John Carlos BaezI 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 @skarthik.
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 an action of the group ℤ/7 on the set of Melakarta ragas... except sometimes when you cyclically permute the notes of a Melakarta raga you 𝑑𝑜𝑛'𝑡 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,𝑥)↦ 𝑔𝑥
such that:
1. for all 𝑥∈𝑋, 1𝑥 is defined and equal to 𝑥.
2. If one of (𝑔ℎ)𝑥 and 𝑔(ℎ𝑥) 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
• the objects are elements of 𝑋
• a morphism 𝑔:𝑥→𝑦 is an element 𝑔∈𝐺 with 𝑔𝑥 = 𝑦
• the composite of morphisms 𝑔 and ℎ 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.
phiofx
Karthik Srinivasan
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