Revision as of 05:47, 29 June 2025 by Lucius Chiaraviglio(talk | contribs)(→Neutral Thirds: Put exact date on demo video because another demonstration video is coming that has the same exact name)
However, Due to the size of the edo, this mapping does not cover all the notes. In addition, like 46edo, 75edo is a leapday system, so the best approximation to 5/4 is a triply-augmented unison, which makes for awkward fingerings. Despite the missing notes, Bryan Deister has demonstrated this layout in improv in 75edo (2025)
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Tetracot
The Tetracot mapping is considerably more efficient at putting harmonics close together, although it also needs to be expanded from 6L 1s to 7L 6s to hit every single note, which puts octaves all over the place
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Fog
To maximise range while covering the whole gamut, giving easy access to lower harmonics and keeping octaves near horizontal, slice the period into thirds to produce the Fog mapping.
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Neutral Thirds
The neutral thirds mapping has slightly less range, but skips fewer notes around the edges and keeps octaves (just over 3½) even closer to horizontal, with scale 7L 3s (step size ratio 9:4). Bryan Deister has demonstrated this mapping, in microtonal improvisation in 75edo (2025-06-24). Like 97edo, 75edo has mainly bad harmonics for its size (but with different harmonics being exceptions), so to find useful intervals, it is necessary to try to stick to primes 3, 5, and 23 (those having the least relative error) and/or take advantage of error cancellation as much as possible. Going right (9\75) 1 key is ~25/23 (error canceling, but starting out with some of the least bad harmonics); right 2 keys (18\75) is a subminor third ~625/529 (no simple ratio maps to this interval in the patent val of 75edo, but it merits mention anyway due to extensive use in the video); right 3 keys (27\75) = ~9/7 (errors only partially cancel, so somewhat flat); and right 7 keys (63\75) = ~34/19 (errors nearly cancel). Going down-right (4\75) 1 key functions as both ~27/26 and ~28/27; down-right 2 keys = ~14/13 (errors largely cancel), 3 keys = ~19/17 (errors largely cancel), and 5 keys = ~5/4 (the 5th harmonic has a small relative error). Going up (5\75) 1 key is ~23/22 (errors only partially canceling, so somewhat sharp).
