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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