The technique of modal-shift of the tonic


Date: 4 May 93 00:30:09 GMT
Message-ID: <krisna.736475409@cs.wisc.edu>
References: <1993May3.190254.22238@bmerh85.bnr.ca>
Subject: Re: Shruti Bheda in Carnatic music

In article <1993May3.190254.22238@bmerh85.bnr.ca> shiv@bnr.ca (Shiv Naimpally) writes:

While searching for a recording in Shubhapantuvaralli for someone, I came across a ragam tanam pallavi of Seshagopalan. I was told he does 'shruti bheda' wherein the notes of the raga are used to perform other ragas. Could someone explain this in more detail and with written example(s)? What is the equivalant in Hindusthani music? I was also told that is quite tricky to do, and there are few ragas that lend themselves to shruti bheda the other being hindolam). Any info is appreciated.

graha-BAEdha is one of the many purely mathematical aspects of Indian classical music. graha-BAEdha works by keeping the inter-note intervals fixed and choosing different notes as the tonic (ADhAra Shadja). This results in the redistribution of the intervals of a scale and hence in different scales, as the tonic shifts along the different notes.

The ancient mUrcanas were supposed to have been discovered/invented by the process of graha-BAEdha. The scale of the sAma gAna (the closest modern rAga is KaraharapreyA) gives rise to hanumathOdE, mAEcakalyANE, harekAmBOjE, naTaBIravE and DhEraSangkarABaraNam. The grahas are enumerated in the table below:

-----------------------------------------------------------------
   graha   rAga                   mAELa   svara
-----------------------------------------------------------------
     s     KaraharapreyA           22     s r2 g2 m1 p  d2 n2
     r     hanumathOdE              8     s r1 g2 m1 p  d1 n2
     g     mAEcakalyANE            65     s r2 g3 m2 p  d2 n3
     m     harekAmBOjE             28     s r2 g3 m1 p  d2 n2
     p     naTaBIravE              20     s r2 g2 m1 p  d1 n2
     d     -                        -     -
     n     DhEraSangkarABaraNam    29     s r2 g3 m1 p  d2 n3
-----------------------------------------------------------------

The DhIvatha graha does not produce a valid mAELa rAga. This turns out to be very common with graha-BAEdha. The inter-note intervals cannot be arbitrarily redistributed to get valid rAgas.

graha-BAEdha on a mAELa rAga will result in either another mAELa rAga or an invalid scale. There is however _no_ mAELa rAga whose graha-BAEdha is itself (except for the trivial case of the Shadja graha). There is also no group of mAELa rAgas which derive valid mAELas on all its grahas---the above example of 6 valid mAELas is the maximum size of a graha-BAEdha clique. There is no other clique of 6 mAELa rAgas related by graha-BAEdha, among the 72 mAELa rAgas.

graha-BAEdha can also be performed on janya rAgas and the results will be other janya rAgas. One of the best known (and quoted) examples is that of mOhanam and friends. The table below enumerates the grahas.

-----------------------------------------------------------------
            graha   rAga               svara
-----------------------------------------------------------------
              s     mOhanam            s r2 g3 p  d2
              r     maDhyamAvathE      s r2 m1 p  n2
              g     hendhOLam          s g2 m1 d1 n2
              p     SudhDha-sAvAErE    s r2 m1 p  d2
              d     SudhDha-dhanyAsE   s g2 m1 p  n2
-----------------------------------------------------------------

There exist certain rAgas which are graha-BAEdha fixed points, i.e. graha-BAEdha on any of their notes will result in the same rAga. It is provable that there can exist only 3 such valid scales. The proof is as follows

graha-BAEdha involves the redistribution of the inter-note intervals of the notes in an octave.
There are 12 notes to an octave.
The number of notes/octave of a graha-BAEdha fixed point must be a divisor of the number of notes in an octave, so that graha-BAEdha on none of its notes will overshoot or undershoot the octave boundaries during the shift of tonic.
The factors of 12 are 12, 6, 4, 3, 2 and 1.
No valid rAga can have 12, 2 or 1 note(s) in its scale.
Therefore, graha-BAEdha fixed points can be 6, 4, 3 note rAgas and these are the only 3 scales that can be fixed points.

The scales of the these three fixed points are enumerated below.

-----------------------------------------------------------------
                  #svaras   scale
-----------------------------------------------------------------
                  6         s r2 g3 m2 d1 n2 S
                  4         s r3/g2 m2 d2/n1 S
                  3         s g3 d1 S
-----------------------------------------------------------------

The 6-svara scale is the rAga gOpreyA, a janya of the 62nd mAELa reShaBapreyA.

Here are some popular graha-BAEdha pairs.

-----------------------------------------------------------------
               rAga           graha   new rAga
-----------------------------------------------------------------
               ABOgE          m       valajE
               valajE         p       ABOgE
               ha~saDhvanE    p       nAgasvarALE
               nAgasvarALE    m       ha~saDhvanE
               SevaranjjanE   r       rAEvathE
               rAEvathE       n       SevaranjjanE
-----------------------------------------------------------------

An interesting tidbit: SevaranjjanE is the same scale as mOhanam with the replacement of the anthara gAnDhara with the sADhAraNa gAnDhAra. The corresponding graha-BAEdha rAgas rAEvathE and maDhyamAvathE differ in their reShaBa notes (SudhDha versus cathuSruthe).

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