There is indeed such a thing... but my brain, my brain, ouch, my brain. Yes, I
think you are right, it is the same thing
There i gaati too --- so that's two brain cell failures for me in one post.
Thing is, when people refer to
tisra nadai or
chatusra nadai (probably the two most common, then what I understand them to be speaking of is the number of subdivisions of the beat, the beat remaining of a fixed time duration.
It only matters when the change is made inside the song. So the point of having 25
*5 talams is that they are completely different talams!
In aniloma and pratiloma, the duration of the avartanam is changed --- but by a fixed fraction/multiple.
So if the talam is kept fixed, the song is sung at first, second, third speed. If the song is kept fixed the talam is changed to first speed, second speed etc.
In this, if one is unable to see the talam, I think it would be possible to put one's own talam and keep it constant throughout --- they are changes of presentation only.
To use your term the invariant --- it is like putting the invariant on the other side of the scales; the result is still the same.
And I agree: I can't really see the point either --- but it is an exercise in rhythmic mastery.
chaturasra nadai 3rd speed is the same as trisra nadai 4th speed (12 syllables per outer beat)
Something wrong here....
ta ka di mi Chatusra Nadai 1st speed One akshara 4 madra (syllables)
ta ka di mi ta ka ju no Chatusra Nadai 2st speed One akshara (8 syllables)
ta ka di mi ta ka ju no ta ka di mi ta ka ju no Chatusra Nadai 3rd speed One akshara (16 syllables)
4th speed tisra would be (3, 6, 12, 24) 24 syllables.
Mmmm... actually, that tisra should probably start at six eg, an adi talam pallavi would count per beat:
1st speed 2
2nd speed 4
Tisra 6
3rd speed 8
Tisra 2nd 12
Whoa.... I'm beginning to get that 'am I right?, or hopelessly confused feeling'!