Lumatone mapping for 87edo: Difference between revisions

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== 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).

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 the near-just ~[[64/63]] (half of the moderately inflated ~[[81/80]]).

== Aureus Pseudo-Meantone ==

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.)

It is possible to lay out [[87edo]] using the generator 14\87, which 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.)

=== 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).

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''] (2026-07-11).

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== Rodan ==

The [[~~Rodan~~]] mapping is quite efficient and 87edo is an essentially perfect tuning for the temperament, which ~~makes~~ it a good ~~option~~. However, the [[1L 4s]] and [[5L 1s]] mappings only cover about half the notes and the [[5L 6s]] one has heavily sloped octaves whichever direction you put the chroma. The reverse chroma one is the only one of these that covers the full gamut in the central octave.

The [[rodan]] mapping is quite efficient and 87edo is an essentially perfect tuning for the temperament, which would make it an excellent option if not for the missing notes problem. Rodan splits the near-just (only slightly sharp) fifth ~[[3/2]] into three mildly sharp septimal whole tones ~[[8/7]], as 17\87 (the gamelisma [[1029/1024]] is tempered out); furthermore, it splits the slightly flat classic major sixth ~[[5/3]] into a two sharp septimal major thirds ~[[9/7]], as 32\87 (the sensamagic comma [[245/243]] is tempered out). Finally, it defines a fine-adjustment interval known as the "quark" (a type of [[Aberrismic theory|aberrisma]]) which may or may not corresond to the MOS chroma; at 2\87, this functions as several intervals including a near-just ~[[64/63]] and a sharp ~[[81/80]] (the aberschisma [[5120/5103]], formerly known as the hemifamity comma, is tempered out). Thus, even though it is rank-2 because the generators are not independent, from the user's point of view it works like a rank-3 temperament, which is good for a large tuning system like 87edo. However, the [[1L 4s]] and [[5L 1s]] mappings only cover about half the notes and the [[5L 6s]] one has heavily sloped octaves whichever direction you put the chroma. The reverse chroma one is the only one of these that covers the full gamut in the central octave.

=== 1L 4s ===

=== 1L 4s reverse chroma ===

In the [[1L 4s]] (19:17 step ratio) reverse chroma version of the rodan mapping, the septimal major second ~8/7 is one key right, and the quark is the MOS chroma; the septimal major third ~9/7 is one key right plus one key up. The range is over six octaves gently sloping up, but many notes are missing.

=== 5L 1s ===

=== 5L 1s forward chroma ===

In the [[5L 1s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is still one key right, and the quark is the small step of the scale (which is still proper since it only occurs once per octave); the septimal major third ~9/7 is two keys right plus one key up. The range is over five octaves gently sloping up, but many notes are missing. Despite the missing notes, [[Bryan Deister]] has used this mapping in [https://www.youtube.com/shorts/wHPyCHQn55A ''87edo improv''] (2026-09-28).

=== 5L 6s ===

=== 5L 6s forward chroma ===

In the [[5L 6s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is now one key right plus one key down-right, and the quark is still the small step of the scale (now no longer proper); the septimal major third ~9/7 is two keys right plus one key down-right. The range is over three octaves severely sloping down, but many notes still are missing.

=== 5L 6s reverse chroma ===

In the [[5L 6s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is still one key right plus one key down-right, and the quark is still the small step of the scale; the septimal major third ~9/7 is one key right plus two keys down-right. The range is over two octaves moderately sloping down; although the middle octave is complete (and even has a fair number of repeated notes), in practice many notes are still missing due to the lopsided nature of the layout causing many notes to be lost at the left and right edges (as well as likely stretching fingers to the limits) even though not cut off vertically.

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The [[4L 7s]] mapping covers the full gamut and keeps octaves fairly horizontal when the small step goes rightward.

== Misty ==

The [[3L 9s]] [[Misty]] mapping is quite efficient at making harmonics easy to play together, gives you a relatively 12edo-like experience and keeps octaves near horizontal, but has a slightly smaller range. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025).

@@ Line 8: / Line 8: @@
 == 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).
+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 the near-just ~[[64/63]] (half of the moderately inflated ~[[81/80]]).
 {{Lumatone EDO mapping|n=88|start=80|xstep=14|ystep=-5}}
 == Aureus Pseudo-Meantone ==
-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.)
+It is possible to lay out [[87edo]] using the generator 14\87, which 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.)
 === 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).
+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''] (2026-07-11).
 {{Lumatone EDO mapping|n=87|start=0|xstep=9|ystep=5}}
@@ Line 23: / Line 23: @@
 == Rodan ==
-The [[Rodan]] mapping is quite efficient and 87edo is an essentially perfect tuning for the temperament, which makes it a good option. However, the [[1L 4s]] and [[5L 1s]] mappings only cover about half the notes and the [[5L 6s]] one has heavily sloped octaves whichever direction you put the chroma. The reverse chroma one is the only one of these that covers the full gamut in the central octave.
+The [[rodan]] mapping is quite efficient and 87edo is an essentially perfect tuning for the temperament, which would make it an excellent option if not for the missing notes problem. Rodan splits the near-just (only slightly sharp) fifth ~[[3/2]] into three mildly sharp septimal whole tones ~[[8/7]], as 17\87 (the gamelisma [[1029/1024]] is tempered out); furthermore, it splits the slightly flat classic major sixth ~[[5/3]] into a two sharp septimal major thirds ~[[9/7]], as 32\87 (the sensamagic comma [[245/243]] is tempered out). Finally, it defines a fine-adjustment interval known as the "quark" (a type of [[Aberrismic theory|aberrisma]]) which may or may not corresond to the MOS chroma; at 2\87, this functions as several intervals including a near-just ~[[64/63]] and a sharp ~[[81/80]] (the aberschisma [[5120/5103]], formerly known as the hemifamity comma, is tempered out). Thus, even though it is rank-2 because the generators are not independent, from the user's point of view it works like a rank-3 temperament, which is good for a large tuning system like 87edo. However, the [[1L&nbsp;4s]] and [[5L&nbsp;1s]] mappings only cover about half the notes and the [[5L&nbsp;6s]] one has heavily sloped octaves whichever direction you put the chroma. The reverse chroma one is the only one of these that covers the full gamut in the central octave.
-=== 1L 4s ===
+=== 1L&nbsp;4s reverse chroma ===
+In the [[1L&nbsp;4s]] (19:17 step ratio) reverse chroma version of the rodan mapping, the septimal major second ~8/7 is one key right, and the quark is the MOS chroma; the septimal major third ~9/7 is one key right plus one key up. The range is over six octaves gently sloping up, but many notes are missing.
 {{Lumatone EDO mapping|n=87|start=24|xstep=17|ystep=2}}
-=== 5L 1s ===
+=== 5L&nbsp;1s forward chroma ===
+In the [[5L&nbsp;1s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is still one key right, and the quark is the small step of the scale (which is still proper since it only occurs once per octave); the septimal major third ~9/7 is two keys right plus one key up. The range is over five octaves gently sloping up, but many notes are missing. Despite the missing notes, [[Bryan Deister]] has used this mapping in [https://www.youtube.com/shorts/wHPyCHQn55A ''87edo improv''] (2026-09-28).
 {{Lumatone EDO mapping|n=87|start=39|xstep=17|ystep=-15}}
-=== 5L 6s ===
+=== 5L&nbsp;6s forward chroma ===
+In the [[5L&nbsp;6s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is now one key right plus one key down-right, and the quark is still the small step of the scale (now no longer proper); the septimal major third ~9/7 is two keys right plus one key down-right. The range is over three octaves severely sloping down, but many notes still are missing.
 {{Lumatone EDO mapping|n=87|start=11|xstep=15|ystep=-13}}
+=== 5L&nbsp;6s reverse chroma ===
+In the [[5L&nbsp;6s]] (19:2 step ratio) forward chroma version of the rodan mapping, the septimal major second ~8/7 is still one key right plus one key down-right, and the quark is still the small step of the scale; the septimal major third ~9/7 is one key right plus two keys down-right. The range is over two octaves moderately sloping down; although the middle octave is complete (and even has a fair number of repeated notes), in practice many notes are still missing due to the lopsided nature of the layout causing many notes to be lost at the left and right edges (as well as likely stretching fingers to the limits) even though not cut off vertically.
 {{Lumatone EDO mapping|n=87|start=31|xstep=2|ystep=13}}
@@ Line 35: / Line 44: @@
 The [[4L 7s]] mapping covers the full gamut and keeps octaves fairly horizontal when the small step goes rightward.
 {{Lumatone EDO mapping|n=87|start=53|xstep=5|ystep=8}}
 == Misty ==
 The [[3L 9s]] [[Misty]] mapping is quite efficient at making harmonics easy to play together, gives you a relatively 12edo-like experience and keeps octaves near horizontal, but has a slightly smaller range. [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025).