सङ्गीतरत्नाकर: प्रथम भाग (शार्ङ्गदेव - स्वरगताध्याय सविस्तार व्याख्या व अनुवाद सहित)

190 Treatment of Svara
fold⁸. The four⁴ beginning with ṣaḍja⁵ and madhyama⁶ are
twofold each, and the other ten are fourfold; thus, these
original series are forty-eight⁷ in all. (40-41)
The seven hundred and twenty⁸, already enumerated,
when multiplied by the original series make thirty-four thou-
sand five hundred and sixty, which forms the total number
of hexatonics⁹. (42-43a)
(cf. note 3 on 35b-36d), when one or more notes are dropped out of
these series in their original order for making permutational note-series,
these mūrcchanā-series in that modified form are called krama-s or
original series. The author, now onwards, is taking into account the
number of these original series in the hexatonic to monotonic series.
Now, if hexatonic note-series are to be formed out of the mūrcchanā-
series, the last note may be dropped in the process, in each case. Thus
we shall, in effect, have two types of note-series viz. inclusive of either
kākalī or antara or exclusive of either of them.
Fourteen mūrcchanā-s have been spoken of (vide verse 15 ante) in
both of the grāma-s. Out of these fourteen, two in each of the grāma-s
(i.e. four) begin with sa and ma and end in ni and ga respectively. In
their case only two alternate types are possible viz., śuddha (standard)
and antara-inclusive, and śuddha (standard) and kākalī-inclusive. The
rest of the ten mūrcchanā-s will have the usual four varieties. Thus we
have four hexatonic note-series, which are two-fold, making eight varie-
ties in all, and ten hexatonic note-series that are four-fold making forty
varieties in all. The total number of the original series, in this way,
comes to forty-eight. (40-41)
Seven hundred twenty permutational hexatonic note-series have
been enumerated (cf. verse 37 ante) for each mūrcchanā. In all forty-
eight mūrcchanā-series have been counted. Thus the total number of
the permutational hexatonic note-series is worked out (720 × 48 = 34560)
to be thirty-four thousand five hundred and sixty.
3. Viz in case ni is dropped, standard and antara-inclusive; and
in case ga is dropped, standard and kākalī-inclusive.
4. The four mūrcchanā-series.
5. The first mūrcchanā (uttaramandrā) of the ṣaḍja-grāma and the
fourth mūrcchanā (śuddhamadhyā) of the madhyama-grāma.
6. The fifth mūrcchanā (matsarīkṛtā) of the ṣaḍja-grāma and the
first (sauvīrī) of the madhyama-grāma.
7. (4 × 2 = 8) + (10 × 4 = 40) = 8 + 40 = 48.
8. 720 permutational hexatonic note-series, formed out of a given
combination.
9. That is, permutational hexatonic note-series, formed out of
48 hexatonic combinations or original series. This number
also includes the original-series (mūlakrama-s).
Section 4 : Grāma, Mūrcchanā, Kūṭatāna ...
7. औडुवकूटतान-सङ्ख्या
, अथ पञ्चस्वरान्ब्रुवे ।
गाद्यौ धाद्यौ निषादाद्यौ चतुर्भेदाः षडौडुवाः ॥४३॥
अष्टावन्ये द्विधेत्येवं चत्वारिंशदिमे क्रमाः ।
सविंशतौ शते तैश्च गुणिते ऽष्टौ शतानि तु ॥४४॥
चत्वारि च सहस्राणि सङ्ख्या पञ्चस्वरेष्विति ।
(7) The enumeration of the pentatonics : 43b-45b
Now, I am to describe the pentatonics.¹ The six
pentatonics² commencing with ga dha and ni are fourfold,
and the eight others are twofold; thus, these (original) series
are forty in all, which when multiplied by one hundred and
twenty³ bring the total number of the pentatonics⁴ to four
thousand and eight hundred. (43b-45b)
¹The pentatonic permutational note-series are formed by dropping
the last two notes of the mūrcchanā-series. In the two grāma-s we have
six series i. e., three in each beginning with ga, dha and ni in which
naturally neither ga nor ni is dropped. So these six series are fourfold;
but in the case of remaining eight series either ga or ni is dropped, and
so we can have only two types. That is how the total number of pen-
tatonic series is found to be forty, i. e. (6 × 4 = 24) + (8 × 2 = 16) = 24 +
16 = 40.

1. The pentatonic permutational note-series.
2. These six are identified to be from the mūrcchanā-s of the two
   grāma-s as under.
   | Commencing with the note | The mūrcchanā-s in the grāmas: Ṣaḍja-grāma (No.) | The mūrcchanā-s in the grāmas: Ṣaḍja-grāma (Name) | The mūrcchanā-s in the grāmas: Madhyama-grāma (No.) | The mūrcchanā-s in the grāmas: Madhyama-grāma (Name) |
   |---|---|---|---|---|
   | ga | 6th | Aśvakrāntā | 2nd | Hariṇāśvā |
   | dha | 3rd | Uttarāyatā | 6th | Pauravī |
   | ni | 2nd | Rajanī | 5th | Mārgī |
   (Chart 18)
   (Showing pentatonic series inclusive of ga and ni)
3. 120 is the number of permutational pentatonic note-series (kūṭa-
   tāna-s) formed out of a given combination or original-series.
4. That is permutational pentatonic, note-series formed out of 40
   pentatonic combinations or original series. The number also
   includes the original series (mūla-krama-s).

192 Treatment of Svara
Again each of the mūrcchanā-series has been spoken of as having one
hundred and twenty permutational pentatonic note-series. We have
forty mūrcchanā-series for pentatonics, and multiplied by one hundred
and twenty, the total number of the permutational pentatonic note-
series comes to four thousand eight hundred including the original
mūrcchanā-series.
8. चतुःस्वरकूटतान-सङ्ख्या
चतुःस्वरेषु न्याद्यौ द्वौ चतुर्धा द्वादशापरे ॥४५॥
क्रमा द्विधेति द्वात्रिंशच्चतुर्विंशतिताडिता ।
शतानि सप्ताष्टषष्ट्या स्याच्चतुःस्वरसंमितिः ॥४६॥
(8) The enumeration of the tetratonics : 45c-46
Among the tetratonics¹, the two commencing with ni
are fourfold, while the other twelve (original) series are two-
fold; these thirty-two² when multiplied by twenty-four³ make
a sum-total of seven hundred and sixty-eight.⁴ (45c-46)
Of the fourteen mūrcchanā-s of the two grāma-s when the tetratonics
or the series of four notes are derived, two of them that begin with ni
(i. e. from the second of the ṣaḍja-grāma, rajanī) and the fifth of the
madhyama-grāma mārgi), include ga as well, and therefore have four
varieties. The other twelve drop either ni or ga and hence have only
two types. So among the tetratonics we have (2 × 4 = 8) + (12 × 2 = 24) =
8 + 24 = 32 (thirty-two) original series.
Twenty-four permutational note-series of the tetratonics have been
enumerated for each of the original series (vide verse 38 ante). So,
thirty-two multiplied by twenty-four yields seven hundred and sixty-
eight tetratonic series.

1. 'Tetratonic' is an abbreviated expression for the permutatio-
   nal note-series of four notes.
2. That is, eight obtained from the two mūrcchanā-s beginning
   with ni and twenty-four from the remaining twelve mūrcchanās,
   having two varieties each.
3. The number of permutational note-series of four tones in each
   combination or original series.
4. That is, permutational tetratonic note-series, formed out of
   32 tetratonic combinations or original series. This number
   also includes the original series (mūla-krama-s).
   SECTION 1 : SVARA
5. त्रिस्वरकूटतान-सङ्ख्या
   त्रिस्वरेषु तु माद्यौ द्वावभेदौ द्वादशापरे ।
   द्विधा षड्विंशतिरिति क्रमास्ते षड्भिराहताः ॥४७॥
   षट्पञ्चाशच्छतं च स्युः,
   (9) The enumeration of the tritones : 47-48a
   But among the tritones¹, the two commencing with ma
   have no variations, while the remaining twelve are twofold :
   thus, these twenty-six (original) series when multiplied by six
   make one hundred and fifty-six² in all. (47-48a)
   When the tritone permutational note-series are formed from the
   fourteen mūrcchanā-s, the last four notes are dropped in each case. In
   this process the two mūrcchanā-s of the two grāma-s beginning with ma,
   i. e., the fifth (matsarīkṛtā) of the ṣaḍja-grāma and the first (sauvīrī) of the
   madhyama-grāma, when deprived of their last four notes, drop out both
   ni and ga, and thus can have no varieties. The other twelve series have
   two varieties each, for they would retain either ni or ga. That is how
   the original series for three tones are counted as twenty-six : 2 + (12 × 2 =

24. = 26.
    Again, each of the original series has six tritone permutational
    note-series (cf. verse 38 ante). Thus, the total number of the tritone
    permutational note-series comes to one hundred and fifty-six (i. e.
    26 × 6 = 156).

10. द्विस्वरैकस्वर-कूटतान-सङ्ख्या
    , द्विस्वरेषु पुनर्द्विधा ।
    रिगधन्यादयो ऽष्टौ स्युः शुद्धास्तदितरे क्रमाः ॥४८॥
    (10) The enumeration of the bitones and the monotones :
    48b-49
    Again among the bitones¹, the (original) series com-
    mencing with ri, ga, dha and ni being twofold are eight, while

1. That is, the permutational note-series of three notes.
2. That is, permutational tritonic note-series, formed out of 26
   tritonic combinations or original series. This number also
   includes the original series (mūla-krama-s).
3. 'Bi-tone' represents the permutational note-series of two notes.
   13

194 Treatment of Svara
द्वाविंशतिस्ते तु चतुश्चत्वारिंशद् द्विताडिताः ।
एकस्वरास्त्वभेदत्वान्मौला एव चतुर्दश ॥४९॥
the others² are śuddha (standard). These twenty-two multi-
plied by two become forty-four³ while the monotones⁴
having no variations are fourteen⁵ krama-s in identity with
the original-series. (48b-49)
The bitone permutational note-series are formed by dropping the
last five notes from the mūrcchanā-series. In this process the bitone series
beginning with ri, ga, dha and ni include either ga or ni and are therefore
twofold, yielding sixteen varieties in the two grāma-s. The remaining
six series, being devoid of both ga and ni, have no variations. Thus the
bitones are twenty-two series. But each mūrcchanā-series has two bitones;
so the total number of bitone permutational note-series is forty-four
(i. e. 16+6=22×2=44).
With the so-called monotone series, there is obviously no question
of variation, so in identity with the fourteen mūrcchanā-s from which
they are derived, they are fourteen in all.
To recapitulate, the total number of the permutational note-series
enumerated hitherto is computed as under :—
(i) The complete series of seven tones each 2,82,240 vide
verse No. 34 ante
(ii) The hexatonic series of six tones each 34,560 ,, ,, 42 ,,
(iii) The pentatonic series of five tones each 4,800 ,, ,, 45 ,,
(iv) The tetratonic series 768 ,, ,, 46 ,,
(v) The tritonic series 156 ,, ,, 48 ,,
(vi) The bitonic series 44 ,, ,, 49 ,,
(vii) The monotones 14 ,, ,, 49 ,,
Total number including original series 3,22,582
2. The remaining six original series i. e., krama-s.
3. This number includes the original series as well.
4. Permutational note series of one note each.
5. That is because the series of mūrcchanā-s are fourteen.

Section 4 : Grāma, Mūrcchanā, Krama and Tāna 195
(iii) पुनरुक्ताः कूटतानाः, तेषामपनये कूटतान-सङ्ख्या
षड्जादेः शुद्धमध्याया भेदकं पञ्चमं विना ।
चतुःस्वरे क्रमद्वन्द्वे ऽष्टाचत्वारिंशदीरिताः ॥५०॥
तानास्त्रिस्वरयोस्त्वेते द्वादश द्विस्वरे द्वयम् ।
एक एकस्वरस्ते त्रिषष्टिरौत्तरमन्द्रकैः ॥५१॥
पुनरुक्ता मतास्तानेन्यादिमार्गीक्रमाः पुनः ।
पञ्चस्वरा ये चत्वारस्तत्तानानां चतुःशती ॥५२॥
अशीत्यभ्यधिका चातुःस्वरी षण्णवतिर्भवेत् ।
द्वादश त्रिस्वरद्वन्द्वे चत्वारो द्विस्वरद्वये ॥५३॥
एक एकस्वरस्तानस्तेषां पञ्चशती त्वियम् ।
त्रिनवत्या युता तानैर्भिन्ना रजनीगतैः ॥५४॥
धैवतादेस्तु पौरव्याश्चत्वारः षट्स्वराः क्रमात् ।
तत्तानानां तु साशीतिः शताष्टाविंशतिर्मता ॥५५॥
औडुवानां चतुर्णां प्रागुक्ता सङ्ख्या चतुःस्वरौ ।
त्रिस्वरौ द्विस्वरावेकस्वरः प्रागुक्तसङ्ख्यकाः ॥५६॥
पञ्चविंशतिसंयुक्ता चतुस्त्रिंशच्छती त्वियम् ।
तानानां सदृशाकारा स्यात्तानैरौत्तरायतैः ॥५७॥
इत्येकाशीतिसंयुक्तं सहस्राणां चतुष्टयम् ।
तानानां पुनरुक्तानां पूर्णपूर्णैः सह क्रमैः ॥५८॥
अपनीयेत चेदेषा कूटतानमितिर्भवेत् ।
लक्षत्रयं सप्तदश सहस्राणि शतानि . च ॥५९॥
नवत्रिंशद्युतानीति,

196 Treatment of Svara
(iii) The repetitions and the total number of permutational
note-series : 50-60
In the absence of the grāma-indicative¹ pañcama of the
śuddha-madhyā² commencing with ṣaḍja, forty-eight note-series
of tetratones, twelve of tritones, two of bitones and one of the
monotone in the two original series,³ making sixty-three in
all are considered to be the repetitions of those pertaining to
uttaramandrā⁴. (50-52a)
In mārgī⁵, which commences with niṣāda, the four⁶
(original-series) provide four hundred and eighty pentatonic
The 3,22,582 permutational series of notes just enumerated not
only include the number of original-series (krama-s) but also many
repetitions in-between the note-series of the two grāma-s. 'K' very ably
introduces this matter as under:
"Even though ni and ga in their kakalī and antara forms determine
and bring about the mūrcchanā-variation, yet while this difference in
relation to other notes is quite notable, it is hardly perceptible within
these notes i. e., with reference to niṣāda and gāndhāra; and that is why
these two (i. e. kākalī and antara) are not considered to be as good as the
standard notes. Again, if as a rule, the different mūrcchanā-s such as
rajanī etc. commencing with ni etc. in the ṣaḍja-grāma come into being by
occupying the position of middle ṣaḍja and those in the madhyama-grāma
such as hariṇāśva etc., commencing with ga etc., occur by occupying the
middle madhyama of the madhyama-grāma, then even where the grāma-
indicator viz. the pañcama is omitted, because of the difference in the

1. Pañcama is considered to be indicative of the grāma in the
   sense that if it is a four-śruti pañcama it indicates the ṣaḍja-
   grāma and if it is a three-śruti pañcama it indicates the
   madhyama-grāma. Literally, the 'bhedaka' of the text means
   'discriminator' implying that it functions to distinguish
   between the two grāma-s.
2. Śuddhamadhyā is the fourth mūrcchanā of the madhyama-grāma
   and runs as 'sa ri ga ma pa dha ni'.
3. The series pertaining respectively to the two types standard
   and antara-inclusive.
4. The first mūrcchanā of the saḍja-grāma which runs as : sa ri ga
   ma pa dha ni.
5. The fifth mūrcchanā of the madhyama-grāma commencing with ni
   and ending with dha.
6. These are—standard, kākalī-inclusive, antra-inclusive and
   inclusive of kākalī-antara.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna 197
   note-series and ninety-six tetratonic series; the two⁷ (original
   series) produce twelve tritone and four bitone note-series,
   which along with the one monotone altogether make five
   hundred and ninety-three note-series that are identical with
   those of rajanī.⁸ (52-54)
   Pauravī⁹, commencing with dhaivata, with its four¹⁰
   (original series) is found to have two thousand eight hundred
   gramic position of the commencing note of the mūrcchanā, there is no
   question of any repetition being made in the permutational note-series
   proceeding from them, since their parent grāma can easily be traced.
   But however, if as a rule, in the two grāma-s the different mūrcchanā-s
   are formed by placing the commencing notes one below the other then
   the note-series of the śuddha-madhyā, mārgī and pauravī mūrcchanā-s of
   the madhyama-grāma have a necessary correspondence with the note-series
   such as hexatonics etc of the uttaramandrā, rajanī, uttarāyatā etc., the
   mūrcchanā-s of the ṣaḍja-grāma; and thus, due to the lack of difference
   in the gramic position of the commencing notes and also because of the
   omission of pañcama there is a definite possibility of their having been
   repeated. Therefore, the author is now to point out such repetitions in
   the note-series of the madhyama-grāma, in order to exclude them from the
   total number."
   Now, this introduction to the topic of repetitions, significant as it
   is, obviously requires some elucidation. Two ways of forming the
   mūrcchanā-s have been mentioned (verse 12-15 ante) viz. (i) by com-
   mencing the succeeding mūrcchanā-s with their opening notes placed in
   the position of ṣaḍja in the ṣaḍja-grāma and madhyama in the madhyama-
   grāma and (ii) by placing their commencing notes one below the other.
   The commentator points out that in the first case there is no possibility
   of repetitions, for then the grāma-distinction is preserved, but in the
   second case the possibility does exist. After all, the only markable
   difference between the two grāma-s is the number of śruti-s of pañcama,
   the other differences being consequent upon it. The process of forming
   note-series is such that successively the last notes are dropped. In this
   process pa, which is indicative of the grāma, is invariably retained in the
7. These are—standard and kākalī-inclusive.
8. The second mūrcchanā of the ṣaḍja-grāma corresponding to the
   mārgī of the madhyama-grāma.
9. Pauravī is constituted as : dha ni sa ri ga ma pa .
10. The four original series are of the four types, viz. standard,
    kākalī-inclusive, antara-inclusive and inclusive of both kakalī
    and antara.

Left Page (Page 202)

202 Treatment of Svara
The number of repetitions

• 22 + 11 = 179\end{aligned}\right}$ | Hexatonic, pentatonic, tetratonic, tritonic, bitonic and monotone series respectively. |
  | 3. | Repeated note-series | 4081 | $\left.\begin{aligned}63 + 593 + 3425
  = 4081\end{aligned}\right}$ | śuddhamadhyā, mārgi pauravi respectively |
  Total number of repetitions : 392 + 179 + 4081 = 4652
  (Chart 19b)
  The correct number of permutational note-series
  | | | |
  | :--- | :---: | ---: |
  | Total number of note-series | ... .. | 3,22,582 |
  | Total number of repetitions | (-) | 4,652 |
  | Correct number of note-series | ... ... | 3,17,930 |
  (Chart 19c)
  (iv) प्रतिमूर्च्छनं कूटतान-सङ्ख्याज्ञानोपायः
  , ज्ञानोपायो ऽत्र कथ्यते ।
  अङ्कानेकादिसप्तान्तानूर्ध्वमूर्ध्वं लिखेत्क्रमात् ॥६०॥
  हते पूर्वेण पूर्वेण तेषु चाङ्के परे परे ।
  एकस्वरादिसङ्ख्या स्यात्क्रमेण प्रतिमूर्च्छनम् ॥६१॥
  (iv) Determining the number of permutational forms in the note-series of each mūrcchanā : 60b-61
  Now, the method of determining them¹ is herewith described : Let the numbers one to seven be written (verti-
  Since the number of permutations making for the note-series enumerated above is enormous, a mathematical device has been devised to determine the number of a particular permutation of a given form

1. This refers to the permutational note-series i. e., the kūṭatāna-s.

Right Page

Section 4 : Grāma, Mūrcchanā, ... ...
cally) one over² another in consecutive order. Every succeeding³ number multiplied by the preceding⁴ one respectively points out the number of note-series such as monotone etc., in each mūrcchanā. (60b-61)
in a particular series and also the particular form of a certain permutation in a particular series, given its proper number. This device is called khaṇḍa-meru, which will be explained subsequently.
Presently the author is trying to explain the method of working out the precise number of the note series or the permutational form of a particular series of notes such as monotone, bitone, tritone series and so on in each mūrcchanā.
The text directs us to write numbers commencing from one upto seven, one over the other i. e., vertically. These numbers incidentally would represent the particular note-series such as monotone, bitone, etc. Keeping in view the directions of the text the numbers would be written from the bottom to the top. 'S' too interprets the text as such.
Number of permutational forms in different note-series in each mūrcchanā

2. This expression does not suggest that the numbers are not to be written serially, for that direction has explicitly been given immediately in the text, it only implies that they are not to be written horizontally as usual.
3. The successive note.
4. The preceding resultant i. e., the number of the permutations of the previous series.

204 Treatment of Svara
Number of permutational forms in different note-series in each mūrcchanā

1. 'sa ri ga ma pa dha ni' is the original order of the seven notes.
   The word krama which has been rendered as 'the mūrcchanā
   series' or sometimes as the original series has a different shade
   of meaning here, though essentially the connotation of the
   word remains the same. If it is a tetratonic series the order
   would be 'sa ri ga ma' for it, and so on. Thus, in this context,
   krama implies the original combinational order of notes of a
   given note-series. But for the sake of convenience we are to
   use the expression "original order" corresponding to the Skt.
   mūlakrama.
2. The repetition of the term 'pūrva' in the text has been effected
   through this expression, since that is the import of the text.
   Section 4 : Grāma, Mūrchhanā, Tāna and ...
   that³ becomes the succeeding note, the preceding one may
   be advanced keeping the subsequent⁴ notes afterwards; and
   describing the method of placing the notes of various types of note-
   series such as complete, hexatonic etc., in a particular and progressive
   sequence of notes, giving all the possible permutations of the series.
   The essential formula consists in the rule that, while forming the 'permu-
   tational calculus' (prastāra) the notes of the series may, in the first
   instance, be placed in their original order, which may be considered to
   be the first note-series. In order, further to derive the other possible
   permutations the preceding note in each of the series has been placed
   below the succeeding note.
   The question is, 'what to do about the remaining notes'?, and the
   author himself explains that the notes following the succeeding note will
   have to be placed after the note transposed, and the notes preceding it
   will be placed before it in their original order. This much is the action
   that is to be taken.
   The next series, and all the other possible series will be formed
   by applying the above rule to their preceding series. The application
   of the rule is demonstrated by the commentators as under with respect
   to the permutational calculus of the tetratonic series.
   (i) The first series is of the original order viz. sa ri ga ma.
   (ii) To form the series the preceding note viz. sa is to be written
   below the succeeding note, viz. ri. The subsequent notes ga and ma
   will be written after (or in front of) ri as in the overhead line (i. e., the
   preceding series), and ri will naturally be placed before i. e., behind it;
   thus what we get is ri sa ga ma.
   (iii) To form the third series, the second series may be taken as
   the top. Now as per the rule, ri may be placed below sa which will
   result in the series sa ri ga ma, a repetition of the original order. The
   purpose of the permutational calculus being the forming of different
   The implication derived by 'K' is that this action can be taken
   only once in each of the note-series of a particular sequential
   order.
3. The pronoun 'that', according to 'S', refers to the preceding
   note i. e., in the first step. This is the second step being
   explained and accordingly the preceding note of the first step
   becomes the succeeding note of the second step.
4. The subsequent notes are the ones that follow the action taking
   place among the preceding notes with reference to the previous
   note-series.

206                      Treatment of Svara                              Section 4 : Grama, Murchhana, ...
the rest (of the notes in the series) be left behind$^5$ according                      Permutational calculus of the tetratonic series
to the original order; such is the 'permutational calculus'$^6$
(prastāra). (62-63b)

5. 'Behind' the placement of the preceding note i. e., to the left of it on the paper.
6. Prastāra is defined as the extension of the series of note-series in a particular permutational sequence, which has been explained above.
   (Chart 21)
   The process of formation is briefly presented in the following chart
   as per the rule stated in the text, and demonstrated up to seventh note-
   series as explained above.
   The process of formation of the permutational calculus of the
   tetratonic series
   | Sl. No. of the note-series | Notes transposed | Series obtained | Remarks |
   | :--- | :--- | :--- | :--- |
   | 1st series | ............ | sa ri ga ma | This being the 1st series no series precedes it and there is no transposition of notes. |
   | 2nd series | sa for ri | ri sa ga ma | |
   | 3rd series | ri for ga | sa ga ri ma | |
   | 4th series | sa for ga | ga sa ri ma | |
   | 5th series | ga for ri | ri ga sa ma | |
   | 6th series | ri for ga | ga ri sa ma | |
   | 7th series | ga for ma | sa ri ma ga | |
   (Chart 22 a)
   If the notes are 1, 2, 3, 4 we get the following picture of the first six series :
   | No. of series | sa ri ga ma | Remarks |
   | :--- | :---: | :--- |
   | 1st series | 1—2—3—4 | Original order |
   | 2nd series | 2—1—3—4 | 1 transposed for 2 |
   | 3rd series | 1—3—2—4 | 2 transposed for 3 |
   | 4th series | 3—1—2—4 | 1 transposed for 3 |
   | 5th series | 2—3—1—4 | 3 transposed for 2 |
   | 6th series | 3—2—1—4 | 2 transposed for 3 |
   (Chart 22 b)

208 Treatment of Svarā Section 2 . Grāma, Mūrcchanā, ...
Thus, the method of forming the permutational calculus of note- (vi) Khaṇḍameru (permutation-indicator) : 63c-71
series has been explained in detail applying the rule to the tetratonic (1) The construction of the permutation-indicator: (63c-66b)
series for illustration. Similarly pentatonic, hexatonic and heptatonic Let there be seven series of squares¹ commencing with
series can also be worked out. Since this is a tedious process, a mathe- seven squares and ending with one², arranged one below
matical graph has been prepared to facilitate the recognition of note- The text of the verses 63c to 66b describes the method of drawing
series with reference to their serial number in the sequential order and the mathematical graph which is devised to indicate naṣṭa and uddiṣṭa.
their particular order of notes. Naṣṭa is a technical name of the musical problem in which what is given
is the number of the note-series in question, and what is to be found out
is its tonal form giving the order of notes in it; while uddiṣṭa denotes the
(vi) खण्डमेरुः musical problem in which the tonal form of the note-series is given and
its serial number is to be found out. This graph, called khaṇḍameru (per-

1. खण्डमेरु-निर्माणम् mutation-indicator), helps to solve these two musical problems jointly
   known as 'naṣṭa-uddiṣṭa'.
   सप्ताद्येकान्तकोष्ठानामधो ऽधः सप्त पङ्क्तयः ॥६३॥ The permutation-indicator is drawn by placing seven series of squares
   तास्वाद्यायामाद्यकोष्ठे लिखेदेकं परेषु खम् । having seven, six, five, four, three, two and one squares each in due order
   वेद्यतानस्वरमितान्न्यस्येत्तेष्वेव लोष्टकान् ॥६४॥ one below the other. The graph is thus formed into the following pattern.
   प्राक्पङ्क्त्यन्त्याङ्कसंयोगमूर्ध्वाधःस्थितपङ्क्तिषु । Permutation-indicator
   शून्यादधो लिखेदेकं तं चाधो ऽधः स्वकोष्ठकान् ॥६५॥
   कोष्ठसङ्ख्यागुणं न्यस्येत्खण्डमेरुरयं मतः ।
   [खण्डमेरु-कोष्ठकम् / Devanagari Permutation Table]
   | स | रि | ग | म | प | ध | नि |
   | :-: | :-: | :-: | :-: | :-: | :-: | :-: |
   | १ | ० | ० | ० | ० | ० | ० |
   | | १ | २ | ६ | २४ | १२० | ७२० |
   | | | ४ | १२ | ४८ | २४० | १४४० |
   | | | | १८ | ७२ | ३६० | २१६० |
   | | | | | ९६ | ४८० | २८८० |
   | | | | | | ६०० | ३६०० |
   | | | | | | | ४३२० |
   [English Permutation-indicator Table]
   | sa | ri | ga | ma | pa | dha | ni |
   | :-: | :-: | :-: | :-: | :-: | :-: | :-: |
   | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
   | | 1 | 2 | 6 | 24 | 120 | 720 |
   | | | 4 | 12 | 48 | 240 | 1440 |
   | | | | 18 | 72 | 360 | 2160 |
   | | | | | 96 | 480 | 2880 |
   | | | | | | 600 | 3600 |
   | | | | | | | 4320 |
   इति खण्डमेरुः Khaṇḍameru
   (Chart 23)

1. Koṣṭha literally means a chamber or an inner apartment, a store-
   room or space surrounded by walls. Here it is used in the sense
   of an enclosed space which in this case is in the form of a square.
2. Only the commencing and the ending number of the squares is
   given. What is implied is that if the first series has seven squares,
   the last will have one i. e., the second will have six, the third will
   have five, the fourth will have four, the fifth will have three, the
   14

210 Treatment of Svara
the other.⁸ Let no. 1 be written in the first⁴ square of the
first⁵ series among them and let there be zero in the other
squares. Pebbles⁶, corresponding to the number of tones in
the note-series⁷ in question, may be placed in those very
squares.⁸ (63c-64)
The figures filled in the squares of the graph have been worked
out and are based on the permutational calculus of the permutational
note-series as demonstrated earlier; for example, it was observed in the
case of the tetratonic series that the twenty-four note-series were clearly
divided into four groups distinguishable from each other on the basis of
the notes sa ri ga ma being placed into the fourth position respectively in
groups of six note-series. Similarly, the monotone series forms a group
by itself, the bitone series forms two groups, so to say i. e., sa-ri and ri-sa.
The tritone series forms three groups viz.,
sa ri ga sa ga ri ri ga sa
ri sa ga ga sa ri ga ri sa
These note-series are formed into three groups where each note
remains in the third position twice. Rest of the notes, it may inciden-
tally be observed, are placed first in their ascending order and then
in their descending order. Similarly, the pentatonic series has five
groups in each of which one particular note viz., pa in the 'sa ri ga ma
pa' series remains in the fifth position, in the fifth group. This is done
sixth will have two, and the seventh only one, as counted from
top to bottom.
3. This order will form into vertical columns of squares disting-
uished from top to bottom and not from bottom to top.
4. The first square, as counted from left to right in the series
of squares drawn on the paper.
5. The first series is the one that has seven squares and is on the
top of the graph.
6. Pebbles are movable objects and are therefore instrumental in
the working process of the graph called khaṇḍameru. The graph
can also be utilised even without using the pebbles, by mark-
ing the squares appropriately. But the system of pebbles
adopted here has the advantage of leaving the graph unaffect-
ed and thus rendering it usable for any number of times.
However, if the squares of the graph are marked instead of
using the pebbles, it could be used only once. Thus, the
movable pebbles make the graph durable.
7. Vedya-tāna is the note-series the tonal form of which has to be
ascertained. However, the type of the series, such as mono-
tone, bitone, tritone, etc. is given, and therefore the pebbles
are to be placed in the squares in accordance with the number
of tones constituting the particular note-series.
8. That is, in the squares of the first series of seven squares,
commencing from the first.

Section 4 : Grāma, Mūrcchanā, Krama and Tāna 211
Let the sum total of the numbers of the last squares⁹
of the antecedent¹⁰ series be written below the zero in the
subsequent series progressively in the respective squares¹¹,
e. g., one (in the first instance); while in the squares below
that¹² in vertical order, let the sum be multiplied by the
from right to left to keep the notes in the ascending order in the first
instance. So also the hexatonics have six groups and the heptatonic or
complete note-series have seven groups.
Now, the seven squares of the first horizontal series of the permu-
tation-indicator represent the seven notes written from left to right,
while the squares taken in their vertical order represent the number of
groups of note-series in their permutational calculus; e. g., the first
column has one square, for the monotone series has only one group; and
similarly, the second column has two squares, for the bitone series has two
groups of note-series. Again the third column has three squares, and so
on with the seventh having seven squares representing its seven groups.
This group indication is useful for solving the problems called naṣṭa and
uddiṣṭa as will be explained presently.
Now, it will be easier to see how the different figures have been
filled into the different squares of the indicator graph and what their
significance is. The first column square represents the monotone series
and has figure one written in it. In other words, in the first squares
of the remaining six vertical columns (from top to bottom) is placed zero
since the number of groups does not become manifest at the very outset
as in the monotone series; e. g., in the bitone series the two groups sa ri
and ri sa become groups only when they are two i. e, they are relative.
The rest of the figures of the graph are arrived at in the following
procedure.
The rule given in the text is that in the series of squares following
or placed one below the other, one may write the total of the figure
indicated in the last squares of the preceding series below the zero given
in the overhead squares. That is to say that, if only one series of
9. The last squares of the series as counted from right to left.
10. Antecedent one or more than one series i. e, all the preceding
series.
11. 'Ūrdhvādhaḥ-sthita-paṅktiṣu' is quite significant and points out the
squares placed one below the other in the series of vertical
columns, below the zero of the respective columns.
12. 'That' refers to the square in which the sum of the numbers of
the last squares of the series formed till then is given.

212 Treatment of Svara
number of the square concerned¹³ and the resultant
written below that.¹⁴ This is considered to be the "permu-
tation-indicator" (khaṇḍameru).¹⁵ (65-66b)
squares is preceding, then the total of the last square of that series be
written in the last square (from right to left) of the succeeding series,
i. e., the 2nd. If however, there are two preceding series, then the total
of the last squares of both the series may be given below the zero in the
third square of the second series of six notes.
However, if there are many preceding series, then the total of the
figures given in their last squares may be added in the squares and
given below the respective zeros in the first series. For example, in the
column-square below the first zero, (1) is written, for the total given in
the last square of its preceding series is one. Then, in the next square
of the series below 2nd zero, we have 2 as the total of (1+1=2) the
last squares of the first and the second series. Again in the next square
of the series below the third zero in the fourth column, we have 6 as the
total of the last numbers of the first, second and the third series
(1+1+4=6). Similarly, the fourth square in the fifth column shows
24 as the total of the preceding last squares (1+1+4+18=24); and so
on That is how the figures of the second series are computed.
For computing the figures of the subsequent series (i. e., from
3rd to the 7th) another method is suggested which cuts short the above
process, for it would be cumbersome if applied further; and it consists
in multiplying the sum thus obtained in the squares of the second series
by the number of the respective column-squares thereunder. For
example, to fill in the first square (from the left) of the 3rd (horizontal)
series, we have figure 2 in the 2nd series above, and this being the 2nd
square from it, multiplying 2×2 we get 4. 2 represents the sum total
of the last squares of the preceding series' which is here multiplied by
the number of the square to be filled in. Similarly, to fill in the next
square in the same series (below ma), we have the figure 6 as the sum of
the last squares of the preceding series which is multiplied by 2 to
yield 12. Likewise, in the 4th vertical column (below ma) the last
13. The square concerned is the one in which the sum arrived at
is given and is to be counted vertically from top to bottom.
14. The square below the one in which the sum of the addition
referred to is given i. e., the last square in the vertical series.
15. Khaṇḍameru is a mathematical graph that indicates what is
known as naṣṭa and uddiṣṭa, the terms that are to be explained
presently. Both naṣṭa and uddiṣṭa are finally related to the
sequential extension of the note-series; thus this graph actually
works as the indicator of note-series.
Section 4 : Grāma, Mūrcchanā, Krama and Tāna 213
square of the 4th series bears the figure 18 in the graph. This is found
as per the same rule, i. e., 6 is obtained in the 2nd series of the same
column as the sum of the last squares of the preceding series which is
multiplied by 3, being the number of the square below the series (indi-
cative of the sum). Other figures can similarly be worked out. This
process can also be understood from the following two charts :
Computation of 2nd series of the indicator (The additive process)
No. of column | Total of the last squares of the preceding series | The figure obtained
square in question left to right | No. of series | Respective figures |
1 | 1 | 1 | 1
2 | 1,2 | 1+1 | 2
3 | 1,2,3 | 1+1+4 | 6
4 | 1,2,3,4 | 1+1+4+18 | 24
5 | 1,2,3,4,5 | 1+1+4+18+96 | 120
6 | 1,2,3,4,5,6 | 1+1+4+18+96+600 | 720
(Chart 24)
Computation of 3-7 series of the indicator
(The process of multiplication)
No. of series | Figure in the 2nd column square | Sl. No. of the square computed vertically from below the zero | Process of multiplication: Multiplication | Process of multiplication: Result | Process of multiplication: No. of the column squares filled
3rd | 2 | 2 | 2×2 | 4 | 3rd
4th | 6 | 2 | 6×2 | 12 | 3rd
| 6 | 3 | 6×3 | 18 | 4th
5th | 24 | 2 | 24×2 | 48 | 3rd
| 24 | 3 | 24×3 | 72 | 4th
| 24 | 4 | 24×4 | 96 | 5th
6th | 120 | 2 | 120×2 | 240 | 3rd
| 120 | 3 | 120×3 | 360 | 4th
| 120 | 4 | 120×4 | 480 | 5th
| 120 | 5 | 120×5 | 600 | 6th
7th | 720 | 2 | 720×2 | 1440 | 3rd
| 720 | 3 | 720×3 | 2160 | 4th
| 720 | 4 | 720×4 | 2880 | 5th
| 720 | 5 | 720×5 | 3600 | 6th
| 720 | 6 | 720×6 | 4320 | 7th
(Chart 25)

Page 214

214 Treatment of Svara
2. उद्दिष्ट-विधिः
स्वरान्मूलक्रमस्यान्त्यात्पूर्वं यावतिथः स्वरः ।।६६।।
उद्दिष्टान्त्यस्तावतिथे कोष्ठे ऽधो लोष्टकं क्षिपेत् ।
लोष्टचालनमन्त्यात्स्यात्त्यक्त्वा लब्धं क्रमो भवेत् ।।६७।।
लोष्टाक्रान्ताङ्कसंयोगादुद्दिष्टस्य मितिर्भवेत् ।
(2) The procedure for finding out the indicated note-series
(uddiṣṭa) : (66c-68b)
Let the pebble be placed in the column-square¹ below²
according to the number of the last note³ of the indicated
note-series⁴ obtaining backwards⁵ with reference to the last
note of its original order.⁶ The pebble movement starts
Having drafted the form of the permutation-indicator, the author
now proceeds to explain its working, laying down the procedure for
arriving at the solution of the two problems known as naṣṭa and
uddiṣṭa. He takes the latter first for the convenience of demonstration.
Uddiṣṭa literally means 'that which is indicated', and in our
context it implies the note-series indicated i. e., given; for example,
“ma ga sa ri” is a tetratonic note-series and its tonal form is given,
i. e., indicated ; but its serial number in the permutational calculus of
the tetratonic series is not given and is to be found out. That is how
the name uddiṣṭa i. e., 'indicated note-series' becomes significant. The
procedure for finding out the serial number of a given note-series is
explained hereunder with reference to the text.
Let the given note-series be “ma ga sa ri”; its original order would
be 'sa ri ga ma.” The last note of the indicated note-series in this
case is ri which is placed 3rd from the last note ma as counted back-
wards in its original order. Since this is a four-tone series, the permu-
tation-indicator would operate only for the first four columns (sa to ma).
¹1. The expression 'column square' is indicative of the vertical
order of the squares of the permutation-indicator.
2. Below the respective zero in the first series.
3. Last note as counted from left to right.
4. Uddiṣṭa is literally the note-series indicated, and what is sought
to be found out from the permutation indicator is its serial
number.
5. That is from right to left.
6. The order of the notes in the original series.

Page 215

Section 4 : Grāma, Mūrcchanā, Krama and Tāna 215
with the last note⁷ and is ordered⁸ by dropping the note
ascertained progressively. The number⁹ of the indicated
The movement of the pebbles commences with the last note i. e. , the last
square which is fourth. Ri is 3rd with reference to the last note of the
original order of the series. Therefore, the pebble of the fourth column
will now be placed in the third square below the zero which indicates
No. 12. To take the next step, the note ascertained i. e., ri is to be
dropped from the original order as well as from the indicated note-
series. So, the original order now obtained is “sa ga ma”; while the
given series is “ma ga sa.” Sa, now is second with reference to the
ma of the original order. So the second pebble, since this series is of
three notes, will now be placed in the second square of the third
column, which indicates 4. In the third step sa will drop away and the
original order obtained will be “ga ma;” while the given series would
be “ma ga.” Its ga is in the first order (i. e., next to) with reference to
the ma of the original order; therefore, the third pebble will be moved
to the first column below zero in the second column since this is a bitone
series now. This square indicates 1. In the fourth step, even ga will drop
away and what remains is the monotone-series of ma. Since in this case
the original order and the indicated note-series are identical, so there
will be no movement of the pebble and there is no square below the
first which indicates 1. Thus, the total of the squares with pebbles is
18 (12+4+1+1). So the serial number of the note-series “ma ga sa ri”
is 18th in the permutational extension of the four-tone series.
This procedure is summarised in the following chart :
Pebble-movement in the permutation-indicator for indicated tetratonic
permutational note-series.
The indicated note-series : “ma ga sa ri”

(Chart 24)
7. The last note of the note-series indicated i.e , from left to right.
8. That is, in the next step the original order is restored by drop-
ping the last note of the note-series ascertained.

216 Treatment of Svara
note-series would be obtained by adding up the figures (of
the squares) covered by the pebbles.¹⁰ (66c-68b)
The number of the indicated series in the permutational extension of
the tetratonic permutational note-series : 12+4+1+1=18.
Thus, the formula for finding out the number of the indicated
note-series can briefly be restated as under in terms of action :
( i ) Find out the relative position of the last note of the indicated
note-series with reference to the last note of its original
order by counting from it backwards i.e., from right to
left including the last note.
( ii ) Move the pebble into the corresponding square down below
according to the result obtained in (i) i.e., as many squares
down below as the number of the last-note obtained in (i).
( iii ) The pebble should be moved from the last note of the
series.
( iv ) Having thus ascertained the numerical value of one note,
that note be dropped progressively from the original order
as well as the indicated note-series and the procedure (i) to
(iii) repeated till the first note.
( v ) Add up the figures of the column squares into which the
pebbles have been moved and the sum total will represent
the serial number of the indicated note-series in its permuta-
tional extension.
This explains the procedure of working out the problem uddiṣṭa.
Now, an example from a complete permutational note-series is taken
for a full demonstration, which is worked out as under in a tabular
form :

1. Uddiṣṭa (the problem), “ma ga sa ri dha ni pa” (i.e., the indi-
   cated note-series).
2. Original order of the series : “sa ri ga ma pa dha ni”.
3. Pebble movement of a complete permutational note-series.
4. Serial No. of the indicated note-series in the permutational
   extension of the complete series : total of column 3(c) i.e.
   (1440+0+0+12+4+1+1)=1458th.
5. This is the serial number of the particular note-series in its
   permutational calculus which is to be ascertained.
6. The total of the numbers of the squares in which the pebbles
   are placed is the required number of the given note-series.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna 217
   | Sl. No. of the steps taken | The relative position of the last note of the indicated series with reference to the last note of the original order i.e., the original-series | The pebble movement and its significance | Results obtained |
   | 1 | 2(a) The last note of the given series | 2(b) The last note of the original order | 2(c) Relative position of 2(a) with reference to (right to left) 2(b) | 3(a) No. of column (left to right) | 3(b) No. of square to which pebble is moved | 3(c) The figure indicated by the occupied square | 4(a) The note omitted from the original order | 4(b) The remaining order of notes in the original series |
   | 1 | pa | ni | 3rd | 7th | 3rd | 1440 | pa | sa ri ga ma dha ni |
   | 2 | ni | ni | 1st | 6th | no movement | 0 | ni | sa ri ga ma dha |
   | 3 | dha | dha | 1st | 5th | —do— | 0 | dha | sa ri ga ma |
   | 4 | ri | ma | 3rd | 4th | 3rd | 12 | ri | sa ga ma |
   | 5 | sa | ma | 3rd | 3rd | 3rd | 4 | sa | ga ma |
   | 6 | ga | ma | 2nd | 2nd | 2nd | 1 | ga | ma |
   | 7 | ma | ma | 1st | 1st | No movement | 1 | ma | nil |
   (Chart 25)
7. नष्टविधिः
   यैरङ्कैनिष्टसंख्या स्यान्मौलेकाङ्कसमन्वितैः ॥६८॥
   तेषु लोष्टं क्षिपेन्मूले लोष्टस्थानमितं भवेत् ।
   नष्टतानस्वरस्थानं ततो यावतिथे पदे ॥६९॥
   अधःक्रमादस्ति लोष्टः स्वरस्तावतिथो भवेत् ।
   क्रमान्तिमस्वरात्पूर्वो लब्धत्यागादि पूर्ववत् ॥७०॥

218 Treatment of Svara
(3) The procedure for finding out the missing note-series
(naṣṭa) : (63c-70)
Let the initial¹ pebbles be placed in the respective²
squares, the sum total of the figures of which would constitute
the number³ of the 'missing note-series'⁴ inclusive of the
figure 1 in the first column square.⁵ The note-positions⁶ of
Naṣṭa is the name given to the note-series, the tonal form of which
is lost sight of and is required to be discovered with the help of the note-
series indicator. The type of the note-series such as complete, hexatonic,
pentatonic etc. indicative of the number of notes in it and its serial
number in the permutational extension is given. Thus, the problem of
naṣṭa essentially consists in discovering the tonal form (i. e. the order of
notes) of a given serial number of the permutational note-series lost to
us. Naṣṭa may therefore be called 'the missing note-series'.
The formula or the rule for discovering the tonal form of the given
note-series implies the following procedure in terms of action :
(i) Place the pebbles in the squares of the first horizontal series
of the indicator in as many squares as there are the notes in
the given series, from left to right.
(ii) Move the pebbles into the column squares down below, so
that the figures indicated by them when added make up the
number of the given series.

1. The word 'initial' qualifying pebbles is indicative of the
   original position of the pebbles in the first series of the permu-
   tation indicator with the figure 1 in the left end first square
   and zeros in the others.
2. That is the squares in the vertical columns, on a pebble being
   placed in each of the series.
3. The serial number of a particular note-series obtaining in the
   permutational calculus.
4. Naṣṭa of the text literally means 'that which is destroyed';
   but the term is figuratively employed in a technical sense.
   Naṣṭa refers to that note-series (tāna), the tonal form of which
   is lost sight of, but in respect of which the serial number and
   the type (i. e., monotone, bitone, hexatone, etc ) is given in
   order that its tonal form may be discovered with the help of
   the note-series indicator. Naṣṭa, therefore, essentially refers
   to a note series which is lost so far as its tonal form is concern-
   ed. That is why it has been rendered into English as the
   'missing note-series.'
5. The first square at the left end of the first series.
6. The expression 'note-position' refers primarily to the relative
   position of notes in the lost note-series; and secondarily to
   the position of the pebbles i. e., the squares in which they
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna
   the missing note-series are to be determined from the pebble-
   squares⁷, the serial number of the squares down below⁸
   being indicative of the relative backward⁹ position of the
   (iii) Every column is indicative of a particular note. The serial
   number of the square into which the pebble is moved from
   the top indicates its relative position in the given series
   when compared to the last note of the original order. So
   place it accordingly.
   (iv) Drop the note ascertained from the original order as well as
   the given note-series out of view and repeat the procedure
   laid down in (i) to (iii), till the position of the first note of
   the series is discovered.
   (v) Place the notes in their relative order, as ascertained and
   the 'missing note-series' is found.
   Now this procedure is demonstrated with an example as follows :
   (i) Given particulars :
   (a) Serial No. 18 of the tetratonic series.
   (b) Note-series indicator with four column-squares.
   (c) Original order : sa ri ga ma.
   (ii) Action :
7. Place the pebbles initially in the squares of the first series
   indicating 1 and 3 zeros.
8. Deduct 1, the figure of the left end square of the first series
   which is essentially to be taken from 18, the serial number of the series
   (18-1) = 17. So we have to add up to 17. Therefore, take 12 from
   the 3rd sq. of column 4 as the next figure exceeds our requirement.
9. Then deduct 12 from 17 and we have 5 yet to add up. Take
   4 from the 3rd square of column 3
10. Then deduct 4 from 5 and we have 1 left to be added. 1 is
    available in square 2 of column 2. Square 1 of column 1 is already taken
    into consideration.
    are moved to indicate the order of notes in the 'missing note-
    series.' Thus, there would be as many pebbles as there are
    notes in the series.
11. The pebble-squares are the squares in which the pebbles are
    initially lodged. Plural expression is used for the abstract
    singular of the text to bring out its actual implication.
12. Down below in the vertical columns i. e., the squares into
    which the pebbles are moved.
13. Backward order is from right to left.

220 Treatment of Svara
respective notes with reference to the last note of the original
order¹⁰; while the rule for dropping the ascertained note
etc.¹¹ would apply as before.¹² (68c-70)
5. We get these numbers in the series 1-2-3-3 corresponding to
the serial number of squares in the respective columns. Now relate this
series to the original order "sa-ri-ga-ma". Place ri as the last note of
the 'missing note-series', for it is 3rd from ma, the last note of the
original series.
6. Drop ri and restore the original order to "sa-ga-ma" and the
numbers in the series to 1-2-3 Then repeat the process.
7. Place sa anterior to ri in the process of finding out the 'missing
note-series' as sa is 3rd from ma.
8. Drop sa and 3 respectively from the original order and the
numerical series, and then restore them to "ga-ma" and 1-2 respectively
and repeat the process.
9. Place ga anterior to "sa-ri" discovered through the process,
for ga is 2nd to ma in the original order and corresponds to the figure 2
in the given numerical series. So we get "ga sa ri".
10. Drop ga from the original order and 2 from the numerical
series. Now we are left with ma which corresponds to 1 of the numerical
series and the left-end square of the 1st series i. e., the first column
square. So put ma anterior to "ga sa ri" already obtained. We get
"ma ga sa ri".
11. The 'lost note-series' is discovered to be "ma ga sa ri" which
corresponds to its given serial number 18 in the tetratonic series.
It will be observed that wherever the original order coincides with
the numerical series the note-position of the tone in question remains
the same.
An example of a complete permutational note-series is also
demonstrated in the following chart.
(i) The problem : To find out the tonal form of the 'missing
note-series' of heptatonic type bearing serial No. 1000.
10. The original order corresponds to the mūrcchanā-series.
11. At every step as in the case of uddiṣṭa, i. e., the process of the
indicated note-series, the note ascertained would be dropped
out of view.
12. 'S' infers that 'etc.' in the text is indicative of the direction
that, with the 'missing note-series' of, say five notes, the
permutation indicator will have five series of square-columns,
that of six notes will have six and that of seven will have
seven
Section 4 : Grāma, Mūrcchanā, ...
(ii) The original order : "sa ri ga ma pa dha ni".
(iii) Procedure in action.
Discovering the 'lost note-series' of complete permutational note-series

222 Treatment of Svara
4. खण्डमेरुत एकस्वरादितानानां सङ्ख्या-परिज्ञानोपायः
तानस्वरमितोध्वर्धःपङ्क्तिगान्त्याङ्कमिश्रणात् ।
एकस्वरादितानानां संख्या संजायते क्रमात् ॥७१॥
(4) The number of permutational note-series in each of the
series as shown by the permutation-indicator : (71)
The number of note-series in monotone¹ etc. is com-
puted by adding up the figures of the last column-squares²
The note-series indicator does not merely show the indicated note-
series and the 'missing note-series', but it also informs us with regard to
the total number of note-series in each of the different types such as
heptatonic, hexatonic, monotonic, etc.
As many notes there are in a given note-series (e. g. seven in the
complete and one in the monotone series) so many end-squares of these
columns will be taken into consideration and figures indicated by them
when added up would in their sum-total indicate the number of note-
series in that type.
The following chart shows the working of this relationship.
The number of note series in each type derived from the
permutation-indicator

1. In each type of the permutational note-series such as mono-
   tone, bitone, hexatonic, complete, etc.
2. The squares at the bottom and of the vertical series arranged
   into the columns of the permutation-indicator.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna
   corresponding respectively³ to the notes in the series.⁴ (71)
   It will be observed that every total incorporates the total of its
   antecedent squares. Thus, in effect, to find out the total number
   of note-series of a particular type what may essentially be taken into
   account is the figures indicated in the squares just below the zero and
   the last squares of the respective columns, as the squares immediately
   below the zero indicate the total number of the preceding series. For
   example, the total number of the complete permutational note-series is
   found to be 5040, but this figure is composed of the figures shown in the
   squares immediately below the zero of the 7th column as the 7th. square
   i. e., 720+4320=5040. So also the figure 720 is composed of
   120+600=720.
   Incidentally, this shows that the zeros placed in the first squares
   of the columns are symbolic of the figures indicated into the last squares
   of the antecedent columns and are therefore identical with them. That
   is how the zeros become significant.
   It may now be observed that the Khaṇḍameru i. e., the permuta-
   tion-indicator has a three-fold function viz., (i) to find out serial
   number of the note-series indicated, (ii) to discover the tonal form of
   a given number of note-series, and (iii) to ascertain the total number
   of note-series in each of the seven types of permutational note-series.
   (vii) शुद्धतान-नामानि
3. षड्जग्रामस्थ-षाडव-शुद्धतानाः
   अथात्र शुद्धतानानां नामानि व्याहरामहे ।
   अग्निष्टोमो ऽत्यग्निष्टोमो वाजपेयश्च षोडशी ॥७२॥
   पुण्डरीको ऽश्वमेधश्च राजसूयस्ततः परः ।
   इति स्युः षड्जहीनानां सप्त नामान्यनुक्रमात् ॥७३॥
   स्विष्टकृद्बहुसौवर्णो गोसवश्च महाव्रतः ।
   विश्वजिद्ब्रह्मयज्ञश्च प्राजापत्यस्तु सप्तमः ॥७४॥
   3 & 4. Each column is indicative of a note i. e., the first column
   square for sa, the second for ri and so on. So the figures
   of these last column-squares only have to be taken into
   account, the notes pertaining to which go to constitute the
   series in question. Series denotes the note-series.

Left Page

क्रमादृषभहीनानां तानानामभिधा इमाः ।
अश्वक्रान्तो रथक्रान्तो विष्णुक्रान्तस्ततः परः ॥७५॥
सूर्यक्रान्तो गजक्रान्तो वलभिन्नगपक्षतः ।
इति पञ्चमहीनानां संज्ञाः सप्त क्रमान्मताः ॥७६॥
चातुर्मास्यो ऽथ संस्थाऽऽख्यः शस्त्रश्चोक्थश्चतुर्थकः ।
सौत्रामणी तथा चित्रा सप्तमस्त्विदुद्भिदाह्वयः ॥७७॥
संज्ञा निषादहीनानां षाडवानामिमाः क्रमात् ।
(vii) The names of the combinational note-series : 72-90
(a) The hexatonic combinational note-series of the ṣaḍja-grāma : (72-78b)
Now, we give herewith the names of the combinational¹ note-series. (72ab)

1. The names of the seven note-series devoid of ṣaḍja respectively² are—agniṣṭoma³, atyagniṣṭoma, vājapeya, ṣoḍaśī, puṇḍarīka, aśvamedha and rājasūya. (72c-73b)
2. The names of the note-series devoid of ṛṣabha respectively are—sviṣṭakṛt, bahusuvarṇa, gosava, mahāvrata, viśvajit, brahmayajña and prajāpati. (73c-75b)
3. The seven names for (the note-series) omitting pañcama are respectively considered to be aśvakrānta, rathakrānta, viṣṇukrānta, sūryakrānta, gajakrānta, valabhit and nāgapakṣaka. (75c-76)
4. The names for the hexatonic (combinational note-series) eliminating niṣāda respectively are cāturmāsya, saṃsthā, śastra, uktha, sautrāmaṇī, citrā and udbhid. (77-78b)
   Having dealt with the permutational note-series in great detail, the author now incidentally proceeds to give the names of the significant combinational note-series with a view to refer to their merit.
   He takes only the hexatonic and the pentatonic note-series of both the grāma-s respectively, and first he gives the names of the hexatonic series and then the pentatonic series.

1. Śuddha of the text.
2. That is in the order of mūrcchanā-series.
3. This and the following names pertain to vedic sacrifices. These names pertain to the note-series of the ṣaḍja-grāma.

Right Page

Section 4 : Grāma, Mūrcchanā, Krama and Tāna 225
2. मध्यमग्रामस्थषाडवशुद्धतानाः
सावित्रो चार्धसावित्री सर्वतोभद्रसंज्ञकः ॥७८॥
आदित्यानामयनञ्च गवामयननामकः ।
सर्पाणामयनः षष्ठः सप्तमः कौणपायनः ।
नामानि षड्जहीनानां तानानामिति मेनिरे ॥७९॥
अग्निचित् द्वादशाहश्चोपांशुः सोमाभिधस्ततः ।
अश्वप्रतिग्रहो वह्निरथाम्भ्युदयसंज्ञकः ॥८०॥
ऋषभेण विहीनानामिति नामानि मन्वते ।
सर्वस्वदक्षिणो दीक्षा सोमाख्यः समिदाह्वयः ॥८१॥
स्वाहाकारस्तनून्नपात्ततो गोदोहनो मतः ।
इति गान्धारहीनानां क्रमात्संज्ञाः प्रचक्षते ॥८२॥
(b) The hexatonic combinational note-series of the madhyama-grāma : (78c-82)
5. The names of the note-series¹ eliminating ṣaḍja are respectively known to be—sāvitrī, ardha-sāvitrī, sarvatobhadra, ādityāyana², gavāmayana³, sarpāyana⁴ and kauṇapāyana.⁵ (78c-79)

1. The (specific) combinational note-series of the madhyama-grāma.
2. The expression ‘ādityānām ayanaḥ’ of the text has been rendered as ādityāyana for semantic considerations; but it needs an elaboration. Āditya has several meanings such as the sun, the dwarf incarnation of Viṣṇu, god in general, and so on. But since here the plural is used we may take it etymologically in the sense of ‘the sons of Aditi’ i. e., the gods. Ayana too has several connotations such as the path, the way, the passage of the sun, and abode etc. of which the last one seems to be appropriate in this context. So the expression as a whole would literally mean ‘the abode of gods’.
3. The expression gavāmayana similarly signifies ‘the abode of cows’.
4. Also the expression sarpāyana is a compound rendering of sarpāṇāmayana in the text which literally means the abode of serpents.
5. Kauṇapa literally means a goblin, a demon. So, kauṇapāyana literally signifies the abode of demons.
   15

226 Treatment of Svara
3. षड्जग्रामस्थौडुवशुद्धतानाः
इडा पुरुषमेधश्च श्येनो वज्र इषुस्ततः ।
अङ्गिराः कङ्क इत्येताः सपहीनाभिधाः क्रमात् ॥८३॥
ज्योतिष्टोमस्ततो दर्शो नान्द्याख्यः पौर्णमासकः ।
अश्वप्रतिग्रहो रात्रिः सौभरः सप्तमः स्मृतः ॥८४॥
एता निषादगान्धारहीनानामभिधाः क्रमात् ।
सौभाग्यकृच्च कारीरी शान्तिकृत्पुष्टिकृत्तथा ॥८५॥
वैनतेयोच्चाटनौ च वशीकरणसंज्ञकः ।
पञ्चमर्षभहीनानां तानानामभिधा इमाः ॥८६॥
6. The names of (the note-series) devoid of ṛṣabha are
believed to be—agnicit, dvādaśāha, upāṁśu, soma, aśvapratigraha,
barhi and abhyudaya. (80-81b)
7. The names of (the note-series) devoid of gāndhāra
are respectively said to be sarvasvadakṣiṇa, dīkṣā, soma, samit,
svāhākāra, tanūnapāt and godohana. (81c-82)
(c) The pentatonic combinational note-series of ṣaḍja-
grāma : (83-86)

1. The names of (the note-series)⁶ eliminating ṣaḍja
   and pañcama respectively are as under :
   Iḍā, puruṣamedha, śyena, vajra, iṣu, aṅgirā and kaṅka. (83)
2. The names of (the note-series) omitting niṣāda and
   gāndhāra respectively are jyotiṣṭoma, darśa, nāndī, paurṇamāsa,
   aśvaprati-graha and saubhara. (84-85b)
3. The names of the note-series leaving out pañcama and
   ṛṣabha respectively are—saubhāgyakṛt, kārīrī, śāntikṛt, puṣṭikṛt,
   vainateya, uccāṭana and vaśīkaraṇa. (85c-86)
4. These note-series pertain to the ṣaḍja-grama.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna 227
5. मध्यमग्रामस्थौडुवशुद्धतानाः
   त्रैलोक्यमोहनो वीरः कन्दर्पबलशातनः ॥८७॥
   शङ्खचूडो गजच्छाया रौद्राख्यो विष्णुविक्रमः ।
   तानानां रिधहीनानां नामान्येतान्यनुक्रमात् ॥८८॥
   भैरवः कामदाख्यश्चावभृथो ऽष्टकपालकः ।
   स्विष्टकृच्च वषट्कारो मोक्षदः सप्तमो मतः ॥८९॥
   संज्ञा निषादगान्धारहीनानामिति संमताः ।
   (viii) शुद्धतानानां वैदिकनाम्नां तत्तद्यज्ञफलम्
   यद्यज्ञनामा यस्तानस्तस्य तत्फलमिष्यते ॥९०॥
   (d) The pentatonic combinational note-series of the
   madhyama-grāma : (87-90b)
6. The names of the note-series eliminating ṛṣabha and
   dhaivata respectively are trailokyamohana, vīra, kandarpabala-
   śātana, śaṅkhacūḍa, gajacchāya, raudra and viṣṇuvikrama. (87-88)
7. The names of the note-series devoid of niṣāda and
   gāndhāra are believed to be bhairava, kāmada, avabhṛtha, aṣṭaka-
   pālaka, sviṣṭakṛt, vaṣaṭkāra and mokṣada. (89-90)
   (viii) The functional purpose of the vedic names of combi-
   national note-series : 90cd
   It is believed¹ that the note-series² bearing the name
   of a particular vedic sacrifice is effectual in bringing about
   the corresponding³ benefit. (90cd)
8. By the ancients, the preceding music-authorities.
9. The combinational note-series are meant.
10. That is, if the note-series bears the name of aśvamedha sacri-
    fice, it would bring about the benefit afforded by that sacrifice,
    provided it is appropriately employed.

228 Treatment of Svara
(ix) गान्धर्वे गाने च शुद्धकूटतानानामुपयोगः
गान्धर्वे मूर्च्छनास्तानाः श्रेयसे श्रुतिचोदिताः ।
गाने स्थानस्य लाभेन ते कूटाश्चोपयोगिनः ॥९१॥
(ix) The specific use of the combinational and permutational note-series in gāndharva and gāna : 91
In the context of gāndharva⁴, the mūrcchanās⁵ and the combinational note-series have been ordained in the vedic lore⁶ for the attainment of beatitude; while in gānam⁷, these along with the permutational note-series are useful in obtaining sthāna⁸. (91)
The commentators do not offer any explanation with regard to the names of various vedic sacrifices having been attributed to the different note-series. However, it seems that it is an extension of the basic atti- tude of the vedic tradition towards all aspects of human activity. This attitude was responsible for the formulation of the four upaveda-s (Āyurveda or Arthaśāstra, Gāndharvaveda, Dhanurveda and Sthāpatyaveda), the fifth veda-s (Mahābhārata and Nāṭyaveda), the six Vedāṅga-s etc.
4. Gāndharvam is the ancient name for what later came to be known as mārga, when employed against the term deśī (region- al music) as explained by 'S'. These terms will be dealt with properly in the subsequent chapters of the text. (cf. Chapter IV 1-4).
5. 'K' thinks that only standard mūrcchanā-s are meant.
6. Śruti in this context signifies any vedic passage. 'K' includes the smṛti-s also in its significance.
7. Gāna is the counterpart of gāndharva, later known as deśī (regional) music which was much more liberal and rich in variety. Since kūṭa-tāna (permutational note-series) has not been spoken of by Bharata, Śārṅgadeva specifically limits it to non-gāndharva music.
8. Sthāna has been interpreted by 'K' as ādhāraśruti (note-śruti) of svara, whereas 'S' interprets it as sthāya (a melodic phrase). The interpretation of 'K' is relevant to mūrcchanā and śuddha- tāna and that of 'S' to kūṭa-tāna which serves as the canvas for melodic diversity.
अथ पञ्चमं साधारणप्रकरणम्
(i) द्विविधं साधारणं, तत्र स्वरसाधारणम्
साधारणं भवेद् द्वेधा स्वरजातिविशेषणात् ।
स्वरसाधारणं तत्र चतुर्धा परिकीर्तितम् ॥१॥
काकल्यन्तरषड्जैश्च मध्यमेन विशेषणात् ।
Section 5
OVERLAPPING (Sādhāraṇa)
(i) Twofold overlapping and the overlapping in note-intervals : 1-2b
Overlapping (sādhāraṇa) is twofold as it pertains¹ to the tones (svara-s) and to jāti-s (the melodic types). Of these two, the overlapping of notes (svara-sādhāraṇa) is said to be
The concept of sādhāraṇa, as applied to music is elucidated by Bharata on the analogy of kāla-sādhāraṇa i. e., the overlapping of seasons: "If one perspires by remaining in the sun and yet feels cold staying in the shade neither the winter has ended, nor (can it be said that) the spring has arrived" (N. S., G. O. S. Vol. IV. 28. 34). This is an instance of the overlapping of the winter and the spring on their borders. Thus, the concept implies the conjunction of the end portion of a prior situation and the beginning portion of the following situation, creating a new situation that partakes partly of the characteristics of both and is thereby different from both of them individually. The application of this concept to the note-intervals is to be explained presently. According to Bharata the overlapping (sādhāraṇa) means the quality of a note rising between two (consecutive) notes (Eng. Trans. M. Ghosh, N. S. Vol. II, XXIII 35, p. 13).
The jāti-sādhāraṇa or the overlapping of melodic types is dealt with in Section VII; here we are concerned only with svara-sādhāraṇa or the overlapping of note-intervals which is related to four notes in particular,

1. Literally, "as it is qualified by svara-s and jāti-s".

230 Treatment of Svara
fourfold as it is concerned² with kākalī (niṣāda) antara
(gāndhāra), ṣaḍja and madhyama. (1-2b)
viz. kākalī-niṣāda, antara-gāndhāra, ṣaḍja and madhyama. Thus, sādhāraṇa
(overlapping) is fourfold as it is qualified by these four notes and is
respectively called kākalī-sādhāraṇa (the overlapping of kākalī), antara-
sādhāraṇa (the overlapping of antara), ṣaḍja-sādhāraṇa (the overlapping of
ṣaḍja) and madhyama-sādhāraṇa (the overlapping of madhyama).
(ii) काकल्यन्तरसाधारणम्
साधारणः काकली हि भवेत्षड्जनिषादयोः ॥२॥
साधारण्यमतस्तस्य यत्तत्साधारणं विदुः ।
अन्तरस्यापि गमयोरेवं साधारणं मतम् ॥३॥
(ii) The overlapping of kākalī and antara: 2c-3
Since the overlapping (sādhāraṇa)¹ kākalī occurs between
niṣāda and ṣaḍja, its³ overlap³ (sādhāraṇya) is known as over-
lapping (sādhāraṇa)⁴. The overlapping of antara is similarly
conceived between gāndhāra and madhyama (2c-3)
In the process, technically known as the overlapping of kākalī, two
notes are involved, and the author is here explaining how they are
affected in this process, which elsewhere has also been referred to as
the kakalisation of niṣāda, and also why it is so called.
In the ṣaḍja-grāma, niṣāda has two śruti-s (21st and 22nd) and
ṣaḍja has four (1st-4th). It has already been explained (I. 3.40 ante)
that in the kakalisation (kākalītva) of niṣāda, kākalī-niṣāda is formed by
the conjunction of niṣāda and ṣaḍja to the extent of two śruti-s of each,
i. e., 21st and 22nd śruti of niṣāda and the 1st and the 2nd śruti of ṣaḍja.
Thus, kākalī is of four śruti-s and is situated equidistant from both
niṣāda as well as ṣaḍja. It partakes of the characteristics of both since
it is constituted by two śruti-s contributed by each and is yet different
from each of them since its note-śruti is different (2nd or 24th).
2. Literally, "as it is qualified by kākalī, antara, ṣaḍja and madhyama".

1. Sādhāraṇaḥ qualifies kākalī.
2. That is, the overlapping in the formation of kākalī.
3. Sādhāraṇya is defined by ‘K’ as “sādhāraṇasya bhāvaḥ” i. e., the essence or the quality of overlapping. It implies the sense of “overlapping-ness”, though such a word is not in usage.
4. Sādhāraṇam stands for the abstract noun sādhāraṇyam.
   Section 5 : Overlapping (Sādhāraṇa) 231
   However, kākalī-sādhāraṇa affects the two notes niṣāda and ṣaḍja
   quite differently; whereas it absorbs the note-śruti of niṣāda, it does not
   affect the note-śruti of ṣaḍja and that is one reason why it is called
   kākalī-niṣāda i. e., because it affects niṣāda in categorically a different
   way in so far as it completely absorbs it.
   Antara-sādhāraṇa i. e., the overlapping of antara is similarly
   conceived between gāndhāra and madhyama, affecting these two notes
   in the same way in the terms explained above; and consequently takes
   the name “antara-gāndhāra” on the pattern of “kākalī-niṣāda”. Thus
   antara is constituted of the two śruti-s (8th and 9th) of gāndhāra and two
   śruti-s (10th and 11th) of madhyama, absorbing the note-śruti of the
   former and leaving the note-śruti of the latter un-affected.
   (iii) काकल्यन्तर-प्रयोगविधिः
   प्रयोज्यो षड्जमुच्चार्य काकलीधेवतौ क्रमात् ।
   एवं मध्यममुच्चार्य प्रयुञ्जीतान्तरर्षभौ ॥४॥
   षड्जकाकलिनौ यद्वोच्चार्य षड्जं पुनर्व्रजेत् ।
   तत्परान्यतमं चैवं मध्यमं चान्तरस्वरम् ॥५॥
   प्रयुज्य मध्यमो ग्राह्यस्तत्परान्यतमो ऽथ वा ।
   अल्पप्रयोगः सर्वत्र काकली चान्तरः स्वरः ॥६॥
   (iii) The application of kākalī and antara: 4-6
   After singing¹ ṣaḍja, kākalī followed² by dhaivata may
   be employed. Likewise after singing³ madhyama, antara
   and ṛṣabha may be used. Alternatively, having sung⁴ ṣaḍja
   and kākalī one may return to ṣaḍja and after that move on to
   any succeeding⁵ note. Similarly, after employing madhyama
   and antara, madhyama should be taken before resorting to any
   succeeding note. Everywhere⁶ the notes kākalī and antara
   are rarely used. (4-6)
   1, 3 & 4 Literally, ‘pronouncing’ or ‘articulating’.
5. Krama implies ‘due order’ which, in this case, is the descending one.
6. According to ‘K’ the succeeding note has to be the immediately succeeding one barring the one that has to be eliminated; but ‘S’ thinks that any succeeding note implies any one of ri, ga, ma, pa and dha in case of the overlapping of kākalī and pa, dha, ni, sa, and ri in case of the overlapping of antara i. e., all the notes subsequent to ṣaḍja and madhyama respectively.
7. ‘S’ interprets it as jāti-rāga-s, etc.

(iv) षड्जमध्यमसाधारणम्
निषादो यदि षड्जस्य श्रुतिमाद्यां समाश्रयेत् ।
ऋषभस्त्वन्तिमां प्रोक्तं षड्जसाधारणं तदा ॥७॥
मध्यमस्यापि गपयोरेवं साधारणं मतम् ।
साधारणं मध्यमस्य मध्यमग्रामगं ध्रुवम् ॥८॥
साधारणे कैशिके ते केशाग्रवदणुत्वतः ।
ते एव कैश्चिदुच्येते ग्रामसाधारणे बुधैः ॥९॥
(iv) The overlapping of ṣaḍja and madhyama: 7-9
If niṣāda takes resort to the first śrutis of ṣaḍja and
ṛṣabha to its last, it is called the overlapping of ṣaḍja.
Similarly, the overlapping of madhyama is conceived in-bet-
ween gāndhāra and pañcama. The overlapping of madhyama
is definitely restricted to madhyama-grāma. (7-8)
These two¹ overlappings, being subtle like the point of
hair, are known as kaiśika² and are also called “the over-
lapping of grāma-s”³ by (some) scholars.⁴ (9)
The overlapping of ṣaḍja and madhyama is analogous to the modi-
fication of these two notes The overlapping of ṣaḍja takes place when it
yields its initial śruti to niṣāda and its final śruti to ṛṣabha retaining its
2nd-3rd śruti-s to itself. Similarly, the overlapping of madhyama takes
place when madhyama yields its first śruti to gāndhāra and its fourth śruti to
pañcama, retaining only two śruti-s to itself. However, as pointed out
by ‘K’, the longest śruti-interval conceived in the grāma-s is that of four

1. That is, the overlapping of ṣaḍja and the overlapping of
   madhyama.
2. Kaiśika literally means hair-like, i. e. very fine pointed, very
   subtle. The word (kaiśika) is applied to ṣaḍja sādhāraṇa and
   madhyama-sādhāraṇa.
3. ‘S’ infers that since the overlapping of madhyama is limited
   to madhyama-grāma, the overlapping of ṣaḍja is also to be under-
   stood as applicable to ṣaḍja-grāma only. That is how he
   comes to justify the expression grāma-sādhāraṇa i. e., the
   overlapping of grāma-s, as signifying the process to be indica-
   tive of grāma-s. However, this does not accord with the
   reason pointed out by ‘K’ for restricting the overlapping of
   madhyama to madhyama-grāma since ṛṣabha or niṣāda do not
   exceed the interval of four śruti-s in the overlapping of ṣaḍja
   if made applicable to madhyama-grāma.
4. This, however, does not seem to be the view of the author.
   śruti-s and therefore a five-śruti interval is considered to be undesirable.
   So, if in the ṣaḍja-grāma the overlapping of madhyama is allowed, its
   pañcama will swell into a five-śruti-interval, which is undesirable.
   Therefore, the overlapping of madhyama is permitted only in madhyama-
   grāma in which pañcama has three śruti-s only.
   (v) जातिसाधारणम्
   एकग्रामोद्भवास्वेकांशासु जातिषु यद्भवेत् ।
   समानं गानमार्गास्तज्जातिसाधारणं जगुः ॥१०॥
   जातिसाधारणं केचिद्रागानेव प्रचक्षते ।
   (v) Overlapping of jāti-s: 10-11b
   Among the jāti-s of the same grāma having the same
   tone as the fundamental note, the identical melodic presenta-
   tion that obtains between them is called the overlapping of
   jāti-s by the noble ones.¹ The overlapping of jāti-s is identi-
   fied by some with rāga-s². (10-11b)
   Jāti-sādhāraṇa is very briefly dealt with by Śārṅgadeva as well as by
   Bharata. ‘K’ thinks that it is based on varṇa-sāmya, the identity in the
   patterns of tonal movement. ‘S’ cites śuddha-kaiśika-madhyama as one of
   the instances of jāti-sādhāraṇa. Some details regarding the practice of
   jāti-sādhāraṇa will be found in Section VII of this chapter.
5. The earlier authors such as Bharata, Mataṅga and others are
   meant.
6. ‘S’ cites rāga śuddha-kaiśika-madhyama as an illustration.