[Joe Monzo]

© 1998 by Joseph L. Monzo

Donald Lentz, in *Tones and Intervals of
Hindu Classical Music*, gives a description
of ancient Indian tuning as a series of
"perfect 4ths" and
"perfect 5ths", which would give a
"Pythagorean"
or 3-Limit system of 22 notes
("srutis") to the "octave". He
then gives a tabulation of the ratios,
but, interestingly, it
includes 5-Limit ratios as replacements for some
which should be 3-Limit.

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, which equals 1). 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-EQ 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 during the procedure, not in order of pitch)*

__PROCEDURE OF TUNING ANCIENT INDIAN SYSTEM:__

FRET | INTERVAL | POWER OF 3 | SEMITONES | RATIO |
---|---|---|---|---|

first series: | ||||

n^{ 0} | ||||

^{ 1} | ||||

^{ 2} | ||||

^{ 3} | ||||

^{ 4} | ||||

^{ 5} | ||||

^{ 6} | ||||

^{ 7} | ||||

^{ 8} | ||||

^{ 9} | ||||

^{ 10} | ||||

^{ 11} | ||||

second series: | ||||

n^{ 0} | ||||

^{ -1} | ||||

^{ -2} | ||||

^{ -3} | ||||

^{ -4} | ||||

^{ -5} | ||||

^{ -6} | ||||

^{ -7} | ||||

^{ -8} | ||||

^{ -9} | ||||

^{-10} |

*(Observing the large-number ratios
at the bottom of each chain, 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 | prime-factor "RATIO" | SEMITONES |
---|---|---|

n^{ 0}) | ||

^{ 5} | ||

^{- 7} | ||

^{ 10} | ||

^{- 2} | ||

^{ 3} | ||

^{- 9} | ||

^{ 8} | ||

^{- 4} | ||

^{ 1} | ||

^{ 6} | ||

^{- 6} | ||

^{ 11} | ||

^{- 1} | ||

^{ 4} | ||

^{- 8} | ||

^{ 9} | ||

^{- 3} | ||

^{ 2} | ||

^{-10} | ||

^{ 7} | ||

^{- 5} | ||

n^{ 0} |

Below is a graph of this Pythagorean system for one "octave".

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
"major 3rd" than the
"Pythagorean third" 3^{4} [= 81/64]. Similarly
for the "minor 3rd"
and the "6ths".

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],
the original starting reference tone,
and 3^{1} [= 3/2], its closest
harmonic relation.

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 at least 10 keys or transpositions.

Arranging the notes into one of my
lattice diagrams
*(modified slightly because of ASCII text limitations)*,
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)

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

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 ofsadja-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 ofmadhyama-grama

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

The whole system gave 10 transpositions (and with the rarely-used two extreme notes, 12) 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, in a similar way, Indian musicians gradually substituted more and more 5-limit ratios, eventually adding enough of them to find some that were good substitutions of some 7-limit ratios.

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

REFERENCES

Donald Lentz, 1961. **Tones and Intervals of Hindu Classical Music**

University of Nebraska Studies, Lincoln.

Joseph Monzo, 1998. **JustMusic: A New Harmony**

The following was sent to me in reference to this page:

Date: Thu, 03 May 2001 21:41:33 +0200

From: Martin Spaink

To: monz@juno.com

Hello,

I'm Martin in Amsterdam (1961). I read some of the article, what I could follow, but I also read some misinformed remarks concerning Indian music.

Of course the exact pitch of a paricular tone in a particular raga has always been a point of debate even among musicians. Also, much depends on the tone-quality, the harmonic content. I study a very old style of vocal music through lessons with a great singer from a family that has been keeping this tradition for 20 generations, and they are noted for extreme meticulousness in intonation and timbre-control. But as their manners of verbal expression, understanding, experiencing akouphenomena differ so much from our own, it is not possible to share the information. Westerners such as myself are in a different postion as they can combine both approaches.

The remarks at the bottom of the article about vadi and samvadi are not correct. It means 1st and 2nd most important tones in a certian raga, which can be Sa and Pa (1st and 5th) but this depends on the character of the raga. In raga Shri it is Pa and very low Re, in Multani the raised 4th and high 7 and so fourth.

Have you ever run cooperation with traditionally Indian-educated Westerners?

best wishes, martin spaink

*More info on Indian tuning:*

- Paul Erlich's
*Tuning List*posting on The sruti system as a periodicity-block. __The Theory of 22 Srutis__, by Vidyasankar Sundaresan (i no longer have a good link for this)- A Gentle Introduction to South Indian Classical (Karnatic) Music by Mahadevan Ramesh
- Sitar Lesson #5,
on fret adjustment, from the Batish Institute's
*RagaNet*e-zine - The Idea of 22 Srutis, by Subhash Kak

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)