Lumatone mapping for 87edo: Difference between revisions

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== Aureus Pseudo-Meantone ==

~~[[Bryan Deister]] has demonstrated a [[11L 5s]] (7:2 step ratio) mapping for~~ [[87edo]] ~~in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025). The~~ generator 14\87, which ~~is two keys right from the root note, is the~~ quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals — the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.) Since this interval is two keys right instead of just one (as needed to avoid missing notes), this version of the temperament can be viewed as a split generator version, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds.

It is possible to lay out [[87edo]] using the generator 14\87, which quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals — the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.)

=== Flipped and Rotated ===

In the flipped and rotated verions with a [[3L 5s]] scale (14:9 step ratio)the above-mentioned 14\87 generator is one key down-right, while key right is 9\87, which functions as a flat tridecimal 2/3-tone ~[[14/13]] and a sharp septimal diatonic semitone ~[[15/14]]. This mapping makes it easy to play melodies that sound very diatonic despite the scale being non-diatonic, with octaves sloping moderately down, and a range of over three octaves (although with large partial octaves chopped off by the upper left and lower right corners), which is a bit too much for 87edo, resulting in some missed notes even in the middle octaves. Despite the missing notes, [[Bryan Deister]] has used this mapping in [https://www.youtube.com/shorts/8mHBYBfRjy4 ''87edo improv''] (2026).

=== Split Generator ===

A split generator version of the above mapping returns to mostly horizontal progression with a [[11L 5s]] scale (7:2 step ratio). As in the previous mapping, the generator 14\87 is the quasi-meantone ~[[19/17]], but in this case, it is two keys right, making this a split generator version of the mapping, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025).

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 == Aureus Pseudo-Meantone ==
-[[Bryan Deister]] has demonstrated a [[11L&nbsp;5s]] (7:2 step ratio) mapping for [[87edo]] in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025). The generator 14\87, which is two keys right from the root note, is the quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals &mdash; the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.) Since this interval is two keys right instead of just one (as needed to avoid missing notes), this version of the temperament can be viewed as a split generator version, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds.
+It is possible to lay out [[87edo]] using the generator 14\87, which quasi-meantone ~[[19/17]], which is composed of highly inaccurate harmonics whose errors nearly cancel out, rendering it near-just; two of them make an essentially just classic major third ~[[5/4]]. (In contrast to actual [[meantone]] temperament, 87edo represents ~19/17, ~[[10/9]], and ~[[9/8]] as distinct intervals &mdash; the syntonic comma [[81/80]] is not tempered out, and instead the aureusma [[1445/1444]] equates two quasi-meantones to a classic major third.)
+=== Flipped and Rotated ===
+In the flipped and rotated verions with a [[3L&nbsp;5s]] scale (14:9 step ratio)the above-mentioned 14\87 generator is one key down-right, while  key right is 9\87, which functions as a flat tridecimal 2/3-tone ~[[14/13]] and a sharp septimal diatonic semitone ~[[15/14]]. This mapping makes it easy to play melodies that sound very diatonic despite the scale being non-diatonic, with octaves sloping moderately down, and a range of over three octaves (although with large partial octaves chopped off by the upper left and lower right corners), which is a bit too much for 87edo, resulting in some missed notes even in the middle octaves. Despite the missing notes, [[Bryan Deister]] has used this mapping in [https://www.youtube.com/shorts/8mHBYBfRjy4 ''87edo improv''] (2026).
+{{Lumatone EDO mapping|n=87|start=0|xstep=9|ystep=5}}
+=== Split Generator ===
+A split generator version of the above mapping returns to mostly horizontal progression with a [[11L&nbsp;5s]] scale (7:2 step ratio). As in the previous mapping, the generator 14\87 is the quasi-meantone ~[[19/17]], but in this case, it is two keys right, making this a split generator version of the mapping, in which each ~19/17 is split into two equal intervals that function as both a small septendecimal semitone ~[[18/17]] and a large undevicesimal semitone ~[[19/18]], both being close to just (and the photisma [[324/323]] is tempered out). Although no second generator is needed to reach all of the notes, for convenience, down-right by one key (2\87) can be used for convenience as a second generator that functions as ~[[49/48]] (inconsistently mapped), ~[[55/54]], ~[[64/63]], ~[[65/64]], and ~[[81/80]], of which the septimal chroma ~64/63 is near-just, and four of these make a near-just classic diatonic semitone ~[[16/15]] (the currently unnamed unnoticeable comma [[5250987/5242880]] is tempered out). The range is somewhat over two octaves (which slant very slightly downwards) with no missed notes and some repeated notes to mitigate vertical wraparounds. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover in 87edo)''] (2025).
 {{Lumatone EDO mapping|n=87|start=6|xstep=7|ystep=-5}}