Lumatone mapping for 87edo: Difference between revisions

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There is an alternate diatonic scale that makes 5-limit chords easy to play, but it has a slight octave stretch. Since every note is available in at least some of the octaves, this may actually be advantageous.

== Pseudo-Isomorphic Quasi-Diatonic ==

For those who do not want the octave stretch, [[Bryan Deister]] has demonstrated a pseudo-isomorphic mapping for [[87edo]] in [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025). This mapping retains a diatonic-style layout for ease of access to common consonant intervals, although it does not include all the notes of the tuning system. Each note 87 is actually another note 0, so as to reset octave pitch down by half of ~[[81/80]] (moderately inflated).

== 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.

== Rodan ==

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 There is an alternate diatonic scale that makes 5-limit chords easy to play, but it has a slight octave stretch. Since every note is available in at least some of the octaves, this may actually be advantageous.
 {{Lumatone EDO mapping|n=87|start=79|xstep=14|ystep=-5}}
+== Pseudo-Isomorphic Quasi-Diatonic  ==
+For those who do not want the octave stretch, [[Bryan Deister]] has demonstrated a pseudo-isomorphic mapping for [[87edo]] in [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025). This mapping retains a diatonic-style layout for ease of access to common consonant intervals, although it does not include all the notes of the tuning system. Each note 87 is actually another note 0, so as to reset octave pitch down by half of ~[[81/80]] (moderately inflated).
+{{Lumatone EDO mapping|n=88|start=80|xstep=14|ystep=-5}}
+== 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.
+{{Lumatone EDO mapping|n=87|start=6|xstep=7|ystep=-5}}
 == Rodan ==