Sangeeta Charcha ... experts discuss various aspects of music

      brings you a new section featuring articles on different aspects of music by experts. These may be topics that are the subject of fundamental interest or heated debates. Or just something that gives food for thought...

In first article of this series, read about why an octave is called an octave, why there are 12 notes in an octave. We will continue with a contemporary discussion of the much-written about, much discussed topic of 22 Sruti-s in Indian music.


Numbers in Music - 7, 12 and 22 - PART 1

- Vidyasankar Sundaresan


The history of human civilization, particularly that of its Indo-European branch, is closely correlated with the history of numbers. If the Samkhya thinkers of ancient India developed their philosophy through a process of enumerating fundamental principles, the Pythagoreans of ancient Greece waxed eloquent about beauty and truth in numbers. Indeed, the Pythagoreans believed that the entire universe was music and that music was fundamentally number! However, it is not well known that early in the history of Indian music too, prominent texts like Bharatas Natyasastra (Sanskrit) and Ilanko Adigals Silappatikaram (Tamil) intricately related numbers to music, at a very basic level.

Many of us Carnatic music aficionados would have heard of the classification scheme of 72 Melakarta-s, while every Carnatic music concert presents amazing displays of numerical complexity in the Tala aspects. At a more fundamental level, Carnatic music shares the following characteristic with almost all systems of music in the world. It is based on a system of seven notes, with twelve unique divisions in an octave. Therefore, the first issue discussed in this article is, Why 7, and why 12?

In the following discussion, the notes in the octave will be denoted as

sa ri Ri ga Ga ma Ma pa dha Dha ni Ni Sa

It is interesting to note that many old Indian texts mention that the octave contains 22 sruti-s. This number is found both in the Natyasastra and in an old commentary on the Silappatikaram. The idea of 22 sruti-s in music enjoyed widespread acceptance, as is evident from the following verse found in a non-musicological text, the ancient classical Sanskrit thesaurus, the Amarakosha -

nrNAm urasi madhyasto dvAvimshatividho dhvaniH |
sa mandraH kanThamadhyasthas tAraH shirasi gIyate ||

Here, the Sanksrit term dvAvimshatividho dhvaniH refers to twenty-two notes. Of course, there have been differences of opinion on the part of individual writers over the centuries, and in modern times, there are many who think that the whole idea of 22 sruti-s is an archaic one, bearing little relevance to actual musical practice. On the other hand, the number 22 is mentioned in almost all the more ancient Sanskrit and Tamil texts, and consequently, in all the modern books on Indian musicology, written in modern Indian languages and in English. However, the reasons for arriving at this number are not well understood, and many mutually contradictory explanations for its origin are mentioned. Hence, the second issue discussed in this article is, Why 22?, I have attempted to discuss the topic from a physical perspective and also keeping in mind what the original musicological texts actually say about it.

7 and 12 - Fundamental Numbers in World Music

With the help of basic results from the physics of vibrating bodies, we can deduce a fundamental explanation for the currently widely accepted structure of the musical octave. A brief synopsis of the important results obtained from an analysis of vibrating strings is sufficient for this purpose. The same general conclusions hold true for the vibrations of air columns, and therefore are applicable to the human voice too.

A string vibrates in a combination of harmonic modes. A mode of vibration of a string can be easily understood in terms of its vibrating length. If the whole length of the string vibrates in one loop, the vibration is said to be in the first mode. The second mode of vibration corresponds to the string vibrating in two loops with each loop corresponding to half the total vibrating length. If the frequency of the first mode of vibration (also called the fundamental frequency) is n, that of the second mode of vibration is 2n. The third mode of vibration gives a frequency 3n, which corresponds to dividing the length of the string into three equal parts. It should be noted that when a string is plucked or bowed, it always vibrates in a combination of modes. The energy of its vibration is present in a number of higher harmonics.

In Indian musical terms, if n is the frequency of sa in the middle octave, 2n corresponds to the higher octave Sa, while 3n is the frequency of pa in the higher octave. If we move this back to the middle octave, the relative value of pa is 3n/2. Thus, given the sa, the relative value of pa arises naturally from the third harmonic. Correspondingly, 4n is the frequency of the sa in the next higher octave. Interestingly, the frequency of ma in the middle octave is 4n/3, which can therefore be related to the fourth harmonic. The fifth harmonic, with a frequency 5n, is also musically very important. The Ga that can be heard clearly from a well-tuned tambura arises from this mode of vibration. The corresponding frequency of Ga in the middle octave is 5n/4. From the sixth harmonic arises a different ga, the frequency of which is 6n/5. One can similarly continue to derive theoretical values for a number of notes in the musical octave. In theory, there are infinite harmonics, so that there should be an infinite possibility of notes in the octave. In practice, the harmonic content of a vibration is strongly dependent on the material of the vibrating body, its structural design, the relative position where it is set into vibration and a number of other variables. In general, the energy content is distributed mostly over the first few harmonics, and decreases in the higher order harmonics.

Therefore, using the natural numerical relationships among the first few harmonic frequencies of a vibrating body, we can derive the relative values of various notes in the musical octave (Table I). The frequency of any given note in the octave should lie between that of the reference sa (n) and the higher octave Sa (2n). Therefore, it becomes easier to carry out the following discussion in terms of frequency ratios, which all lie between the numbers 1 and 2. We can relate these ratios to the vibrating length of a string in an instrument like a Veena or Violin. If the open string sounds sa, then Sa (in the higher octave) is located at half the length of the string. The location of pa is such that the length of the freely vibrating string is 2/3rds of that of the open string. Similarly, ma is located at 3/4ths of the length of the open string, and so on.

Table I

Note Length Ratio Frequency Ratio
sa 1 1
Sa 1/2 2
pa 2/3 3/2
ma 3/4 4/3
Ga 4/5 5/4
Dha 3/5 5/3
ga 5/6 6/5

However, we know that there are many more notes in the octave than tabulated above, but it is not immediately obvious how to correlate them with the higher harmonic frequencies. A different procedure to derive notes is to cyclically apply the sa-pa relationship. This is historically well attested in the practice of music, and is referred to as panchama-bhva or the vdi-savdi relationship in the Sanskrit texts, ii-kramam in the Tamil texts and the cycle of fifths in Western music.

As the fifths relationship is characterized by the frequency ratio 3/2, we can derive numerical ratios for the relative value of each note in the octave as a geometric series with characteristic ratio 3/2. However, the repeated application of this panchama relationship often results in a note in the higher octave, giving a value greater than 2. In such cases, we simply revert to the corresponding note in the middle octave, by dividing the corresponding value by 2. Thus, given sa (= 1), we know that Sa = 2 and pa = 3/2. In the next round, we apply the panchama relationship to pa, and obtain Ri = 9/8 (= (3/2)2 x ). In the third round, we apply the panchama relationship to this Ri, and so on. The first few terms in this cycle are listed below.

sa pa Ri Dha Ga Ni
1 3/2 9/8 27/16 81/64 243/128
  1 x 3/2 (3/2)2 x (3/2)3 x (3/2)4 x (3/2)5 x

The general term in the above sequence is 3n/2m, where n and m are integers, and 1 < 3n/2m < 2. Thus, pa = 31/21, Ri = 32/23, Dha = 33/24, Ga = 34/26, and Ni = 35/27. Notice that the power to which 3 is raised (in the numerator) is the same as the ordinal place of the corresponding note in the above cycle. The power to which 2 is raised (in the denominator) is higher, because every time the cycle gives a value greater than 2, we have divided that number by 2, in order to keep all the ratios between 1 and 2. In principle, we can extend the above sequence to finally reach Sa in the higher octave, but it can be easily seen that this procedure will never give the true value of 2 for Sa. This is simply because 2 and 3 are both prime numbers, and therefore, this cycle will always give fractional values for the relative frequencies of notes.

We can, however, find an approximate solution to the equation 3n/2m = 2, such that n and m are integers. The smallest numbers that satisfy all the above conditions are n = 12 and m = 18. This means that 312/218 is the first term in the cycle that is closest to 2, i.e. Sa in the higher octave. It also means that this term will be the 12th one in the cycle, giving a total of 12 notes derived in this cycle. This gives us a fundamental explanation why there are 12 notes between sa and Sa in the musical octave!

This approximate solution corresponds to going through the following additional steps -

Ma ri dha ga ni ma Sa

Thus, the entire sequence of notes derived from this cycle is as follows,

sa pa Ri Dha Ga Ni Ma ri dha ga ni ma Sa

An equivalent procedure to derive notes is the cycle of fourths, which uses the sa-ma relationship instead of the sa-pa relationship. Now, the sa-ma relationship is the complement of the sa-pa relationship. This is because, if instead of sa, we take pa as the reference note, then the corresponding fourth note would be Sa. In other words, the pa-Sa relationship is equivalent to the sa-ma relationship. When expressed in terms of simple numerical ratios, 3/2 x 4/3 = 2, and as we have already seen, 2 is the value of Sa in the higher octave relative to the reference sa. As sa = 1 and ma = 4/3, we cyclically apply this relationship in the following manner

sa ma ni ga dha ri
1 4/3 16/9 32/27 128/81 256/243
1 1 x 4/3 (4/3)2 (4/3)3 x 1/2 (4/3)4 x 1/2 (4/3)5 x 1/4

As 4 = 22, each term in the cycle of fourths can be represented as 2(2n-m)/3n. In this sequence too, we will never get the true value of 2 for Sa in the higher octave, but as in the cycle of fifths, an approximate solution can be found for the equation 2(2n-m)/3n = 2, such that n and m are integers. The lowest values of n and m that satisfy these conditions are n = 12 and m = 4. Again, the 12th term in this cycle also is Sa, giving us an equivalent explanation why there are 12 notes in the octave.

This approximate solution corresponds to going through the following steps,

Ma Ni Ga Dha Ri pa Sa

with the entire sequence in this cycle being,

sa ma ni ga dha ri Ma Ni Ga Dha Ri pa Sa

Note that the cycle of fourths derives the notes in the reverse order as compared to that derived by the cycle of fifths, but gives different values for each note as compared to the cycle of fifths. This is a consequence of the fact that the sa-pa and sa-ma relationships complement each other.

In the cycle of fifths, from the above approximate solution, we have 312/218 ~ 2. If we divide again by two, we get 312/219 ~ 1. We may remember that the power of 2 in the denominator is higher than the power of 3 in the numerator, because of having to divide by 2 whenever we reached a note in the higher octave. Now, 19 12 = 7. In other words, starting from sa, to get back to sa (approximately) through the cycle of fifths, we will have jumped to the higher octave and back to the middle octave 7 times. Another way of thinking about it is that 12 musical fifths approximately equals 7 octaves. Therefore, the 12 values within one octave, as derived in the cycle of fifths, may be paired up preferentially, to give 7 notes in the octave. A similar reasoning will hold true for the cycle of fourths too, as it is complementary to the cycle of fifths.

The defining notes of the two cycles are sa, pa and ma. The cycle of fifths first gives the sequence of the major notes, Ri - Dha - Ga - Ni, while the cycle of fourths first gives the sequence of minor notes, ni - ga - dha - ri. These can therefore be conveniently grouped in pairs, consisting of the major and minor varieties of each note. Both cycles give rise to Ma, as the note where the transition occurs, from the major notes to the minor notes or vice versa. Thus, by conventionally grouping the values derived above into pairs, we go from 12 distinct notes to 7 kinds of notes in the octave.

To be continued


Posted on May 27, 2002


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