaiKXx (]cMiq>osers aiul rousicians \vhdle thegr
ioiprcrvise areirKdiiM xi to fo llo w clcK w d^f lüie tvw) hiemuxdiies eaduMinj; iw iidrytluniB id
pitches.
3.8 Practical applications of generation algorithm
The set o f generation rules described in section? can be extended to other
rzyg as in C)ne w ould retpiire assimihidng tb e t}]p ica l ncüetrairshicMis jpexniUar
Ik) tbc ]niga iri ackihioii h) its scales, inadi arwi sanrvarh. T T h is caui tx: aclufrved Iby
listening to several conq)ositions in the raga and extracting Aequently occurring
phrases. It w ould also be w orth noting the held notes and the ornamentations
associated w ith certain pitches. Ornamentations applied to a pitch vary w ith the raga
and at tim es, w ith in the raga itse lf.
The generative model can be conq)ared w ith a performance in real-tim e to
observe the extent o f correlation between the tw o. This technique could be extended
to serve as an educational tool to students o fN IC M .
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68
References
1.Bharucha. J, C astellano. M , Krumhansl Carol (1984). "7b»a/ jïzerw cAza; m fAe
M « ;c q/"jVbrtA TW fo", Journal o f Experim ental Psychology, V o l 113, No. 3, 394-
412.
2.Cohn, Richard (1998). "Introduction to Neo-Riemannian Theory: A Survey and
H istorical Perspective," VdwrMo/ q/"TAeory, 42: 2, pp. 167-180.
3.Courtney D avid, "E lectronic aids in Indian music education"
http://m usic.utsa.edu/tdm l/conf-l/I-Courm ev/I-Courtnev.htm 1
4.Jasani. Tarun, "Rogar q/"/Ae A/bm m g", A /(p ./ W nv.amc.org'.
Raga — the soul o f classical music: AAp. /ywMw.fwrdAwaMf.co7M/««^ rang. Atm/
5 .Krishnaswamy, A rvindh (2003). "A pp licatio n o f Pitch Tracking to South Indian
Classical M usic," Proceet/m g; q/^tAe ZEEE JA/erMariona/ Commence on
AAf/rimaAa arw/ Baltim ore, M D , July 1999.
6.Classical M usic", Proceedings o f the Stockholm M usic Acoustics Conference,
August 2003.
V.Lewin, D avid (1982). "Generalized M usical Intervals and Transform ations," Yale
U niversity Press: New Haven, CT.
8 Jx)nguet-Higgins, H.C. & Steedman, M .J. (1971). "O n Interpreting Bach," In
MzcAine JAfe/%g»ce, V ol. 6, B. M eltzer and D . M ichie (Eds.) Edinburgh
U niversity Press.
9.Sm ith Nicholas A ., Schmuckler M ark A . , "P itch-distributional effects on A c
perception o f to n a lity", ICMPC 2000 Proceedings paper.
lO.Sriram Ganesh, "D istinguishing Features o f Indian Classical M usic", Iow a State
U niversity, Department o f Chemical Engineering.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
ll.Y a id i .S. ,Chew Elaine, "G iving Ragas the Time o f Day: L in kin g Structure,
Em otion and Per&rmance Time in N orth Indian Classical M usic Using the
Harm onic N etw ork", to appear in the ICM PC proceedings, August 2004.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Yardi, Shivani (author)

Core Title Raga structure: Geometric and generative models

Contributor Digitized by ProQuest (provenance)

Degree Master of Science

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,music,OAI-PMH Harvest

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Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-316699

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