A Model for Implied Intonations in Classical हिन्दूस्तानी (Hindustani) Music - Sidebars

Sidebars

The following are additional ruminations on my model of scales used in Classical हिन्दूस्तानी Music.

Weights-based grouping

An intriguing picture arises if we arbitrarily assign weights to the intervallic atoms in terms of increasing interval size:

1 = s (25:24)
2 = S (16:15)
4 = T (9:8)

Note that T is assigned 4 because T > { S, s } = 3.

These weights are also obliquely inspired by मुनि भरत's २२-शृति model in which three different interval sizes, possibly corresponding to the major tone, minor tone, and semitone, were assigned शृति values 4, 3, and 2. It is thought that these values were coarse approximations to the size of the associated interval.

Now, identify each of the notes in the model with the sum of the weights of the intervallic atoms which comprise the interval. This gives the following grouping:

00 = { 1/1 } = { S }
01 = { 25/24 } = { r1 }
02 = { 16/15 } = { r2 }
03 = { 10/9 } = { R1 }
04 = { 9/8, 256/225 } = { R2, R3 }
05 = { 75/64, 32/27 } = { g1, g2 }
06 = { 6/5} = { g3 }
07 = { 5/4 } = { G1 }
08 = { 81/64, 32/25 } = { G2, G3 }
09 = { 675/512, 4/3 } = { m-, m }
10 = { 27/20, 25/18 } = { m+, M1 }
11 = { 45/32, 64/45 } = { M2, M3 }
12 = { 36/25, 40/27 } = { M4, P- }
13 = { 3/2, 1024/675 } = { P, P+ }
14 = { 25/16, 128/81 } = { d1, d2 }
15 = { 8/5 } = { d3 }
16 = { 5/3 } = { D1 }
17 = { 27/16, 128/75 } = { D2, D3 }
18 = { 225/128, 16/9 } = { n1, n2 }
19 = { 9/5 } = { n3 }
20 = {15/8 } = { N1 }
21 = { 48/25 } = { N2 }
22 = { 2/1 } = { S* }

Note that some notes end up in the same weight group. These notes are precisely those which differ from each other by less than a Comma; by 2048:2025 (~19.55c) in fact, if you discount the four 'false' notes (m-, m+, P-, P+).

22 Weight Groups = २२ शृतिs ?

That there are 22 weight groups in the octave is no accident. It is a natural consequence of having chosen the weight values 1, 2, and 4 for the three intervallic atoms s, S, and T. As a result, the octave totals up to a weight of 22. However, what is indeed striking is that there are no weight groups which are empty, and many of them have a single member (especially if you discount the four 'false' notes).

What is also interesting is that the r, R, g, G, d, D, n, and N स्वर-स्थानs differentiate into two sub-groups each. Following the कोमल/तीव्र terminology employed earlier for each स्वर, one could consider a nomenclature of कोमल-तर, कोमल, तीव्र, and तीव्र-तर. Thus, r1 = कोमल-तर ऋषभ, r2 = कोमल ऋषभ, R1 = तीव्र ऋषभ, and R2/R3 = तीव्र-तर ऋषभ. Ditto for the other notes. The कोमल मध्यम is the only exception due to the peculiar consonant position it enjoys in the scale - and we assign it only one location (m). On the other hand, we consider the तीव्र मध्यम as differentiated into तीव्र मध्यम (M1), तीव्र-तर मध्यम (M2/M3), and तीव्र-तम मध्यम (M4).

For those who wish to view हिन्दूस्तानी scales in terms of a २२-शृति model, the mapping above may be useful. Note however that each शृति (weight group) is an equivalence class of notes and can be multivalued (e.g. weight group 11). This is consistent with the school of thought which holds that शृति positions are not necessarily precise interval values but denote a 'range' of intonational possibilities.

Scale Taxonomy

This section exists out of academic curiosity only and can be safely skipped in the first reading. It has no direct relevance to the rest of this work.

The current कर्नाटकी पद्धती's मेल taxonomy is extended as follows:

1. The first note is the Unison (1/1).
2. The second note is either a कोमल ऋषभ, तीव्र ऋषभ, or कोमल गाँधार.
3. The third note is either a तीव्र ऋषभ, कोमल गाँधार, or तीव्र गाँधार.
4. The fourth note is either the कोमल मध्यम or a तीव्र मध्यम.
5. The fifth note is either a तीव्र मध्यम or the पंचम.
6. The sixth note is either a कोमल धैवत, तीव्र धैवत, or कोमल निषाद.
7. The seventh note is either a तीव्र धैवत, कोमल निषाद, or तीव्र निषाद.
8. The selected notes have to be strictly in increasing order with a traversal path between each pair of neighbouring notes.

Each scale thus created is called a मेल following the current कर्नाटकी terminology. Note that some of the मेलs will not have the पंचम! However, at least one of the कोमल मध्यम or the पंचम is always present. No मेल has all three of the कोमल मध्यम, तीव्र मध्यम, and पंचम present because there is no such traversal path which encompasses all three clusters.

Each मेल can be given a unique identifier by concatenating the identifier of the individual notes that occur in the मेल.

The 42 options for the पूर्वाँग are listed below (the implicit S is dropped from the naming):

r1R1m
r1R1M2
r1g1m
r1g1M2
r1g2m
r1G1m
r1G1M2
r2R1m
r2R1M2
r2R3m
r2R3M4
r2g2m
r2g3m
r2g3M2
r2g3M4
r2G1m
r2G1M2
r2G3m
r2G3M4
R1g2m
R1G1m
R1G3M2
R2g1m
R2g1M2
R2g3m
R2g3M2
R2g3M4
R2G1m
R2G1M2
R2G2M2
R2G2M4
R2G3m
R2G3M4
R3g2m
R3G3m
R3G3M4
g1G1m
g1G1M2
g3G1m
g3G1M2
g3G3m
g3G3M4

The 42 options for the उत्तराँग are listed below:

M1d1D1
M1d1n1
M1d1n2
M1d1N1
M1d2n2
M1D1n2
M1D1N1
M1n1N1
M3d2n2
M3d3D1
M3d3D3
M3d3n2
M3d3n3
M3d3N1
M3d3N2
M3D1n2
M3D1N1
M3D3n2
M3D3N2
M3n3N1
M3n3N2
Pd1D1
Pd1n1
Pd1n2
Pd1N1
Pd3D1
Pd3D3
Pd3n2
Pd3n3
Pd3N1
Pd3N2
PD1n2
PD1N1
PD2n1
PD2n3
PD2N1
PD2N2
PD3n2
PD3N2
Pn1N1
Pn3N1
Pn3N2

The 5 'glue' options which connect the पूर्वाँग to the उत्तराँग are listed below:

mM1 - 21 पूर्वाँग options leading m, 8 उत्तराँग options trailing M1. Total 168.
mM3 - 21 पूर्वाँग options leading m, 13 उत्तराँग options trailing M3. Total 273.
mP - 21 पूर्वाँग options leading m, 21 उत्तराँग options trailing P. Total 441.
M2P - 13 पूर्वाँग options leading M2, 21 उत्तराँग options trailing P. Total 273.
M4P - 8 पूर्वाँग options leading M4, 21 उत्तराँग options trailing P. Total 168.

In all, this indicates that the TOTAL number of मेलs is 1323.


थाट् Taxonomy

This section exists out of academic curiosity only and can be safely skipped in the first reading. It has no direct relevance to the rest of this work.

The large number of मेलs can also be collapsed into a coarser taxonomy of 108 थाट्s. Following Bhatkhande's plan, this is done by not discriminating between the individual members of each स्वर cluster.

The 6 पूर्वाँग options are listed below:
rR
rg
rG
Rg
RG
gG

The 6 उत्तराँग options are listed below:
dD
dn
dN
Dn
DN
nN

The 3 मध्यम पंचम options are listed below:
mM
mP
MP

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A Model for Implied Intonations in Classical हिन्दूस्तानी (Hindustani) Music

What follows is a collection of my scattered meditations on modelling the scales used in classical हिन्दूस्तानी (Hindustani) music. A key difference is that my work uses a purely combinatorial approach instead of the more conventional transpositions (e.g. cycle of fifths, pitch-translation of harmonic scales, etc.) used by earlier efforts.


Disclaimer

The स्वरs of classical हिन्दूस्तानी music are not fixed intonational positions; and there is much to be said about the handling ('attack', 'release') and ornamentation of स्वरs during राग विस्तार which I don't deal with at all. There is also a certain 'flexibility' inherent in the use of स्वरs depending on the performer's personal taste and mood. Nevertheless, it is my belief that each स्वर can be assigned one or more implied intonations because रागs can be identified even when there is wide deviation in the performed intonation. This suggests that the psycho-acoustical perception of intervals involves pattern matching to an ideal intonational centre.


Abstract

This work outlines a new model for scales pertinent to the present day practice of classical हिन्दूस्तानी music. The model is based on intervallic atoms that arise naturally in the overtone series implied by a tonic. Whereas the current nomenclature in classical हिन्दूस्तानी music hems to a chromatic sequence of twelve स्वर­-स्थानs, it is shown that the स्वर­-स्थानs are equivalence classes of notes which represent the implied intervals across all रागs. The value of each note is derived from relatively simple first principles.


On शृति and स्वर

There has been considerable controversy on the subjects of शृति and स्वर in the sphere of classical हिन्दूस्तानी musicology. Most, if not all, of the discussions on this are in vain because the two crucial terms, शृति and स्वर, are not always defined clearly; and when they are, the definition varies from person to person. Further more, with respect to the हिन्दूस्तानी पद्धती, it has been demonstrated by Jairazbhoy [Jai1963, Jai1995] and Levy [Lev1981] (and more recently by Krishnaswamy [Kri2003] for the कर्नाटकी पद्धती) that the range of intonation of a स्वर was variable between:

1. Two performers exploring the same राग.
2. Two intonations of the same स्वर by a given performer in the same राग.

Clearly then, a स्वर represents a cluster of intonational possibilities, possibly around an implied tonal centre, which varies according to
(i) the melodic context (including the attack/release specific to the स्वर / राग at hand); and
(ii) the intonation implied by the performer based on her personal interpretation.

The topic of शृति is best avoided since no two practitioners can agree on what these are.

The present work is partly motivated by an internal resistance to the oft-repeated २२-शृति model [Bha1984] which has no basis in actual practice, at least in modern times. Whether it had a basis in earlier times is also in question: it is impossible that the २२-शृति system, as described by मुनि भरत, could have existed without being equally tempered [Bha1984]. It is more likely that मुनि भरत's description of the २२-शृति system was a happy approximation (given the constraints of measurement and mathematics during his time) or a thought-experiment and should not be taken too literally.


The Overtone Scale and the Intervallic Atoms

Intonational variability in actual performance notwithstanding, it is instructive to consider whether specific intervals are implied either in the acoustic or the psycho­-acoustic domain. A tonic, established in the background by a drone, provides the beginnings of a justly intoned scale based on the (6­-limit) overtone series:

1/1, 6/5, 5/4, 4/3, 3/2, 5/3, 2/1

where the intervals between successive notes are:

6:5, 25:24, 16:15, 9:8, 10:9, 6:5

The intervals between the notes provide the intervallic atoms which can be treated as first class entities for the purposes of the present model:

25:24 - Small Semitone - s
16:15 - Just Semitone - S
9:8 - Major Tone - T

Note, that the 10:9 (Minor Tone) is not an atom since it is composed of the { s, S } pair. The 6:5 (Just Minor Third) similarly is composed of the { S, T } pair.

It is instructive that with a तान-पुरा tuned to the tonic and its पंचम (Perfect Fifth), or मध्यम (Just Fourth), the set of intervallic atoms obtained is { 25:24, 16:15, 27:25 }. This set of atoms eventually leads to a 'dense chromatic' scale with 67 notes to the octave which is rather unwieldy and impractical to reason with. In such a model, "anything goes". The situation is worse with a 7-limit overtone series for even just a tonic: we get { 36:35, 25:24, 21:20, 16:15 } as intervallic atoms and end up with a really dense chromatic scale with 207 notes to the octave ...

The span of an octave can then be represented by the interval sequence:

{ S, T }, s, S, T, { s, S } , { S, T }

A natural consequence of identifying these intervallic atoms is to wonder about the possibility of new scales created by shuffling the positions of these intervallic atoms in the span of an octave.


Scale Generation using the Intervallic Atoms

The following questions naturally arise:

• What kinds of genera can be constructed by bin-packing the intervallic atoms into the span of an octave?
• Which of these genera are permissible in the हिन्दूस्तानी पद्धती?

As seen above, an octave can be spanned by a set of { 3 Major Tones, 4 Just Semitones, and 2 Small Semitones }. In fact, this is the only set which can do so (see The Fundamental Theorem of Intervallic Atoms). This yields 9C3 * 6C4 = 1260 genera. However, 684 of these have neither the Just Fourth nor the Perfect Fifth. The latter genera are ruled out as potential constructors of 'मेलs' (parent scales) in the हिन्दूस्तानी पद्धती. The remaining 576 genera can be used to construct candidate मेलs.

Another way of modelling this is to consider the octave as two overlapping Perfect Fifth Intervals. The first one spans the Unison (1/1) to the Perfect Fifth (3/2), the second one spans the Just Fourth (4/3) to the Octave (2/1). The interval of overlap is the Major Tone (9:8). The Perfect Fifths are spanned by a unique set of { 2 Major Tones, 2 Just Semitones, 1 Small Semitone }. The complementary Just Fourths are spanned by a unique set of { 1 Major Tone, 2 Just Semitones, 1 Small Semitone }.

Bin-packing the Perfect Fifth yields 5C2 * 3C2 = 30 unique genera.
Bin-packing the Just Fourth yields 4C1 * 3C2 = 12 unique genera.

Thus, bin-packing the Octave in the manner described above yields 30*12 + 12*30 - 12*1*12 = 576 unique genera.

The image on the left shows the graphical view of the 576 genera. Each node represents a possible note (its interval ratio indicated by the label) and an arrow from one node to another represents the interval between the corresponding notes. A traversal from the Unison (1/1) to the Octave (2/1) maps a unique genera.

Note that the notes naturally cluster together into 12 groups (counting the Unison and the Octave in the same group) - which reflects the modern day classification of १२ स्वर-स्थानs. Interestingly enough, some notes in a given cluster differ from their nearest neighbour by the interval of a Comma (81:80, ~21.5c) or less. At worst, the difference between two nearest neighbours in a cluster is no greater than a Quartertone of Equal Temperament (50c).

Each cluster is given a name, and, inspired by 19th century terminology, they are:

षड्ज = { 1/1, 2/1 }
कोमल ऋषभ = { 25/24, 16/15 }
तीव्र ऋषभ = { 10/9, 9/8, 256/225 }
कोमल गाँधार = { 75/64, 32/27, 6/5 }
तीव्र गाँधार = { 5/4, 81/64, 32/25 }
कोमल मध्यम = { 675/512, 4/3, 27/20 }
तीव्र मध्यम = { 25/18, 45/32, 64/45, 36/25 }
पंचम = { 40/27, 3/2, 1024/675 }
कोमल धैवत = { 25/16, 128/81, 8/5 }
तीव्र धैवत = { 5/3, 27/16, 128/75 }
कोमल निषाद = { 225/128, 16/9, 9/5 }
तीव्र निषाद = { 15/8, 48/25 }

Note that the term शुद्ध is avoided completely because it suggests the existence of a 'standard' scale. Instead, कोमल is used to indicate 'flat' notes and तीव्र to indicate the corresponding 'sharp' ones.

It can't be stressed enough that these notes represent the tonal centres around which there is a certain 'flexibility' (or sloppiness?) allowed during the exploration of a राग. Thus, in actual practice, there would be significant variation in the exact note intoned by a performer. The model above represents an ideal for the implied intonation.

Adjustments for Contemporary Practice

In contemporary हिन्दूस्तानी music, the "शुद्ध" मध्यम is (acoustically & musically) always the Just Fourth (4/3), and the पंचम is always the Perfect Fifth (3/2). Now, the 40/27 and 1024/675 notes are a Comma, or less, away from the intonationally strong Perfect Fifth (3/2) and will always be construed as a mis-tuned Perfect Fifth (or, as a 'false' fifth) by any listener since the पंचम plays a vital role via it's strong consonance in the scale. Similarly, the 675/512 and 27/20 are a Comma, or less, away from the Just Fourth (4/3) and will always be construed as a mis-tuned Just Fourth (or, as a 'false' fourth) for the same reasons. In practice, no scale which contains any of these four 'false' notes as न्यास स्थानs (rest locations) would be viable in the context of present-day हिन्दूस्तानी music. [This does not mean that these 'false', or wolf, intervals wouldn't exist between two arbitrary notes in a राग. For instance, a 27:20 interval exists in a राग, like मेघ (?), which employs the 10/9 ऋषभ and the 3/2 पंचम.] However, these notes could definitely exist as leading grace embellishments to the पंचम or मध्यम.

On Septimal Notes

Clements suggests that septimal notes were inducted into हिन्दूस्तानी music as a result of Islamic influence [Cle1913]. Specifically, based on the work of Raosaheb Krishnaji Ballal Deval, Clements identifies the following notes: 21/20, 21/16, 63/40, and 7/4. However, I'm not entirely convinced that Indian musicians have an innate understanding of septimal harmonies/intervals. Rather, I'm certain that they naturally converge to the equivalent quintal/tertian interval.

In the model above, the equivalents for the septimal notes are: 25/24, 675/512, 25/16, and 225/128. Each of these notes is less than 14c away from its corresponding septimal note, and thus is approximated by the septimal note effectively.


Notation for the Notes

Each cluster is notated using a one letter roman symbol which represents its (transliterated) name. कोमल clusters are denoted by a lower-case letter, and the corresponding तीव्र clusters by the upper-case letter. Notes within a cluster are identified using integers suffixed to the cluster symbol in increasing order of interval size. The four 'false' notes are identified with -/+ signs as appropriate. The interval value in cents is noted in parenthesis.

षड्ज Cluster
S = 1/1 (0.00c), 2/1 (1200.0c)

कोमल ऋषभ Cluster
r1 = 25/24 (70.67c)
r2 = 16/15 (111.73c)

तीव्र ऋषभ Cluster
R1 = 10/9 (182.40c)
R2 = 9/8 (203.91c)
R3 = 256/225 (223.46c)

कोमल गाँधार Cluster
g1 = 75/64 (274.58c)
g2 = 32/27 (294.13c)
g3 = 6/5 (315.64c)

तीव्र गाँधार Cluster
G1 = 5/4 (386.31c)
G2 = 81/64 (407.82c)
G3 = 32/25 (427.37c)

कोमल मध्यम Cluster
m- = 675/512 (478.49c)
m = 4/3 (498.04c)
m+ = 27/20 (519.55c)

तीव्र मध्यम Cluster
M1 = 25/18 (568.72c)
M2 = 45/32 (590.22c)
M3 = 64/45 (609.78c)
M4 = 36/25 (631.28c)

पंचम Cluster
P- = 40/27 (680.45c)
P = 3/2 (701.96c)
P+ = 1024/675 (721.51c)

कोमल धैवत Cluster
d1 = 25/16 (772.63c)
d2 = 128/81 (792.18c)
d3 = 8/5 (813.69c)

तीव्र धैवत Cluster
D1 = 5/3 (884.36c)
D2 = 27/16 (905.87c)
D3 = 128/75 (925.42c)

कोमल निषाद Cluster
n1 = 225/128 (976.54c)
n2 = 16/9 (996.09c)
n3 = 9/5 (1017.60c)

तीव्र निषाद Cluster
N1 = 15/8 (1088.27c)
N2 = 48/25 (1129.33c)


Forward Consonances

All notes in the model have a note with which they form a consonance of the मध्यम (Just Fourth) or the पंचम (Perfect Fifth) with the following exceptions:

• G2 (81/64), m- (675/512), m+ (27/20), M2 (45/32), M4 (36/25) don't have a corresponding पंचम but they do have a मध्यम.
• Reciprocally, M1 (25/18), M3 (64/45), P- (40/27), P+ (1024/675), d2 (128/81) don't have a corresponding मध्यम but they do have a पंचम.
  

Each note in the model is thus a candidate for a वादी-संवादी pair for those who still adhere to the वादी-संवादी theory.


Future Work

Given any model for हिन्दूस्तानी music, one way to validate its practicality in reasoning about रागs is to synthesize specific राग based melodies from the model and subjectively assess whether they are identifiable as the intended राग. However, before that, there needs to be agreement on whether the choice of notes for a given राग is indeed accurate. Both these issues are closely related topics which need more analysis and research. As an example, consider the two scales below which can be generated from the model above:

1/1, 25/24, 75/64, 45/32, 3/2, 25/16, 15/8, 2/1

1/1, 16/15, 6/5, 36/25, 3/2, 8/5, 48/25, 2/1

Or, using the notation:

S, r1, g1, M2, P, d1, N1, S*

S, r2, g3, M4, P, d3, N2, S*

Both the above scales fall under the aegis of the modern तोड़ी ठाट (SrgMPdNS). However, does the first scale represent the canonical तोड़ी while the second मुल्तानी?

Another issue to resolve is exactly how much deviation around the tonal centres is allowable before advanced practitioners conclusively detect that a performer has become बे-सुर with respect to the राग being explored.

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Sidebars

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There are additional ruminations not directly related to the main text of this post. The section on "Weights-based grouping" is especially interesting.

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References

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[Bha1984] Pt. Vishnu Narayan Bhatkhande, "Music Systems in India: A Comparative Study of Some of the Leading Music Systems of the 15th, 16th, 17th, & 18th Centuries", South Asia Books, 1984.

[Cle1913] Ernest Clements, "Introduction To The Study Of Indian Music: An Attempt To Reconcile Modern Hindustani Music With Ancient Musical Theory And To Propound An Accurate And Comprehensive Method Of Treatment Of The Subject Of Indian Musical Intonation", Longmans, Green, and Co., 1913.

[Jai1963] Nazir Ali Jairazbhoy, A. W. Stone, "Intonation in Present­ Day North Indian Classical Music", Bulletin of the School of Oriental and African Studies, University of London, 1963. pp. 119­-132.

[Jai1995] ^^Electronic analysis has confirmed that there is variation in intonation from one musician to another, as well as for a single musician during the course of a performance^^, Nazir Ali Jairazbhoy, "The Rags of North Indian Music: Their Structure and Evolution", Popular Prakashan, 1995, pg 34.

[Kri2003] Arvindh Krishnaswamy, "Pitch Measurements versus Perception of South Indian Classical Music", Proceedings of the Stockholm Music Acoustics Conference (SMAC 03), 2003.

[Lev1981] Mark Levy, "Intonation in North Indian Music: A Select Comparison of Theories with Contemporary Practice". Biblia Impex Pvt. Ltd., 1981.

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