Lumatone mapping for 75edo: Difference between revisions

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== Diatonic ==

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.

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 [https://www.youtube.com/watch?v=sdN-5y3jhDY&lc=UgxpTCf_D53IW3ACl1Z4AaABAg ''improv in 75edo''] (2025)

== 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~~

The [[Tetracot]] mapping is considerably more efficient at putting harmonics close together.

=== Standard (6L 1s) Tetracot ===

However, 75edo is too large for the standard [[Lumatone mapping for Tetracot]] to include all of the notes.

=== Compressed (Reversed-Chroma 1L 5s) Tetracot ===

[[Bryan Deister]] has demonstrated a compressed [[1L 5s]] mapping of [[75edo]] in [https://www.youtube.com/watch?v=5-G2KkYfKLs&list=WL&index=343&pp=gAQBiAQB8AUB ''microtonal improvisation in 75edo''] (2025-06-22). This combines the temperament of [[Tetracot]] with the idea of the compressed [[Lumatone mapping for Porcupine]], by reversing the chroma. (This also contains the Tetracot [[6L 1s]] scale, but playing that scale in this layout requires backtracking.) This expands the overall range to more than five octaves, but too much so for anything so large as 75edo, thereby causing missed notes (although it demonstrates how such a mapping could shine for a smaller EDO or a hypothetical XL-sized Lumatone that has room for all of the notes).

=== Expanded (7L 6s) Tetracot ===

To cover all of the notes in 75edo, Tetracot needs to be expanded from [[6L 1s]] to [[7L 6s]], which puts octaves all over the place

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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''] (2025-06-24). 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).

== 18L 1s MOS subset pseudo-diatonic ==

[[Bryan Deister]] has demonstrated a pseudo-isomorphic pseudo-diatonic mapping for [[75edo]] in [https://www.youtube.com/shorts/nlurS-3VYkA ''75edo improv''] (2025) that deliberately uses non-coprime generators on all axes with the numbering of [[76edo]] to use a subset of 75edo to make 19 non-equal divisions of the octave in the form of a [[18L 1s]] MOS scale with a 4:3 step ratio. Note 75 is actually a duplicate of note 0; due to missing notes, it does not actually appear here.

Removing the octave correction turns this into an approximation of [[Carlos Beta]].

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 == Diatonic ==
-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.
+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 [https://www.youtube.com/watch?v=sdN-5y3jhDY&lc=UgxpTCf_D53IW3ACl1Z4AaABAg ''improv in 75edo''] (2025)
 {{Lumatone EDO mapping|n=75|start=6|xstep=13|ystep=-8}}
 == Tetracot ==
-The [[Tetracot]] mapping is considerably more efficient at putting harmonics close together, although it also needs to be expanded from [[6L&nbsp;1s]] to [[7L&nbsp;6s]] to hit every single note, which puts octaves all over the place
+The [[Tetracot]] mapping is considerably more efficient at putting harmonics close together.
+=== Standard (6L&nbsp;1s) Tetracot ===
+However, 75edo is too large for the standard [[Lumatone mapping for Tetracot]] to include all of the notes.
 {{Lumatone EDO mapping|n=75|start=45|xstep=11|ystep=-2}}
+=== Compressed (Reversed-Chroma 1L&nbsp;5s) Tetracot ===
+[[Bryan Deister]] has demonstrated a compressed [[1L&nbsp;5s]] mapping of [[75edo]] in [https://www.youtube.com/watch?v=5-G2KkYfKLs&list=WL&index=343&pp=gAQBiAQB8AUB ''microtonal improvisation in 75edo''] (2025-06-22). This combines the temperament of [[Tetracot]] with the idea of the compressed [[Lumatone mapping for Porcupine]], by reversing the chroma. (This also contains the Tetracot [[6L&nbsp;1s]] scale, but playing that scale in this layout requires backtracking.) This expands the overall range to more than five octaves, but too much so for anything so large as 75edo, thereby causing missed notes (although it demonstrates how such a mapping could shine for a smaller EDO or a hypothetical XL-sized Lumatone that has room for all of the notes).
+{{Lumatone EDO mapping|n=75|start=17|xstep=11|ystep=9}}
+=== Expanded (7L&nbsp;6s) Tetracot ===
+To cover all of the notes in 75edo, Tetracot needs to be expanded from [[6L&nbsp;1s]] to [[7L&nbsp;6s]], which puts octaves all over the place
 {{Lumatone EDO mapping|n=75|start=0|xstep=9|ystep=-7}}
@@ Line 16: / Line 24: @@
 {{Lumatone EDO mapping|n=75|start=42|xstep=6|ystep=7}}
+== 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''] (2025-06-24).  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}}
+== 18L&nbsp;1s MOS subset pseudo-diatonic ==
+[[Bryan Deister]] has demonstrated a pseudo-isomorphic pseudo-diatonic mapping for [[75edo]] in [https://www.youtube.com/shorts/nlurS-3VYkA ''75edo improv''] (2025) that deliberately uses non-coprime generators on all axes with the numbering of [[76edo]] to use a subset of 75edo to make 19 non-equal divisions of the octave in the form of a [[18L&nbsp;1s]] MOS scale with a 4:3 step ratio. Note 75 is actually a duplicate of note 0; due to missing notes, it does not actually appear here.
+{{Lumatone EDO mapping|n=76||start=68|xstep=12|ystep=-4}}
+Removing the octave correction turns this into an approximation of [[Carlos Beta]].
+{{Lumatone EDO mapping|n=75||start=67|xstep=12|ystep=-4}}
 {{Navbox Lumatone}}