2023-11-13
https://www.youtube.com/watch?v=eNZW_REeKY8
More on possible relations between tuning systems and Coxeter groups. Just intonation involves a group homomorphism from ℤ² to GL(1,ℝ), sending the first generator to 5/4 (a just major third) and the second to 6/5 (a just minor third). Similarly equal temperament involves a group homomorphism ℤ² to GL(1,ℝ) sending the first generator to 2^(1/3) (an equal-tempered major third) and the second to 2^(1/4) (an equal-tempered minor third).
Similar concepts led to the 'Fokker periodicity blocks':
https://en.wikipedia.org/wiki/Fokker_periodicity_block
which are related to the Tonnetz:
https://en.wikipedia.org/wiki/Tonnetz
A hexagon of triads containing a single note. How the PLR group is a quotient of the Coxeter group {∞,∞,∞} with three generators which acts as isometries on the hyperbolic plane preserving the infinite-order triangular tiling:
https://en.wikipedia.org/wiki/Infinite-order_triangular_tiling
This group maps onto the Coxeter group {3,∞} which is the symmetry group of the infinite-order triangular tiling.
(2/n)
John Carlos Baez /
npub17u…8pd6m
2025-01-10 18:21:59
in reply to nevent1q…tf33
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npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6mSeen on
wss://relay.nostr.bandPublished at
2025-01-10 18:21:59Event JSON
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"content": "2023-11-13\n\nhttps://www.youtube.com/watch?v=eNZW_REeKY8\n\nMore on possible relations between tuning systems and Coxeter groups. Just intonation involves a group homomorphism from ℤ² to GL(1,ℝ), sending the first generator to 5/4 (a just major third) and the second to 6/5 (a just minor third). Similarly equal temperament involves a group homomorphism ℤ² to GL(1,ℝ) sending the first generator to 2^(1/3) (an equal-tempered major third) and the second to 2^(1/4) (an equal-tempered minor third).\n\nSimilar concepts led to the 'Fokker periodicity blocks':\n\nhttps://en.wikipedia.org/wiki/Fokker_periodicity_block\n\nwhich are related to the Tonnetz:\n\nhttps://en.wikipedia.org/wiki/Tonnetz\n\nA hexagon of triads containing a single note. How the PLR group is a quotient of the Coxeter group {∞,∞,∞} with three generators which acts as isometries on the hyperbolic plane preserving the infinite-order triangular tiling:\n\nhttps://en.wikipedia.org/wiki/Infinite-order_triangular_tiling\n\nThis group maps onto the Coxeter group {3,∞} which is the symmetry group of the infinite-order triangular tiling.\n\n(2/n)",
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