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 (npub1qtkβ¦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 πππ'π‘ 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 πΊ on a set π is a partially defined map
πΊΓπβπ
(π,π₯)β¦ ππ₯
such that:
1. for all π₯βπ, 1π₯ is defined and equal to π₯.
2. If one of (πβ)π₯ and π(βπ₯) 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 π
β’ a morphism π:π₯βπ¦ is an element πβπΊ with ππ₯ = π¦
β’ the composite of morphisms π and β is their product in the group πΊ.
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.
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 πππ'π‘ 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 πΊ on a set π is a partially defined map
πΊΓπβπ
(π,π₯)β¦ ππ₯
such that:
1. for all π₯βπ, 1π₯ is defined and equal to π₯.
2. If one of (πβ)π₯ and π(βπ₯) 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 π
β’ a morphism π:π₯βπ¦ is an element πβπΊ with ππ₯ = π¦
β’ the composite of morphisms π and β is their product in the group πΊ.
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.