Lumatone mapping for 75edo: Difference between revisions

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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 [https://www.youtube.com/shorts/QflMtKRmlSI ''microtonal improvisation in 75edo''] (2025). 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).

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 [https://www.youtube.com/shorts/QflMtKRmlSI ''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).

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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&nbsp;3s]] (step size ratio 9:4). [[Bryan Deister]] has demonstrated this mapping, in [https://www.youtube.com/shorts/QflMtKRmlSI ''microtonal improvisation in 75edo''] (2025).  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).
+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&nbsp;3s]] (step size ratio 9:4). [[Bryan Deister]] has demonstrated this mapping, in [https://www.youtube.com/shorts/QflMtKRmlSI ''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).
 {{Lumatone EDO mapping|n=75|start=2|xstep=9|ystep=-5}}
 {{Navbox Lumatone}}