According to (1), "if we want to study the details of the ancient Indian tone system, its scales and the relationship between the individual notes, we have to bear in mind that whole musical system is based on the concept of Nada-Brahman, sound as fundamental cosmic vibration, identified with the creative principle of the universe.
As Anahata Nada it is the unmanifested sound; as ahata nada, manifested and aesthetically enjoyable sound, it becomes music. Making music is considered to be a creative process comparable to yoga, the religious exercise repeating cosmic creation on a human level. However, the Nada-Brahman concept is not an abstract philosophical principle, a kind of general metaphysical background of the Indian music, but it works out in the details of its musical system. The three octaves, the three registers of the human voice, are compared with three particular cakra of yoga.
The division of the octave and the size of the intervals between the main notes (svara) are calculated by means of smallest audible units of sound (sruti), in conformity with an ancient system of mathematics that was already used in the construction of the Vedic sacrificial places.
The relationship between the individual notes was based on the same principles of harmony as the ancient Hindu temple. Hindu musical theory developed an intricate system of microcosm-macrocosm identifications in which every note was associated with a particular form of existence (god, demon, human being, animal), a phenomenon (time, color, pitch, poetical meter), topography (world, country), psychology (sentiment), and physiology (parts of the human body)."
The ancient theory of Gandharva tone-system, described in Natya Sastra, Dattilam, Brihadesi, Sangitaratnakara, Sangitasiromani, Sangitasuddha, etc. contains three major elements:
Bharata defines two basic gremas, sadjagrama and madhyamagrama. Madhyama grama can be achieved by lowering pa in sadja grama by one sruti (3 sruti pa or a.pa).
|grama||svaras||intervals in sruti||consonants (samvadin)|
|sadjagrama||sa ri ga ma pa dha ni sa||(4 3 2 4 4 3 2)||(sa-ma, sa-pa)|
|madhyamagrama||sa ri ga ma a.pa dha ni sa||(4 3 2 4 3 4 2)||(ri-pa, ri-dha)|
Madhyamagrama could also be achieved by the transposition method, i.e. by starting the sadjagrama on pa and rising ga by two sruti (antara ga or a.ga), and then renaming the intervals in the following manner:
|grama||svaras||intervals in sruti||consonants (samvadin)|
|sadjagrama||pa dha ni sa ri a.ga ma||(4 3 2 4 3 4 2)||(pa-sa, sa-ma)|
|madhyamagrama||sa ri ga ma pa dha ni sa||(4 3 2 4 3 4 2)||(ri-pa, ri-dha)|
In this case the minor third (suddha ga) of Sadjagrama had to be tuned to a major third (antara ga) in order to become the dha of Madhyamagrama.
In musical practice the latter method might have been easier, since lowering the fifth (pa) by one sruti must have been more difficult than rising the third (ga) by a semitone.
The difference between the four-sruti fifth of Sadja grama and three-sruti fifth of Madhyama grama is considered to be the Pramana sruti, i.e. the standard sruti.
The musical treatises of Matanga, Nanyadeva and the authors of Sangitasiromani use special rectangles (mandala) to show 22 sruti and their distribution over the seven main intervals in sadja, madhyama and ganhdara grama. The authors of Sangitasiromani (15 cen AD) refer to the ancient sadjagrama with its 22 sruti, which they illustrate by simple square and line diagrams.
|Sadja grama mandala||Madhyama grama mandala||Gandhara grama mandala|
The third tone-system, gandhara-grama, which fell into disuse, was still in use in Bharatas days, since Nanyadeva, the eleventh century commentator on the Natyasastra, explains it is his commentary Bharatabhasya. Dattila also mentions Gandhara grama in Dattilam, but without giving particular definition of intervals and corresponding murchanas. He explains that Gandhara grama belongs to heavenly regions and should not be used on the earth. Saringadeva in Sangitaratnakara defines intervals between the notes of Gandhara grama (in srutis). Using transposition method, i.e. renaming the notes of Sadja-grama, we can get the following scale:
In case of Gandhara grama, suddha ga of Sadja grama (s.ga) has to be raised by one sruti to become the trisruti pa of Gandhara-grama.
|grama||svaras||intervals in sruti||consonants (samvadin)|
|sadja grama||dha ni sa ri s.ga ma pa||(3 2 4 3 3 3 4)||(sa-ma, sa-pa)|
|gandhara grama||sa ri ga ma pa dha ni||(3 2 4 3 3 3 4)||(ri-pa, ri-dha)|
Through sadja and madhyama gramas were regarded as basic scales, a larger number of scales were known in the days of Natyasastra. According to Bharata, six different scales should be used in the six main places (sandhi) of dramatic performance, viz. sadava in purvaranga (preliminaries), madhyama grama in mukha (opening), sadja grama in pratimukha (progression), sadharita in the central part (grabha), pancama in the reflective part (avamarsa) and kaisikamadhyama in the conclusion (nirvahana). In Natyasastra only two gramas are explained. In other Sanskrit works seven gramas, the previous six plus one more scale names kaisika, are mentioned as principal melodic forms.
According to Bake (2) and Jairazbhoy (3) the author of Natya Sastra tried to describe intervals which secular music had inherited from the older religious music in a systematic way and merely used the sruti as convenient numbers to determine the older unequal intervals of Vedic chant (see Levy (4), the Vedic theory).
This theory is based on a late Vedic textbook of recitation, the Naraditya Siksa, which compares the musical intervals of samagana, i.e. chanting of the Samaveda, with the intervals of the flute (venu), representing secular music.
The authors of Sangitasiromani mention the traditional sadja and madhyama grama murchanas with their religious associations, omitting the ancient, mythical gandharagrama and its murcana, which we can find in Sangitaratnakara, Naradiya Siksa and in Naradas Sangitamarakanda.
Sangitasiromani text refers to an older tuning system which is apparently no longer understood by contemporary musicologists. It is an uncommon theory regarding the three ancient gramas (Nandyavarta, Jimuta and Subhadra) not found in most of the treatises on music. The compilers of Sangitasiromani may have borrowed it from the early medieval work Gitalamkara written by an author who calls himself Bharata.
According to Danielou (9) Nandyavarta, Jimuta and Subhadra are basic tetrachords and forerunners of the three traditional grama. These three peculiar series of notes may represent some practical tuning devises for the ancient sadja, madhyama and gandhara gramas.
Tuning has always been a compromise between the "additive" and the "divisive" principle, or rather between a linear and a harmonic method. Since the ancient days musical theorists, philosophers and mathematicians have been searching for a simple unit and an additive series of numbers instead of the ratio and its multiplications to calculate the intervals of their tine-systems. Whereas the Greek theoretician Aristoxenos used the whole tone for this purpose, Bharata devised a much smaller, but still "audible" unit, the sruti, a mathematically calculated sruti of 54.5 cents corresponds to the ratio 32/31, an interval which in nineteenth century the English scholar A.J. Ellis found be string division, when he was searching for the smallest audible unit. In ancient Greek musical theory this ratio represented one of the two enharmonic quartertones (30:31:32).
Proportional measurement was an important aspect of ancient Indian temple architecture. We may assume that Bharata was familiar with the Hindu temple concept of vastupurusamandala, the ground plan of the temple, which represents the circular phenomenal world in square form with proportional divisions representing the divine paws of space and time.
In this concept number 32 is a significant one, since the 32 squares in the outer border of the diagram are the sites of the 32 protective deities.
Indian sruti as well as European cents are a linear representation of a complex non-linear sound phenomenon, a simplification resulting from an attempt to visualize musical intervals.
Many modern scholars could not understand the peculiar ancient Indian division of the octave into twenty-two units, because they tried to prove its existence in modern Indian music practice. They did not realize that the Indian tone system and temperament had changed in medieval times, between 13th and 16th century. The introduction of fretted stringed instruments entitled a transition to a system in which the octave was divided into twelve semitones, a system which is irreconcilable with a division of the octave into 22 srutis. Kallinatha, fifteenth century author who wrote a commentary on Sangitaratnakara, notices differences between contemporary practice and traditional theory.
The ancient Indian master-musician must have been able to distinguish these fine intonations by ear. The Naradiya Sisksa says: "The one who is not able to distinguish between the sruti cannot be a called a teacher". Form a vocalists point of view the calculation of twenty-two sruti per octave is only relevant as a additive concept, this is to say, twenty-two as the sum total of pitches which in musical practice may have been used as tonal colors, or rather, as finer intonations expressing a particular emotion. This is probably the reason why in the Natya Sastra the sruti are not only mentioned in connection with the tone-systems (grama) but also in the context of the musical figures (alamkara).
As the present day Indian musician, the ancient Indian musicians were able to tune their instruments by ear and be remembering the peculiar intonations of their music system. No musician in the world thinks about ratios, when he tunes his instrument. He simply tunes the strings in the scales of his own musical culture, which he knows by heart. He may only use a few harmonic cross-relationship between the notes to check his tuning. This is also the procedure described in the Cilappatikaram. The harp player Madhavi first plays all the fourteen notes of the classical scale from ulai (ma) in the lower octave up to kaikkilai (pa) in the higher octave. Then she checks the tuning of the second (pa), the fifth (sa), the sixth (ri), the third (dha) and the fourth (ni) string. In the same way one may understand Bharata's consonance relationship between certain notes in the sadjagrama and the madhyamagrama.
On stringed instruments, however, notes can be made audible as well as visible, since pitches can be related to a particular string length. As a result, the instrumentalists approach to the problem of intonation is a different one. In his view, the relationship between the notes is of special importance, because he uses the principle of consonance, when he tunes the strings. In many musical cultures tonal relationships based on simple mathematic ratios, the so called "harmonic intervals", produced by dividing a string into two, three, four, five or more parts, are considered to be consonant.
As elements of such consonance theory are also found in Bharatas system, musicologists have tried to explain the ancient Indian tone-system accordingly. Unfortunately, many scholars, following Fox Strangways example (5) used just temperament (1:2:3:4:5) to this purpose. This is why until today most Western as well as Indian scholars take the major whole tone (ratio 9/8) and the minor whole tone (10/9) of that system as equivalents of the ancient Indian intervals of four sruti and three sruti respectively, although in the case of the latter interval the ratio 11/10 would be a much better equivalent.
Selecting a number of harmonics that might represent the main intervals of the ancient Indian tone-system and their micro-tonal alterations inevitable leads to unequal sruti.
Lute playing naturally requires division of the playing strings by the left hand, but the strings of the ancient arched harp show a proportional length, which was certainly not overlooked by scientists of those days. Already in the Vedic scriptures we may find such a notion of proportions. The Jaiminiya Brahmana describing the victory of Prajapati, the Father of Creation, over death, probably contains the oldest description of the proportional length of the seven strings of Vedic arched harp.
A similar passage can be found in Aitareya Aranyaka, which mentions an instrument with seven or ten strings. In both cases the strings are compared to a series of poetical meters, increasing in length by four syllables. The fourteen century commentator Sayana tells us which Vedic meters were probably meant. In this way we may reconstruct the following set of meters and corresponding strings:
|Poetical meter||Syllables||String length||Tonal ratio|
In this scheme the numbers of the syllables in the poetical meters provide us with a series of numbers representing the proportional length of the strings. In the corresponding tonal ratios we may recognize an interesting series of harmonic intervals.
The general form of the ancient Indian arched harp, a boat shaped resonator covered with hide and a curved neck, shows clear differences in string length. But tonal pitch cannot only be determined by the length of the strings. It depends a great deal on the tension of the strings. By alteration and adjustment of this tension tuning is achieved. On the more primitive arched harps the tension of the strings could only be changed by moving the loops by which the strings were attached to the neck upwards and downwards. To facilitate the tuning the more sophisticated types of this instrument had special rings.
The harps Bharata used for his demonstration of the sruti may have had the same size as the large mukhya vina, the main harp of the ancient Indian theater, which in common practice was called mattakokila. But, whereas the twenty one string og the latter instrument may have produced the seven main notes in three octaves, the twenty two strings of Bharatas special vina were tuned to twenty two pitches within the octave.
The best instrument to show Bharatamuni consonance (samvadin) theory is the sakota yal, the arched harp with fourteen strings, reaching from ma in the lower octave to ga in the higher octave, mentioned in the Tamil classic Cilappatikaram of Prince Ilanko Atikal (third century AD).
(where major consonants are: ma1-sa1, sa1-pa2, dha1-ri2, ni1-ga2)
In the sadjagrama, in which sa functions as a tonic, only the intervals sa-ma, sa-pa, ri-dha and ga-ni are consonant (samvadin). If we explain the samvadin as a perfect fifth or a perfect fourth relationship, the major whole tone between ma and pa can be established as follows: 3/2 : 4/3 = 9/8. Until this point I agree with most scholars. But to take ma-dha as a natural major third from the series of harmonics, i.e. as ratio 5/4, seems to me far-fetched. It is exactly this misinterpretation which leads to the generally accepted, but in my opinion incorrect calculation of the three-sruti intervals sa-ri and pa-dha as ration 10/9. This becomes clear from the following chart of harmonic intervals:
The major third and the minor third are more tolerant than the perfect fourth, the perfect fifth and the octave, and permit various intonations.
For this reason it must have been quiet easy to tune the third ma-dha a little lower to achieve a minor whole tone between pa and dha of ratio 11/10 (=165 cents), which is nearer to the mathematic three-sruti interval of 163.5 cents. Combining the harmonic major third (5/4) with perfect fourths (4/3) and perfect fifths (3/2), as is done in just temperament, never leads to an acceptable sruti value and makes Bharatas experiment with the two vina useless.
It is possible to make an exact calculation of Bharatas standard (Pramana) sruti by using simple harmonics. Excepting Modak (6), most scholars have not paid much attention to the harmonic seventh (7/4), which may have been an important interval in the ancient Indian music, since the octave was not conceived as series of eight notes, but as a series of seven intervals (saptaka). The basic scales of Vedic chant and ancient Tamil music show the importance of the seventh.
In her standard work on the Hindu temple Stella Kramrisch (7) informs that the body (atman) of the bird-shaped Agni, i.e. the basis of the fire-place, had an area of four man's lengths (purusa) square in a larger area of seven man lengths square.
By twice using the musical ratio 7/4 (968.83 cents) in combination with two perfect fifths (3/2 = 701.96 cents) one obtains an interval which is precisely one sruti (54.5 cents) lower then the fourth octave:
7/4 x 3/2 x 3/2 x 7/4 x 3/2 x 3/2 = 4745.5 cents = 4800-54.5 cents
This is a theoretically correct method of calculating the standard sruti, because as against other cyclic theories such as Kolinski method (8), it does not contain more then two joint fifths (ma-sa-pa), in conformity with Bharatas theory of consonance.
There is a doubt, whether a complicated cyclic method as shown above was used in musical practice. The three-sruti pa was most probably not obtained by lowering the four-sruti pa-string, but simply by renaming the ri-string as part of the transposition process by which the notes of the sadjagrama became madhyamagrama notes.