It's time I learn about Indian music. Hindustani? Carnatic? Shrutis? Let's go.
Here's what I think I know. The basical musical notes are called swaras. Their solfege goes "Sa Ri/Re Ga Ma Pa Dha Ni Sa". They're short for "Shadja, Rishabh, Gandhar, Madhyam, Pancham, Dhaivat, Nishad" in North India, or those words with an additional "-am" in south India. I don't know why "Re" is an option as an abbreviation for "Rishabh", but I'd say it's a more common abbreviation than "Ri".
The scales are called ragas. They're tuned in 5-limit just intonation.
When you sing a raga in solfege, you sing the Ga for the third scale degree whether it's a major third or a minor third. Carnatic music distinguishes textually at least between Ga1, a minor third, and Ga2, a major third. Here are some options for each swara:
(P1 # 1/1) :: Sa
(Grm2 # 256/243) or (m2 # 16/15) :: Re1
(M2 # 10/9) or (AcM2 # 9/8) :: Re2
(Grm3 # 32/27) or (m3 # 6/5) :: Ga1
(M3 # 5/4) or (AcM3 # 81/64) :: Ga2
(P4 # 4/3) or (Ac4 # 27/20) :: Ma1
(AcA4 # 45/32) or (AcAcA4 # 729/512) :: Ma2
(P5 # 3/2) :: Pa
(Grm6 # 128/81) or (m6 # 8/5) :: Dha1
(M6 # 5/3) or (AcM6 # 27/16) :: Dha2
(Grm7 # 16/9) or (m7 # 9/5) :: Ni1
(M7 # 15/8) or (AcM7 # 243/128) :: Ni2
You can see that usually "1" means minor and "2" means major. The only time we break this pattern is with the fourth scale degree, Ma, for which "1" means perfect and "2" means augmented.
With only one option for ^1 (=Sa) and for ^5 (=Pa), we get 22 options all together. I've sometimes seen (Gr5 # 40/27 @ 680c) discussed in texts on indian scale intonation - I think "mentioned in Bharata's "Natya Shastra" but not used in modern Indian music" is probably a concise explanation of that.
I've also seen a notation system which gives three options each for Ri, Ga, Dha, Ni, but then equates one of each set of three options to another interval. I think the system is basically this:
S ~ P1 or perfect unison,
R1 ~ m2 or minor second,
R2 ~ M2 or major second,
R3 ~ A2 or augmented second,
G1 ~ d3 or diminished third,
G2 ~ m3 or minor third,
G3 ~ M3 or major third,
M1 ~ P4 or perfect fourth,
M2 ~ A4 or augmented fourth,
P ~ P5 or perfect fifth
D1 ~ m6 or minor sixth
D2 ~ M6 or major sixth
D3 ~ A6 or augmented sixth
N1 ~ d7 or diminished seventh
N2 ~ m7 or minor seventh
N3 ~ M7 or major seventh
Even though there are 16 lines in this table, it's more like 12 distinct sound levels because they recognize these equivalences:
Ri2 = Ga1 :: m2 ~ d3
Ri3 = Ga2 :: A2 ~ m3
Dha2 = Ni1 :: M6 ~ d7
Dha3 = Ni2 :: A6 ~ m7
I don't like this system at all. In addition to not distinguishing between differences of Ac1 (the options in the previous system), it doesn't distinguish between intervals separated by diminished 2nds, i.e. different letter names. So it's just 12-TET. And it doesn't even use the same numbers as the previous system. They could have done 0 for diminished, 1 for minor, 2 for major, 3 for augmented. Or something. Let's not talk of it again.
The 22 notes from the previous sytem, all the 5-limit options, are called shrutis. There are supposedly some different accounts of the shrutis, but I think the above is the most widely accepted. I have trouble finding differing accounts at least. Sometimes you'll see long names for each individual shruti. I've seen enough disagreement that I decided to just use my own interval names for now, but I think I'll figrue it out and give fairly standard long names for shrutis from both Carnatic and Hindustani music shortly.
...
Some old Indian music theorists talked like the 22 shrutis were equally spaced - 9/8 is a distance of 4 shrutis, 4/3 is a distance of 9 shrutis, 3/2 is a distance of 13 shrutis. This is weird, but actually consistent with a 22-EDO (or 44-EDO or 66-EDO) interpretation. In fact, if you tune all 22-shrutis in 22-EDO, you just get a list of ascending integers.
P1: 0\22
Grm2: 1\22
m2: 2\22
M2: 3\22
AcM2: 4\22
Grm3: 5\22
m3: 6\22
M3: 7\22
AcM3: 8\22
P4: 9\22
Ac4: 10\22
AcA4: 11\22
AcAcA4: 12\22
P5: 13\22
Grm6: 14\22
m6: 15\22
M6: 16\22
AcM6: 17\22
Grm7: 18\22
m7: 19\22
M7: 20\22
AcM7: 21\22
P8: 22\22
People don't use any of those EDOs for indian music these days, if they ever did. And they might have? But not now. We'll come back to this.
I think it's exciting that the Chinese guqin and the indian shrutis are both 5-limit; in addition to some 5-limit just intonation efforts in the west, there are multiple independent well developed 5-limit composition systems in this world. If you were ever not sure what to do with yourself in 5-limit, or if you weren't sure of the importance of 5-limit just intonation to humankind, just look around for inspiration. It's everywhere.
Ragas are sometimes separated into two tetrachords - the purvanga ("first part") from Sa to Ma, and the uttaranga ("second part") from Pa to high Sa.
A raga has an ascending form, "arohana", and a descending form, "avarohana". Often the same. A raga has a primary melodic note, "vadi", and a secondary melodic note, "samvadi", which is usually P4/P5 away from vadi.
Not all ragas have 7 notes. Often a note is skipped. The skipped note is called varjya, ("avoided, excluded, forbidden"). I think it's correct to describe skipped scale degrees as varjya, but also a varjya note might tell you to avoid one option for a scale degree when another one is present in the scale - e.g. don't ever play a minor third in melodic ornaments for a certain raga that contains a major third.
The raga called Mohanam is pentatonic, and it skips ^4 and ^7, i.e. Ma and Ni, in both ascending and descending forms. Perhaps we could call these varjya swaras (skipped scale degrees) in contrast to varjyra shrutis (forbidden notes).
A raga will also often have an associated melodic phrase, a pakad, which helps people to identify it.
The cyclic permuation of a scale to get a new mode is called Graha Bhedam in Carnatic music or murchhana in Hindustani music.
The primary ragas of Carnatic music are called Melakarta, and there are 72 of them. I don't think they're all used - they just form the full combinatoric space of 7 note scales with certain nice properties. Other Carnatic ragas are considered to be derived from the primary ones, and are called Janya ragas. I believe that Carnatic music has pretty well settled into a finite closed set of ragas at this point, possibly as a consequence of detailed codification, like the 72 melakarta raga system.
Somewhat similarly to the 72 melakarta ragas of Carnatic music, Hindustiani music has Thaats, which are like families of Ragas. Hindustani ragas are definitely not a closed set, but we can still look the most popular ones.
Let's look at some scales.
I learned about five Hindustanti ragas from the authors of the ShrutiSense program, or at least Rajarshi Ghosh (the code repository is in his name). Here are their shruti names:
P1: Sa
Grm2: Re1
m2: Re1_
M2: Re2
AcM2: Re2_
Grm3: Ga1
m3: Ga1_
M3: Ga2
AcM3: Ga2_
P4: Ma1
Ac4: Ma1_
AcA4: Ma2
AcAcA4: Ma2_
P5: Pa
Grm6: Dha1
m6: Dha1_
M6: Dha2
AcM6: Dha2_
Grm7: Ni1
m7: Ni1_
M7: Ni2
AcM7: Ni2_
P8: Sa_
Some of the intervals get an underscore if they're a syntonic comma off from a more ?common? interval. Also the octave gets an underscore. The interval with a 3-limit just tuning is the more common / unadorned option when the interval has a minor sonority and the interval with a 5-limit tuning is the more common / unadorned option when the interval has a major sonority.
All five ragas from the ShrutiSense authors have the same ascending and descending forms. We'll first look at the ragas Yaman and Kalyan. They have identical scales:
[Sa, Re2, Ga2, Ma2, Pa, Dha2, Ni2, Sa_]
which is
[P1, AcM2, M3, AcA4, P5, M6, M7, P8] # [1/1, 9/8, 5/4, 45/32, 3/2, 5/3, 15/8, 2/1]
They also have the same vadi or primary melodic note, namely Ga2 ~ M3, and the same samvadi of Ni2 ~ M7. And since
M3 + P5 = M7
that looks fine. The two ragas also have the same avoid-note / forbidden-tone / varjya shruti, namely Ma1 ~ P4. The shruti Ma1 isn't in the scale, but they want you to be sure you don't play it even a little bit in melodic ornaments.
These two ragas, it seems, only differ in their distinguishing melodic motif, their pakad. Here's how the ShrutiSense authors share pakads for ragas:
Yaman pakad: [[Ni2, Re2, Ga2], [Ma2, Pa, Dha2]]
Kalyan pakad: [[Sa, Re2, Ga2, Ma2], [Pa, Ma2, Ga2]]
Those don't quite look like melodies to me - the notes don't have duration, and I don't know why they're bracketed that way. But supposedly those two lines hold the key for disambiguating these two ragas.
Let's look at raga Bilaval next. Its scale is simply a 5-limit major scale:
[Sa, Re2, Ga2, Ma1, Pa, Dha2, Ni2, Sa_]
or
[P1, M2, M3, P4, P5, M6, M7, P8]
Its vadi is Sa ~ P1, its samvadi is Pa ~ P5. It has no varjya to avoid. Its pakad is
[[Sa, Re2, Ga2], [Pa, Dha2, Ni2]]
Nice. At some point I'll listen to all the pakads and find out what structure they possess, if any.
Next let's look at raga Khamaaj. Here is its scale:
[Sa, Re2, Ga2, Ma1, Pa, Dha2, Ni1, Sa_]
or
[P1, M2, M3, P4, P5, M6, Grm7, P8]
The vaid is Pa ~ P5, the samvadi is Sa ~ P1. The varjya to avoid is Ni2 ~ M7. And the pakad is
[[Sa, Ga2, Ma1, Pa], [Pa, Dha2, Ni1]]
.
Finally here is raga Bhairavi. Its scale is
[Sa, Re1, Ga1, Ma1, Pa, Dha1, Ni1, Sa_]
or
[P1, Grm2, Grm3, P4, P5, Grm6, Grm7, P8]
The vadi is Ma1 ~ P4, the samvadi is Sa ~ P1, there is no varjya, and the pakad is
[[Sa, Re1, Ga1], [Ma1, Pa, Dha1]]
You've heard that before, right? Classic raga Bhairavi pakad.
I've seen other ragas described in other sources, but at the level of 12-TET swaras, not at the level of shrutis. I'm going to keep looking.
...
Here's some more data from Dinesh S. Thakur in "The Notion of Twenty-Two Shrutis". He has his own shruti notation:
S: 1/1
R11: 256/243
R12: 16/15
R21: 10/9
R22: 9/8
G11: 32/27
G12: 6/5
G21: 5/4
G22: 81/64
M11: 4/3
M12: 27/20
M21: 45/32
M22: 729/512
P: 3/2
D11: 128/81
D12: 8/5
D21: 5/3
D22: 27/16
N11: 16/9
N12: 9/5
N21: 15/8
N22: 243/128
Using this, he describes two pentatonic ragas, namely raga Bhoop and raga Deskar.
Bhoop: [S, R21, G21, P, D21, S']
Deskar: [S, R22, G22, P, D22, S']
These are
Bhoop: [P1, M2, M3, P5, M6, P8]
Deskar: [P1, AcM2, AcM3, P5, AcM6, P8]
Nice. These are just 5-limit and 3-limit versions of the major pentatonic scale. Easy.
He also decsribes raga Todi in 12-EDO, and then tries to infer the shrutis from knowing that ^2 should be its lower variant, R11, and then assuming that notes of the scale are related by factors of 3/2 and 4/3 as much as possible. This is interesting. I'm not sure that it's right, but here's what he came up with:
[S, R11, G11, M21, D11, N21, S']
That's a six note scale, but he says P is also used sometimes. But keeping with the six note scale, we get these intervals:
[P1, Grm2, Grm3, AcA4, Grm6, M7, P8]
I don't think he's very confident in this, but it's something.
...
Vidyadhar Oke of 22shruti.com describes lots of ragas in term of shrutis. He has his own notation:
S = 1/1
r1 = 256/243
r2 = 16/15
R1 = 10/9
R2 = 9/8
g1 = 32/27
g2 = 6/5
G1 = 5/4
G2 = 81/64
M1 = 4/3
M2 = 27/20
m1 = 45/32
m2 = 729/512
P = 3/2
d1 = 128/81
d2 = 8/5
D1 = 5/3
D2 = 27/16
n1 = 16/19
n2 = 9/5
N1 = 15/8
N2 = 243/128
Upper and lower case letters, just one number. I like this fine.
He describes these on the page "Topic 5":
Raga Bhoop: [S, R1, G1, P, D1]
Raga Bageshri: [S, R1, g1, m1, P, D1, n1]
Raga Darabari: [S, R1, g1, M1, P, d1, n1]
Raga Bhairav: [S, r1, G1, M1, P, d1, N1]
Raga Saraswati: [S, R2, m1, P, D2, n2]
On the page "Topic 53", he describes these hindustani 100 ragas. Why does he have 100 when other people have 2 to 5? I don't know. Maybe he made them all up. Maybe he's just that good. From his writing, I wouldn't guess that he's all that good, but he does have those 100 ragas. Let's dive in. I started by trascribing all of them, but quickly saw how many were 4 or 5 note scales and got disinterested. I'm going to focus on more complete ragas.
1. Raga Abhogi: [S, R1, g1, M1, D1]
2. Raga Adana: [S, R2, g2, M2, P, d2, n2]
3. Raga Ahirabhairav: [S, G1, M1, P, D1, n1] // He also describes 4. Ahirabhairav as having a larger set of swaras: [S, r2, G1, M1, P, D1, n1].
4. Raga Amritavrshini: [S, G1, m1, P, n1]
5. Raga Arunakouns: [S, G1, M1, N1]
6. Raga Bageshri: [S, R1, M1, P, D1, n1]
7. Raga Bageshrikouns: [S, M1, D1, n1] // More swaras: [S, g1, M1, D1, n1].
8. Raga Bahar: [S, R2, M1, P, D1, N1] // More swaras: [S, R2, G1, M1, P, D1, n1, N1]. Both n1 and N1, yes.
9. Raga Bairagi: [S, r2, M1, n1]
10. Raga Barawa: [S, R1, G1, M1, P, D1, n1] // More swaras: [S, R1, g1, G1, M1, P, D1, n1]. This looks a little fake. He lists g1 and then G1, but those aren't in increasing order - g1 is higher than G1.
11. Raga Basant:
12. Raga Basantbukhari:
13. Raga Bhairav:
14. Raga Bhairavi: [S, R2, g2, M2, P, d2, n2]
15. Raga Bhatiyar:
16. Raga Bheemapalas:
17. Raga Bihag: [S, R2, G1, M1, P, D1, N1]
18. Raga Bihagada: [S, R2, G1, M1, P, D1, N1]
19. Raga Bhinnakouns:
20: Raga Bhinnashadja:
21: Raga Bhoop: [S, R1, G1, P, D1]
22: Raga Bhoopal Todi: [S, r2, g2, P, d2]
23: Raga Bibhas:
24: Raga Bilashkhani Todi: [S, r2, g2, M1, P, d2, n2]
25: Raga Bilawal: [S, R2, G1, M1, P, D1, N1]
26: Raga Brindavani Sarang:
27: Raga Champakali: [S, R2, G1, m1, P, d2]
28: Raga Chandrakouns:
29: Raga Charukeshi: [S, R2, G1, M1, P]
30: Raga Chhayanata: [S, R2, G1, M1, P, D1, N1]
31: Raga Darabari: [S, R1, M1, P, n1]
32: Raga Desh: [S, R2, G1, M1, P, D1, N1]
33: Raga Deshakar:
34: Raga Dhani:
35: Raga Durga:
36: Raga Garudadhwani: [S, R2, G1, M1, P, D1]
37: Raga Gorakha Kalyan: [S, R1, M1, D1, n1]
38: Raga Gouda Malhar: [S, R2, G1, M1, P, D1, N1]
39: Raga Gujari Todi: [S, r1, g1, m2, d1, N2]
40: Raga Gunakouns: [S, R2, G1, M1, P]
41: Raga Gunakri: [S, r2, M1, P, d2]
42: Raga Hamir: [S, R2, G1, m1, P, d2, N1]
43: Raga Hansadhwani: [S, R2, G1, P, N1]
44: Raga Hindol:
45: Raga Janasammohinini: [S, R1, G1, P, D1, n1]
46: Raga Jayajayavanti: [S, R2, G1, M1, P, D1, N1]
47: Raga Jayat: [S, G1, P, D2]
48: Raga Jog:
49: Raga Jogakouns:
50: Raga Jogiya
51: Raga Jounapuri: [S, R2, g2, M2, P, d2, n2]
52: Raga Kafi: [S, R2, M1, P, D1]
53: Raga Kalavati:
54: Raga Kalingada:
55: Raga Kamod: [S, R2, G1, m1, P, D2, N1]
56: Raga Kedar: [S, R2, G1, m1, P, D2, N1]
57: Raga Keeravani:
58: Raga Khamaj: [S, R1, G1, M1, P, D1, N1]
59: Raga Kolahas:
60: Raga Lalat:
61: Raga Madhamat Saranga:
62: Raga Madhukouns:
63: Raga Madhuvanti: [S, R2, m1, P, D1, N1]
64: Raga Malakouns: [S, g2, M1, d2, n1]
65: Raga Malhar: [S, R2, M1, P, D1, N1]
66: Raga Mand: [S, R2, G1, P, D2, N1]
67: Raga Marawa:
68: Raga Marubihag: [S, R2, G1, m1, P, D2, N1]
69: Raga Multani: [S, r2, g2, m2]
70: Raga Nand: [S, R2, G1, m1, P, D2, N1]
71: Raga Nata Bhairav: [S, R2, G1, M1, P, N1]
72: Raga Nayaki Kanada: [S, R2, g2, M2, P, n2]
73: Raga Paraj: [S, G1, P, N1]
74: Raga Patadeep: [S, R2, m1, P, D2, N1]
75: Raga Parameshwari: [S, g1, M1, D1, n1]
76: Raga Pooriya: [S, G1, D1, N1]
77: Raga Pooriyadhanashree: [S, G1, P, N1]
78: Raga Poorvi: [S, G1, M1, P, N1]
79: Raga Rageshri: [S, R1, G1, M1, D1, n1]
80: Raga Rajakalyan: [S, R2, G1, m1, D1, N1]
81: Raga Ramadasi Malhar: [S, R2, g1, G1, M1, P, D1, n1, N1] // Both g1 and G1, both n1 and N1. Pretty crazy.
82: Raga Ramakali: [S, G1, M1, P, D1, N1]
83: Raga Salagavarali: [S, r2, g2, P, n2]
84: Raga Saraswati: [S, R2, m1, P, D2]
85: Raga Shahana Kanada: [S, R1, M1, P, D1, n1]
86: Raga Shankara: [S, G1, P, D1, N1]
87: Raga Shree: [S, r2, G2, m2, d2, N2]
88: Raga Sohani: [S, G1, D1, N1]
89: Raga Soora Malhar: [S, R2, G1, M1, P, D1, N1]
90: Raga Sorath: [S, R2, M1, P, D1, N1]
91: Raga Shivaranjani: [S, R2, g2, P]
92: Raga Shuddha Kalyan: [S, R2, G2, m1, P, D2, N1]
93: Raga Shuddha Saranga: [S, R2, m2, P, D2, N1]
94: Raga Shyama Kalyan: [S, R2, G1, m1, P, D2, N1]
95: Raga Tilakakamod: [S, R2, G1, M1, P, D1, N1]
96: Raga Tilanga: [S, R2, G1, M1, P, N1]
97: Raga Todi: [S, r1, g1, m2, d1, N2]
98: Raga Yaman: [S, R2, G1, m1, P, D2, N1]
99: Raga Yaman Kalyan: [S, R2, G1, m1, P, D2, N1]
100: Raga Zinzhoti: [S, R1, G1, M1, P, D1, n1]
Nice.
Let's look at only the full 7-note scales. There are a lot of repetitions:
[P1, AcM2, AcM3, AcA4, P5, AcM6, M7] _ Shuddha Kalyan
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Kamod
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Kedar
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Marubihag
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Nand
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Shyama Kalyan
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Yaman
[P1, AcM2, M3, AcA4, P5, AcM6, M7] _ Yaman Kalyan
[P1, AcM2, M3, AcA4, P5, m6, M7] _ Hamir
[P1, AcM2, M3, P4, P5, M6, M7] _ Bihag
[P1, AcM2, M3, P4, P5, M6, M7] _ Bihagada
[P1, AcM2, M3, P4, P5, M6, M7] _ Bilawal
[P1, AcM2, M3, P4, P5, M6, M7] _ Chhayanata
[P1, AcM2, M3, P4, P5, M6, M7] _ Desh
[P1, AcM2, M3, P4, P5, M6, M7] _ Gouda Malhar
[P1, AcM2, M3, P4, P5, M6, M7] _ Jayajayavanti
[P1, AcM2, M3, P4, P5, M6, M7] _ Soora Malhar
[P1, AcM2, M3, P4, P5, M6, M7] _ Tilakakamod
[P1, AcM2, m3, Ac4, P5, m6, m7] _ Adana
[P1, AcM2, m3, Ac4, P5, m6, m7] _ Bhairavi
[P1, AcM2, m3, Ac4, P5, m6, m7] _ Jounapuri
[P1, M2, M3, P4, P5, M6, Grm7] _ Barawa
[P1, M2, M3, P4, P5, M6, Grm7] _ Zinzhoti
[P1, M2, M3, P4, P5, M6, M7] _ Khamaj
[P1, m2, m3, P4, P5, m6, m7] _ Bilashkhani Todi
There are only 8 distinct scales if we look at them this way, but maybe they have different vadis, samvadis, varjyras, pakads. There's an Ionian and a Phrygian mode, no Aeolian. That's interesting.
It's entirely possible I have some transcription errors, but I'd think that would tend to create more scales, not a mere 8 distinct ones.
...
His Raga Bageshri from the first set of 5 ragas doesn't match his Bageshri from the set of 100 ragas. And his Darabari doesn't match itself. And his Bhairav doesn't match itself. Starting to regret transcribing so much data from this guy. Maybe there would be more agreement if I transcribed his swaras instead of his shrutis. But shrutis are supposed to be more precise than swaras. Why would I do that? Why do some ragas have more swaras than shrutis in his writing?
...
Okay, I have a few ideas of how to proceed.
1) Look at Bharata's "Natya Shastra" for information about ancient ragas. He used shrutis, so these should be microtonal.
2) Look at 12-TET data about ragas and compare it to Vidyadhar Oke's data. Maybe it will show that his shrutis or his swaras are not as terrible as they seem.
3) Just keep looking for sources that describe 2 to 5 ragas at a time in terms of shrutis and hope that I can get a picture of indian music at that rate.
...
I was reading about texts like Natya Shastra and needed a little break. I decided to check if the Huygens Fokker list of musical modes had some ragas. It has tons, and they're specifed at the level of shrutis. Like with all the modes on that list, I don't know who authored them or where they get their ideas, but they're often quite good when they're historic / ethnographic. Let's have a look.
First the modes are described as coming from this set of frequency ratios: [1/1, 256/243, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 4/3, 27/20, 45/32, 729/512 (or 64/45), 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 243/128]. The only thing to notice here is the 64/45 option. This is the just tuning for the grave diminished fifth, Grd5, and it's the octave complement of the acute augmented fourth that we all know and love from other shruti texts, 45/32. I'm not going to use 64/45 for a bunch of reasons.
First, it's tuned by 22-EDO to 11 steps, not 12 like 729/512. So it's at least in the wrong place on this list.
Second, I've never heard of anyone using it in shruti analysis. I've seen 7/5 as a tritone in "‘Yati’ – Decorative mathematical patterns in Carnatic music" by Srividhya Balaji, 2020. I've seen 40/27 mentioned in discussions of Bharata's "Natya Shastra". Give me a historic reference or a famous ethnographer or at least a guy with an Indian name who I can cite.
Third, 64/45 is a kind of altered 5th interval, and Indian music only has the pure fifth at Pa. To be fair, 40/27 from Bharata's "Natya Shastra" is also an altered fifth interval, but I'm not using that in my analysis of modern ragas either. So there.
Fourth, I wouldn't know when to use it in preference to the ratios which are more standard.
Anyway, I don't like that suggestion of a ratio, but I still have a lot of hope for the modes. They're all written like this:
"Raga Deskar": [3, 4, 6, 3, 6],
To decode that, we take a running sum and then use the results as indices in the frequency ratio list. If the scales were in 22-EDO, you could also think of them as relative steps in the scale. But we're not doing that yet. So let's accumulate steps, figure out frequency ratio references, figure out justly associated intervals, and look at a whole bug of ragas (and thats and murchhanas and gramas).
Here we go:
[P1, AcM2, AcM3, AcA4, P5, AcM6, M7, P8]: Murchhana Asvakranta
[P1, AcM2, AcM3, AcAcA4, P5, AcM6, AcM7, P8]: Raga Yaman
[P1, AcM2, Grm3, P4, P5, Grm6, Grm7, P8]: Raga Jaunpuri
[P1, AcM2, Grm3, m3, P4, P5, Grm6, m6, m7, P8]: Raga Darbari Kanada
[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8]: Murchhana Sauviri, Kalyan That
[P1, AcM2, M3, P4, AcA4, P5, M6, M7, P8]: Raga Suddha Kalyan
[P1, AcM2, M3, P4, AcAcA4, P5, AcM6, AcM7, P8]: Raga Bihag
[P1, AcM2, M3, P4, AcAcA4, P5, AcM6, M7, P8]: Raga Gaud Sarang
[P1, AcM2, M3, P4, AcAcA4, P5, M6, AcM7, P8]: Raga Yaman Kalyan
[P1, AcM2, M3, P4, AcAcA4, P5, M6, Grm7, AcM7, P8]: Raga Chayanat
[P1, AcM2, M3, P4, P5, AcM6, Grm7, P8]: Khamaj That
[P1, AcM2, M3, P4, P5, AcM6, M7, P8]: Shadja Grama, Murchhana Uttaramandra, Suddha Swara Saptaka
[P1, AcM2, M3, P4, P5, M6, Grm7, AcM7, P8]: Raga Khamaj
[P1, AcM2, M3, P4, P5, M6, Grm7, P8]: Murchhana Suddhasadja, Raga Harikambhoji, Palaiyazh
[P1, AcM2, M3, P4, P5, M6, M7, P8]: Madhyama Grama, Bilaval That, Murchhana Suddhamadhya
[P1, AcM2, M3, P4, P5, M6, P8]: Raga Kambhoji
[P1, AcM2, M3, P5, AcM6, P8]: Raga Bhupali
[P1, AcM2, M3, P5, M7, P8]: Raga Hamsadhvani
[P1, AcM2, P4, P5, AcM6, P8]: Raga Durga
[P1, AcM2, m3, Ac4, P5, M6, m7, P8]: Gandhaara Grama (Narada der. sa-grama)
[P1, AcM2, m3, Ac4, P5, m6, m7, P8]: Gandhaara Grama (Narada der. ma-grama), Murchhana Pauravi
[P1, AcM2, m3, AcA4, P5, AcM6, M7, P8]: Raga Madhuvanti, Ambika
[P1, AcM2, m3, AcM3, P4, P5, AcM6, m7, AcM7, P8]: Raga Ramdasi Malhar
[P1, AcM2, m3, P4, P5, AcM6, m7, AcM7, P8]: Raga Mian Ki Malhar
[P1, AcM2, m3, P4, P5, AcM6, m7, P8]: Raga Kharaharapriya, Bhimpalasi
[P1, AcM2, m3, P4, P5, m6, AcM6, m7, P8]: Raga Anandabhairavi
[P1, AcM2, m3, P4, P5, m6, Grm7, P8]: Asavari That
[P1, AcM2, m3, P4, P5, m6, m7, P8]: Gandhaara Grama (Damodara), Raga Darbari, Darbari Kanada
[P1, Grm2, Grm3, AcA4, Grm6, AcM7, P8]: Raga Gurjari Todi
[P1, Grm2, Grm3, AcAcA4, P5, Grm6, AcM7, P8]: Todi That
[P1, Grm2, Grm3, AcAcA4, P5, Grm6, M7, P8]: Raga Varali
[P1, Grm2, Grm3, P4, P5, Grm6, AcM7, P8]: Raga Rampurmat Pilu
[P1, Grm2, Grm3, P4, P5, Grm6, Grm7, P8]: Raga Bilashkhani Todi
[P1, Grm2, Grm3, P4, P5, Grm6, m7, P8]: Raga Asavari
[P1, Grm2, M3, P4, Ac4, P5, Grm6, M7, P8]: Raga Ramkali
[P1, Grm2, M3, P4, AcA4, Grm6, AcM7, P8]: Raga Lalita
[P1, Grm2, M3, P4, P5, Grm6, AcM7, P8]: Bhairav That
[P1, Grm2, M3, P4, P5, Grm6, M6, M7, P8]: Raga Saurashtra
[P1, Grm2, M3, P5, M6, P8]: Raga Vibhas (marva)
[P1, M2, Grm3, P4, AcAcA4, Grm6, Grm7, P8]: Gandhaara Grama (Sarngadeva der. ma-grama)
[P1, M2, Grm3, P4, AcAcA4, Grm6, Grm7, P8]: Murchhana Uttarayata [17]
[P1, M2, Grm3, P4, AcAcA4, M6, Grm7, P8]: Murchhana Kalopanata [4]
[P1, M2, Grm3, P4, AcAcA4, m6, Grm7, P8]: Gandhaara Grama (Sarngadeva der. sa-grama)
[P1, M2, Grm3, P4, P5, M6, Grm7, P8]: Murchhana Abhirudgata, Kafi That, Raga Bageshri
[P1, M2, Grm3, P4, P5, m6, M6, Grm7, P8]: Raga Mukhari
[P1, M2, Grm3, m3, M3, P4, AcAcA4, P5, M6, Grm7, m7, M7, P8]: Old Indian gamut
[P1, M2, M3, P4, P5, M6, Grm7, P8]: Murchhana Hrishyaka
[P1, M2, M3, P4, P5, M6, M7, P8]: Raga Jogiya
[P1, M2, M3, P5, M6, P8]: Raga Deskar
[P1, M2, P4, P5, M6, P8]: Raga Suddha Malhar
[P1, M3, P4, P5, Grm7, M7, P8]: Raga Tilang
[P1, m2, AcM2, m3, M3, P4, AcA4, P5, m6, AcM6, m7, M7, P8]: Modern Indian gamut
[P1, m2, Grm3, P4, P5, m6, Grm7, P8]: Raga Kanakangi
[P1, m2, M3, AcA4, P5, AcM6, M7, P8]: Raga Marva
[P1, m2, M3, AcAcA4, P5, M6, M7, P8]: Raga Puriya Kalyan
[P1, m2, M3, AcAcA4, P5, m6, M7, P8]: Purvi That, Raga Shri
[P1, m2, M3, P4, AcAcA4, P5, M6, M7, P8]: Raga Bhatiyar
[P1, m2, M3, P5, m6, M7, P8]: Raga Bauli
[P1, m2, P4, P5, Grm6, P8]: Raga Saveri
[P1, m2, m3, Ac4, P5, m6, m7, P8]: Gandhaara Grama (Popley), Murchhana Matsarikrita
[P1, m2, m3, AcAcA4, P5, Grm6, AcM7, P8]: Raga Multani
[P1, m2, m3, P4, AcA4, m6, Grm7, P8]: Murchhana Margavi [20]
[P1, m2, m3, P4, AcA4, m6, m7, P8]: Murchhana Rajani [20]
[P1, m2, m3, P4, AcAcA4, m6, m7, P8]: Gandhaara Grama (Somanatha)
[P1, m2, m3, P4, P5, m6, m7, P8]: Murchhana Harinasva, Bhairavi That
[P1, m2, m3, P5, m6, P8]: Raga Bhoop
[P1, m3, P4, P5, m6, M7, P8]: Raga Takka
[P1, m3, P4, m6, Grm7, P8]: Raga Malkauns
There are sixty eight lines here, and some of them are associated with more than one scale.
Let's learn about some of these scales. We said before than a murchhana is a mode of another raga in hindustani music - which ragas are these murchhanas modes of? Do the Thats have similarities to each other, or differences from the ragas? Why do some of the scales have numbers in brackets, like [17], [20], [4]? How many of these are Hindustani, how many are Carnatic, and how many ancient enough to be neither? When these modes have the same names as modes described by other sources, how closely do they agree?
One mode from Huygens-Fokker has "(Popley)" in its name, so that gives us one hint about sources. It's probably Herbet Popley who wrote "The Music of India" in 1921. Fortunately that one has been digitized and archived online.
Another mode is credited to Damodara. Presumably this is Damodara Misra/Chatura Damodara Pandita who wrote "Sangita Darpana" around 1625 CE.
Another is attributed to Somanatha. Presumably the same Somanatha who wrote the Raga-Vibodha analyzing Carnatic music around 1610 AD.
I've got some reading to do if I want to catch up.
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I started reading! The Music Of Indian by Popley is very good. I only had to scroll to page 52 to get some data about scales. Here's what I've learned so far.
The two foundational scales in the Natya Sastra were Sa-grama and Ma-grama. The Sa-grama goes
[4, 3, 2, 4, 4, 3, 2] : [0, 4, 7, 9, 13, 17, 20, 22]
i.e.
[P1, AcM2, M3, P4, P5, AcM6, M7, P8]
The Ma-grama differs in having a sharpened fourth:
[4 3 4 2 4 3 2] : [0, 4, 7, 11, 13, 17, 20, 22]
i.e.
[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8]
.
Each of these has seven modes, called jatis in some places, or palai in Tamil.
Let's construct these ourselves. Here are 7 modes of Sa-grama, all distinct:
1: [P1, AcM2, M3, P4, P5, AcM6, M7]
2: [P1, M2, Grm3, P4, P5, M6, Grm7]
3: [P1, m2, m3, Ac4, P5, m6, m7]
4: [P1, AcM2, AcM3, AcA4, P5, AcM6, M7]
5: [P1, AcM2, M3, P4, P5, M6, Grm7]
6: [P1, M2, Grm3, P4, Gr5, Grm6, Grm7]
7: [P1, m2, m3, P4, Grd5, m6, m7]
And here are the 7 modes of Ma-grama, all distinct from each other and from the 7 modes of Sa-grama:
1: [P1, AcM2, M3, AcA4, P5, AcM6, M7]
2: [P1, M2, M3, P4, P5, M6, Grm7]
3: [P1, AcM2, m3, Ac4, P5, m6, m7]
4: [P1, m2, m3, P4, Grd5, m6, Grm7]
5: [P1, AcM2, M3, P4, P5, M6, M7]
6: [P1, M2, Grm3, P4, Gr5, M6, Grm7]
7: [P1, m2, m3, P4, P5, m6, m7]
Among these 14 modes, we have two notes that are not found in the standard modern set of shrutis: Ma-grama mode 6 has a Gr5, justly tuned to 40/27. We've heard before that that would show up. Nice. Also mode 7 of Sa-grama dn mode 4 of Ma-grama have Grd5, justly tuned to 64/45, which was mentioned as a possibility in the Huygens Fokker data. Pretty cool.
Now, in 12-TET, these 14 scales become 7 distinct scales - the major scale and its modes. So there isn't a huge amount of variety above. If you can't hear a syntonic comma, poor thing, they'll all sound like western music. In particular, mode 1 of Sa-grama sounds like mode 5 of Ma-grama, and then the correspondences move around the circle, 2 with 6, 3 with 7, et cetera. But even if you're comma deaf, poor thing, it's still nice intellectually to have a bunch of historically important 5-limit just intonation scales, don't you think? I think it's nice.
I've read that there were actually 18 jatis, 7 based on Sa-grama and 11 based on Ma-grama. I'm excited to learn how the extra jatis are formed.
After describing Sa-grama and Ma-grama, Popley next describes raga Bilaval as the core modern Hindustani raga, and raga Mukhari (or Kanakangi) as the core Carnatic scale. He also says that the first is a major scale and the second is a minor scale.
Bilaval: [4, 3, 2, 4, 3, 4, 2]
Kanakangi: [2, 3, 4, 4, 2, 3, 4]
After seeing how cyclic permutations of Sa-grama and Ma-grama produce intervals that aren't found in the standard modern gamut of shrutis, I'm a little wary of looking at a scale in terms of shruti "steps" and replacing them with gamut tones - that procedure wouldn't give us the correct modes of Sa-grama and Ma-grama, so why should it give us the correct intervals here?
Fundamentally, if we want to take the gamut as fundamental, then we can't have exact cyclic permutation of scales. And if we want to rotate scales, then we can't have that particular gamut which gives only one option per step of 22-EDO. So which is more fundamental and which one should budge? I don't know.
Anyway, the step patterns for the two ragas above match the data from Huygens Fokker, so we've already found a possible detempering of them, restricted to the modern gamut. And that's not so bad.
I don't work with 22-EDO much, so I looked up its properties. Its canonical definition over rank-3 interval space leads to it tempering out GrA1, justly tuned to 250/243, and the Grd2, justly tuned to 2048/2025. So, uh, if we think of the shruti system as a 5-limit detempering of 22-EDO, then we shouldn't be shocked to find pairs of intervals that differ by GrA1 or Grd2 being treated identically. I guess. I'm not sure that really tells me anything of value.
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The rest of Popley details many modern Ragas, but not at the level of shrutis. I think it's effectively 12-EDO. I don't think there's any method we can come up with which will detemper each 12-EDO scale to a single 5-limit scale which is correct in the Indian system, since e.g. the sa-grama and ma-grama modes tempered down to the same set of 7 modes, so those 7 modes each have two valid detemperings.
Sa-grama and Ma-grama both have only [AcM2, M2, and m2] as relative intervals. It would be nice if we could use that as a constraint in detempering.
If we make scales out of [AcM2, M2, m2] as relative degrees, and we require that every scale degree is separated from another one by one of [P8, P5, M3, m3] # [2/1, 3/2, 4/3, 5/4, 6/5], and we also have to hit P5 and P8 exactly, then there are only 38 ideal theoretical ragas:
[P1, AcM2, AcM3, Ac4, P5, AcM6, M7, P8]
[P1, AcM2, AcM3, Ac4, P5, AcM6, m7, P8]
[P1, AcM2, AcM3, AcA4, P5, AcM6, M7, P8]
[P1, AcM2, AcM3, AcA4, P5, AcM6, m7, P8]
[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8]
[P1, AcM2, M3, AcA4, P5, AcM6, m7, P8]
[P1, AcM2, M3, AcA4, P5, M6, M7, P8]
[P1, AcM2, M3, AcA4, P5, m6, m7, P8]
[P1, AcM2, M3, P4, P5, AcM6, M7, P8]
[P1, AcM2, M3, P4, P5, AcM6, m7, P8]
[P1, AcM2, M3, P4, P5, M6, Grm7, P8]
[P1, AcM2, M3, P4, P5, M6, M7, P8]
[P1, AcM2, M3, P4, P5, m6, Grm7, P8]
[P1, AcM2, M3, P4, P5, m6, m7, P8]
[P1, AcM2, m3, Ac4, P5, AcM6, M7, P8]
[P1, AcM2, m3, Ac4, P5, AcM6, m7, P8]
[P1, AcM2, m3, Ac4, P5, M6, M7, P8]
[P1, AcM2, m3, Ac4, P5, m6, m7, P8]
[P1, AcM2, m3, P4, P5, AcM6, M7, P8]
[P1, AcM2, m3, P4, P5, AcM6, m7, P8]
[P1, AcM2, m3, P4, P5, M6, Grm7, P8]
[P1, AcM2, m3, P4, P5, M6, M7, P8]
[P1, AcM2, m3, P4, P5, m6, Grm7, P8]
[P1, AcM2, m3, P4, P5, m6, m7, P8]
[P1, M2, Grm3, P4, P5, M6, Grm7, P8]
[P1, M2, Grm3, P4, P5, m6, Grm7, P8]
[P1, M2, M3, AcA4, P5, M6, M7, P8]
[P1, M2, M3, P4, P5, M6, Grm7, P8]
[P1, M2, M3, P4, P5, M6, M7, P8]
[P1, M2, M3, P4, P5, m6, Grm7, P8]
[P1, M2, M3, P4, P5, m6, m7, P8]
[P1, m2, Grm3, P4, P5, M6, Grm7, P8]
[P1, m2, Grm3, P4, P5, m6, Grm7, P8]
[P1, m2, m3, Ac4, P5, m6, m7, P8]
[P1, m2, m3, P4, P5, M6, Grm7, P8]
[P1, m2, m3, P4, P5, M6, M7, P8]
[P1, m2, m3, P4, P5, m6, Grm7, P8]
[P1, m2, m3, P4, P5, m6, m7, P8]
Not all the modes of Sa-grama and Ma-grama were self-consonant in this way, but that's okay. Removing modes of Sa-grama and Ma-grama, there are 28 new ideal theoretical ragas here.
The full set of 38, if we remove cyclic permutations, can be reduced to 10 theoretical ideal ragas:
[P1, AcM2, M3, P4, P5, AcM6, M7, P8]
[P1, AcM2, M3, AcA4, P5, AcM6, M7, P8]
[P1, M2, M3, P4, P5, M6, M7, P8]
[P1, AcM2, AcM3, Ac4, P5, AcM6, M7, P8]
[P1, M2, M3, AcA4, P5, M6, M7, P8]
[P1, AcM2, m3, P4, P5, AcM6, M7, P8]
[P1, AcM2, m3, P4, P5, M6, M7, P8]
[P1, m2, m3, P4, P5, M6, M7, P8]
[P1, AcM2, m3, Ac4, P5, AcM6, M7, P8]
[P1, AcM2, m3, Ac4, P5, M6, M7, P8]
I might have normally selected the lexicographically least scale from all the cyclic permuations for the representative member, but I noticed that each cyclic equivalence class had just one self-consonant scale with a M7, and since Sa-grama, Ma-grama, and the western 5-limit just intonation major scale all have M7, it seemed a natural choice to use that one as the type-species. I didn't try to select for it, but this set of 10 scales happens to have no intervals with a grave minor quality, like Grm3 or Grm7 (though their rotations do).
None of that is real Indian music theory, I just thought it was interesting. If you want to detemper a 12-TET raga, and you have nothing else to go on, you could do worse than finding an option among the set of 38 theoretical ragas (if any of the set of 38 theoretical ragas tempers down to your scale).
...
Okay, time to read more primary and secondary sources.
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