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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Scale Taxonomy in the कर्नाटकी पद्धती

The कर्नाटकी पद्धती is often claimed to be faithful to the music of ancient (pre-Islamic) India. In some respects this is reflected even today in the musicological terms used. Specifically, terms like शुद्ध, चतुःशृति, षट्शृति, etc. have a provenance in the २२-शृति model of मुनि भरत् documented in the नाट्यशास्त्र and in the संगीत्-रत्नाकर् by शारंग्देव्. I've always thought that the use of this terminology in the कर्नाटकी पद्धती was somewhat curious. But before I elaborate on what I mean, it is worth taking a quick look at the षड्ज् ग्राम् of मुनि भरत्.

Recovering the षड्ज् ग्राम्

According to the नाट्यशास्त्र, the षड्ज् ग्राम् is defined by the allocation of the २२ शृतिs (whatever the term शृति means) into an octave as ४/३/२/४/४/३/२:

नि - - - स - - र - ग - - - म - - - प - - ध - नि

This is also the division in the चतुर्दंडीप्रकाशिका of व्यंकतमुखी (1560 A.D.) indicating continuity with the tradition of the नाट्यशास्त्र. The names given by व्यंकतमुखी to these स्वर्s were:

०० - शुद्ध निषाद् / पंचशृति धैवत्
०१ - कैशिक निषाद् / षट्शृति धैवत्
०२ - काकली निषाद्
०४ - षड्ज्
०७ - शुद्ध ऋषभ्
०९ - शुद्ध गाँधार् / पंचशृति ऋषभ्
१० - साधारण् गाँधार् / षट्शृति ऋषभ्
११ - अंतर् गाँधार्
१३ - शुद्ध मध्यम्
१६ - वराळी मध्यम्
१७ - पंचम्
२० - शुद्ध धैवत्
२२ - शुद्ध निषाद् / पंचशृति धैवत्

[The modern terminology however uses चतुःशृति ऋषभ् (शृति ८) and चतुःशृति धैवत् (शृति २१) in place of the पंचशृति versions. This has the advantage of avoiding १-शृति intervals between notes in a मेल्. The मध्यम् terminology is also different. It is not clear to me when this change took place.]

The शुद्ध scale above (i.e. षड्ज् ग्राम्) corresponds to the काफ़ी mode of the modern हिन्दूस्तानी पद्धती when referred to the षड्ज् based tonality. [However, in the निषाद् based tonality, it really represents the बिलावल् mode.]

In the षड्ज् ग्राम्, the intervals between the notes are either ४, ३, or २ शृतिs. Clearly the २-शृति intervals are akin to modern semitones, and the ४-शृति intervals to modern tones. For the purposes of the current discussion the exact numerical sizes of the intervals don't matter.

Discussion on the notes

With reference to a षड्ज् based tonality, the (३-शृति) शुद्ध ऋषभ् of the षड्ज् ग्राम् represents an interval larger than a २-शृति 'semitone'. However, present-day कर्नाटकी पद्धती definitely employs a शुद्ध ऋषभ् which is a 'semitone' away from the षड्ज्. In other words, contemporary practice seems to no longer have any relation with theory. It appears that the modern शुद्ध ऋषभ् is more representative of a २-शृति interval.

The षट्शृति ऋषभ् presents its own problem. When this note is used in a मेल्, the choice of the following गाँधार् is then restricted to the अंतर् गाँधार् which however forms a १-शृति interval with the षट्शृति ऋषभ्. १-शृति intervals are meant to be avoided in theory because of their alleged dissonance. Choosing the next higher शृति for the गाँधार् is also not possible because that location forms a १-शृति interval with the शुद्ध मध्यम्. This suggests that the षट्शृति ऋषभ् cannot be a viable note in the modern कर्नाटकी पद्धती.

Redefining the notes

Given what has been said above, the whole note naming in the कर्नाटकी पद्धती needs to be revisited. The only rule to be followed is that two adjacent notes cannot be closer than २-शृतिs. Thus, the शुद्ध ऋषभ् can be no lower than the २-शृति position. This puts the शुद्ध गाँधार् at the ४-शृति position. Similarly, the highest गाँधार् cannot be higher than the ७-शृति position which puts the highest ऋषभ् at the ५-शृति position. The ऋषभ् then can range from the २-शृति to the ५-शृति position while the गाँधार् can range from the ४-शृति to the ७-शृति position. Similar considerations apply to the positions of the धैवत् and निषाद्.

The individual positions for each स्वर् are then better identified by the following qualifiers: शुद्ध, तीव्र, तीव्रतर्, and तीव्रतम्.

Final Taxonomy

Based on all the discussions above, a new taxonomy for the notes in the कर्नाटकी पद्धती is proposed:

०० - षड्ज् (S)
०२ - शुद्ध ऋषभ् (r1)
०३ - तीव्र ऋषभ् (r2)
०४ - तीव्रतर् ऋषभ् (r3) / शुद्ध गाँधार् (g1)
०५ - तीव्रतम् ऋषभ् (r4) / तीव्र गाँधार् (g2)
०६ - तीव्रतर् गाँधार् (g3)
०७ - तीव्रतम् गाँधार् (g4)
०९ - शुद्ध मध्यम् (m1)
१० - तीव्र मध्यम् (m2)
११ - तीव्रतर् मध्यम् (m3)
१२ - तीव्रतम् मध्यम् (m4) [Unused]
१३ - पंचम् (P)
१५ - शुद्ध धैवत् (d1)
१६ - तीव्र धैवत् (d2)
१७ - तीव्रतर् धैवत् (d3) / शुद्ध निषाद् (n1)
१८ - तीव्रतम् धैवत् (d4) / तीव्र निषाद् (n2)
१९ - तीव्रतर् निषाद् (n3)
२० - तीव्रतम् निषाद् (n4)

The possible options for the ऋषभ् and गाँधार् give 10 'genera':
r1 g1
r1 g2
r1 g3
r1 g4
r2 g2
r2 g3
r2 g4
r3 g3
r3 g4
r4 g4

The possible options for the मध्यम् and पंचम् give 3 'genera' (assuming that P must be present):
m1 P
m2 P
m3 P

The possible options for the धैवत् and निषाद् give 10 'genera':
d1 n1
d1 n2
d1 n3
d1 n4
d2 n2
d2 n3
d2 n4
d3 n3
d3 n4
d4 n4

In all, this gives 10 * 3 * 10 = 300 'genera' in all. Each of them is a valid candidate 'मेल्'.

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.