utkramas represent changed order or permutations distinct from kramas
"and utkramas 54
(changed order or permutations)"
79
TRANSLATION
(Anu. 56)
The Kūṭa-tānas are five thousand and thirty-three in total. Dattila has also said-
"The pūrņa (complete i.e. heptatonic) (tānas) are five thousand and thirty-
three in each grāma; now they speak about the method of their numeration or
calculation."
[ Dattilam 39 ]
Here the calculation is (thus) — the earlier tāna-prayoga 51 (number of
tānas) is multiplied by the later or next tāna-prayoga 52 (number of svaras
constituting the tāna). Similarly has said Dattila -
"One should multiply the earlier (numbers) (with the later number of the
svaras of a tana), taken without any omission, the resulting number would be
constituted of kramas 53 (original order or combinations) and utkramas 54
(changed order or permutations) (of the later i.e. last number of the series);
out of this (aggregate number obtained by multiplication) the kramas (tānas in
the original order) are equal to the multiplicator 55 (guṇakāra or guṇaka i.e. the
last number of the series) and the remainder will be the number of utkramas 56
(permutations)".
[ Anu. 57 ]
Now he (the author) speaks about the process of multiplication—one
should multiply the number one (of the) earlier one i.e. of the first tana of one
svara with the number two (of the) later (tana of two svaras) and the result is
two itself. Those two multiplied by three (the number of svaras in the next tana)
become six. Those [ six ] multiplied by four (the number of svaras in the next
tāna) become twenty-four. That (group of) twenty-four multiplied by five (the
number of svaras in the next tana) becomes one hundred and twenty. This
(group of) one hundred and twenty, multiplied by six (the number of svaras in
the next tana) becomes seven hundred and twenty. The (seven hundred and
twenty) multiplied by seven (the number of svaras in the next tana) become five
thousand and forty.
[ Anu. 58 ]
Out of these (5040), subtracting the seven that are made up of krama
(original order), the kūṭa-tānas (permutations) born of the use of seven svaras
are five thousand and thirty-three.
The prastāra is thus -
Out of them (seven svaras) the svara şadja has only one order as sa. ṛṣabha
(added to sadja) has only one krama (original order) as sa - ri; the second is
the vyutkrama (reverse order) as n - sa. Gândhāra (added to sa - n) has one