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Paul Erlich (TD #1562, Topic #3) gives a diagram
which does indeed represent exactly the ratios
given by Donald Lentz in "Tones and Intervals of
Hindu Classical Music". I would argue, however,
that the theoretical implications of the ancient
Indian tuning would allow even more 5-limit ratios
than Lentz's description.

| Before I discuss Indian tuning, a |
| note about my somewhat idiosyncratic |
| nomenclature: |
| |
| In my prime-factor notation, 1/1 is |
| referred to as n^0 (that is, every |
| prime to the 0th power). I use a |
| measurement of "semitones" to two |
| decimal places rather than cents, |
| simply because anything less than |
| a cent is usually insignificant, at |
| least for purposes of music-theory, |
| and it relates better to our familiar |
| old 12-equal scale. (It's really the |
| same as cents, just with a decimal |
| point.) |
| |
| Now, on with the show: |

Lentz gives the most ancient Indian fretting as
a series of "perfect 4ths and 5ths". This makes
a lot of sense, as it could have been easily
accomplished by ear. It can be measured for
one "octave" on a single string in two separate
procedures, as follows (frets are numbered in
order of placement, not order of pitch):

PROCEDURE OF TUNING ANCIENT INDIAN SYSTEM:

POWER
FRET INTERVAL OF 3 SEMITONES RATIO

open string: n^0 0.00 1/1
1. up a 3/2 to: 3^1 7.02 3/2
2. down a 4/3 to: 3^2 2.04 9/8
3. up a 3/2 to: 3^3 9.06 27/16
4. down a 4/3 to: 3^4 4.08 81/64
5. up a 3/2 to: 3^5 11.10 243/128
6. down a 4/3 to: 3^6 6.12 729/512
7. down a 4/3 to: 3^7 1.14 2187/2048
8. up a 3/2 to: 3^8 8.16 | c'mon - do |
9. down a 4/3 to: 3^9 3.18 | you really |
10. up a 3/2 to: 3^10 10.20 | need these |
11. down a 4/3 to: 3^11 5.22 | ratios? |

open string: n^0 0.00 1/1
12. up a 4/3 to: 3^-1 4.98 4/3
13. up a 4/3 to: 3^-2 9.96 16/9
14. down a 3/2 to: 3^-3 2.94 32/27
15. up a 4/3 to: 3^-4 7.92 128/81
16. down a 3/2 to: 3^-5 0.90 256/243
17. up a 4/3 to: 3^-6 5.88 | forget |
18. up a 4/3 to: 3^-7 10.86 | about |
19. down a 3/2 to: 3^-8 3.84 | these |
20. up a 4/3 to: 3^-9 8.82 | ratios, |
21. down a 3/2 to: 3^-10 1.80 | already. |

| (If you're at all familiar with |
| the large-number ratios I omitted, |
| the advantages of prime-factor |
| notation should be obvious.) |

Arranging these into a 1-"octave" scale, the
way the frets would appear on the fingerboard,
gives the following (open string at the bottom):

TABULATION OF SRUTIS IN ANCIENT INDIAN SYSTEM:

SRUTI "RATIO" SEMITONES

22. (n^0) 12.00
21. 3^5 11.10
20. 3^-7 10.86
19. 3^10 10.20
18. 3^-2 9.96
17. 3^3 9.06
16. 3^-9 8.82
15. 3^8 8.16
14. 3^-4 7.92
13. 3^1 7.02
12. 3^6 6.12
11. 3^-6 5.88
10. 3^11 5.22
9. 3^-1 4.98
8. 3^4 4.08
7. 3^-8 3.84
6. 3^9 3.18
5. 3^-3 2.94
4. 3^2 2.04
3. 3^-10 1.80
2. 3^7 1.14
1. 3^-5 0.90
n^0 0.00

It would have been evident to any musician
with even a half-decent ear that 3^-8 (only
a schisma [= 2 cents] flatter than 5/4) gave
a much more consonant "third" than the
"Pythagorean third" 3^4 [= 81/64]. Similarly
for the "minor third" and the "sixths".

Looking closely at the semitone values in the
tabulation of srutis, it can be seen that there
are 10 pairs of notes a Pythagorean comma
[= 24 cents] apart (this is only a schisma
larger than the all-important syntonic comma).

The remaining two notes, which appear alone
without a comma-away relative, are n^0 [= 1/1]
and 3^1 [= 3/2], the two notes which would be
sounded by the tamburas as drones throughout
the entire piece.

In fact, the "schismatic equivalents" provided
ancient Indian musicians with the ability
to play their 7-tone basic scale in a
pseudo-5-limit in 12 keys or transpositions.

Rearranging Paul Erlich's diagram a bit to
conform with my lattice diagrams, it is easy
to see that schismatic equivalence gave a
neat box-like structure to the Indian system:

| (I give _ratios_ for the notes |
| implied by the tuning and probably |
| in practical or at least theoretical |
| use, and a few of the _powers-of-3_ |
| for the actual notes tuned, to show |
| schismatic substitution. Semitones |
| are given below the ratio or power. |
| Ratios in parentheses are schismatic |
| substitutions, of notes just beyond |
| either end of the series of 3/2s and |
| which Lentz says were ocasionally |
| used in the tuning.) |

MONZO LATTICE DIAGRAM OF
5-LIMIT IMPLICATIONS OF ANCIENT INDIAN SYSTEM:

etc.
/
3^6
6.12
/
/
3^5
11.10
/
/
405/25-----81/64----(81/80)
7.94 4.08 0.22
/ / /
/ / /
135/128----27/16-----27/20
0.92 9.06 5.20
/ / /
/ / /
45/32------9/8-------9/5
5.90 2.04 10.18
/ / /
/ / /
15/8-------3/2-------6/5
10.88 7.02 3.16
/ / /
/ / /
5/4-------1/1-------8/5
3.86 0.00 8.14
/ / /
/ / /
5/3-------4/3------16/15
8.84 4.98 1.12
/ / /
/ / /
10/9------16/9------64/45
1.82 9.96 6.10
/ / /
/ / /
(40/27)----32/27----256/135
6.80 2.94 11.08
/
/
3^-4
7.92
/
/
3^-5
0.90
/
/
3^-6
5.88
/
etc.

The basic ascending scale with the syllables
sa-ri-ga-ma-pa-dha-ni-(sa) was possible
in two forms:

In the sadja-grama scale, "pa" formed a
3/2 above (or 4/3 below) "sa".

In the madhyama-grama scale, "pa" formed
a 3/2 below (or 4/3 above) "ri".

Also, in some of the older Indian theoretical
treatises, there is the confusing statement
that the scale began on the 4th sruti (rather
than the open string), which is 3^2 [= 9/8]
in my tabulation above. I have made sense out
of this by interpreting the basic scale
(with both "pa"s) as follows:

MONZO LATTICE DIAGRAM
OF BASIC SCALE OF ANCIENT INDIAN SYSTEM:

27/16
9.06
PA
of sadja-grama
/
/
9/8
2.04
SA
/
/
15/8--------3/2
10.88 7.02
DHA MA
/ /
/ /
5/4--------1/1
3.86 0.00
RI NI
/ /
/ /
5/3-------4/3
8.84 4.98
PA GA
of madhyama-grama

This gives a sadja-grama basic scale
which resembles our "dorian mode", and
a madhyama-grama basic scale which
resembles our "mixolydian mode".

If I were able to outline this scale on the
larger diagram (which ASCII text would have
made even more difficult and time-consuming
than what I've already done), it could be
easily seen that the whole system gave
transpositions of these basic scales.

Of course, there were many other notes
outside these scales, which could be used
in a raga as ornamentation or melodic inflection
for expressive purposes.

I'm sure, based on what has happened in
Western music, that similarly, Indian musicians
gradually substituted more and more 5-limit
ratios, eventually adding enough of *them*
to find some that were good substitutions of
7-limit ratios.

If this process continued, they would likewise
eventually find 7-limit ratios that were good
approximations of 11- and 13-limit ratios.

But even in the very oldest Indian treatises,
which are quite old indeed, a 5-limit system
is already implied.

| (All of my books are still on the |
| opposite side of the country from me, |
| thus I can't give specifics as to dates |
| and authors. Any help here? Paul?) |
| |
| (PS - this will eventually go up on my |
| website, but the *really good* quality |
| diagrams are in my book, all you |
| potential buyers...) |
| |
| (PPS - I'm disgusted with the slowness |
| of my Hotmail account, so write to me |
| once again c/o Juno.) |

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
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koto/guitar concert

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