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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