Derivation of the 22 Srutis

Transliteration scheme : -

----------------------

a A i I u U e ai o au ( the anuswara and visarga do not figure in this

discussion).

k kh g gh

ch chh j jh

T Th D Dh N

t th d dh n (n gets modified as in tank, lung, lunch, enjoy )

p ph b bh m

y r l v (w) S sh s h

The Tamizh words are transliterated as pronounced, with the use of E and O

for the long vowels. The consonants denoted by zh, L and R are unique to

Tamizh.

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Murali Sharma posted a primer on Hindustani music (written by Dinesh

Prabhu) for inclusion in the FAQ some time ago. The question of the

number of SrutIs in an octave has arisen again. As Dinesh says, the

amount of confusion and controversy regarding the no. of SrutIs in an

octave is huge. However, I do not think that trying to make some sense

out of all the conflicting claims, is a non-issue with respect to music as

it is practised today. The frequency ratio values that one derives may

well be open to question, but the fact remains that some basis set is used

in both Hindustani and Carnatic music as of today. Out of the various

numbers 12, 16, 22, 24, 44, etc. to infinity that have been proposed, let

us examine each number and see what can be acceptable.

If one accepts a working definition of Sruti as "the raise in pitch for

the note to be identified as another one", the rigorous answer to the

question is infinity. On the other hand, there are some problems which

this leaves unanswered. Say a musician slips out of pitch at the panchama.

What is the means by which one identifies this as a 'slip from the

panchama', rather than as a dhaivata or as a madhyama? Such a problem

forces one to try to identify musical intervals in a more precise way.

Both Hindustani and Carnatic music today, use the concept that shaDja and

panchama are invariant notes, while the other five notes admit of

variations. It would therefore seem that Hindustani music uses 12 (two

varieties each for r, g, m, d and n), while Carnatic music uses 16 (three

varieties each of r, g, d and n, and two varieties of m). The answer is

not that simple however, because the various shades of the same note that

are used in different ragas are not accounted for by these numbers.

Furthermore, 16 arises only from a superposition of four notes with some

from the set of 12, so that 12 is the number of swarasthanas that both

systems use. 16 is pertinent only for the construction of the 72

meLakarta system.

The number 22 has been used historically by Bharata and in ancient Tamizh

music as seen from the SilappadikAram references. At least for the sake

of historical evolution, 22 is of some interest. 24 is suggested from an

argument that each of the 12 semitones should have a higher and a lower

variety. This does not make sense as far as sa and pa are concerned, so

that if we reject two varieties for these two notes, we again end up with

the number 22. I am not conversant with the arguments in favour of or

against the higher numbers like 44, 49 etc. which have been suggested, so

I am not going to talk of them.

One possible significance for the number 22 has been suggested on the

basis of pi, so that 22 SrutIs are distributed among 7 swaras.

Personally, I do not think there is much to this theory except the

remarkable coincidence. I have yet to see a convincing argument for this,

unless one accepts some sort of mystic significance, which cannot be

scientifically validated nor falsified.

In the following posts, I shall attempt to rationalize the number 22 as

used in ancient writings based on certain first principles. I hope we can

generate a discussion that will clarify matters enough so that we can have

a comprehensive account in an FAQ.

S. Vidyasankar

We all know that a vibration of a string or an air column can be studied

in terms of the fundamental modes which give rise to the various

harmonics. The 2nd harmonic is the octave (called stAyi or iNai) while

the third harmonic is the perfect fifth in the higher octave. This

directly gives us a value of 3/2 for the perfect fifth in the middle

octave, with respect to the tonic as 1. The octave (tAra stAyi shaDja)

has a value 2. Right at this stage, we have thus fixed three numbers (1,

3/2 and 2) which are fundamental to our musical perception. The fourth

harmonic gives rise to the perfect fourth, which thus gets the value 4/3.

The fifth harmonic is clearly important as it can be heard distinctly from

a perfectly tuned tambUra. This now gives the value 5/4 for the perfect

third. In all these cases, these numbers can also be interpreted as the

reciprocals of the lengths of a vibrating string, as measured from the

leftmost node.

The sixth harmonic gives again a note related to the perfect fifth, while

the seventh harmonic is clearly unpleasant. (This is something like "from

this it follows that" in a math exam when you don't know how to complete

the proof!) Let us see how we can derive other frequency ratio values

just based on these numbers i.e. 1, 5/4, 4/3, 3/2 and 2.

In the subsequent discussion, I shall use the following symbols for the

twelve semitones

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

for the sake of convenience. We have the following relationships between

the various notes.

Table I

1. sa : Sa = 1 : 2,

2. sa : pa = 1 : 3/2

3. sa : ma = 1 : 4/3

4. sa : Ga = 1 : 5/4

The other notes can be derived from a cycle of fifths (called iLikramam in

Tamizh texts).

Table II

1. sa's panchama = pa

2. pa's panchama = Ri

3. Ri's panchama = Dha

4. Dha's panchama = Ga

5. Ga's panchama = Ni

6. Ni's panchama = Ma

or from a cycle of fourths.

Table III

1. sa's madhyama = ma

2. ma's madhyama = ni

3. ni's madhyama = ga

4. ga's madhyama = dha

5. dha's madhyama = ri

In each case, whenever the cycle takes you to the higher octave, i.e.

gives you a number greater than 2, divide by 2 to get back to the middle

octave. This gives the following ratio values for the various notes.

Table IV

sa = 1

ri = 256/243

Ri = 9/8

ga = 32/27

Ga = 81/64

ma = 4/3

Ma = 729/512

pa = 3/2

dha = 128/81

Dha = 27/16

ni = 16/9

Ni = 243/128

If you notice the value calculated for Ga from the fifths cycle, it is

higher than 5/4 by a factor of 81/80. This is the value that Bharata

calls pramANa Sruti. The SilappadikAram characterizes this as 'pakai'

meaning enmity. This value 81/80 is the famous Pythagorean comma, that

comes from the seven ancient Greek modes. This ratio of 81/80 has

relevance if one has an instrument like the yaazh (like a harp), and is

trying to tune the gandhara string. The question that the musician then

faces is whether to tune the gandhara string to be in tune with the fifth

harmonic arising from the shaDja string (5/4), or to tune it as per the

panchama cycle. In addition to this, if one back calculates for Dha and

Ri based on the value of 5/4 for Ga, one gets the values Dha = 5/3,

instead of 27/16 and Ri = 10/9, instead of 9/8. Similarly, the

calculation for Ni gives Ni = 15/8, instead of 243/128. In each case,

this calculation is identical to applying the pramANa Sruti value of 81/80

to the values already derived. As an aside, it may be mentioned that the

value 81/80 does not arise as a factor that relates the fifths cycle to

the fourths cycle, as popularly thought. It arises from the discrepancy

introduced by carrying out the fifths cycle, as compared to the note

arising from the fifth harmonic.

This may be well and good as far as the notes related immediately to Ga =

5/4 by the fifths cycle is concerned. Is there any basis for applying the

same ratio to the other notes also i.e. the ones obtained from the fourths

cycle, and to the panchama itself?

We have calculated a set of 12 values for the 12 semitones, starting from

basic principles of vibrating strings. We have also seen the possibility

of generating two different values for the same semitone (Ga or Ri or Dha

or Ni) so that now the problem is one deciding between two different

values for the same semitone. If Indian music were to be based on

straight notes, without any modification of the notes by means of

oscillations and shakes (kampita gamaka), or by means of varying the

stress on the note (spurita or pratyahata) etc. then the question of

choosing one of the two values for a semitone is a tough problem. Since

we have the facility to accomodate a variety of frequency ratio values for

the same semitone, let us go ahead with all the values which we have

calculated now.

The values of the 22 srutis can be obtained by simply applying the ratio

81/80 to all the notes obtained here. In the case of the cycle of fifths,

divide by this ratio, while in the case of the cycle of fourths, multiply

by 81/80 i.e. augment the notes from the fourths cycle and flatten the

notes from the fifths cycle. Leave sa and Ma alone (yes Ma, not pa!). If

this is done, the set of numbers so obtained is the same as that described

by Bharata. The seemingly odd ratio 40/27 that is obtained for the

flattened fifth was definitely described by Bharata in the madhyama grAma.

What is the basis for doing this? The answer lies in the fact that the

various ragas in use in ancient times were derived by a technique of graha

bheda (tamizh 'kural tiripu'). For example, Bharata's shaDja grAma was

the same scale as that of kharaharapriya (kAfi thAT)

sa Ri ga ma pa Dha ni = shaDja grAma

1 10/9 32/27 4/3 3/2 5/3 16/9.

The other scales were obtained by graha bheda from shadja grAma.

Similarly the basic scale of ancient tamizh music was pAlaiyAzh, the same

scale as of harikambhoji (khamaj thAT), and the other pAlais (scales as

opposed to paN = rAga) were obtained by kural tiripu from this scale.

sa Ri Ga ma pa Dha ni = pAlaiyAzh

1 9/8 5/4 4/3 3/2 5/3 16/9.

Seven scales can be obtained by shifting the reference to another note.

The scales obtained are identical irrespective of whether we start from

shaDja grAma or from pAlaiyAzh. This is because these two scales

themselves are derivable from each other by shifting the reference. Thus

pAlaiyAzh with pa as the tonic gives arumpAlai, which is the same as

shaDja grAma. Similarly the scale obtained from shaDja grAma, by shifting

the tonic to ma is the same as pAlaiyAzh. As one goes through the process

of shifting the tonic, the only new ratios one generates are related to

the values generated till now, by the ratio of 81/80. This is because

both scales contain two notes that deviate by this factor from the cycle

of fifths value. If we had used 9/8 and 27/16 instead of 10/9 and 5/3

respectively in shaDja grAma, all ratios are related by the cycle of

fifths and we would obtain no new ratio values by doing graha bheda.

The ratios that are generated by this process are tabulated below.

Table V

sa = 1

ri = 256/243, 16/15 = (81/80) x (256/243)

Ri = 10/9, 9/8 = (81/80) x (10/9)

ga = 32/27, 6/5 = (81/80) x (32/27)

Ga = 5/4, 81/64 = (81/80) x (5/4)

ma = 4/3, 20/27 = (81/80) x (4/3)

Ma = 64/45 , 45/32, 729/512 = (81/80) x (45/32)

pa = 40/27, 3/2 = (81/80) x (40/27)

dha = 128/81, 8/5 = (81/80) x (128/81)

Dha = 5/3, 27/16 = (81/80) x (5/3)

ni = 16/9, 9/5 = (81/80) x (16/9)

Ni = 15/8, 243/128 = (81/80) x (15/8)

Note that we have a total of 24 values. Surely all these values were

calculated by ancient musicologists such as Bharata, because he clearly

derives scales by graha bheda. Still he talks of 22 SrutIs. Similarly,

the SilappadikAram talks only of 22 'mAttirais', though all the scales

such as viLarippAlai, mErcempAlai etc. are derived by kural tiripu. Did

these people make a mistake when they counted 22 instead of 24? I should

think not. Before we try to rationalize the 22 vs. 24 conflict let us

have another look at the way these numbers are obtained.

The cycle of fourths and the cycle of fifths are based on two key

consonances. The fifth consonance (tamizh iLi) is based on the ratio 3/2

and the fourth consonance (tamizh natpu) is based on the ratio 4/3. Note

that these consonances are reciprocal to each other. sa : pa = 3/2, but pa

: sa (higher octave) = 4/3. Bharata combines these in the phrase

vadi-samvadi. Thus the cycle of fourths is not independent of the cycle

of fifths. If we start from ni, and proceed by a cycle of fifths, we get

ni - ma - sa - pa - Ri - Dha - Ga, i.e. we have obtained all seven notes

in an octave. The other notes are obtained by carrying through this cycle

once more to get

Ga - Ni - Ma - ri - dha - ga.

This may have been the basis for reckoning an octave from ni rather than

sa in olden times. In fact the tamizh scales pAlaiyAzh, marutayAzh etc.

were obtained by precisely this method. However one must remember that

these cycles can theoretically be carried on ad infinitum, because they

will never give the correct value of 2 for the octave. They are based on

a geometric series with a ratio 3/2 (or 3/4 in order to keep each term

between 1 and 2). As such the number 2 is the limit of these cycles.

Which is why the two cycles stop making practical sense beyond the first

few terms. How many is few? The very fourth and fifth terms in the

fifths cycle generate a dissonance with the notes obtained from the fifth

harmonic, so that beyond the seventh or eighth term, we should probably

not stick to this cycle.

In order to figure out the number of SrutIs in an octave, let us list the

numbers we have obtained in two groups, in terms of the pUrvanga (sa to

ma) and the uttarAnga (pa to Sa). As of now, let us ignore the other

values from the table. Thus the two groups are

I. sa ri Ri ga Ga ma

1_(256/243_16/15)_(10/9_9/8)_(32/27_6/5)_(5/4_81/64)_4/3

II. pa dha Dha ni Ni Sa

3/2_(128/81_8/5)_(5/3_27/16)_(16/9_9/5)_(15/8_243/128)_2

Here the interval between two consecutive ratio values is marked by a _ .

>From this format, we see that

(1) the ma (4/3) is 9 intervals away from sa, in keeping with both

traditions. Other such 9 interval distances are -

ni (lower octave) to ga (32/27 : 8/9 = 4/3); Dha (lower octave) to Ri

(9/8 : 27/32 = 4/3) and pa to Sa (2 : 3/2 = 4/3), thus allowing us to say

9 SrutIs <=> 4/3.

(2) the intervals are not of equal distances. In that sense one Sruti

could mean different values in different contexts. However between two

values for the same note, the distance corresponds to 81/80 of the lower

value. This is what is called as 1 mAttirai or pramANa Sruti.

(3) the following 4 interval distances are equal.

sa - Ri = 9/8

Ri - Ga = 81/64 : 9/8 = 9/8

pa - Dha = 27/16 : 3/2 = 9/8

Dha - Ni = 243/128 : 27/16 = 9/8

ni - Sa = 2 : 16/9 = 9/8

Therefore, 4 SrutIs <=> 9/8.

This is the reason for using the term ChatuSruti rishabha. The Ri (9/8)

is four SrutIs away from sa. Similarly, the ChatuSruti Dhaivata (27/16)

is four SrutIs away from pa.

The relationship between ni (of the lower octave) and ma is that

of a fifth. How many intervals are there between the two notes? Since

the number of intervals is simply additive, there are 4 (ni to sa) + 9 (sa

to ma) = 13 intervals. Similarly, the interval from pa to Ri (in the

higher octave) is 9 (pa to Sa) + 4 (Sa to Ri) = 13 intervals. The

relationship is again that of the fifth. Similarly, for Dha to Ga (higher

octave) and for dha to ga (higher octave), the distance is again 13

intervals and the relationship is that of the fifths. Thus '13 intervals'

is characterized by the consonance ratio 3/2.

i.e. 13 SrutIs <=> 3/2

Therefore between sa and pa, there must be 13 intervals. Since there are

9 intervals between sa and ma, there must be 13 - 9 = 4 intervals between

ma and pa. Is this valid? The relationship between ma and pa is 3/2 :

4/3 = 9/8. In (4) above, we saw that 4 srutis = 9/8. Thus the

postulation of 13 srutis between sa and pa is consistent with the

postulation of 9 srutis between sa and ma. The number 22 for the number

of srutis between sa and Sa can now be easily explained. As there are 13

srutis between sa and pa, and 9 srutis between pa and Sa, there are 13 + 9

= 22 srutis between sa and Sa. In the interval from sa to Ri, there are

the following ratios - 1, 256/243, 16/15, 10/9 and 9/8. Thus 9/8 is the

fourth ratio from sa, which is why 9/8 represents a 4 sruti interval. On

the other hand, between ma = 4/3 and pa = 3/2, we have calculated the

following ratio values in Table V.

ma Ma pa

4/3 x 27/20 x 45/32 x 64/45 x 729/512 x 47/20 x 3/2

Clearly we have calculated more ratios in this interval than in the sa to

Ri interval. Yet the sa to Ri and the ma to pa intervals are both

characterized by the ratio 9/8. This means that some of the values

obtained for Ma need to be discarded. Which were the values that Bharata

discarded? It is clear that Bharata retained 40/27 as a valid ratio,

because it appears in his madhyama grAma. Hence he must have discarded

some of the values calculated for Ma. Is there a basis for selectively

discarding some values of Ma? To clarify these issues, let us look at

other relationships between the various ratios. Since we have now

satisfactorily shown that an octave does indeed contain 22 srutis, in the

sense that 4 srutis = 9/8 and 9 srutis = 4/3, we can easily see the

following relationships. In these, addition of sruti intervals

corresponds to multiplication of ratio values.

sa - Sa = sa - ma + ma - pa + pa - sa

<=> 9 + 4 + 9

= 4/3 x 9/8 x 4/3

= 2.

sa - Sa = sa - pa + pa - sa

<=> 13 + 9

= 3/2 x 4/3

= 2.

In the tamizh pAlaiyaazh,

sa-Sa = sa-Ri + Ri-Ga + Ga-ma + ma-pa + pa-Dha + Dha-ni + ni-Sa

<=> 4 + 3 + 2 + 4 + 3 + 2 + 4

= 9/8 x 10/9 x 16/15 x 9/8 x 10/9 x 16/15 x 9/8 = 4/3 x 4/3 x 9/8

= 2.

In shaDja grAma

sa-Sa = sa-Ri + Ri-ga + ga-ma + ma-pa + pa-Dha + Dha-ni + ni-Sa

<=> 3 + 2 + 4 + 4 + 3 + 2 + 4

= 10/9 x 16/15 x 9/8 x 9/8 x 10/9 x 16/15 x 9/8 = 4/3 x 9/8 x 4/3

= 2.

Since the other major scales were all obtained by shifting the

tonic from shadja grAma or from pAlaiyaazh, they all preserve this

division of the octave into three 4 Sruti intervals, two 3 Sruti intervals

and two 2 Sruti intervals. When we talk of a 4 Sruti interval, we mean a

ratio of 9/8. Similarly a 3 Sruti interval refers to 10/9 and a 2 Sruti

interval refers to 16/15. In all cases, the distance is with respect to

sa. This is because 9/8 is the 4th Sruti from sa, 10/9 is the 3rd Sruti

from sa and 16/15 is the 2nd Sruti from sa. When we talk of a certain

number of SrutIs, we always mean that ratio whose position is that many

intervals away from sa. Thus 1 sruti is 256/243. Can the value of 1

sruti also be 81/80? This is context specific, because the number 81/80

refers to the relative distance not from sa, but between two values of the

same note. It is to distinguish this fact that Bharata has used the word

pramANa Sruti for the value 81/80. Thus the number 22 for the number of

srutis in an octave arises not from some mysterious association with a

circle's circumference and its diameter, or from a mindless repetition of

the cycles of fourths and fifths, but from criteria based on valid musical

principles of consonance and dissonance, coupled with the technique of

shifting the tonic to obtain different melodic scales. The use of Sruti

intervals gives numbers that can be added instead of using ratios that

needed to be multiplied, without resorting to logarithms. For example, if

the number 22 is broken up as a sum of two numbers, in all possible

combinations, we have

22 = 1 + 21 (<=> 256/243 x 243/128 = 2)

= 2 + 20 (<=> 16/15 x 15/8 = 2)

= 3 + 19 (<=> 10/9 x 9/5 = 2)

= 4 + 18 (<=> 9/8 x 16/9 = 2)

= 5 + 17 (<=> 32/27 x 27/16 = 2)

= 6 + 16 (<=> 6/5 x 5/3 = 2)

= 7 + 15 (<=> 5/4 x 8/5 = 2)

= 8 + 14 (<=> 81/64 x 128/81 = 2)

= 9 + 13 (<=> 4/3 x 3/2 = 2).

In order to account for the sums 10 + 12 and 11 + 11, we first need to

decide what the 10th, 11th and 12th srutis are. In Bharata's system, we

are certain that the 12th sruti is 40/27, because that is the value of the

panchama in his madhyama grAma, and it is deficient with respect to the

perfect panchama (3/2) by one pramANa Sruti. If that were the case, in

order to satisfy 10 + 12 = 22, the 10th sruti has to be 27/20 i.e. the

value obtained by augmenting the ma by one pramANa Sruti. The 11th Sruti

can rigorously satisfy the relationship 11 + 11 = 22, only if the 11th

sruti is given a value 2^(1/2). Thus

22 = 10 + 12 (<=> 27/20 x 40/27 = 2)

= 11 + 11 (<=> 2^(1/2) x 2^(1/2) = 2).

If we now go back to the ratio values obtained between 4/3 and 3/2

in the table, we find the numbers 27/20 and 40/27. However, 2^(1/2) is

impossible to find there, simply because 2^(1/2) is not a ratio. It is an

irrational number. 2^(1/2) is the value for Ma in the tempered scale of

Western classical music. We however have three different values 45/32,

64/45 and 729/512 for Ma in our table. 45/32 and 64/45 are actually very

close approximations to 2^(1/2). Are there any musical guidelines by

which we can choose one of these to be the Ma? Is this supported by

actual musical practice? The answer is yes, because we do not see many

shades of Ma in either Hindustani or Carnatic music. Is there a

sufficiently strong argument to discard some values of Ma and retain only

one? It would seem so, because although the sa-Ri interval and the ma-pa

interval are mathematically equivalent, there are some crucial differences

between them. The human ear can distinguish at least two distinct shades

of ri (as in sAveri and Ahir Bhairav) and Ri (as in KharaharapriyA and

SankarAbharaNam) in the sa-Ri interval, whereas between the ma and pa,

one is not able to distinguish that many shades of prati (tIvra) madhyama.

This is because the melodic experience is conditioned by the pitch of the

tonic. The ability or inability to distinguish various values of a note,

is a direct consequence of relating that note to the shadja. In

Hindustani music, and in panchama-varja Carnatic ragas like Ranjani, where

the prati madhyama is a plain note, the vadi-samvadi relationship

projected is that between Ni and Ma. In most Carnatic ragas, the prati

madhyama is handled in conjunction with the panchama. In no raga is the

fourths relationship between ri and Ma prominent. In either case, the

prati madhyama is a very weak note, that uses some other note as a crux,

in order to define itself. All the more reason why it does not make much

sense in practical terms to count all the calculated ratio values for Ma

as defining valid Sruti intervals. This uneasy position of the prati

madhyama in the octave, may be one reason why there are so few ragas using

this note that are in current practice, as compared to the profusion of

rAgAs using the suddha madhyama. However, it is not possible to

confidently assert that Bharata used this or that value and rejected the

others. What we are certain about is that he did retain only one value

out of the three calculated. As for the ancient Tamil music, the ratios

40/27, 27/20, 729/512, 45/32 and 64/45 all figure in the scales derived by

kural tiripu. Still as only 22 mAttirais are talked of in one octave, two

of these ratios were definitely discarded for practical purposes. It is

possible that the 10th and 12th srutis in the Tamil system were also

27/20 and 40/27, in order to be consistent with the sum 10 + 12 = 22.

This is not so sacrosanct however. The description of 22 srutis or

maattirais was based on contemporary practice and not just on dry

arithmetic. Consequently, it is difficult to say with any degree of

certainty, which ratios were actually used. The other definition of the

word Sruti is helpful in understanding this. A Sruti has been described as

the raise in pitch necessary before the note can be distinguished as

another note. Consequently, even if two different values are used for a

prati madhyama, the higher value is still not identifiable as the

panchama. Thus it does not make much sense to talk of a musical interval

between two values of the prati madhyama. On the other other hand,

because of the perception of the fifth harmonic, it does make sense to

distinguish between two values for the Ga (5/4 and 81/64), because that is

definitely perceived as a dissonant interval. Therefore, the grouping of

all possible Ma values under one musical interval is still consistent with

actual practice.

It is pertinent to close this discussion with a note of how these

22 SrutIs affect our musical perception. As an individual's sensitivity

grows, the ability to detect minute variations in pitch becomes better.

Consequently, even a small change in pitch sounds dissonant and

unpleasant. In that context, the number of Sruti intervals is infinite.

In all the vikrti swaras, the usage of kampita and other gamakas gives us

the flexibility to actually produce all possible ratios that we derived,

and possibly other intermediate values as well. However in the case of

the prakrti swara pa, what role does a number such as 40/27 play? The

answer lies in the fact that the concept that pa is a prakrti swara with

no variations is a later evolution in the history of Indian classical

music. Similarly, the ratio 81/64 is used in ragas like kalyANi, where

the Ga is higher than in say SankarAbharaNam, but then the dissonance is

masked by the use of a large shake given to the Ga. In general, we tend

to use 5/4 for the Ga, in ragas which have Ri, Ga and ma, while we use

81/64 also in ragas with ri, Ga and ma or Ri, Ga and Ma. Similarly 5/3 is

used for the Dha in ragas in which the immediately following note is ni,

whereas if it is Ni, the higher value of Dha (27/16) is prominent. The ma

in a rAga like bEgaDa in Carnatic music is definitely higher in pitch than

the perfect fourth, and it corresponds to a value of 27/20. Thus which

specific srutis are used in a given raga, is a matter of tradition and

aesthetics.

On the other hand, the ratio 81/80 is a dissonant interval arising

naturally out of the laws which govern the very production of musical

sound. It is the recognition of this natural phenomenon which led our

ancient musicologists to derive the concept of 22 SrutIs. The 22 Sruti

concept represents a non-logarithmic approach to deal with simple numbers

which can be added instead of ratios which need to be factored or

multiplied. It was by means of this concept that the "just" intonation in

Indian music was represented in terms of simple numbers without recourse

to logarithmic values or equally tempered scales. Furthermore, it is the

Sruti concept which takes into account the fact that our perception of any

note is always with respect to the reference sa. This is a basic

prerequisite in a musical system that is based on melody rather than

harmony. 22 SrutIs arise naturally out of the application of the concepts

of 1) consonance and dissonance to the swaras in a scale, and 2)

derivation of new scales by shifting the tonic. The shifting of the tonic

is a powerful method to obtain new scales from existing ones, and has been

used later in the history of Carnatic music, in the formulation of the

meLakarta scheme. The rationalization of the 22 sruti concept, as shown

here, shows us how the successive application of the fourth or fifth

relationship can be carried only upto a point. Thus the fifths cycle is

used only upto the prati madhyama (sa-pa-Ri-Dha-Ga-Ni-Ma) and the fourths

cycle upto the suddha rishabha (sa-ma-ni-ga-dha-ri). Even upto this

point, the two cycles generate dissonance because of the closeness of the

notes thus obtained, to those arising out of the 5th, 6th, 8th and 9th

harmonics. This is a consequence of our constant perception of the

reference sa, which has components of all these harmonics, as seen from

the physical laws of vibrations, which hold for vibrating strings and for

air columns e.g. the human voice or the flute. The concept of 22 srutis

is thus an extremely neat technique of systematizing these consonance and

dissonance relationships. This is very important both for Hindustani and

for Carnatic music, and represents an important, early development in the

history of a melody based musical system.

As for contemporary practice, since we have essentially discarded

the possibility of another value for pa, we need to discard the ratio

40/27. That would bring the number of SrutIs to 21. If we however take

into account that we actually use two Ma's in practice, a plain Ma as in

ranjani or hamsAnandi and a higher pitched Ma as in varALi, the number

goes up to 22 again. We will thus still have 13 SrutIs from sa to pa and

9 SrutIs from sa to ma. In contemporary Hindustani music, ratio values

like 81/64 and 243/128 are coonspicuous by their absence. 5/4 and 15/8

are the only values used for Ga and Ni in Hindustani music today. Which

is why some people feel that the nishAda in Hamsadhwani as rendered by a

Hindustani musician is much more prominent than in Carnatic music.

The alternative derivation of 22 SrutIs as explained by taking

twelve terms from each of the fifths and the fourths cycles and discarding

the value of the panchama obtained from the cycle of fourths does not make

much sense for reasons stated before. Clearly the origin of the concept

is related to the accounting for the 5th harmonic also, which these cycles

do not take into consideration at all.

S. Vidyasankar