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John Carlos Baez /
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2024-01-15 18:45:34

John Carlos Baez on Nostr: I don't post much about serious math here anymore. But here's a little piece of ...

I don't post much about serious math here anymore. But here's a little piece of category theory connected to music. I dreamt this up thanks to a comment from npub1qtkdu9s8mnygvkcvycw0wtmp85t95v25v8eyavtkx2q0nhnuxmnq9txqrw (npub1qtk…xqrw).

In Carnatic music there are 72 seven-note scales called 'Melakarta ragas'. By cyclically permuting the notes of a Melakarta raga you sometimes get another Melakarta raga. This process is called 'graha bhedam':

https://en.wikipedia.org/wiki/Graha_bhedam

This would give be an action of the group β„€/7 on the set of Melakarta ragas... except sometimes when you cyclically permute the notes of a Melakarta raga you π‘‘π‘œπ‘›'𝑑 get another Melakarta raga, since there are constraints on which 7-note scales count as Melakarta ragas. So we only have some sort of partially defined group action. Let's make up a definition:

Let's say a 'partial group action' of a group 𝐺 on a set 𝑋 is a partially defined map

𝐺×𝑋→𝑋
(𝑔,π‘₯)↦ 𝑔π‘₯

such that:

1. for all π‘₯βˆˆπ‘‹, 1π‘₯ is defined and equal to π‘₯.
2. If one of (π‘”β„Ž)π‘₯ and 𝑔(β„Žπ‘₯) are defined, then so is the other, and they are equal.

Now the interesting thing is that any partial group action gives a groupoid where

β€’ the objects are elements of 𝑋
β€’ a morphism 𝑔:π‘₯→𝑦 is an element π‘”βˆˆπΊ with 𝑔π‘₯ = 𝑦
β€’ the composite of morphisms 𝑔 and β„Ž is their product in the group 𝐺.

This construction is famous for ordinary group actions:

https://ncatlab.org/nlab/show/action+groupoid

but it works fine for partial group actions!

So, there's a groupoid of Melakarta ragas. The Wikipedia article shows that Carnatic music theory is concerned with the connected components of this groupoid: that is, bunches of Melakarta ragas related to each other by graha bhedam.
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