Date: Sun, 30 May 1993 05:58:26 GMT
Message-ID: <krisna.738741506@cs.wisc.edu>
Subject: penta-tonic scales
From a purely "existential" point of view, it is not too difficult to compute the number of possible pentatonic scales---it is simply a matter of proper enumeration/counting.
The counting can be done in 3 ways:
Whatever the categorization, there will be only one type of "p". Clearly, the three categories will yield numbers that are strictly non-decreasing, for the number of possible scales, since each progressively encompasses more possibilities than the previous.
Combinatorially, the problem is one of selecting 4 notes from among 6 ie. from "rgmpdn". There are (6-choose-4) = 15 distinct "groups" of rAgas (and the current discussion on mOhanam and friends pertains to just _one_ of these 15 groups, to wit, the "rgpd" group). Due to the different combinations of notes, each of these 15 groups will yield different numbers of possible scales. The numbers are summarized in the table below [the groups are named by the svaras that they are comprised of and their "yields" for each of the three categories defined above is given. I have numbered the groups arbitrarily].
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Group# Name Category-1 Category-2 Category-3
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1 rgmp 8 12 12
2 rgmd 16 24 36
3 rgmn 16 24 36
4 rgpd 8 12 18
5 rgpn 8 12 18
6 rgdn 16 36 36
7 rmpd 8 8 18
8 rmpn 8 8 18
9 rmdn 16 24 36
10 rpdn 8 12 18
11 gmpd 8 8 18
12 gmpn 8 8 18
13 gmdn 16 24 36
14 gpdn 8 12 18
15 mpdn 8 12 12
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total all 160 204 348
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The mOhanam-group is #4 in the above table. The above figures are applicable only to upAngga (not BAShAngga), simple (the same notes occur in the ArOhaNa and avarOhaNa), non-vakra (monotonically increasing and decreasing scales) rAgas. If vakras are permitted, the number of possible scales is infinite (by an application of the musical version of the "pumping lemma"). If BAShAngga and non-simple scales are permitted, then the total number of possible scales is simply the square of the number of scales as computed above (vis-a-vis the SudhDha-meSra mAELa rAgas). Therefore, the total possible number of pentatonic scales of any kind (but without vakra) is a staggering number: 25600 (1602), 41616 (2042) or 121104 (3482) [depending on your categorization].
Below is a table listing popular rAgas (at least for which I have recordings available). In this list, I have not specified BAShAngga rAgas (except for mohanAnggE), non-simple rAgas (ex. AndhOLekA) or vakra rAgas. I have also excluded multiple scales having the same name (ex. the many names for candhrakAUns; lathAnthapreyA is also called karNAtaka-SudhDha-sAvAErE; rAEvagupthE is also called rAEgupthE; SudhDha-sAvaerE is also called dhurgA/dhAEvakreyA ...).
The list below is annotated with the rAga group (as defined in the above table).
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Scale rAga Group# Category
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s r2 g2 m1 d2 s+ ABOgE 2 1
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s r1 g3 m1 n3 s+ mAEGa-ranjjanE 3 1
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s r1 g2 p d1 s+ BUpALam 4 1
s r1 g3 p d1 s+ rAEvagupthE 4 1
s r1 g3 p d2 s+ raseka-ranjjanE 4 1
s r2 g2 p d2 s+ Seva-ranjjanE 4 1
s r2 g3 p d1 s+ jana-sammOdhenE 4 1
s r2 g3 p d2 s+ mOhanam 4 1
s r3 g3 p d2 s+ mOhanAnggE 4 2
-----------------------------------------------------------------
s r2 g3 p n3 s+ ha~sadhvanE 5 1
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s r2 g3 d2 n3 s+ nerOShTa 6 1
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s r1 m1 p d1 s+ lathAntha-preyA 7 1
s r2 m1 p d2 s+ SudhDha-sAvAErE 7 1
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s r1 m1 p n2 s+ rAEvathE 8 1
s r2 m1 p n2 s+ madhyamAvathE 8 1
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s r2 m1 d1 n3 s+ preyadharShenE 9 1
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s g3 m1 p d2 s+ nAgasvarALE 11 1
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s g2 m1 p n2 s+ SudhDha-dhanyASE 12 1
s g2 m2 p n2 s+ samudhra-preyA 12 1
s g3 m1 p n2 s+ sAvethrE 12 1
s g3 m1 p n3 s+ gamBEra-nAtta 12 1
s g3 m2 p n3 s+ amRtha-varSheNE 12 1
-----------------------------------------------------------------
s g2 m1 d1 n2 s+ hendhOLam 13 1
s g2 m1 d1 n3 s+ candhrakAUns 13 1
s g2 m1 d2 n2 s+ sUrya 13 1
s g3 m2 d2 n3 s+ sunAdha-venOdhenE 13 1
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s g3 p d2 n2 s+ valajE 14 1
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s m1 p d1 n3 s+ dhAEva-ranjjE 15 1
s m1 p d2 n3 s+ kunthala-varALE 15 1
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I do not have examples of any popular rAgas of the "rgmp" (#1) group (I received information about preyadharShenE, of the "rmdn" (#9) group, from Vasudevan Vellambi). Of the 29 rAgas listed above, I believe at least 15-17 are popular in hendhusthAne music as well.
[ Indian Classical Music | Krishna Kunchithapadam ]
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