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

176 Treatment of Svara Section 4 : Grāma, Mūrcchanā, Krama and Tāna 177
order. These⁴ are mūrcchanā-series (krama-s) and their num-
ber is three hundred and ninetytwo. (19-20b).
These series of tonal order do not stand in the descending order and
are therefore called krama-s or mūrcchanā-krama-s-i.e., mūrcchanā-series.
Since each of the fiftysix mūrcchanā-s has seven series, the total number of
such sub-varieties comes to (56 x 7 = 392) three hundred and ninetytwo.
The purport is that each one of the fiftysix mūrcchanā-s spoken of
above, has seven series of mūrcchanā-s of its own which are formed by
commencing the mūrcchanā with the first, the second, the third, the
fourth, the fifth, the sixth and the seventh note, keeping the śruti-value
of the notes unaltered. Now, 'S' poses the problem that, if the mūrc-
chanā is commenced with the second note and so on, how can it have
seven notes in it ? The text resolves the difficulty by suggesting that
the last notes may be sung in the first instance and the preceding notes
may be sung in their respective order. Moreover, it is notable that in
this scheme the basic scale is not altered and so the mūrcchanā does not
change, what changes is the order of notes in it, and actually each
mūrcchanā has seven possibilities of such a variety in the order. These
series are therefore called krama-s, literally 'orders'; they are useful in
the computation of kūṭatāna-s (permutational note-series). The series of
mūrcchanā-s are therefore sub-varieties of the mūrcchanā-s and are distin-
guished from the mūrcchanā-s proper by the fact that they are formed only
in the ascending order of notes, whereas the mūrcchanā-s are formed in
the descending order as well. For instance uttaramandrā, the first
mūrcchanā, of the ṣaḍja-grāma is sa ri ga ma pa dha ni : ni dha pa ma
ga ri sa; but the series of this mūrcchanā are recorded as follows :-
These krama-s, named as mūrcchanā-series are considered to be
mūrcchanā-s only in so far as the notes in their ascending order are placed
consecutively and exhibit some sort of order in their arrangement.
Mataṅga, following Bharata, classifies the seven-tonal mūrcchanā-s
in slightly a different way, viz. pūrṇā (complete), ṣāḍavitā (hexatonic),
auḍuvitā (pentatonic) and sādhāraṇī (mixed), the first one of which
incorporates all the seven notes, the second is composed of six notes, the
third is formed with five notes only, while the fourth includes kākalī
or/and antara forms of niṣāda and gāndhāra.
Bharata has defined mūrcchanā as a group of seven notes placed
in order, and he derives mūrcchanā-tāna-s (also called only tāna) from
them by the omission of one or two notes (cf. N. S., G. O. S. IV.
32-33, p. 25). Śārṅgadeva too, basically follows this classification
but technically he applies the term tāna to all those modifications of
mūrcchanā-s that are formed by the omission of one to six notes in the
combinational and permutational note-series (śuddha and kūṭa tāna-s).
(vii) मूर्च्छनानां देवताः
यक्षरक्षोनारदाब्जभवनागाश्विपाशिनः ॥२०॥
षड्जग्रामे मूर्च्छनानामेताः स्युर्देवताः क्रमात् ।
ब्रहेन्द्रवायुगन्धर्वसिद्धद्रुहिणभानवः ॥२१॥
स्युरिमा मध्यमग्राममूर्च्छनादेवताः क्रमात् ।
(vii) The presiding deities of the mūrcchanā-s : 20c-22b
The presiding deities of the mūrcchanā-s of the ṣaḍja-
grāma respectively¹ are yakṣa², rakṣa³, Nārada⁴, Brahmā⁵,

1. That is, from the first mūrcchanā-series to the seventh series.
2. Yakṣa is a class of demigods who are considered to be the
   guards of the treasures of Kubera, the god of wealth.
3. Rakṣa is the name given to evil spirits 'K. R.' in his note
   on verse 20 considers rakṣa-s along with yakṣa-s to be semi-
   celestial beings. He does not elaborate on their function.
4. Nārada is a celestial figure, a sage elevated to be a divine
   roaming minstrel of the three worlds (i.e. the heaven being the
   abode of gods, the earth being the abode of mortals and the un-
   der world below the earth, being the abode of nāga-s or reptiles).
5. The creative aspect of the Hindu trinity.
   this verse; they may be understood as subvarieties of the seven
   mūrcchanā-s, called krama-s in the text. The word krama is used
   in the sense of series in this particular context.
   12

178 Treatment of Svara
nāga,⁶ aśvin⁷ and Varuṇa⁸; and those of the madhyama-grāma
respectively are Brahmā, Indra⁹, Vāyu¹⁰, gandharva,¹¹ siddha¹²
Śiva and sun.¹³ (20c-22b)
(viii) नारदकथितानि मूर्च्छना-नामानि
तासामन्यानि नामानि नारदो मुनिरब्रवीत् ॥२२॥
मूर्च्छनोत्तरवर्णा ऽऽद्या षड्जग्रामे ऽभिरुद्गता ।
अश्वक्रान्ता च सौवीरी हृष्यका चोत्तरायता ॥२३॥
रजनीति समाख्याता ऋषीणां सप्त मूर्च्छनाः ।
आप्यायनी विश्वकृता चन्द्रा हेमा कपर्दिनी ॥२४॥
मैत्री चान्द्रमसी पित्र्या मध्यमे मूर्च्छना इमाः ।
नन्दा विशाला सुमुखी चित्रा चित्रवती सुखा ॥२५॥
आलापा चेति गान्धारग्रामे स्युः सप्त मूर्च्छनाः ।
ताश्च स्वर्गे प्रयोक्तव्या विशेषान्न नोदिताः ॥२६॥
(viii) Nārada's nomenclature for the mūrcchanā-s : 22c-26
The sage Nārada has given¹ another nomenclature for
them² viz. the first in the ṣaḍja-grāma is uttara-varṇā and
then³ abhirudgatā, aśvakrāntā, sauvīrī, hṛṣyakā, uttarāyatā and
Nārada is a celebrated authority of ancient India and this
subject is dealt with in two works ascribed to him, viz. Nāradīya-śikṣā
and Saṅgīta-makaranda. The nomenclature ascribed to Nārada in the
text tallies more or less with the one given in Nāradīya-śikṣā, but is
slightly different from that of Saṅgīta-makaranda (cf. N. Śikṣā 1.2.9-14
6. Nāga, a class of beings inhabiting the world below the earth,
who are likened to serpents.
7. Aśvins are the twin brothers, the sons of the sun and the physi-
cians of the gods.
8. A god of the vedic pantheon and puranic mythology.
9. The chief of the gods.
10. The god of wind.
11. Celestial chorister, a singer in general.
12. A class of demi-gods with supernatural powers, or a perfected
human being.

1. Literally, it would be translated as 'said'.
2. 'Them' refers to the seven mūrcchanā-s of each of the grāma-s.
3. The sense of 'then' is implied in the context.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna 179
   rajanī; these are the seven mūrcchanā-s of the seers. In the
   madhyama-grāma the mūrcchanā-s are: āpyāyanī, viśvakṛtā, candrā,
   hemā, kapardinī, maitrī and cāndramasī belonging to the manes.
   Nandā, viśālā, sumukhī, citrā, citravatī, sukhā and ālāpā are the
   seven mūrcchanā-s of gāndhāra-grāma; but they have not been
   spoken of in detail as they are to be employed in heaven
   only. (22c-26)
   and S. Mak. 1.2.10-15). However, it is notable that the name uttara-
   varṇā given to the very first mūrcchanā of the ṣaḍja-grāma is not found in
   both of them ; and what is found in its place is uttara-mandra. Besides,
   there are a few other minor variations. That shows that Śārṅgadeva
   probably had access to a different MS. of Nāradīya-śikṣā.
   Another notable point is the distribution of the mūrcchanā-s of the
   three grāma-s among the gods, the seers (who are human beings) and
   the manes. This is in keeping with the Brahminical tradition of
   symbolic characterisation already spoken of. It may however be
   mentioned that, since the seven mūrcchanā-s of the ṣaḍja-grāma only are
   allocated to the seers (ṛṣi-s) these mūrcchanā-s only are considered to
   be of use in this world by the Nāradīya-śikṣā (cf. 1.2.13-14).
   ग. तानाः
   (i) शुद्धताननिर्माणविधिः, सङ्ख्या च
4. शुद्धतानलक्षणम्
   तानाः स्युर्मूर्च्छनाः शुद्धाः षाडवौडुवितीकृताः ।
   C. The tāna-s : 27-90
   (i) The formation and the number of the śuddha-tāna-s
   (specific combinational note-series) : 27-30
   (1) The definition of the śuddha-tāna : (27ab)
   The śuddha¹ (standard) mūrcchanā-s, by dropping one
   Tāna, which has been rendered as 'note-series' is essentially a
   mūrcchanā. The word tāna is derived from the Sanskrit root tan 'to
   spread, to stretch, or to expand, enlarge, develop.
5. Śuddha literally means pure i. e., unmodified or natural. In
   this context, śuddha implies an ascending and descending
   series of seven standard notes placed in their regular order in
   the ṣaḍja or madhyama grāma; and that is why it has been
   rendered here as 'standard'.

Page 180

180 Treatment of Svara
or two notes become śuddha-tāna-s² (specific combinational
note-series). (27ab)
The mūrcchanā-s have been classified into four types (cf. verse 16
ante), viz. standard (śuddha, kākalī-inclusive, antara-inclusive and
inclusive of both. Broadly speaking, these four types fall into two
categories, viz. composed of standard notes only and composed of
standard and modified notes. A tāna is formed by dropping one or
two notes from any standard mūrcchanā. Thus, a śuddha-tāna is nothing
but a standard mūrcchanā deprived of one or two notes.
The word śuddha qualifies both the words mūrcchanā and tāna.
Tāna-s are also of two types śuddha and kūṭa, i. e. combinational and
permutational. 'K' argues that if the word śuddha of the text is taken
merely to qualify the word tāna, then śuddha-tāna-s would proceed from
all the fiftysix mūrcchanā-s already described, and in that case the
number of hexatonic and pentatonic tāna-s would not tally with the
text; while on the other hand, if it qualifies only 'mūrcchanā' then even
kūṭa i. e. permutational tāna-s would also be possible from the standard
mūrcchanā-s, which is contrary to the fact. Thus it is that, śuddha
(standard) mūrcchanā-s when deprived of one or two notes become what
is called the hexatonic and the pentatonic combinational note-series.
Mataṅga, however, points out another difference between mūrcchanā and
tāna. He says : "mūrcchanā proceeds in the ascending order, while tāna
proceeds in the descending order only." So tāna, as differentiated by
him, is restricted to the descending order of notes only.

2. Tāna, in the ancient terminology, is a series of five to six notes
   in a specific combinational order called śuddha-tāna (specific
   combinational note-series). Though śuddha-tāna is the full
   expression, when writing about it in its context, the word
   śuddha is taken for granted and mere tāna is used instead.
   Accor dingly, the expression specific combinational note-series
   is also sometimes replaced by the simple term note-series (in
   due context). But tāna as such has to be distinguished from
   kūṭa-tāna (permutational note-series) to be defined presently.
   However, in modern music tāna, as defined by Ravi Shankar,
   is a musical phrase sung on vowels, syllables or words drawn
   out or stretched by expressive passages. (cf. Glossary of
   technical words, My Music My Life).

Page 181

Section 4 : Grāma, Mūrcchanā, Krama and Tāna 181

2. षाडवतानाः

षड्जगाः सप्त हीनाश्चेत्क्रमात्स्रिरिपसप्तमैः ॥२७॥
तदा ऽष्टाविंशतिस्ताना मध्यमे सरिगोर्झताः ।
सप्त क्रमाद्यदा तानाः स्युस्तदा त्वेकविंशतिः ॥२८॥
एते चैकोनपञ्चाशदुभये षाडवा मताः ।
(2) The hexatonic note-series : (27c-29b)
The seven (mūrcchanā-s) of the ṣaḍja-grāma, by an indi-
vidual omission of sa, ri, pa and ni¹ in their respective turn,
make twenty-eight (note-series). Similarly, the seven
(mūrcchanā-s) of the madhyama-grāma, by the omission of sa, ri
and ga in turn respectively form twentyone (note-series).
Both of them put together, are considered to be fortynine
hexatonic note-series.² (27c-29b)
Though the combinational (śuddha) note-series (tāna-s) are derived
from the standard (śuddha) mūrcchanā-s, the latter is distinguished by the
absence of modified notes, the former is distinguished not only
by the absence of kākalī and antara notes, but also by the absence
of any change in the regular arrangement of notes. If the note-series
is set in the permutational form then it is called kūṭa-tāna i. e. permu-
tational note-series. The note-series are formed by the omission of one
or two notes. If one of the seven notes is omitted, what we get is a
hexatonic note-series and if two out of the seven are omitted we get
pentatonic note-series.
'K' raises the question that, "since pañcama determines the grāma,
if it is omitted, how can it discharge this function ?" He resolves the
difficulty by pointing out that the pañcama can only be omitted in the
ṣaḍja-grāma; therefore, even its omission is indicative of that particular
grāma.
'S' explains at length how we get twentyeight note-series in the
ṣaḍja-grāma and twentyone note-series in the madhyama-grāma. When
the seven mūrcchanā-s of the ṣaḍja-grāma are deprived of ṣaḍja, then we
get seven hexatonic note-series, likewise when deprived of ri, pa and ni
respectively, we get seven other such series in each case, making twenty-
eight in all. Similarly, in the madhyama-grāma when the standard

1. Saptama—literally means the seventh (note) i. e. ni.
2. That is, 'ṣāḍava-tāna-s'.

182 Treatment of Svara
mūrcchanā-s are deprived of sa, ri and ga respectively, they form into
twentyone note-series. Taken together, we get forty-nine hexatonic
note-series. That is how the standard mūrcchanā-s being deprived of
one or two notes become combinational note-series.
An illustration of the hexatonic note-series with the omission of
sa in the ṣaḍja-grāma
No. Name of the mūrcchanā Form of the mūrcchanā
1 Uttaramandrā x ri ga ma pa dha ni
2 Rajanī ni x ri ga ma pa dha
3 Uttarāyatā dha ni x ri ga ma pa
4 Śuddhaṣaḍjā pa dha ni x ri ga ma
5 Matsarīkṛtā ma pa dha ni x ri ga
6 Aśvakrāntā ga ma pa dha ni x ri
7 Abhirudgatā ri ga ma pa dha ni x
(Chart 16*)
Note :—x denotes omitted sa.
* Adopted from S. Raj. Vol. I, p. 132.
3. औडुवतानाः
सपाभ्यां द्विश्रुतिभ्यां च रिपाभ्यां सप्त वर्जिताः ॥२९॥
षड्जग्रामे पृथक्ताना एकविंशतिरोडुवाः ।
रिधाभ्यां द्विश्रुतिभ्यां च मध्यमग्रामगास्तु ते ॥३०॥
हीनाश्चतुर्दशैव स्युः पञ्चत्रिंशत्तु ते युताः ।
(3) The pentatonic note-series : (29c-31b)
Besides, the seven (standard mūrcchanā-s) of the ṣaḍja-
-grāma being deprived of sa and pa, ga and ni¹ and ri and pa
respectively form twentyone pentatonic note-series; while
those of the madhyama-grāma being short of ri and dha and ga
and ni² respectively become fourteen (tāna-s): altogether,
they make thirtyfive. (29c-31b)
Just as the standard mūrcchanā-s by dropping one note become
hexatonic note-series, by dropping two notes they become pentatonic
note-series. In ṣaḍja-grāma, by the omission of sa and pa we get one set
of seven pentatonic note-series, by the omission of ga and ni, another set
of seven and by the omission of ri and pa the third set of seven, making
twentyone in all. Similarly, in the madhyama-grāma by the omission of
1 & 2. Dviśruti of the text literally means a bi-śrutic note; ga
and ni are the two notes with two śruti-s each.
Section 4 : Grāma, Mūrcchanā, Krama and Tāna 183
4. षाडवौडुवतान-मिलितसङ्ख्या
सर्वे चतुरशीतिः स्युर्मिलिताः षाडवौडुवाः ॥३१॥
(ii) कूटताननिर्माणं, तेषां सङ्ख्या च

1. कूटतानलक्षणम्
   असम्पूर्णाश्च सम्पूर्णा व्युत्क्रमोच्चारितस्वराः ।
   मूर्च्छनाः कूटतानाः स्युः,
   (1) Total number of hexatonic and pentatonic combinational
   note series (śuddha-tāna-s) : (31cd)
   However, the hexatonic and the pentatonic (specific
   combinational) note-series make eightyfour in all. (31cd)
   (ii) The formation and the number of kūṭa (permutational)
   tāna-s (note-series) : 32-49
   (1) The definition of kūṭa-tāna (permutational) note-
   series) : (32 abc)
   The complete or incomplete mūrcchanā-s with their notes
   produced in a permuted form become permutational note-
   series (kūṭa-tāna-s). (32a-32c)
   ri and dha we get one set of seven pentatonic series, and by the omission
   of ga and ni, the second set, making fourteen in all. Thus the pentatonic
   note-series in both the grāma-s are counted as thirtyfive and the number
   of both the hexatonic and the pentatonic note-series in both the grāma-s
   is considered to be (49+35 = 84) eightyfour. An illustration of a penta-
   tonic note-series is given as under. We know that mūrcchanā is in the
   ascending as well as in the descending order of notes.
   An illustration of the pentatonic note-series with the omission
   of sa and pa in the ṣaḍja-grāma
   No | Name of the mūrcchanā | Form of the mūrcchanā
   1 | Uttaramandrā | x ri ga ma • dha ni
   2 | Rajanī | ni x ri ga ma • dha
   3 | Uttarāyatā | dha ni x ri ga ma •
   4 | Śuddha-ṣaḍjā | • dha ni x ri ga ma
   5 | Matsarīkṛt | ma • dha ni x ri ga
   6 | Aśvakrāntā | ga ma • dha ni x ri
   7 | Abhirudgatā | ri ga ma • dha ni x
   (Chart 17*)
   Note:—x denotes sa and • denotes pa.
   *Adopted from S. Raj. Vol. I, p. 136.

184 Treatment of Svara
A mūrcchanā is said to be complete when it has seven notes, and it
is considered incomplete when it has less than seven notes i.e. from six
to one; although one note does not make a series, but that is theoretical.
Actually, when one or more 'notes are omitted from a mūrcchanā, it
becomes a tāna (as already explained in the comments on 19-20b ante),
a note-series. But, here, the author is talking in terms of incomplete
mūrcchanā-s (as well) giving rise to permutational note-series, because
such note-series are derived from the mūrcchanā-series, and not because
on this account, they continue to be mūrcchanā-s. It has already been
said that Bharata has used the expression mūrcchanā-tāna for hexatonic
and pentatonic combinational note-series.
We have already come across the fourfold classification of mūr-
cchanā-s and this is a twofold classification viz. complete and incom-
plete. Now, the purport of the text is that, a mūrcchanā, whether
complete or incomplete, becomes a permutational note-series if the
normal order of its notes is disturbed, and in that case even a descend-
ing movement to a certain extent is no bar. It may be pointed out that
ordinarily the notes of the mūrcchanā-series (krama-s) from which the
note-series are derived are in the ascending order.
2. पूर्णकूटतानानां सङ्ख्या
तत्सङ्ख्यामभिदध्महे ॥३२॥
पूर्णाः पञ्च सहस्राणि चत्वारिंशद्युतानि तु ।
एकैकस्यां मूर्च्छनायां कूटतानाः सह क्रमैः ॥३३॥
षट्पञ्चाशन्मूर्च्छनास्थाः पूर्णाः कूटास्तु योजिताः ।
लक्षद्वयं सहस्राणि द्व्यशीतिर्द्वे शते तथा ॥३४॥
चत्वारिंशच्च सङ्ख्याता
(2) The enumeration of the complete permutational note-
series : (32d-35a)
We calculate their number as follows. Including the
mūrcchanā-series (krama-s) there are five thousand and forty
(5040) complete permutational note-series in each of the
Mataṅga has spoken of five thousand thirtythree permutational
note-series, and Śārṅgadeva has added the seven mūrcchanā-series to
this number, for each of the fiftysix mūrcchanā-s does have seven series
of its own. Dattila also does not include the seven mūrcchanā-series
(krama-s) in the total number of permutational note-series, but that
Section 4 : Grāma, Mūrcchanā, Krama and Tāna 185
mūrcchanā-s. Thus, the total number of the complete permu-
tational note-series subsisting in the fiftysix mūrcchanā-s comes
to two lakh eightytwo thousand and two hundred forty
(2,82,240). (32d-35a)
makes no difference in principle. The number 5040 is arrived at in
the following manner :
With 1 note the number of possible permutational series of
of the notes will be 1 × 1 = 1
With 2 notes ,, ,, ,, ,, 1 × 2 = 2
With 3 ,, ,, ,, ,, ,, 2 × 3 = 6
With 4 ,, ,, ,, ,, ,, 6 × 4 = 24
With 5 ,, ,, ,, ,, ,, 24 × 5 = 120
With 6 ,, ,, ,, ,, ,, 120 × 6 = 720
With 7 ,, ,, ,, ,, ,, 720 × 7 = 5040
To demonstrate let us take items with 2 and 3 notes to begin
with :—

1. Two-note-series : sa ri, ri sa
2. Three-note-series : sa ri ga } group A with ga constant
   ri sa ga }
   sa ga ri } group B with ri constant
   ga sa ri }
   ri ga sa } group C with sa constant
   ga ri sa }
   Now, it will be observed that the two-note-series is a simple one
   and does not involve any complicated procedure in the working out of
   the different permutations as in the case of the three-note-series, and
   much more so in all the subsequent series. The permutations in the
   three-note-series are so arranged that an attempt is made as a rule to
   keep the last note of the series constant as long as possible. This is
   demonstrated in the four-note-series worked out hereunder.
   S. No. Group A Group B Group C Group D
3. sa ri ga ma sa ri ma ga sa ga ma ri ri ga ma sa
4. ri sa ga ma ri sa ma ga ga sa ma ri ga ri ma sa
5. sa ga ri ma sa ma ri ga sa ma ga ri ri ma ga sa
6. ga sa ri ma ma sa ri ga ma sa ga ri ma ri ga sa
7. ri ga sa ma ri ma sa ga ga ma sa ri ga ma ri sa
8. ga ri sa ma ma ri sa ga ma ga sa ri ma ga ri sa
   It will be observed that on the basis of the above principle, the
   24 permutations of the four-note series have been classified and arranged

186 Treatment of Svara
in four groups in each of which the notes ma, ga, ri and sa respectively
are constant, while the position of other notes keeps on changing. This
gives us the clue to find out the number of possible permutations.
Similarly the 120 permutations of the pentatonic series, 720 of the
hexatonic series and 5040 of the complete series can be worked out
easily. Another principle involved in these calculations is that the
number of permutations of the preceding series denotes the number of
times each note of the succeeding series can be kept constant. Thus, for
example, the permutations of the two-note-series are 2, therefore in the
three-note-series all the three notes can be kept constant twice, leading
us to (2 × 3 = 6) six permutations in all. Likewise, in the four-note-
series, each note can be kept constant for 6 times since the preceding
three-note series has six permutations, leading us to (4 × 6 = 24) twenty-
four permutations in all. That is how the pentatonics are 5 × 24 = 120,
the hexatonics are 6 × 120 = 720, and the complete series are
7 × 720 = 5040.
3. अपूर्णकूटतान-निर्माणम्
,अथापूर्णान्प्रचक्ष्महे ।
एकैकान्त्यान्त्यविरहाद्भेदाः षट् षट्स्वरादयः ॥३५॥
एकस्वरो ऽत्र निर्भेदो ऽप्युक्तो नष्टादिसिद्धये ।
क्रमा अकूटतानत्वे ऽप्युक्तास्तेषूपयोगिनः ॥३६॥
(3) The formation of the incomplete permutational
note-series : (35b-36)
Now, we are to describe the incomplete (permutational
note-series). In each (of the mūrcchanā-s) six varieties are
formed, viz. hexatonic¹ etc. by successively dropping the
last note step by step. Although the monotone has no sub-
The incomplete permutational note-series are formed by the simple
device of dropping the last note from seventh to the second; thus, by
dropping the seventh we get a hexatonic tāna, by dropping the sixth we
get a pentatonic tāna, by dropping the fifth we get a tetratonic note-
series, by dropping the fourth we get a tri-tone note-series, by dropping
the third we get bi-tone series and by dropping the second we get the
mono-tone: these are the six varieties.

1. Hexatonic, i. e., having six tones, the other varieties being
   pentatonic, tetra-tonic series, tritone-series, bitone-series and
   monotone-series.
   Section 4 : Grāma, Mūrcchanā, Krama and Tāna 187
   varieties, it is also counted for the working of naṣṭa.² The
   original-series (krama-s³) which are unrelated to the permu-
   tational note-series have also been included as they will be
   useful with regard to them. (35b-36)
   The number of note-series in each of them will now be described.
   The terms naṣṭa and uddiṣṭa will shortly be explained in their proper
   context, but for the time being the author has pointed out the reason
   for enumerating the mono-tone as a note-series even though it can have
   obviously, no varieties. Similarly, the krama-s or the mūrcchanā-series
   (precisely called here as, the original-series) too have been enumerated
   for technical reasons which will soon be clear.
2. अपूर्ण कूटतानानां सङ्ख्या
   स्युः षाडवानां विशत्या सह सप्त शतानि तु ।
   औडुवानां तु विशत्या सहितं शतमिष्यते ॥३७॥
   चतुःस्वराणां कूटानां चतुर्विंशतिरीरिताः ।
   त्रिस्वराः षड् द्विस्वरौ द्वावेकस्त्वेकस्वरो मतः ॥३८॥
   (4) The enumeration of the different permutational
   note-series : 37-38
   The hexatonic series formed are seven hundred and
   twenty¹, and the pentatonic series are one hundred and
   Ārcika literally means relating to the Ṛgveda (Ṛc + veda). Ṛc is a
   hymn, and the hymns employed in the performance of sacrifices, it
   seems, were sung in the monotone, though the accent system was also
   there with its three accents viz., udātta (raised), anudātta (unraised) and
   svarita (intermediate note). 'K' also says, "Yajñaprayogeṣu ṛcamekasvara-
   śrayatvāt, tatsambandhād ārcikaḥ" i. e., hymns while they are employed in
   (2. Naṣṭa will be explained later.
3. Krama in this context refers to the original (mūla) form of the
   mūrcchanā-s in their ascent i. e., ‘sa ri ga ma pa dha ni’ for
   uttamamandrā. However, the term mūrcchanā-series applied to the
   392 series spoken of vide 19-20 ab ante is not precisely appli-
   cable in the context of the calculation of permutational note-
   series, the krama-s for which may consist of six to one notes
   each in the original order of the mūrcchanā-series concerned.
   Thus, the krama-s in this context have been rendered as the
   original series, as the author himself names them as mūla-
   krama-s somewhere else (verse 40 of this section).
4. This includes the original series (krama-s) as well.

188 Treatment of Svara Section 1 : Grāma, Mūrcchanā, ...
Although later in the classical period gāthā represents non-vedic
5. एकस्वरादि-कूटतानचतुष्टस्य नामानि religious verse, from the above context and also from the following verse
it seems, here it refers to the Yajur-veda. The fact that this tonal distri-
आर्चिको गाथिकश्चाथ सामिको ऽथ स्वरान्तरः । bution is related only to those hymns that are meant to be employed in
the sacrificial acts is also evidenced by the following verse of this work:
एकस्वरादितानानां चतुर्णामभिधा इमाः ॥३९॥ "Ṛk sāmayajuṣāṅgāni ye yajñeṣu prayuñjate, avijñānāddhi śāstrāṇām
teṣāṁ bhavati visvaraḥ." (1.1.4), i e, 'Those parts of the Ṛg. Yajur and the
twenty, the tetratonic permutational note-series are known Sāma-veda-s that are applied in sacrificial acts, due to the lack of scienti-
to be twentyfour, the tritone series are six, the bitone series fic understanding, are recited in the wrong tone (i. e. in an improper
are two and the monotone is one, of course. (37-38) and unprecise way)' Here, the word Yajur seems to have been used for
gāthā. However, in the Vedic literature the word gāthā is also used in
(5) The names of the first four permutational note-series : 39 the sense of gāna i. e., song Swami Prajñānānanda, after giving the
The names for the first four note-series such an mono- illustrations of its use in the Ṛigveda, concludes that "gāthā is, gīyate-iti
tone etc. respectively are ārcika, gāthika, sāmika and svarāntara. gāthā i. e., that which is sung, or is suitable for singing is gāthā". (cf.
(39) Historical Development of Indian Music, pp. 46, 47).
the sacrifices take resort to the monotone, and so being related to them,
the monotone is called ārcika. 6. षाडवकूटतान-सङ्ख्या
Similarly gāthika, is the one that is related to gāthā, i. e nonvedic उक्ताः शुद्धादिभेदेन निगयुक्ताश्चतुर्विधाः ।
religious verse. Likewise the tritone is associated with the Sāmaveda: तयोरेकैकहीनास्तु द्वेधा मूलक्रमा मताः ॥४०॥
the hymns of this particular veda were sung in three tones when employ- षड्जौ च मध्यमौ च चत्वारः स्युर्द्विधा द्विधा ।
ed for sacrificial purposes. However, the Sāmaveda otherwise bears ample
evidence that its music was based on seven notes, though we need not चतुर्धा ऽन्ये दशेत्यष्टाचत्वारिंशदमी क्रमाः ॥४१॥
go into this subject in this context. But it is worthwhile to quote सविंशतिः सप्तशतो प्रागुक्ता गुणिता क्रमैः ।
Swami Prajñānānanda who writes, "The sāmagāna was of various types चतुस्त्रिंशत्सहस्राणि षष्ट्या पञ्च शतानि च ॥४२॥
with different numbers of tone in different vedic recensions (śākhā-s), इति षाडवसङ्ख्या स्यात्,
and they had different methods of rendering. Usually three, four or
five notes were used in the sāmagāna-s, but in the kauthuma recension (6) The enumeration of the hexatonics : 40-43a
(śākhā) the sāmagāna was presented with seven vedic notes. The vedic The original series, inclusive of ga and ni have been
notes were in the downward movement (avarohaṇa-krameṇa), whereas the mentioned¹ to be fourfold viz. śuddhā (standard) etc., but if
modern classical notes are used in the upward movement (ārohaṇa- deprived of either² of them, they are considered to be two-
krameṇa)", (cf. Historical Development of Indian Music, pp. 46, 47). It
may also be observed that the bulk of the hymns of the Samaveda are The mūrcchanā-s such as 'sa ri ga ma pa dha ni : ni dha pa ma ga ri sa'
taken verbatim from the Ṛgveda with a view to their application in the etc. have been explained as fourfold vide verse 16 ante. The four
context of certain sacrifices. It is interesting to refer to the Nāradīya- types enumerated are the standard, the kākalī-inclusive, the antara-
śikṣā in this context which also gives the four names exactly as in the inclusive and the twin-inclusive (i.e. inclusive of both kākalī and antara).
text : "Ārcikaṁ gāthikaṁ caiva sāmikaṁ ca svarāntaram" (1.1.2), and then Corresponding mūrcchanā-series are derived from out of these, taking the
it goes on to explain as under : seven notes in their ascending order. However, as already explained
"Ekāntaraḥ svaro hyṛkṣu gāthāsu dvyantaraḥ svaraḥ : Samasu tryantaram 1. The fortysix mūrcchanā-series mentioned vide 16-17 ante.
vidyādetavat svarato-antaram" (1.1.3) i. e., 'The hymns of the Ṛigveda are 2. That is, if either ni or ga is dropped.
to be sung with a single-space tone, those of the gāthā with the double-
space-tone, while those of the Sāmaveda with the triple-space tone; such
is the scheme of tonal spacing.'

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.