In an earlier essay, I had detailed the derivation of the 72 `standard' mAELa rAgas from a minimal set of starting points using only iterated applications of the operations of graha-BAEdha and the interchange of the SudhDha and prathe maDhyamas. The conclusions of the essay (part I) were that:
Part of the reason why the derivation was not `comprehensive' enough is because the graha-BAEdha operation produces many invalid scales. Another reason is, of course, that the operations chosen for the derivation, namely graha-BAEdha and maDhyama interchange, are not `powerful' enough.
This essay (part II) will focus on the first aspect i.e. that of expanding the scope of what is considered a valid mAELa rAga and reworking the derivations using the same operations as before.
The notions of posets and closures, and the nature of the graha-BAEdha and maDhyama interchange operations (with examples) were explained in part I and will not be repeated here. The extention of the graba-BAEdha (operation `gB') and SudhDha-prathe maDhyama interchange (operation `Sp') to dhve-maDhyama mAELas should be obvious and straight-forward.
As commonly defined and understood, the mAELa rAgas are sampUrNa-sampUrNa, avakra, avarjya i.e. they have 7 svaras in the ArOhaNa and avarOhaNa, the svaras are in their canonical order of increasing frequency, and the full complement of the saptha svaras occur in the scale. There are 6 possible `r-g' combinations, 6 again for `d-n' and two for `m-p', giving a total of 72 (6 x 6 x 2) mAELa rAgas.
However, musicologists have proposed relaxing the requirement that mAELa rAgas always contain a maDhyama and a panjcama, and that mAELa rAgas with both SudhDha and prathe maDhyamas, but without a panjcama, be allowed as legitimate janaka rAgas. The use (although fairly rare) of janya rAgas with both maDhyamas (and in some cases without a panjcama) offers some justification, based on musical practice, for this theoretical scheme.
Allowing dhve-maDhyama-panjcama-varjya mAELa rAgas into the scheme brings the total of mAELa rAgas to 108 (with 3 instead of 2 `m-(m|p)' combinations (where the notation indicates either `m-m' or `m-p' combinations) and the usual 36 possibilities for `r-g' and `d-n' combined). The nomenclature of these dhve-maDhyama mAELas is due to thanjjAvUr s.kalyANarAman, and conforms to the katapayAdhe sangKya (when a shift of 72 is added to the computed numbers; the new mAELas have sangKyas from 73 to 108 by convention).
The rationale for including the 36 additional mAELa rAgas in the graha-BAEdha and maDhyama interchange derivations is that many times the product of graha-BAEdha on a `standard' mAELa rAga is dhve-maDhyama mAELa rAga (although at other times it is not). Such scales would not be considered as valid points for further derivation in the 72 mAELa rAga scheme but would be subject to derivation in the 108 mAELa rAga scheme.
However, the 36 new scales are 36 new `problems', points of potential failure in the derivation. While the new mAELas may allow the derivation to proceed in some cases, they do not help any in other cases and may even cause additional problems in the derivation (since the new mAELas themselves need to be derived in turn). It is not obvious a priori (atleast to me) how (if at all) the derivation forest and the clique structure of the iterated graha-BAEdha and maDhyama interchange operations would be affected by the new mAELa rAgas.
The curious reader should pause now to think about what might happen to the derivation forest in the 108 mAELa scheme, before looking ahead to the answers. Incidentally, none of the derivations presented here were done by hand---the search (breadth-first), the graha-BAEdha and maDhyama interchange operations, and the pretty-printing of the derivation forest were all done by computer programs. I am also fairly confident that there are no errors in the derivations for the very reason that I did not do them by hand.
Starting rAga: kanakAnggE
Tree depth: 13
Partition size: 78
kanakAnggE
|
|-> mAyAmALavagAULa
| |
| |-> kEravANE
| | |
| | |-> KaraharapreyA
| | | |
| | | |-> DhEraSangkarABaraNam
| | | |-> BavapreyA
| | | | |
| | | | |-> hanumathOdE
| | | | |-> cethrAmbarE
| | | | | |
| | | | | |-> naTaBIravE
| | | | | |-> nagAnandhenE
| | | | | |-> SuBapanthuvarALE
| | | | | | |
| | | | | | |-> DhAENukA
| | | | | | |-> rasekapreyA
| | | | | | +-> mAyASrE
| | | | | |
| | | | | |-> kOSalam
| | | | | +-> vakuLaSrE
| | | | |
| | | | |-> nAsekABUShaNE
| | | | | |
| | | | | |-> nAtakapreyA
| | | | | | |
| | | | | | |-> harekAmBOjE
| | | | | | |-> ShadvedhmArgenE
| | | | | | |-> DharmavathE
| | | | | | | |
| | | | | | | |-> gAUremanOharE
| | | | | | | |-> rAmapreyA
| | | | | | | | |
| | | | | | | | |-> cakravAham
| | | | | | | | |-> JangkAraSrE
| | | | | | | | +-> gAUreSrE
| | | | | | | |
| | | | | | | |-> lathAnggE
| | | | | | | | |
| | | | | | | | |-> sarasAnggE
| | | | | | | | |-> gavAmBOdhE
| | | | | | | | | |
| | | | | | | | | |-> sAEnAvathE
| | | | | | | | | |-> veSvAmbarE
| | | | | | | | | | |
| | | | | | | | | | |-> hAtakAmbarE
| | | | | | | | | | |-> JangkAradhvanE
| | | | | | | | | | | |
| | | | | | | | | | | |-> sUryakAntham
| | | | | | | | | | | |-> jalArNavam
| | | | | | | | | | | | |
| | | | | | | | | | | | |-> rathnAnggE
| | | | | | | | | | | | +-> nAgaSrE
| | | | | | | | | | | |
| | | | | | | | | | | |-> SyAmalAnggE
| | | | | | | | | | | +-> DhEraSrE
| | | | | | | | | | |
| | | | | | | | | | |-> JAlavarALE
| | | | | | | | | | | |
| | | | | | | | | | | |-> gAnamUrthe
| | | | | | | | | | | +-> calaSrE
| | | | | | | | | | |
| | | | | | | | | | +-> SUlaSrE
| | | | | | | | | |
| | | | | | | | | |-> gAnaSrE
| | | | | | | | | +-> vAgaSrE
| | | | | | | | |
| | | | | | | | |-> gamanaSrama
| | | | | | | | |-> rathnaSrE
| | | | | | | | +-> hareSrE
| | | | | | | |
| | | | | | | |-> sunAdhaSrE
| | | | | | | +-> KaraSrE
| | | | | | |
| | | | | | |-> reShaBapreyA
| | | | | | | |
| | | | | | | |-> cArukAEsE
| | | | | | | |-> suvarNAnggE
| | | | | | | | |
| | | | | | | | |-> kOkelapreyA
| | | | | | | | +-> saraSrE
| | | | | | | |
| | | | | | | +-> cAruSrE
| | | | | | |
| | | | | | +-> naTaSrE
| | | | | |
| | | | | |-> vAgadhESvarE
| | | | | +-> kEraNaSrE
| | | | |
| | | | +-> DhAENuSrE
| | | |
| | | |-> ShaNmuKapreyA
| | | |-> hAEmavathE
| | | |-> vAcaspathE
| | | |-> mAEcakalyANE
| | | +-> hanumaSrE
| | |
| | |-> SUlenE
| | |-> nAmanArAyaNE
| | | |
| | | |-> vakuLABaraNam
| | | +-> kOkelaSrE
| | |
| | |-> se~hAEndhramaDhyamam
| | +-> nAtakaSrE
| |
| |-> calanAtta
| |-> kAmavarDhanE
| |-> kanakaSrE
| +-> cakraSrE
|
|-> sAlagam
| |
| +-> hataSrE
|
+-> sUryaSrE
Starting rAga: vanaspathe
Tree depth: 3
Partition size: 6
vanaspathe
|
|-> navanEtham
+-> kAnthAmaNE
|
|-> mAraranjjanE
+-> pAvanE
|
+-> mAnavathE
Starting rAga: thAnarUpE
Tree depth: 1
Partition size: 2
thAnarUpE
|
+-> raGupreyA
Starting rAga: rUpavathE
Tree depth: 2
Partition size: 4
rUpavathE
|
|-> dhevyamaNE
| |
| +-> gangaSrE
|
+-> rAgaSrE
Starting rAga: gAyakapreyA
Tree depth: 4
Partition size: 11
gAyakapreyA
|
|-> gAnggAEyaBUShaNE
| |
| |-> varuNapreyA
| | |
| | |-> nEthemathE
| | +-> jyOthesvarUpeNE
| | |
| | +-> rAgavarDhanE
| |
| |-> DhAthuvarDhanE
| +-> gAyakaSrE
|
|-> DhavaLAmbarE
| |
| +-> mAnaSrE
|
+-> vanaSrE
Starting rAga: yAgapreyA
Tree depth: 1
Partition size: 2
yAgapreyA
|
+-> sucarethra
Dead-ends (partitions of size 1):
thAnaSrE
rUpaSrE
varuNaSrE
mAraSrE
yAgaSrE
Starting rAga: kanakAnggE
Tree depth: 14
Partition size: 78
kanakAnggE
|
|-> sAlagam
| |
| |-> kAmavarDhanE
| | |
| | |-> mAyAmALavagAULa
| | |-> se~hAEndhramaDhyamam
| | | |
| | | |-> vakuLABaraNam
| | | | |
| | | | |-> nAmanArAyaNE
| | | | +-> kOkelaSrE
| | | |
| | | |-> kEravANE
| | | |-> hAEmavathE
| | | | |
| | | | |-> hanumathOdE
| | | | | |
| | | | | |-> nagAnandhenE
| | | | | | |
| | | | | | |-> DhAENukA
| | | | | | | |
| | | | | | | |-> calanAtta
| | | | | | | |-> SuBapanthuvarALE
| | | | | | | +-> mAyASrE
| | | | | | |
| | | | | | |-> SUlenE
| | | | | | |-> ShaNmuKapreyA
| | | | | | |-> cethrAmbarE
| | | | | | +-> vakuLaSrE
| | | | | |
| | | | | |-> vAgadhESvarE
| | | | | | |
| | | | | | |-> ShadvedhmArgenE
| | | | | | | |
| | | | | | | |-> nAtakapreyA
| | | | | | | |-> gAUremanOharE
| | | | | | | | |
| | | | | | | | |-> cakravAham
| | | | | | | | | |
| | | | | | | | | |-> rAmapreyA
| | | | | | | | | |-> JangkAraSrE
| | | | | | | | | +-> gAUreSrE
| | | | | | | | |
| | | | | | | | |-> sarasAnggE
| | | | | | | | | |
| | | | | | | | | |-> sAEnAvathE
| | | | | | | | | | |
| | | | | | | | | | |-> hAtakAmbarE
| | | | | | | | | | | |
| | | | | | | | | | | |-> gAnamUrthe
| | | | | | | | | | | | |
| | | | | | | | | | | | |-> JAlavarALE
| | | | | | | | | | | | +-> calaSrE
| | | | | | | | | | | |
| | | | | | | | | | | |-> veSvAmbarE
| | | | | | | | | | | |-> SyAmalAnggE
| | | | | | | | | | | | |
| | | | | | | | | | | | |-> rathnAnggE
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |-> jalArNavam
| | | | | | | | | | | | | +-> nAgaSrE
| | | | | | | | | | | | |
| | | | | | | | | | | | |-> JangkAradhvanE
| | | | | | | | | | | | |-> gamanaSrama
| | | | | | | | | | | | +-> DhEraSrE
| | | | | | | | | | | |
| | | | | | | | | | | +-> SUlaSrE
| | | | | | | | | | |
| | | | | | | | | | |-> gavAmBOdhE
| | | | | | | | | | |-> gAnaSrE
| | | | | | | | | | +-> vAgaSrE
| | | | | | | | | |
| | | | | | | | | |-> sUryakAntham
| | | | | | | | | |-> lathAnggE
| | | | | | | | | |-> rathnaSrE
| | | | | | | | | +-> hareSrE
| | | | | | | | |
| | | | | | | | |-> DharmavathE
| | | | | | | | |-> sunAdhaSrE
| | | | | | | | +-> KaraSrE
| | | | | | | |
| | | | | | | |-> cArukAEsE
| | | | | | | | |
| | | | | | | | |-> kOkelapreyA
| | | | | | | | | |
| | | | | | | | | |-> suvarNAnggE
| | | | | | | | | +-> saraSrE
| | | | | | | | |
| | | | | | | | |-> reShaBapreyA
| | | | | | | | +-> cAruSrE
| | | | | | | |
| | | | | | | |-> vAcaspathE
| | | | | | | +-> naTaSrE
| | | | | | |
| | | | | | |-> nAsekABUShaNE
| | | | | | +-> kEraNaSrE
| | | | | |
| | | | | |-> BavapreyA
| | | | | +-> DhAENuSrE
| | | | |
| | | | |-> naTaBIravE
| | | | |-> KaraharapreyA
| | | | |-> harekAmBOjE
| | | | |-> DhEraSangkarABaraNam
| | | | |-> mAEcakalyANE
| | | | +-> hanumaSrE
| | | |
| | | |-> kOSalam
| | | +-> nAtakaSrE
| | |
| | |-> rasekapreyA
| | |-> kanakaSrE
| | +-> cakraSrE
| |
| +-> sUryaSrE
|
+-> hataSrE
Starting rAga: vanaspathe
Tree depth: 4
Partition size: 6
vanaspathe
|
+-> navanEtham
|
+-> mAraranjjanE
|
|-> mAnavathE
| |
| +-> pAvanE
|
+-> kAnthAmaNE
Starting rAga: thAnarUpE
Tree depth: 1
Partition size: 2
thAnarUpE
|
+-> raGupreyA
Starting rAga: rUpavathE
Tree depth: 2
Partition size: 4
rUpavathE
|
|-> dhevyamaNE
| |
| +-> rAgaSrE
|
+-> gangaSrE
Starting rAga: gAyakapreyA
Tree depth: 5
Partition size: 11
gAyakapreyA
|
|-> DhavaLAmbarE
| |
| |-> DhAthuvarDhanE
| | |
| | |-> gAnggAEyaBUShaNE
| | |-> nEthemathE
| | | |
| | | |-> varuNapreyA
| | | +-> rAgavarDhanE
| | | |
| | | +-> jyOthesvarUpeNE
| | |
| | +-> gAyakaSrE
| |
| +-> vanaSrE
|
+-> mAnaSrE
Starting rAga: yAgapreyA
Tree depth: 1
Partition size: 2
yAgapreyA
|
+-> sucarethra
Dead-ends (partitions of size 1):
thAnaSrE
rUpaSrE
varuNaSrE
mAraSrE
yAgaSrE
From the previous two derivation forests, it can be seen that the order of application of the graha-BAEdha and maDhyama interchange operations does not affect the parition sizes or membership (it merely changes the tree structure in some cases). It is also interesting to compare the derivation forest for the 108 mAELa rAgas with that for the 72 mAELa rAgas. In some cases (e.g. the `vanaspathe' derivation tree) nothing new is gained from the new mAELas.
The addition of the dhve-maDhyama mAELas also leaves 5 irreducible residues ("dead-ends" as I have called them) which cannot be profitably subjected to graha-BAEdha (they can however be subjected to maDhyama interchange, which is quite useless since, if it has not become apparent by now, the application of `Sp' to a dhve-maDhyama mAELa is the identity operation since these mAELas have both maDhyamas).
In some sense, each mAELa rAga can be pictured as sitting in its own little "well". Through the application of the `gB' and `Sp' operations, the mAELas climb out of their well and travel to other wells in their "quest" to visit every little well in their "universe". The `standard' mAELa rAgas can potentially use both `gB' and `Sp' to climb out of their wells (`Sp' always works for every one of these 72 mAELa rAga, even those of the notorious `RShe' cakra that are impervious to `gB' because of their 4-semitone `g-m' interval). The dhve-maDhyama mAELas, unlike the `standard' ones, cannot use the `Sp' operation at all to do any climbing since the identity operator simply returns them to their own well. If the `gB' operator fails on them, as it does so on the residues above, the rAgas will define singleton sets.
The following table summarizes the results of the derivation trees. The partitions (or rAga cliques) are named by the starting mAELa that I have used. There is nothing significant about my starting points, these mAELas happen to be the ones with the smallest katapayAdhe number of the poset partition to which they belong.
---------------------------------------------------------------- clique-name (gB o Sp) depth (Sp o gB) depth partition-size ---------------------------------------------------------------- kanakAnggE 13 14 78 vanaspathE 3 4 6 thAnarUpE 1 1 2 rUpavathE 2 2 4 gAyakapreyA 4 5 11 yAgapreyA 1 1 2 ---------------------------------------------------------------- thAnaSrE 0 0 1 rUpaSrE 0 0 1 varuNaSrE 0 0 1 mAraSrE 0 0 1 yAgaSrE 0 0 1 ----------------------------------------------------------------
Quite clearly, the total number of rAgas in all partitions put together works out to the full poset size of 108.
In this purely mathematical sense, even the expanded set of 108 mAELa rAgas is not elegant enough (or equivalently, the graha-BAEdha and SudhDha-prathe maDhyama interchange operations are not sufficiently powerful) to form a complete structure. I.e. it is not possible to derive all the 108 mAELa rAgas from a single starting point by the iterated application of the graha-BAEdha and SudhDha-prathe maDhyama interchange operations alone. The 108 mAELa scheme also reveals the existence of residues that cannot be related to any other mAELa rAga by the use of the `gB' and `Sp' operations.
[ Indian Classical Music | Krishna Kunchithapadam ]
Last updated: Sun Jun 27 17:00:19 PDT 2004
URL: http://geocities.datacellar.net/krishna_kunchith/rmic/rmic.1996.08.01.html