Metrics and combinatorics: naṣṭaḥ or “lost”

We saw that one way to determine all of the possible combinations of n items taken k at a time is to list them according to a certain procedure, called prastāraḥ or “spreading out.” This procedure assigns a serial number to each of the possible combinations. Now supposing you don’t want to do the entire prastāraḥ, but simply want to figure out the combination that occurs at a given serial number. For instance, among the combinations of 4 syllables, each of which can be either light (।) or heavy (ऽ), what is the exact combination that occurs at serial number 13 in the prastāraḥ?

The procedure for finding out this “lost” (naṣṭaḥ) pattern is as follows:

  1. Start with the serial number of the lost pattern (r).
  2. For each position in the pattern:
    1. if the number is even, the syllable in the position is light; divide this number by 2, and return to step 2;
    2. if the number is odd, the syllable in the position is heavy; add one to this number, then divide the sum by two, and return to step 2.

Hence for r = 13 in the prastāraḥ of four positions (k = 4):

  • First syllable: 13 is odd, hence heavy (ऽ).
  • Second syllable: (13+1)/2 = 7 is odd, hence heavy (ऽ).
  • Third syllable: (7+1)/2 = 4 is even, hence light (।).
  • Fourth syllable: 4/2 = 2 is even, hence light (।).

Hence the lost pattern is ऽऽ।।, which can be verified in the prastāraḥ given as an example in the previous post.

Puṣpadanta and Bharata

iha paṭhitam udāraṁ vācakair gīyamānaṁ
iha likhitam ajasraṁ lēkhakaiś cāru kāvyaṁ
gatavati mitrē mitratāṁ puṣpadantē
bharata tava gr̥hē ’smin bhāti vidyāvinōdaḥ

Here readers recite properly in song.
Here scribes are always writing out beautiful poems.
Since you’ve become friends with Puṣpadanta, Bharata,
the diversions of learning are taking place at your house.

A mālinī verse in Puṣpadanta’s Mahāpurāṇu (Apabhramsha, 965 CE), describing the house of his patron, Bharata, a minister of the Rāṣṭrakūṭa king Kr̥ṣṇa III, which was presumably in Malhed. It praises the poet as much as his patron.

Līlāvaī 858

ता सुक्कपाअवब्भंतरुग्गओ हुअवहो व्व मह मअणो ।
चिंतासमीरणुद्दीविओ व्व हिअअम्मि पज्जलिओ ॥

Then, like a fire that has shot up
into the branches of a dry tree,
desire ignited within my heart,
inflamed by the wind of anxious thought.

From Kōūhala’s Līlāvaī, when the titular character tells her friend, Vicittalēhā, about the effect that a sexual dream has had on her.

Metrics and combinatorics: prastāraḥ or “spreading out”

There were “six combinatorial methods” (ṣaṭ pratyayāḥ, lit. ‘six notions’) that South Asian thinkers applied to metrical forms (as well as other combinatorial problems), and Hēmacandra’s discussion of them, in the seventh chapter of his Chandōnuśāsanam, was nicely explained by Ludwig Alsdorf (“Die Pratyayas: Ein Beitrag zur indischen Mathematik” in Zeitschrift für Indologie und Iranistik 9 [1933]: 97–157, downloadable here).

The general question that these methods were devised to answer is: given a certain number of syllabic positions (k), in which each syllable can either be light or heavy (n = 2), how many distinct combinations of syllables are possible? Having figured out that it is nk, Indian scholars started to ask other questions about these combinations.

The first method is called prastāraḥ or “spreading out.” This is a brute-force method that involves writing out each combination, but it has two important features: (1) each combination is assigned a serial number, and (2) each combination is derived from the previous one in a consistent, algorithmic manner. It is:

  1. On the first line (serial number 1), write k heavy syllables.
  2. For each subsequent line:
    1. take the first heavy syllable of the previous line and change it into a light syllable;
    2. for all of the syllables that follow, copy them exactly as they were in the previous line;
    3. for all of the syllables that preceded, restore them to their original (i.e., heavy) form.
  3. Proceed until you are left with all light syllables.

When k = 4 (the class of meters called pratiṣṭhā), the following prastāraḥ is made (resulting in 16 = 24 possibilities), using the symbols ऽ for “heavy” and । for “light”:

1. ऽऽऽऽ
2. ।ऽऽऽ
3. ऽ।ऽऽ
4. ।।ऽऽ
5. ऽऽ।ऽ
6. ।ऽ।ऽ
7. ऽ।।ऽ
8. ।।।ऽ
9. ऽऽऽ।
10. ।ऽऽ।
11. ऽ।ऽ।
12. ।।ऽ।
13. ऽऽ।।
14. ।ऽ।।
15. ऽ।।।
16. ।।।।

As Alsdorf explains, the same procedure can be applied to mora-counting meters like the gāthā. In such cases, however, the number of moras in a certain position must be kept constant, so when a heavy syllable is replaced in the prastāraḥ with a light syllable, another light syllable must be added (so a heavy syllable is effectively replaced by two light syllables). These “facultative” light syllables are marked in gold in the table below. Generally they do not count as syllables in the prastāraḥ. Hence he gives the following prastāraḥ for the first group of moras (gaṇa-) in the gāthā:

1. ऽऽ
2. ।ऽ
3. ऽ।
4. ।।।

Note that the third combination is ruled out by the rules of the gāthā, which state that the pattern ।ऽ। must not occur in even-numbered positions. Hence from combination no. 2 we change the first heavy syllable (i.e., the last syllable) into a light syllable, revert the initial (non-facultative) light syllable into a heavy, and add a facultative light syllable between them, yielding ऽ।।.

You might have noticed a pattern, which Hēmacandra also describes. In the first column, representing the first syllabic position, you can write ऽ and । alternating with each other for the odd and even serial numbers. Then, in the second column, write ऽ and ऽ on top of each other (for serial numbers 1 and 2) and then । and । on top of each other (for serial numbers 3 and 4). Then, in the third, write ऽ, ऽ, ऽ and ऽ on top of each other (serial no. 1–4) and ।, ।, ।, and । on top of each other (serial numbers 5–8). And so on and so forth. Thus write 2r-1 heavy syllables on top of 2r-1 light syllables in position r of k total positions.

Dhanapāla’s Satyapurīya-śrīmahāvīra-utsāhaḥ

Dhanapāla, the Sanskrit and Prakrit poet associated with the Paramara kings, is known to have written a few Apabhramsha works as well. One of them, the Satyapurīya-śrīmahāvīra-utsāhaḥ (in Apabhramsha: Saccaüri-vīra-ucchāhu), is cited somewhat frequently in secondary literature—not because of any great scholarly interest in the image of Vardhamāna that once stood at Sanchore in Rajasthan, but because it is one of the very few texts (maybe the only one?) to refer to the campaigns of Mahmud of Ghazni written by someone who was alive at the time (unlike, say, Jinaprabha Sūri) and who was not attached to Mahmud’s court (unlike, say, Garzidi). The work was edited from a single manuscript in Patan by the industrious Jinavijaya Muni and published in the third issue of the third volume of Jaina Sāhitya Saṁśōdhaka in 1927. The entire issue can be download from Jain e-Library. Since I have never seen any verses actually quoted and translated from this text, I will do so here.

The text is a hymn to Mahāvīra (Vardhamāna), whose image was installed at Satyapurī, modern Sanchore, in Rajasthan. It is called an utsāhaḥ, which is a specific type of metrical form and genre in Apabhramsha. Interestingly it differs from the utsāhaḥ that is defined by metrical authorities (Svayambhū, Hēmacandra, Rājaśēkhara, etc.), but that can be a topic for another day. The basic idea is that multiple rulers—but above all the turukka or Turks, led by Mahmud of Ghazni—attempted to take away or destroy the image, and failed. They succeeded, though, with other images. Here is verse 3:

bhañjēvi ṇu sirimāladēsu anu aṇahilavāḍaüṁ,
    caḍḍāvalli sōraṭṭhu bhaggu dēulavāḍaüṁ;
sōmēsaru sō tēhi bhaggu jaṇamaaāṇandaṇu,
    bhaggu na siri saccaüri-vīru siddhatthaha nandaṇu.

They destroyed the region of Bhinmal and Anhilwad,
Caḍḍāvalli, Saurashtra, and Dilwara, destroyed,
That Someshwar they destroyed, who brought joy to people’s hearts,
But they did not destroy Śrī Satyapurī-Vīra,
delight of those who have reached their aims.

And here is verse 4, which uses the word turukka:

bahuēhi vi tārāyaṇēhi ravipasaru kiṃ bhijjaï,
    bahuēhi vi visaharēhi milivi kiṁ guruḍu galijjaï;
bahu kuraṅga āruṭṭha karahi kiri kāi mayandaha,
    pūṇihi bahuya turukka kāi saccaüri-jiṇindaha.

Is the light of the sun destroyed constellations, however numerous they may be?
Is Garuḍa felled when he meets with serpents, however numerous they may be?
Do antelopes, however numerous they may be, menace the Lord of Beasts?
Can the Turks, however numerous they may be, touch the Jinēndra of Satyapurī?

I suggest reading pūsihi in the last line (pūsa- = “touch”), since pūṇihi doesn’t mean anything to me.

Since the raid on the Somanath Temple took place in 1024 CE, more than 50 years after Dhanapāla’s first available work (the Pāiyalacchī, a Prakrit dictionary from 972 CE), he was probably around 75 when he composed this hymn. I would guess that it was one of his “retirement projects,” after he completed his Tilakamañjarī for King Bhōja.

Pearls and Black Pepper (repost)

Thi is a repost, upon request, from my old blog at Harvard, which I have since taken down, since I did not find the ‘OpenScholar’ software easy to use. I have corrected a few mistakes and added some more information.

The most common metaphor for a “good” mixture of language in South Asia is “jewel-coral” (maṇi-pravāla-), which has a very long history. Part of what is “good” about this mixture is that the elements retain their individuality in one sense, but in another form a homogenous whole, as do (red) jewels and (red) coral strung on a (red) necklace. What about “bad” mixtures of language?

Nāgavarman, also known as Kavitāguṇōdaya, included the following verse in his Kāvyāvalōkanam (around 1040). There it stands (v. 55 in Dēvīrappa’s edition) as an example of the substitution of k by g. Probably it was not his own composition. The verse was later quoted by the grammarian Keśava or Keśirāja (1260), where I encountered it. For those who are interested there is a short glossary for this verse at the bottom of the post, which also explains the phonological changes that served as Nāgavarman’s excuse for including this verse in his book.

ಪೞಗನ್ನಡದಂ ಪುದುಂಗೊಳೆ
ಕೊೞೆಸಕ್ಕದಮಂ ತಗುಳ್ಚಿ ಜಾಣ್ಗಿಡೆ ಮುತ್ತಂ
ಮೆೞಸುಂ ಕೋದಂತಿರೆ ಪೇ
ೞ್ವೞಿಗವಿಗಳ ಕವಿತೆ ಬುಧರನೆರ್ದೆಗೊಳಿಸುಗುಮೇ

paḻagannaḍaṁ puduṅgoḷe
koḻesakkadamaṃ taguḷci jāṇgiḍe muttaṃ
meḻasuṁ kōd’ antire pē-
ḻv’ aḻigavigaḷa kavite budharan erdegoḷisugum ē?

The poetry of those horrible poets
who write by putting worn-out Sanskrit
together with old Kannada for want of skill,
as if stringing pearl and black pepper—
are skilled readers really taken in?

I think this is probably a counter-metaphor to maṇipravāla: if “jewel-coral” minimizes the contrasts, then “pearl-pepper” maximizes them, since pearl is white and pepper is black. Nāgavarman’s usage is notable here, since it kind of embodies the qualities he is criticizing: muttaṃ is a Sanskrit word, and would have been recognized as such (although modified slightly from its Sanskrit form muktā), but meḻasu has no Sanskrit cognate. (Although probably the Sanskrit word muktā- was itself borrowed from the Dravidian word *muttu; see Dravidian Etymological Dictionary s.v.)

Glossary and notes

  • paḻagannaḍaṁ = paḻa, an adjectival stem meaning ‘old’ + kannaḍaṁ (n.) ‘the Kannada language.’ Note the lenition of the initial k in the seam of an attributive (adjective-substantive) compound. The word is in the nominative case because it is the subject of the converb puduṅgoḷe.
  • puduṅgoḷe = the ‘absolute converb’ (I will explain this term in a later post) for pudu-ṁ-goḷ, a compound verb meaning literally ‘taking (koḷ) entry (pudu).’ The absolute converb is formed by adding -e directly to the verbal root. Note that -ṁ has been added to the nominal element, and the initial k- of koḷ has been lenited to g. The entire verb phrase means ‘[in such a way that] Old Kannada enters into [it].’
  • koḻesakkadamaṃ = koḻe ‘old, dirty, worn-out’ + sakkadaṁ (n.) ‘the Sanskrit language.’ Another attributive compound. This is in the accusative case because it is the object of the serial converb taguḷci.
  • taguḷci = the past-tense serial converb of taguḷcu ‘having caused to join’ (the transitive version of taguḷ ‘join’). The entire verb phrase means ‘having joined the worn-out Sanskrit language [with it].’
  • jāṇgiḍe = the absolute converb of jāṇ ‘cleverness, skill’ + kiḍu ‘decay, waste.’ If the reading is jāṇgiḍe rather than jāṇ kiḍe, this is another example of the lenition of an initial k- in the seam of a compound. This is an adverbial verb phrase meaning ‘[in such a way that their] cleverness deteriorates.’
  • muttaṁ (n.) ‘pearl.’ In the nominative because it is the co-subject (with meḻasu) of the verb kōdantire.
  • meḻasuṁ = meḻasu (n.) ‘black pepper’ + uṁ conjunctive particle.
  • kōdantire = an ‘absolute converb’ indicating comparison, from kōdu ‘be threaded, be on a string’ + ant’ ire ‘in that way.’ The entire verb phrase means ‘in the same way that pearl[s] and black pepper[-corns] might be threaded [on a string].’
  • pēḻva = a non-past ‘relative participle’ (peyareccam) from pēḻ ‘speak, say, compose,’ modifying (as usual) the constituent to its right, namely, aḻigavigaḷa kavite ‘the poetry of bad poets, which [they] compose…’
  • aḻigavigaḷa = aḻi ‘bad, terrible’ + kavi ‘poet’ (from Sanskrit kavi-). Note again the lenition of initial k- to g- in the seam of an attributive compound. The word is in the genitive plural (pl. ending -gaḷ, gen. ending -a) because it modifies the constituent to its right.
  • kavite ‘poetry’ (from Sanskrit kavitā-). The word is in the nominative because it is the subject of the verb erdegoḷisugum.
  • budharan = budha-r-an, the accusative (-aṁ) of the animate plural (-r) of budha- ‘wise’ (= Sanskrit). It is the object of the verb erdegoḷisugum.
  • erdegoḷisugum = the imperfective non-past form of erde-koḷisu, from erde ‘heart’ + koḷisu ‘cause to take,’ hence ‘enchant, captivate.’ This verb is the causative of koḷ. Note, one last time, the lenition of k- into g- in a compound verb. Its equivalent in Sanskrit would be manō harati.
  • ē an interrogative particle that turns the sentence into a yes-or-no question, with an implied answer of ‘no.’

Dhanapāla’s Contradictions (vv. 27–28)

These verses, the culmination of Dhanapāla’s contradictory praise, constitute a syntactically-connected pair (a yugalam) that involves a number of references to the Jain scriptures.

How is it that this is your teaching, Teacher of the World?
It is adorned by the beauty of the glances from an even number of eyes,
yet it has the glances of an odd number of eyes.
In it endless numbers of supplicants can be seen,
yet it is seen with only ten supplicants.
Devoid of reasoning, it is heavy with reasoning.
Though made of alcohol, it takes away delusion.
Teeming with antelopes, though without deer.
    [How is it that this is your teaching, Teacher of the World?
    It is adorned by the beauty of the doctrines of Śramaṇas,
    and contains the difficult Dr̥ṣṭivāda.
    It shines with the Daśavaikālika,
    and illuminates the endless perspectives of the right path.
    It is heavy with the reasoning of the Niryuktis.
    It makes the mind joyful, and removes delusion.
    It is full of the essential aṅgas,
    and by it, carelessness is removed.]

samaṇayaṇavāyasōhāvihūsiyaṁ visamadiṭṭhivāyaṁ pi
dasavēyāliyapayaḍaṁ pi payaḍiyāṇaṁtamaggaṇayaṁ [27]
nijjutti juttiguruaṁ jayaguru maïrāmayaṁ pi mōhaharaṁ
sāraṁgasaṁgayaṁ gayamayaṁ pi kaha sāsaṇaṁ tumha [28]

समणयणवायसोहाविहूसियं विसमदिट्ठिवायं पि ।
दसवेयालियपयडं पि पयडियाणंतमग्गणयं ॥ २७ ॥
निज्जुत्ति जुत्तिगुरुअं जयगुरु मइरामयं पि मोहहरं ।
सारंगसंगयं गयमयं पि कह सासणं तुम्ह ॥ २८ ॥

The dr̥ṣṭivāda is the twelfth aṅga, or part of the Jain scriptures, that was reportedly lost before the entire set of scriptures was committed to writing. The Daśavaikālika is one of the Jain canonical works (considered a mūlasūtram) that gives guidelines for ethical and monastic conduct. The Niryuktis are versified commentaries (really lists of topics for oral explication) that came to be attached to the scriptural texts in the beginning of this era.

Dhanapāla’s Contradictions (vv. 25–26)

How is it that your voice is the abode of marine life,
though it is not full of the beauty of sea-monsters,
not rich in fish, without cranes or ducks,
and always without many conches?
    [How is it that your speech is attended by the thunder of the clouds,
    filled with streams of nectar, unconsumed by jealous men,
    in which the primary sound is not lost, and which is always without agitation?]

amayarasōhāliddhā amacchariddhā asārasarahaṁgā
kaha niccam asaṁkhōhā jalayaravasahī vi tuha vāṇī [25]

अमयरसोहालिद्धा अमच्छरिद्धा असारसरहंगा ।
कह निच्चं असंखोहा जलयरवसही वि तुह वाणी ॥ २५ ॥

How is it that satisfaction is reached, Greatest of Jinas,
by crowds of various kinds, betaking themself to the gods —
crowds of beings who have never come to your samavasaraṇa?
    [How is it, Greatest of Jinas, that your lotus feet,
    which are worshipped by the gods, create an unparalleled satisfaction
    for those beings who come to your samavarasaṇa?]

kaha kīraü paḍiōsō asamōsaraṇāgayāṇa jaṁtūṇaṁ
surasēviēṇa tuha jiṇavariṁda pāyāraviṁdēṇa [26]

कह कीरउ पडिओसो असमोसरणागयाण जंतूणं ।
सुरसेविएण तुह जिणवरिंद पायारविंदेण ॥ २६ ॥

A samavasaraṇa- is the event at which a realized Jina preaches the dharma for the first time.

Dhanapāla’s Contradictions (vv. 22–24)

How it is, Conqueror, that you make available to living beings in this world
an auspicious happiness which is both wholesome and not,
in which there is both eternal knowledge and no knowledge at all?
    [How is it, Conqueror, that you make available a cure
    to living beings in this world who are sick and depressed:
    a cure consisting in auspicious happiness and eternal knowledge?]

kaha sāsayaviṇṇāṇaṁ nivviṇṇāṇaṁ bhavammi jīvāṇaṁ
sivasukkhamakallāṇaṁ kallāṇaṁ jiṇa paṇāmēsi [22]

कह सासयविण्णाणं निव्विण्णाणं भवम्मि जीवाणं ।
सिवसुक्खमकल्लाणं कल्लाणं जिण पणामेसि ॥ २२ ॥

Lord, how is it that you make known a dharma
that is sweet because of its ripeness,
despite being the basis for the twenty diseases and poisoned food,
which is accepted by the sages, yet originates from women?
    [Lord, how is it that you make known a dharma
    that is sweet because of its ripeness,
    a place for those who are dejected to take confidence,
    nectar to the ears, and accepted by the sages?]

nāha pariṇāmamahuraṁ visannavīsāmaṭhāmabhūaṁ pi
kaha pannavēsi kannāmayaṁ pi muṇisaṃmayaṁ dhammaṃ [23]

नाह परिणाममहुरं विसन्नवीसामठामभूअं पि ।
कह पन्नवेसि कन्नामयं पि मणिसंमयं धम्मं ॥ २३ ॥

Take on your independence, seeing everything else to be trivial.
Although you have conquered all of the sense faculties,
your power is still weak, and that does not make sense.
    [With awareness, take on pure compassion as if it were your total supreme self.
    You have conquered all of the sense faculties,
    and you are no longer attached. Is that at all strange?]

sacchandayaṁ samuvvaha savvaṁ param appayaṁ va picchantō
jaṁ puṇa jiasayalakkhō vi appasattō si tamajuttaṁ [24]

सच्छंदयं समुव्वह सव्वं परमप्पयं व पिच्छंतो ।
जं पुण जिअसयलक्खो वि अप्पसत्तो सि तमजुत्तं ॥ २४ ॥

Dhanapāla’s Contradictions (19–21)

How is it that you are devoted to the utterance of sāmans,
yet are an enemy of the triple Vedas?
    [How is it that you are given to making to encouraging statements,
    and are the single enemy of the three kinds of sexual desire?]

How indeed are you known by the title of Brahmā,
yet are also Śiva?
    [How is it that you are said to be
    the undisputed lord of all beings?]

sāmapaüttipasattō kahaṁ si vēattaïkkasattū vi
kaha ṇu visaṅkō vi payāvaï tti payaḍaṁ vahasi saddaṁ [19]

सामपउत्तिपसत्तो कहं सि वेअत्तइक्कसत्तू वि ।
कह णु विसंको वि पयावइ त्ति पयडं वहसि सद्दं ॥ १९ ॥

The three kinds of sexual desire in the case of Jainism are sexual desire for a man, sexual desire for a woman, and “non-binary” sexual desire (napuṁsakavēda-).

How is it, you who ward off the attacks of the wicked,
that you produce a pack of tigers,
attended by vultures and demons, when your feet are grasped?
    [How is it, you who ward off the attacks of the wicked,
    that when your feet are grasped, you produce a cluster of lotuses,
    each containing a thick layer of pollen on top?]

kaha maṁsaluddharayaṇiaraparigayaṁ puṇḍarīariñchōliṁ
vāriaasaïpariggaha pariggahē kuṇasi calaṇāṇa [20]

कह मंसलुद्धरयणिअरपरिगयं पुंडरीअरिंछोलिं ।
वारिअअसइपरिग्गह परिग्गहे कुणसि चलणाण ॥ २० ॥

How is it that you, despite being the leader of the tax-revenue office,
have no concern at all with regions and villages?
    [How is it that you no longer have any eagerness for all sensory objects,
    despite being the leader of all of the senses?]

How is it, Lord, that despite being master over your faculties,
you do not restrain your faculties?
    [How is it, Lord, that you are master over your faculties,
    and restrain the mind?]

gayavisayagāmatattī kahaṁ si nēā vi akkhavaḍalassa
indiapahū vi kaha pahū nōindianiggahaṁ kuṇasi [21]

गयविसयगामतत्ती कहं सि नेआ वि अक्खवडलस्स ।
इंदिअपहू वि कह पहू नोइंदिअनिग्गहं कुणसि ॥ २१ ॥

The first half involves the technical meaning of akṣapaṭala- as the “house of public records,” which is discussed in Kauṭilya’s Arthaśāstram.