Musical Mathematics

on the art and
science of acoustic instruments

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CHAPTER 11: *
WORLD TUNINGS*

Part III: Indian Music

Ancient Beginnings

Section 11.20

One of the oldest and most revered texts on Indian music is a work
entitled *Natyasastra*, written by Bharata (early centuries
a.d.). Although large portions of
Bharata’s treatise recount performance practices of the theater and dance,
Volume 2, Chapters 28–33, deal exclusively with music. In *Natyasastra 28.21,
*Bharata begins his description of the classical 22-*sruti* scale by
giving the names of seven *svaras*, translated below as *notes*, and
also interpreted in this discussion as *tones* and *scale degrees*.

* * ^{↓} * *

*Nat. 28.21*:
The seven notes [*svaras*] are: *Sadja* [Sa], *Rsabha* [Ri], *
Gandhara* [Ga], *Madhyama* [Ma], *Pañcama* [Pa], *Dhaivata *
[Dha], and *Nisada *[Ni].[1]
(Text in brackets mine.)

* * ^{↑} * *

Bharata then defines
the musical qualities of four different kinds of sounds, and specifies the
consonant and dissonant intervals contained in two different scales called *
Sadjagrama *(*Sa-grama*) and *Madhyamagrama* (*Ma-grama*).

* * ^{↓} * *

*Nat. 28.22*:
[According] as they relate to an interval of [more or less] *Srutis*, they
are of four classes, such as Sonant (*vadin*), Consonant (*samvadin*),
Assonant (*anuvadin*), and Dissonant (*vivadin*).

That which is an *
Amsa* [note] anywhere, will in this connection, be called there Sonant (*vadin*).
Those two notes which are at an interval of nine or thirteen *Srutis* from
each other are mutually Consonant (*samvadin*), e.g., *Sadja* and *
Madhyama*, *Sadja* and *Pañcama*, *
Rsabha* and *Dhaivata*, *Gandhara* and *Nisada *in the
*Sadja* *Grama*. Such is the case in the *Madhyama Grama*, except
that *Sadja* and *Pañcama* ** are not** Consonant, while

*Nat. 23*:
In the *Madhyama* *Grama*, *Pañcama* and *Rsabha* are
Consonant while *Sadja* and *Pañcama* are so in the *Sadja* *
Grama* [only].

The notes being at
an interval of [two or] twenty *Srutis* are Dissonant, e.g., *Rsabha*
and *Gandhara*, *Dhaivata* and *Nisada*.

. . . As a note
[prominently] sounds it is called Sonant; as it sounds in consonance [with
another] it is Consonant; as it sounds discordantly [to another] it is
Dissonant, and as it follows [another note] it is called Assonant. These notes
become low or high according to the adjustment of the strings . . . of the *
Vina* . . .[2]
(Bold italics mine.)

* * ^{↑} * *

With this general background information — which will prove crucial in
constructing the scales — Bharata then quantifies these seven scale degrees
according to how many srutis (from the Sanskrit *sru*, lit. *to hear;*
*sruti* in music, *an interval*) are contained between each degree.

* * ^{↓} * *

*Nat. 28.23*:
. . . Now, there are two *Gramas:* *Sadja* and *Madhyama*. Each
of these two (lit. there) include twenty-two *Srutis* in the following
manner:

*Nat. 24*:
*Srutis* in the *Sadja* *Grama* are shown as follows: ** three**
[in Ri],

In the *Madhyama*
*Grama*, *Pañcama* should be made deficient in one *Sruti*. The
difference which occurs in *Pañcama* when it is raised or lowered by a *
Sruti* and when consequential slackness or tenseness [of strings] occurs,
will indicate a ** typical** (

* * ^{↑} * *

Next, Bharata describes a demonstration on two *vinas*, each equipped with
seven strings, and tuned exactly alike to the *Sa-grama*. The tuning of one
*vina* remains unchanged. Bharata gives directions for changing the tuning
of the other *vina* in four separate steps. Each step requires the lowering
of all seven degrees by increments of one *sruti*. Consequently, after the
first step, or after lowering all the strings by 1 *sruti*, no two degrees
of the two *vinas* match because the smallest interval of the *Sa-grama*
consists of 2 *srutis*. After the second step, or after lowering Ga by 2 *
srutis*, it will sound the same tone as Ri of the *unchanged vina;* and
after lowering Ni by 2 *srutis*, it will sound the same tone as Dha of the
unchanged *vina*. Similarly, after the third step, or after lowering Ri by
3 *srutis*, it will sound the same tone as Sa of the unchanged *vina;*
and after lowering Dha by 3 *srutis*, it will sound the same tone as Pa of
the *unchanged vina*. Finally, after the fourth step, or after lowering Ma
by 4 *srutis*, it will sound the same tone as Ga of the *unchanged vina;*
after lowering Pa by 4 *srutis*, it will sound the same tone as Ma of the
*unchanged vina;* and after lowering Sa by 4 *srutis*, it will sound
the same tone as Ni of the *unchanged vina*. In a passage translated by N.
A. Jairazbhoy,[4] Bharata
states, “ . . . lower again, in exactly this manner . . .” (*punarapi
tadvadevapakarsat*),[5] which
means that this experiment was intended to prove that all *sruti* intervals
are exactly equal in size. Bharata implies that *only* controlled decreases
by identical *srutis* will produce the scale degrees on the changed *vina*
that exactly match the degrees of the unchanged *vina*. In this context,
the unchanged *vina* represents a scientific control, or an aural reminder
of the changed *vina* before it was lowered.

Bharata then summarizes

* * ^{↓} * *

*Nat. 28.25–26*:
In the *Sadja* *Grama*, *Sadja* includes four *Srutis*, *
Rsabha* three, *Gandhara* two, *Madhyama* four, *Pañcama*
four, *Dhaivata* three, and *Nisada* two.

*Nat. 27–28*:
[In the *Madhyama* *Grama*] *Madhyama* consists of four *Srutis*,
*Pañcama* three, *Dhaivata* four, *Nisada *two, *Sadja*
four, *Rsabha* three, and *Gandhara* two *Srutis*. [Thus] the
system of [mutual] intervals (*antara*) has been explained.[6]

* * ^{↑} * *

In the absence of clearly defined length ratios and interval ratios,[7] this mixture of numerical and verbal terms seems completely open to interpretation. However, a historically accurate analysis reveals that the musical possibilities contained in this text are extremely limited and point toward only one plausible explanation. Before we examine this interpretation of Bharata’s text, let us first eliminate two possibilities.

In *Natyasastra* *28.24*, Bharata distinguishes between the *
Sa-grama* and the *Ma-grama* by stating that in the former scale, Pa
contains 4 *srutis*, and in the latter scale, Pa contains only 3 *srutis*.
He defines this difference based on a *typical sruti*, or a *pramana
sruti*. Bharata goes on to describe his experiment with two *vinas*,
which only works if the *pramana sruti* is a standard interval, or an
interval of a constant size. All these formulations lead to one possibility,
namely, that Bharata was contemplating a *geometric division* of the
“octave” into 22 equal parts. To achieve such a
“division”
requires the calculation of a complicated irrational number called a *common
ratio*,[8] which,
in this case, leads directly to 22-tone equal temperament. Recall that in 1584
and 1585, Chu Tsai-yü[9] and
Simon Stevin,[10] respectively,
solved for the “semitone,” or the *common ratio* of 12-tone equal
temperament, when they independently discovered simplified solutions for the
twelfth root of 2. They were able to calculate this complicated constant without
logarithms because composite number 12 is a product of primes 3 × 2 × 2.[11]
This factorization enabled Tsai-yü and Stevin to *effectively* extract the
required root in two different ways:

A similar technique
for the *pramana sruti* of 22-tone equal temperament does not yield
favorable results because a factorization of composite number 22 yields prime
numbers 11 and 2. Here a “simplified” solution would require the extraction of
the eleventh root of the square root of 2:

Because 11 is a
relatively large prime number, this equation ** cannot** be solved
without logarithms. Consequently, even though Bharata was under the impression
that the

Although there is no evidence of irrational length ratios in ancient Indian music, the intriguing question remains, “Is there a mathematical theory Indian musicians could have contemplated that explains the origin of 22 divisions per “octave?” This question probably has no definitive answer. However, we may attempt an explanation by examining the rational interval ratio 32/31, which very closely approximates the irrational interval ratio :

Now, raise 32/31 to
the 22nd power, and observe that twenty-two successive rational *pramana
srutis* exceed one “octave” by only 9 ¢:

This discrepancy is
less than half of the spiral of twelve “fifths,” which exceeds seven “octaves”
by the *ditonic comma*, known as the *comma of Pythagoras*, or by 23.5
¢.[12]

Unfortunately, to actually tune a scale through 22 powers of 32/31 is not any
easier than through 22 powers of
:
both methods require the precision and control provided by a monochord or canon
with moveable bridges. Again, no human being can accurately tune so many
successive rational or irrational intervals by ear. Since there is no historical
evidence of the construction of monochords in ancient India,[13] these
“theories” could *not *have been realized in the tuning of ancient
Indian instruments. The only difference is that the rational approximation could
have been contemplated in numerical terms, whereas the irrational division could
not have been contemplated in numerical terms. We conclude, therefore, that a
literal interpretation of Bharata’s

Section 11.21

The *vina* in Bharata’s text is* not* the kind of stringed
instrument that appeared in the second half of the first millennium
a.d. The latter

Since no specimens survived, two critical aspects about the internal
construction of the harp-*vina* remain unknown. (1) We do not know whether
the strings were attached to a strip of wood (called a string holder), which in
turn was anchored to the inside walls of the resonator, or whether the curved
arm extended all the way through the resonator and, thereby, provided a
continuous structure that held the strings at both ends. (2) We also do not know
by what mechanism the strings were tuned. Ancient sculptures and reliefs show no
tuning pegs in the upper part of the arm. Therefore, the strings were probably
adjusted with tuning-cords, very much like the modern *saùng-gauk* of
Burma, also an arched harp with a boat-shaped resonator.[16]
However, if the arm passed through the resonator, the remote possibility exists
that tuning pegs were located in the lower part of the arm. In either case,
adjusting the tension of the strings must have been tedious at best because
neither design facilitates the demanding process of precision tuning. It is
difficult to achieve fine incremental adjustments with tuning-chords because
such lashings are subject to creep and slippage; similarly, it is difficult to
manipulate tuning pegs inside the resonator because they are situated in a
confined and awkward location. Finally, note carefully that the *vina* had
*no *post to give it structural rigidity between the lower open end
where the resonator terminates in a round corner, and the upper open end where
the arm terminates in a scroll. This open semi-circular design severely limited
the tuning possibilities of the instrument.

* * ^{↓} * *

It is nevertheless very difficult to understand how such harps . . . could be tuned or kept in tune; not so much because no tuning devices are to be seen in the representations, as because it would be impossible to make an instrument with a curved frame and no post, were the frame even of steel, so rigid that a change of tension in one string would not alter that of all the others.[17]

* * ^{↑} * *

We conclude,
therefore, that it is extremely unlikely that the ancient *vina* was used
to tune technically difficult scales. Bharata’s demonstration was at best a
“thought experiment” designed to illustrate the distribution of *sruti*
intervals. The slightest motion between open ends would have obliterated the
subtle intonational differences of scale degrees lowered (or strings loosened)
in 1-*sruti* increments.[18]
As on most folk harps, the *vina* was probably tuned to “octaves,”
“fifths,” “fourths,” and “thirds.”

Section 11.22

Among modern Indian and European writers, the greatest controversy surrounding
Bharata’s text consists of two fundamentally different interpretations regarding
the distribution of *sruti* intervals in both the *Sa-grama* and the
*Ma-grama*. A careful reading of *Natyasastra* *28.24–28* does
not reveal whether these intervals come before (below) or after (above) the
indicated tones. However, if we take into consideration Bharata’s description of
consonant and dissonant intervals, only one correct interpretation emerges. I
can categorically say that all the writers who advocate the incorrect
interpretation never take Bharata’s interval descriptions in *Natyasastra*
*28.22–23* into consideration; it is as if the original text does not
exist. Table 11.15 gives the correct and incorrect interpretations of the *
Sa-grama* and *Ma-grama*.[19]

In the correct version of the *Sa-grama*, the
*sruti* intervals — 4, 3, 2, 4, 4, 3, 2 — come before the indicated tones,
and in the incorrect version, after the indicated tones. Only the correct
version complies with Bharata’s demonstration on the two *vinas*. For
example, recall that in the fourth step, Bharata requires that after *lowering*
Sa by 4 *srutis*, it will have the same tone as Ni of the unchanged *vina*.
In the *Sa-grama*, such a retuning would be impossible if a 4-*sruti*
interval did not *precede* Sa.

In *Natyasastra* *28.22*, Bharata specifically defines consonant
intervals as containing either nine or thirteen *srutis*. Since 9 *srutis*
+ 13 *srutis* = 22 *srutis*, let us express 9 *srutis* as ratio
4/3, 13 *srutis* as ratio 3/2, and 22 *srutis* as ratio 2/1, because
4/3 × 3/2 = 2/1. Therefore, if Ma is a “fourth,” and if Pa is a “fifth,” then
the interval between Ma and Pa is 3/2 ÷ 4/3 = 9/8. Now, since in the *Sa-grama*
the interval between Ma and Pa contains 4 *srutis*, we conclude that all
such intervals in the scale represent a “tone,” ratio 9/8. Refer to Figure
11.22(a), Row 1, and notice that Sa, Ma, Pa, and Sa' are vertically aligned with
ratios 1/1, 4/3, 3/2, and 2/1, respectively, in Row 4; and that all 4-*sruti*
intervals in Row 2 are vertically aligned with interval ratios 9/8 in Row 3.
With these values, compute two *svara* ratios in Row 4: Ni = 2/1 ÷ 9/8 =
16/9, and Ga = 4/3 ÷ 9/8 = 32/27. Next, suppose that a 3-*sruti* interval
represents a “small whole tone,” ratio 10/9, and that a 2-*sruti* interval
represents a “semitone,” ratio 16/15. With the latter ratio, calculate one more
*svara* ratio in Row 4: Dha = 16/9 ÷ 16/15 = 5/3. Finally, according to
this interpretation, interval ratio 10/9 occurs between Sa–Ri and Pa–Dha by
default; and interval ratio 16/15 occurs between Ri–Ga by default.

To confirm these assumptions, return to *Natyasastra* *28.22* and
observe that Bharata insists that in the *Sa-grama*, intervals Sa–Ma,
Sa–Pa, Ri–Dha, and Ga–Ni are consonant. A sequence of solid brackets in Figure
11.22(a) confirms that these tones span interval ratios 4/3, 3/2, 3/2, and 3/2,
respectively. In the last sentence of *Natyasastra 28.22*, and in the first
sentence of *Natyasastra 28.23*, Bharata juxtaposes the *Ma-grama* and
*Sa-grama* and observes the following opposite conditions: in the *
Ma-grama*, Sa–Pa [or C–G'] is not consonant while Pa–Ri [or G–D] is
consonant, and in the *Sa-grama*, Sa–Pa [or C–G] is consonant while Pa–Ri
[or G–D'] is not consonant. (With respect to the *Ma-grama*, in Figure
11.22(b) the tone G' is immediately above F', and with respect to the *
Sa-grama*, in Figure 11.22(a) the tone D' is immediately above C'.)
Although Bharata does not explicitly define Pa–Ri as a dissonant interval in the
*Sa-grama*, we may deduce this description based on a logical analysis of
his juxtaposition. In Figure 11.22(a) the dashed bracket shows that the
inversion of Pa–Ri — interval Ri–Pa — is a dissonant “sharp fourth,” as in 3/2 ÷
10/9 = 27/20 = 519.6 ¢, which means that Pa–Ri is a dissonant “flat fifth,” as
in 20/9 ÷ 3/2 = 40/27 = 680.4 ¢.

Further evidence that these distributions of *srutis* in the *Sa-grama*
and *Ma-grama* are authentic may be found in a text entitled *Dattilam*,
written by Dattila (early centuries a.d).
Bharata refers to Dattila as an authority on music (*Natyasastra 1.26*),
but Dattila does not mention Bharata. Two English translations of the *
Dattilam* exist, one by E. Wiersma-Te Nijenhuis,[20] and
the other by Mukund Lath.[21]
The latter states, “The whole testimony shows that Dattila was at least as
ancient as Bharata and that the *Dattilam* is almost certainly his
authentic creation.”[22]
In *Dattilam 12–14*, the author explicitly states that in the *Sa-grama*,
*Sadja* is the first degree, and that three *srutis* higher, *Rsabha*
is the second degree, etc. Similarly, in the *Ma-grama*, *Madhyama* is
the first degree, and three *srutis* higher, *Pañcama* is the second
degree, etc. The Nijenhuis translation reads

* * ^{↓} * *

*Dat. 12*:
The sound (*dhvani*), which is indicated by the term *Sadja* is [the
starting point] in the *Sadjagrama*. From this one the third [*sruti*]
upwards is, no doubt, *Rsabha*.

*Dat. 13*:
From this one the second [*sruti*] is *Gandhara*, from this one the
fourth [*sruti*] is *Madhyama*. From *Madhyama* in the same way
*Pañcama;* from this one the third [*sruti*] is *Dhaivata*.

*Dat. 14*:
From this one the second [*sruti*] is *Nisada*; from this one the
fourth [*sruti*] is *Sadja*. In the *Madhyamagrama*, *Pañcama*
is the third [*sruti*] from *Madhyama*.[23]

* * ^{↑} * *

With respect to the
*Madhyamagrama*, the Lath translation reads

* * ^{↓} * *

*Dat. 14*:
In the *Madhyama*-*grama*, *Pañcama* is the third higher [*sruti*]
commencing with *Madhyama*.[24]

* * ^{↑} * *

We turn now to the *Ma-grama*, which Bharata describes in *Natyasastra*
*28.24* as having a Ma–Pa interval reduced by one *pramana sruti*.
Given that in the *Sa-grama*, the interval between Ma–Pa contains 4 *
srutis*, in the *Ma-grama* it therefore contains 3 *srutis*. Since
4 *srutis* represents ratio 9/8, and 3 *srutis*, ratio 10/9, calculate
the difference between these two intervals by dividing the larger ratio by the
smaller ratio: 9/8 ÷ 10/9 = 81/80 = 21.5 ¢. In Western tuning theory, this
discrepancy is called the *syntonic comma*, or the *comma of Didymus*.[25]
Unfortunately, 81/80 is less than half the size of 32/21 [55.0 ¢], the closest
rational approximation of the common ratio of 22-TET. If Bharata either heard or
thought of 81/80 (or some similar microtonal interval) as a *pramana sruti*,
it is understandable given the structural imperfections and mathematical
limitations of the ancient harp-*vina*. In any case, the same shift that
decreases the Ma–Pa interval by one *sruti*, increases the Pa–Dha interval
by one *sruti;* therefore, in the *Ma-grama*, the interval Pa–Dha
contains 4 *srutis*.

Refer to Figure 11.22(b), and note that it shows the *Ma-grama* based on
the same organizational principles as the *Sa-grama* in Figure 11.22(a).
Again, to verify the authenticity of these ratios, return to the second
paragraph of *Natyasastra* *28.22*. Here, Bharata clearly indicates
that in the *Ma-grama*, intervals Sa–Ma or inversion Ma–Sa, Ri–Dha or
inversion Dha–Ri, Ga–Ni or inversion Ni–Ga, and Pa–Ri are consonant. The solid
brackets in Figure 11.22(b) confirm that these tones span interval ratios 3/2,
4/3, 4/3, and 3/2, respectively. And in same sentence, Bharata observes that the
interval Sa–Pa is* not* consonant. In Figure 11.22(b), the dashed
bracket shows that this interval is a dissonant “sharp fourth,” ratio 27/20.

If we now assign Sa, ratio 1/1, to C, Figure 11.22(a), Row 5, shows that in
Western music theory, the *Sadjagrama* is a kind of minor scale, which
includes a Pythagorean “minor seventh” [996 ¢], ratio 16/9 [Bb], that sounds a
“fifth” below Ma' [8/3 ÷ 3/2 = 16/9], and a Pythagorean “minor third” [294 ¢],
ratio 32/27 [Eb], that sounds a “fifth” below Ni [16/9 ÷ 3/2 = 32/27].[26]
In contrast, if we assign Ma, ratio 1/1, to F, Figure 11.22(b), Row 5, shows
that the *Madhyamagrama* is a kind of major scale, which includes a 5-limit
“major third” [386 ¢], ratio 5/4 [A], and the “minor seventh,” ratio 16/9 [Eb]
as before.

Finally, let us reflect on the meaning of some ancient Indian terms. Although
Western musicians may think of *Sa-grama* and *Ma-grama* as scales,
the word *grama* in the purest sense of the word does* not*
mean scale. The abstract concept of
“scale”
remained unknown in India until the 19th century
a.d. (See Section
11.32.)
We should understand the word

Section 11.23

One of the most important discussions in Bharata’s book occurs in *Natyasastra
28.38–149*. Here Bharata describes a highly organized system of melodic modes
called *jatis*. At the end Chapter 28, and almost as an aside, Bharata
explains the functions of the *jatis:*

* * ^{↓} * *

*Nat. 28.150–151*:
These are the *Jatis* with their ten characteristics [*laksana*].
These should be applied in the song (*pada*) with dance movements (*Karanas*)
and gestures suitable to them (lit. their own). I shall now speak of their
distinction in relation to the Sentiments (*rasa*) . . .[27]
(Text in brackets mine.)

* * ^{↑} * *

Dattila confirms the
monumental significance of the *jatis* in a single unequivocal statement:

* * ^{↓} * *

*Dat. 97*:
. . . anything which is sung is based on the *jatis*.[28]

* * ^{↑} * *

Bharata divides his *jatis* into two different technical categories: *
suddha* (pure) and *vikrta* (modified). The following description of the
*suddha* *jatis* is from a major treatise entitled *Sangitaratnakara*
by Sarngadeva (1210–1247). (See Sections 30–31.)

* * ^{↓} * *

*San.*
*Ch. I, Sec. 7, A. (i) (b)*: The definition of *
suddhata:* To define *suddhata*, it is stated that the *jatis*,
which have their ** denominative **note [descriptive note] as the [1]
final note (

* * ^{↑} * *

*Dattilam 62*
gives a similar description. The text (translation?) of Bharata is less precise
because it excludes the Semi-final note. Therefore, in the next passage we will
assume that a single tone — which acts as Initial, Prominent, Semi-final, and
Final note — is the denominative note of a given *suddha jati*.[30]
(See Figure 11.24.)

Quote I

* * ^{↓} * *

*Nat. 28.44*:
. . . In the *Sadja* *Grama* the pure (*Jatis*) are *Sadji*
[after *Sadja*], *Arsabhi* [after *Rsabha*], *Dhaivati*
[after *Dhaivata*] and *Naisadi* [after *Nisada*] and in the *
Madhyama* *Grama* they are *Gandhari* [after *Gandhara*], *
Madhyama* [after *Madhyama*] and *Pañcami* [after *Pañcama*].
‘Pure’ (*suddha*) in this connection means having *Svaramsa* ( = *
Amsa*), *Graha*, and *Nyasa* consisting of all the seven notes
(lit. not deficient in notes). When some of these *Jatis* lack two or more
of the prescribed characteristics except the *Nyasa*, they are called
‘modified’ (*vikrta*) . . .[31]
(Text in brackets mine.)

* * ^{↑} * *

Here, as in
Dattila’s and Sarngadeva’s texts, the terms *amsa*, *graha*, *nyasa*,
and *apanyasa* belong to the *laksana*, or to ten technical
characteristics, which, when uniquely applied to the *jatis*, give each
mode its distinctive *technical property*. Bharata describes the *laksana*
thus:

* * ^{↓} * *

*Nat. 28.74*:
Ten characteristics [*laksana*] of the *Jatis* are: *Graha*
[Initial note], *Amsa *[Prominent note], *Tara* [High register], *
Mandra* [Low register], *Nyasa* [Final note], *Apanyasa*
[Semi-final note] . . . *Alpatva* [Rare note] . . . *Bahutva* [Copious
note] . . . *Sadava* [Hexatonic mode], and . . . *Audava* [Pentatonic
mode].[32]
(Text in brackets mine.)

* * ^{↑} * *

The seven *suddha jatis* are by definition heptatonic modes. However, in *
Natyasastra 28.103–149*, Bharata describes eighteen more *jatis*. Seven
of these have the ** same names **as the

* * ^{↓} * *

*Nat. 28.56*:
Of these, four [are] heptatonic (*saptasvara*) . . . ten [are] pentatonic (*pañcasvara*)
. . . and . . . four [are] hexatonic (*satsvara*).[34]
(Text in brackets mine.)

* * ^{↑} * *

An analysis of the
text shows that all four heptatonic modes belong to the *sankara jatis*,
six of the ten pentatonic modes belong to the *vikrta jatis*, and therefore
only one of the four hextonic modes to the *vikrta jatis*.

We will not discuss the eleven *sankara jatis* in full detail,[35] but
we will analyze Bharata’s seven *suddha jatis* and seven *vikrta jatis*.
To do this, we must first familiarize ourselves with an extremely important
development in ancient Indian music: the utilization of auxiliary notes called
*svarasadharana*, translated into English as Overlapping notes. Bharata
defines *sadharana* and *svarasadharana* as

Quote II

* * ^{↓} * *

*Nat. 28.34*:
. . . The Overlapping (*sadharana*) means the quality of a note rising
between two [consecutive] notes [in a *Grama*].

*Nat. 35*:
. . . [1] The *Kakali* and [2] the transitional note (*antarasvara*)
are the Overlapping notes (*svarasadharana*). Now if *two* *
Srutis* are added to *Nisada*, it is called *Kakali*** ***
Nisada *and not *Sadja;* as it is a note rising between the two (pure *
Nisada *and *Sadja*), it becomes Overlapping. Similarly, [the *two*
*Srutis* being added to it] *Gandhara* becomes transitional *
Gandhara* { *Antara* *Gandhara* } and not *Madhyama*, because
it is a transitional note (*antarasvara*) between the two (*Madhyama*
and *Gandhara*). Thus the Overlapping notes [occur].[36]
(Numbers in brackets, bold italics, and text in braces mine.)

* * ^{↑} * *

And again, Dattila gives a terse description:

* * ^{↓} * *

*Dat. 16*:
*Nisada *is called *Kakali*, when [the note] is raised by two *
srutis*. Similarly, *Gandhara* is called *Antarasvara*.[37]

* * ^{↑} * *

For convenience, we will simply refer to *Antara* *Gandhara* as “An,”
and to *Kakali* *Nisada *as “Ka.” With respect to the *Sa-grama*,
calculate the *svara* ratio of An by first increasing the interval between
Ri and Ga from 2 *srutis* to 4 *srutis*, and then multiplying the *
svara* ratio of Ri times a 4-*sruti* interval, which gives: An = 10/9 ×
9/8 = 5/4. Similarly, calculate the *svara* ratio of Ka by first increasing
the interval between Dha and Ni from 2 *srutis* to 4 *srutis*, and
then multiplying the *svara* ratio of Dha times a 4-*sruti* interval,
which gives: Ka = 5/3 × 9/8 = 15/8. With respect to the *Ma-grama*,
calculate the *svara* ratio of Ka by first increasing the interval between
Dha and Ni from 2 *srutis* to 4 *srutis*, and then multiplying the *
svara* ratio of Dha times a 4-*sruti* interval, which gives: Ka = 5/4 ×
9/8 = 45/32. Similarly, calculate the *svara* ratio of An by first
increasing the interval between Ri and Ga from 2 *srutis* to 4 *srutis*,
and then multiplying the *svara* ratio of Ri times a 4-*sruti*
interval, which gives: An = 5/3 × 9/8 = 15/8. (See Figure 11.23.)

Bharata describes the general implementation of the *svarasadharana* in *
Natyasastra 28.36–37*. Because the meaning of this passage in the Ghosh
translation is not clear, consider now an alternate translation found in the
commentary of Mukund Lath’s *A Study of* *Dattilam*. (See Section
11.22.)
Lath translated this passage from a different source and identifies it as *
Natyasastra 28.35–36:*

* * ^{↓} * *

*Nat. 28.35–36*:
The *antara-svara* [An] should always be associated (with the *jati*)
when making an ascending movement; its use should be exceedingly spare and never
in making descending movements. If the *antara-svara* be used in descending
movements, whether sparingly or with profusion, it destroys the *sruti* and
the *jati-raga*.[38]
(Text in brackets mine.)

* * ^{↑} * *

Lath then adds the
following interpretation of this passage based on a famous commentary of the *
Natyasastra* by Abhinava Gupta, (fl. *c*.
a.d. 1000):

* * ^{↓} * *

Abhinava points out
that the word *antara-svara* in these verses denoted not only the *
auxiliary ga* [An] but also the *kakali ni* [Ka] and the maxim applies
equally to both the auxiliary notes: “*antarasvarasabdena catra kakalyapi
samgrhita iti krtopyayameva kramah*” (A.B. on N.S. 28, 36).[39]
(Text in brackets mine.)

* * ^{↑} * *

Bharata describes the specific implementation of the *svarasadharana* with
respect to the *jatis* in *Natyasastra 28.38*. Again, consider this
translation by Lath, which he cites as *Natyasastra 28.37:*

Quote III

* * ^{↓} * *

*Nat. 28.37*:
There are three *jatis* which are connected with the use of the *
sadharana svaras* (i.e., *antara ga* and *kakali ni*), namely, *
Madhyama*, *Pañcami*, and *Sadjamadhya*.[40]

* * ^{↑} * *

*Madhyama*
and *Pañcami* are classified as both *suddha* and *vikrta jatis*,
and *Sadjamadhya*, as a *sankara jati*.

Section 11.24

Before we begin a ratio analysis of the *suddha jatis* and the *vikrta
jatis*, consider first a tuning sequence designed to produce the *Sa-grama*
on an ancient harp-*vina* with nine open strings. In the illustration
below, a progression of white notes represents the required scale degrees, and
black notes indicate previously tuned degrees in the sequence. Notice that this
procedure only requires tuning “octaves,” “fifths,” “fourths,” and one “major
third.”

Refer now to Figure 11.23(a), Rows 1 and 2, which give the ratios and note
names, respectively, of the *Sa-grama* from the lowest tone Sa, ratio 1/1,
on String 1 to the highest tone Ri', ratio 20/9, on String 9. Now, if we were to
tune this *vina* to the *Ma-grama* as illustrated in Figure 11.22(b),
Row 5, we would (1) change the frequency of the lowest tone from C4 to F4, and
(2) retune all the remaining strings. However, given the structural instability
of the *vina*, such an increase in tension would pose significant
mechanical difficulties. It is far more likely that musicians in ancient India
rendered the *Ma-grama* as a *modulated *scale

Recall from the discussion in Section
11.22
that the *principal* difference between the *Sa-grama* and the *
Ma-grama* is that the Ma–Pa interval contains 4 *srutis* in the former
tuning, and 3 *srutis* in the latter tuning. An arrow that points from
Figure 11.23(a) to Figure 11.23(b) indicates this difference. To reduce this
interval, and thereby produce a modulated *Ma-grama* tuning based on the
original *Sa-grama* tuning, requires two simple steps. (1) In Figure
11.23(b), Row 1, identify the tone of String 5 of the *Ma-grama* tuning by
multiplying the tone of String 4 of the *Sa-grama* tuning times a 3-*sruti*
interval: String 5 of *Ma-grama* = 4/3 × 10/9 = 40/27. (2) Tune the *new
*Pa, ratio 40/27 on String 5 as a “fourth,” ratio 4/3, above String 2; that
is: 10/9 × 4/3 = 40/27 = 680.4 ¢. Figure 11.23(b), Row 3, shows that such a
simple retuning modulates the *old tonic* Sa, ratio 1/1, on String 1 to the
*new tonic* Ma, ratio 1/1, on String 4. However, this shift does*
not* render the

In Figure 11.23(a), I arbitrary chose C4 as the fundamental frequency of the *
Sa-grama* tuning. Furthermore, observe that the tones in Figure 11.23(b) and
11.23(c), Rows 1, represent ** modes** of the

In Quote I, Bharata explains that the *suddha* *jatis* called *
Madhyama* and *Pañcami* are derived from the *Ma-grama*, and in
Quote III, that these two *jatis* “. . . are connected with the use of the
*sadharana svaras* . . .” In other words, Bharata does* not*
associate the two Overlapping notes with

[1]Ghosh,
M., Translator (Vol. 1, Ch. 1–27, 1950; Vol. 2, Ch. 28–36, 1961). *The
Natyasastra*, by Bharata, *Volume 2*, p. 5. Bibliotheca Indica, The
Asiatic Society, Calcutta, India.

[2]*Ibid*.,
pp. 5–7.

[3]*Ibid*.,
p. 7.

[4]Jairazbhoy,
N.A. (1975). An interpretation of the 22 *srutis*. *Asian Music*
**VI**, Nos. 1–2, pp. 38–59.

[5]*Ibid*.,
p. 41.

[6]*The
Natyasastra*, *Volume 2*, pp. 8–9.

[7]See Chapter 3, Section 12, and Chapter 9, Section 2.

[8]See Chapter 9, Sections 1 and 8.

[9]See Section 11.11.

[10]See Chapter 10, Sections 32–33

[11]See Chapter 10, Section 1.

[12]See Chapter 10, Section 22.

[13]*Asian
Music* **VI**, 1975, p. 41.

[14]Sadie,
S., Editor (1984). *The New Grove Dictionary of Musical Instruments,
Volume 3*, pp. 729–730. Macmillan Press Limited, London, England.

[15]Coomaraswamy,
A.K. (1930). The parts of a *vina*. *Journal of the American Oriental
Society ***50**, No. 3, pp. 244–253.

[16]*The
New Grove Dictionary of Musical Instruments, Volume 3*, pp. 304–305.

[17]*Journal
of the American Oriental Society ***50**, No. 3, p. 250.

[18]To avoid significant cumulative errors in tuning scales with many small intervals requires sophisticated instruments that are physically stable and acoustically accurate. In contrast, tuning scales with only a few large intervals on simple instruments may produce errors that are less objectionable.

[19]Bhandarkar,
R.S.P.R. (1912). Contribution to the study of ancient Hindu music. *The
Indian Antiquary* **XLI**, pp. 157–164, 185–195, 254–265.

In
this remarkably thorough study, Bhandarkar traces the inaccuracies found in
works on ancient Indian music by both Asian and European writers. He also
stresses correct interpretations of *sruti* distributions and *grama*
constructions.

[20]Nijenhuis,
E.W., Translator (1970). *Dattilam: A Compendium of Ancient Indian Music*.
E. J. Brill, Leiden, Netherlands.

This translation of the *Dattilam* spans pp. 17–61, whereas Nijenhuis’
commentary spans pp. 62–425. Throughout the commentary, Nijenhuis includes
many translated excerpts from the works of Bharata, Narada, Matanga, and
Sarngadeva. On several occasions, footnotes refer the reader to these latter
translations in Nijenhuis’ *Dattilam*.

[21]Lath,
M., Translator (1978). *A Study of Dattilam: A Treatise on the Sacred
Music of Ancient India*. Impex India, New Delhi, India.

[22]*Ibid*.,
p. x.

[23]*Dattilam*,
p. 19.

[24]*A
Study of Dattilam*, p. 218.

[25]See Chapter 10, Section 27.

[26]See Chapter 10, Section 22.

[27]*The
Natyasastra*, *Volume 2*, p. 28.

[28]*Dattilam*,
p. 33.

[29]Shringy,
R.K., and Sharma, P.L., Translators (Vol. 1, Ch. 1, 1978; Vol. 2, Ch. 2–4,
1989). *Sangitaratnakara*, by Sarngadeva, *Volume 1*, p. 267. *
Volume 1*, Motilal Banarsidass, Delhi, India; *Volume 2*, Munshiram
Manoharlal, New Delhi, India.

[30]Rowell,
L. (1981). Early Indian musical speculation and the theory of melody. *
Journal of Music Theory* **25.2**, pp. 217–244.

The following quotations appear on pp. 232–235 in Rowell’s excellent article:

“The most vital choices are those which direct the course of a melody into
one of the prescribed *jatis*, the ancestors of the modern concept of
*raga*. The word *jati* is one of those bland words so useful in
musical terminology; it is a past passive participle of the verbal root *
jan* [cognate with the Greek word *genesis*] meaning “to be born,
arise,” and thus its developed meaning: “kind, type, species.” A glance at
the standard ten characteristics of *jati* discloses most of the
familiar standards by which a mode is recognized in Medieval Western theory
— incipit, final, confinal, and ambitus (high and low): 1. *graha*,
initial; 2. *amsa*, prominent, usually called “sonant” by Indian
authors; 3. *tara*, high; 4. *mandra*, low; 5. *sadava*,
hexatonic; 6. *auduvita*, pentatonic; 7. *alpatva*, scarce, weak;
8. *bahutva*, copious; 9. *nyasa*, final; 10. *apanyasa*,
confinal, an internal cadence tone.

. . . *Amsa*, according to the *Natyasastra*, had its own list of
ten *laksanas*: it is the generating tone, it determines not only the
low tone but the interval between low and high tones, it is the tone most
frequently heard, it determines the initial, the final, the three types of
confinals, and is the tone which all the others follow. *Amsa*, one
gathers, is no trivial concept.

. . . *Vadi* or “sonance” is treated formally as a subtonic of *grama*
(scale), but the concept first becomes operational when applied to the
structure of an individual *jati*. Here are four possibilities: 1. *
vadi*, sonant, “ruling note” (*amsa*); 2. *samvadi*, consonant,
harmonic affinity; 3. *vivadi*, dissonant, distorted; 4. *anuvadi*,
neutral. The commentator Abhinava quotes an old analogy:
‘*Vadi*
is the king, *samvadi* is the minister who follows him, *vivadi*
is like the enemy and should be sparingly employed, and *anuvadi*
denotes the retinue of the followers.’”

[31]*The
Natyasastra*, *Volume 2*, pp. 15–16.

[32](**a**)
*The Natyasastra*, *Volume 2*, pp. 18–19.

(**b**)
*Dattilam*, p. 177.

[33]*Dattilam*,
p. 168.

[34]*The
Natyasastra*, *Volume 2*, p. 17.

[35]Because
the *sankara* *jatis* have no recognizable symmetries and no
internal patterns, they have little theoretical or practical value for
tracing the

evolution of Indian music.

[36]*The
Natyasastra*, *Volume 2*, p. 13.

[37]*Dattilam*,
p. 19.

[38]*A
Study of Dattilam*, p. 227.

[39]*Ibid*.

[40]*Ibid*.,
p. 228.

[41]See Chapter 3, Section 6.