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What is the logic behind dividing an octave into 22 notes?

3 Answers 3

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To use Indian musical terminology, there are 7 swara, 12 swara-prakaar, and 22 shruti in an octave.

The seven swara in Indian music are the equivalents of the Western do re mi fa so la ti. In Indian music, they are called sa re ga ma pa dha ni.

Of the above, sa and pa are given special status as fixed notes in an octave and have only one variant each; re, ga, dha, and ni have natural and flat variants; and ma has natural and sharp variants. This gives us a total of 12 swara variants, or "swara-prakaar." So, when we use the word "notes" in an Indian context, we are usually referring to the 12 swara-prakaar.

For an elementary or intermediate-level understanding of Indian classical music, swara and swara-prakaar are all you need to know about. However, going back all the way to the Vedas (Chandogya Upanishad, 6-8th century BCE), many ancient Indian texts on music talk about an octave being divided into 22 shrutis.

What are these shrutis?

Understanding shrutis involves delving deeper into the physics and mathematics behind musical notes and their frequencies relative to the tonic.

To begin very simply, if you choose a particular frequency for the tonic (Sa), then the other 11 notes in the octave exist not at precise frequencies relative to the tonic, but in small bands of frequencies relative to the tonic. And the upper and lower bounds of each note's frequency band are taken as two distinct shrutis for that note because there is an audible difference in pitch between the two.

However, perhaps in an effort to stabilize the octave and keep the discrepancies from adding up, the frequencies of the the fifth note (Pa) and the octave (Sa') are set at fixed ratios to the tonic, at 1 : 1.5 : 2. The other 10 notes are allowed to exist in their respective frequency bands bound by the lower and higher shrutis for each note. This is what results in 22 shrutis in an octave.

How do we know that the notes exist in frequency bands rather than fixed frequencies relative to the tonic? The easiest way to know and perceive this is by playing the notes on a string. You can hear the different notes at different locations on the string, but each note can be played quite tunefully (consonantly) not just at a single, precise location, but within a small region of the string.

The presence of small bands of frequencies for each note can also be mathematically explained.

Throughout history, people have tried to understand and explain the physics and the mathematics behind musical notes in many different ways. There are different formulae that can be used to arrive at all the 12 notes in an octave from a single starting note.

For instance, if you have a length of string, it will produce a certain pitch when plucked. But plucking two-thirds of that string will produce a pitch that is exactly a fifth higher. So, if the whole string produces the tonic (Sa), two-thirds of it will produce the fifth note (Pa); i.e., you can derive the perfect fifth from the tonic. Now, if you use the perfect fifth as your new tonic, you can derive the perfect fifth of the perfect fifth, which is the major second of the next octave Re'(2'). And if you continue this way, you can discover all the 12 notes in an octave before arriving back at the starting note Sa(1). This is called the "circle of fifths," and the sequence of notes discovered in the circle of fifths is:

Sa(1) -> Pa(5) -> Re(2) -> Dha(6) -> Ga(3) -> Ni(7) -> Ma(#4) -> re(♭2) -> dha(♭6) -> ga(♭3) -> ni(♭7) -> ma(4) -> Sa(1)

A similar way of deriving all the notes is using the circle of fourths. The principle behind the circle of fourths is that if you have a length of string that produces a certain pitch when plucked, plucking three-fourths of that string will produce a pitch that is exactly a fourth higher. So, if the whole string produces the tonic (Sa), three-fourths of it will produce the fourth note (ma). The sequence of notes discovered using the circle of fourths is the exact reverse of that discovered using the circle of fifths:

Sa(1) -> ma(4) -> ni(♭7) -> ga(♭3) -> dha(♭6) -> re(♭2) -> Ma(#4) -> Ni(7) -> Ga(3) -> Dha(6) -> Re(2) -> Pa(5) -> Sa(1)

Similarly, if you pluck four-fifths of a string, it will produce the major third (Ga). However, there is no "circle of thirds," and only three notes in an octave can be discovered using this method: Sa(1) -> Ga(3) -> dha(♭6). But if you already know ma(4), you can additionally discover ma(4) -> Dha(6) -> re(♭2); if you already know Pa(5), you can additionally discover Pa(5) -> Ni (7) -> ga(♭3), and so on.

Having marked locations where each note can be played along a string, we can derive the relative frequency of each note to the tonic using the formula "f = 1/L" (where L is the relative length of the string required to produce that note). In this way, the location of musical notes along a string and their frequencies relative to the tonic can be discovered using a variety of methods.

The interesting thing to note here, though, is that the frequencies of the notes arrived at using the different relationships all differ, if only slightly, from each other. Here is a chart I made to show the frequencies arrived at using the circle of fourths vs. the circle of fifths.

Circle of 4ths vs. 5ths frequency comparisons

This seems to be what gave rise to the idea that musical notes in an octave exist within a small band of frequencies rather than exact frequencies relative to the tonic, which then gave rise to the concept of having an upper and lower shruti for each note. However, to keep the octave stable, the frequencies of Sa(1), Pa(5), and Sa'(8) were fixed.

Musical systems around the world use different relative frequencies for different notes depending on their priorities and requirements. The equally tempered scale used in Western music, for instance, is a compromise necessitated by the use of keyboard instruments. By contrast, just intonation uses the simplest frequency ratios to the tonic possible in an effort to find the most consonant intervals.

The shruti frequencies chosen for the lower and upper bounds of each note in Indian music are based on Sa(1)-Pa(5), Sa(1)-ma(4), and Sa(1)-Ga(3) relationships between the notes as these are considered to be the most pleasing intervals. The chart below shows what relationships are used to arrive at the higher and lower shrutis for each note.

Shruti frequencies and frequency ratios

Different ragas use different shrutis for different notes, but within a raga, there is internal consistency and a logic to which set of shrutis is used (fascinating but complex subject that is a different discussion altogether).

The mathematical rationale behind having 22 shrutis was lost for many hundreds of years and is only now beginning to be rediscovered, but an innate understanding of the use of shrutis has remained alive and well in Indian classical music though the ages, handed down from one generation of musicians to another through the guru-sishya (oral) tradition of learning music.

Dr. Vidyadhar Oke's website www.22shruti.com has a great deal of useful information on this topic.

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  • Do I interpret this correctly, this is "just" another (albeit IMO nice) solution to the Pythagorean comma?
    – D. Kovács
    Dec 25, 2021 at 6:25
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    Okay, I think I finally got what you're saying. Yes, fixing the frequencies of Sa(1), Pa(5), and Sa'(8), I think, does solve the problem of the discrepancies arising over the space of several octaves. Within an octave, of course, Indian music celebrates the pitch variances of notes rather than view them as discrepancies or problems. I was quite confused there for a while (lol). Please let me know if I'm still making no sense.
    – Sadhana
    Dec 25, 2021 at 12:09
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    you are making sense, your last comment just confirmed my understanding. Thanks for taking the time!
    – D. Kovács
    Dec 25, 2021 at 15:48
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    So... it's functionally a 12-tone system, except that a pair of alternate tunings is provided for each note besides the tonic and perfect fifth? I had been wondering, because I'd heard of the 22-shruti thing, but actual 22-tone music (i.e. where the tuning is anywhere remotely in the neighbourhood of 22edo) sounds radically different. Dec 28, 2021 at 7:33
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    As an aside, the equal temperament system is definitely not necessitated by keyboard instruments, and in fact through much of the common practice era, quarter-comma meantone tuning was more common. The point of equal temperament (12edo) is that it allows for transposing melodies freely without changing any of the interval tunings. Pop music often transposes the melody upward one or two half-steps for a final chorus, so it's particularly relevant there. Dec 28, 2021 at 7:37
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So while I really appreciate @Sadhana's answer, I'd also like to point out this article from Xenharmonikon, Tuning, Tonality, and 22-Tone Temperament written by Paul Erlich which approaches the shruti in a more analytic way, connecting modern JI music theory with (IIRC) the Natya Shastra (which I read in translation several years ago out of curiosity sparked by the paper). Beginning around page 17 of the linked PDF, he notes that the sruti is defined implicitly by the equivalences:

  • 4 sruti = 9/8
  • 3 sruti = 10/9
  • 2 sruti = 16/15

from these, the rest of the scale degrees in the Natya Shastra are then defined, producing smaller sets of scale tones based on 22 srutis in an ocatve. Now, Erlich's main interest in the paper is not in the rendering of Indian classical music, but rather in the theoretical properties of 22 TET, especially in relationship to music from outside the western canon, but it really opened my eyes to the possibilities inherent in Just Intonation as well as seeing it's practical use in relationship to a music that I already liked rather a lot.

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  • While you could tune to 22TET, it would be a quite poor approximation of the intended JI ratios. 9/8 is much closer to 10/9 (that's just a syntonic comma, ~21.5 cents) than 10/9 is to 16/15 (that's an entire chromatic semitone, ~70.7 cents). 22TET actually approximates the perfect fourth/fifth reasonably well, but other intervals can be quite jarring or foreign-sounding. 3 steps of 22TET are much closer to 11/10 (not a 5-limit tone) than 10/9, and 4 steps are quite sharp of 9/8. Dec 28, 2021 at 7:53
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@Sadhana's answer is excellent, but I believe there's a way to explain the concept of 22 shrutis in North Indian / Hindustani classical music (HCM) without relying on what she calls:

the idea that musical notes in an octave exist within a small band of frequencies rather than exact frequencies relative to the tonic, which then gave rise to the concept of having an upper and lower shruti for each note.

Or at least, it's worth examining what leads to this "small band of frequencies"—why certain notes have variability in their ratio to the tonic.

To begin with, as Sadhana points out:

The shruti frequencies chosen for the lower and upper bounds of each note in Indian music are based on Sa(1)-Pa(5), Sa(1)-ma(4), and Sa(1)-Ga(3) relationships between the notes as these are considered to be the most pleasing intervals.

These relationships are referred to as:

  • Sha.Daj - pa.ncham bhaav (the relationship of the perfect fifth, 3:2)
  • Sha.Daj - madhyam bhaav (the relationship of the perfect fourth, 4:3)
  • Sha.Daj - ga.ndhaar bhaav (the relationship of the major third, 5:4)

So for any given tonic Sa, the perfect fifth Pa is in the frequency ratio of 3:2, the perfect fourth ma in the ratio 4:3, the major third Ga 5:4. But these are true only for that given Sa. This means that ratios within the other notes of the scale with that tonic are not congruent with each other.

Take the major sixth or shuddh Dhaivat of a given scale. It can be seen as either in shadaj-pancham bhaav from shuddh Re (a perfect fifth from the major second), or in shadaj-gandhaar bhaav from shuddh ma (a major third from the perfect fourth). However, 3:2 (Pa) of 9:8 (Re) is not equal to 5:4 (Ga) of 4:3 (ma). There is a 1/50 difference between 27/16 and 20/12. This means, therefore, that the range of frequencies for the major sixth of a given tonic admits a variation of at least 1/50.

Hindustani classical music accounts for these minuscule differences by talking of shrutis: we say that the re of Bhairav is a different shruti from the re of Todi, or that the Dha of Marwa is different from the Dha of Yaman, etc. That is to say, HCM acknowledges that the exact pitch of a given note is variable depending on what relationships one wants to highlight between the notes of a raga. Or to put it in another way: in HCM, the internal, horizontal relationships between the notes of a given raga are the most important thing, not their absolute frequencies relative to the tonic.

So for instance, in Marwa, the vaadi or sonant note is identified as komal re (the minor second), and the samvaadi or consonant as shuddh Dhaivat (the major sixth). The minor second is of course a major third above the major sixth, and re and Dha are therefore said to be in Sha.Daj-ga.ndhaar bhaav. Consequently, the komal re of Marwa ends up pitched a few microtones higher (at a higher shruti) than the minor second of such ragas as Bhairav, which is differently derived.

Consequently, when musicians speak of the 22 different shrutis, they speak of them the context of particular ragas. Unlike the seven notes or twelve tones, shrutis are rarely discussed in practical contexts except in terms of where a given note is pitched in a given raga. Most practicing musicians, teachers, and students do not aim to identify or reproduce specific shrutis outside of the context of a given raga. A teacher will explain, “the ga.ndhaar (minor third) of Todi is like this while the ga.ndhaar of Multani is like this and that of Darbari is like this” and train the student to reproduce those particular microtonal / intonational differences in the context of those particular ragas. To explain why these differences exist, she is likely to hand-wave toward the theory that there are 22 different shrutis, but very unlikely to present a mathematical chart describing the possible ratios and frequency intervals, or even to explain (as I've tried to do here) that the shruti concept helps highlight the specific relationship that a given note in the raga bears to its corresponding note in the other tetrachord.

Finally, all of the above is just about North Indian or Hindustani classical music. South Indian or Carnatic classical music has a different way of dividing the octave into notes. In Carnatic music too there are seven notes, 12 swarasthaanas, and 22 shrutis, but Re, Ga, Dha, and Ni are considered to have three versions each, not two. Just as in Western music, the same note can be called C# (C sharp) or D♭ (D flat) depending on how it is used in a scale, in Carnatic music the same swarasthaana can be called chatushruti rishabh or shuddh gandhaar depending on the scale. So the details of how the scale is described are not the same as given above; ymmv.

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    Thank you! That's an extremely useful answer and goes to the heart of the logic behind using specific shrutis in different ragas.
    – Sadhana
    Mar 22, 2023 at 2:47
  • glad you liked it! Yours was very comprehensive and informative.
    – verbose
    Mar 22, 2023 at 3:18

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