Lumatone mapping for 81edo: Difference between revisions

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=== Chromatic ===

[[Bryan Deister]] has demonstrated the m-chromatic ([[7L 5s]] with 8:5 step ratio) mapping for [[81edo]] in [https://www.youtube.com/shorts/5O6Cyawjkd8 ''''microtonal improvisation in 81edo''] (2025). The rightward generator 8\81 functions as ~[[15/14]] and ~[[16/15]]; six of them make the alternate sharp fifth. The down-right generator 5\81 functions as ~[[23/22]] and ~[[24/23]]; two of them make an Alpharabian tendoneutral second (~[[12/11]]); four of them make the near-just octave-reduced 19th harmonic ~[[19/16]]; eleven of them make a near-just classic minor sixth ~[[8/5]]. The patent fifth 47\81 is four rightward generators plus three down-right generators. All that said, the patent fifth itself works perfectly well as a generator of the 7L 5s scale, as well as for the entire layout, as befits (rank-2) [[Meantone]], so that the use of dual generators is a matter of convience rather than a requirement. The range is just over three octaves, with no missed notes, and the octaves slant down moderately.

== Unnamed high-limit rank-3 temperament or split and rotated meantone ==

[[Bryan Deister]] has demonstrated a [[5L 9s]] (9:4 step ratio) layout for [[81edo]], in [https://www.youtube.com/shorts/yY0TzTiNeB8 ''81edo improv''] (2025). Both the rightward generator (4\81) and the down-right generator (9\81) map to high-limit intervals: 4\81 maps to ~[[29/28]], ~[[30/29]], and ~[[31/30]]; while 9\81 maps to ~[[40/37]]. The down-right generator is [[Saturation, torsion, and contorsion|contorted]] by itself; in principle, the first generator could be used by itself for an unnamed rank-2 temperament, but it is so small that the second generator is needed for convenience. Alternatively, one could make a rank-2 temperament using the combination of these generators, which is 13\81, which maps to the quasi-meantone ~[[19/17]] and to both sizes of classic whole tone ~[[9/8]] and ~[[10/9]], as befits 81edo being a [[meantone]] tuning, thereby producing a rotated and split diatonic layout yielding a range close to 2½ octaves (with some repeated notes to mitigate vertical wraparounds) having a slight downward slant.

== Porcupine ==

@@ Line 8: / Line 8: @@
 === Chromatic ===
 [[Bryan Deister]] has demonstrated the m-chromatic ([[7L&nbsp;5s]] with 8:5 step ratio) mapping for [[81edo]] in [https://www.youtube.com/shorts/5O6Cyawjkd8 ''''microtonal improvisation in 81edo''] (2025). The rightward generator 8\81 functions as ~[[15/14]] and ~[[16/15]]; six of them make the alternate sharp fifth. The down-right generator 5\81 functions as ~[[23/22]] and ~[[24/23]]; two of them make an Alpharabian tendoneutral second (~[[12/11]]); four of them make the near-just octave-reduced 19th harmonic ~[[19/16]]; eleven of them make a near-just classic minor sixth ~[[8/5]]. The patent fifth 47\81 is four rightward generators plus three down-right generators. All that said, the patent fifth itself works perfectly well as a generator of the 7L&nbsp;5s scale, as well as for the entire layout, as befits (rank-2) [[Meantone]], so that the use of dual generators is a matter of convience rather than a requirement. The range is just over three octaves, with no missed notes, and the octaves slant down moderately.
+{{Lumatone EDO mapping|n=81|start=76|xstep=8|ystep=-3}}
-{{Lumatone EDO mapping|n=81|start=76|xstep=8|ystep=-3}}
+== Unnamed high-limit rank-3 temperament or split and rotated meantone ==
+[[Bryan Deister]] has demonstrated a [[5L&nbsp;9s]] (9:4 step ratio) layout for [[81edo]], in [https://www.youtube.com/shorts/yY0TzTiNeB8 ''81edo improv''] (2025). Both the rightward generator (4\81) and the down-right generator (9\81) map to high-limit intervals: 4\81 maps to ~[[29/28]], ~[[30/29]], and ~[[31/30]]; while 9\81 maps to ~[[40/37]]. The down-right generator is [[Saturation, torsion, and contorsion|contorted]] by itself; in principle, the first generator could be used by itself for an unnamed rank-2 temperament, but it is so small that the second generator is needed for convenience. Alternatively, one could make a rank-2 temperament using the combination of these generators, which is 13\81, which maps to the quasi-meantone ~[[19/17]] and to both sizes of classic whole tone ~[[9/8]] and ~[[10/9]], as befits 81edo being a [[meantone]] tuning, thereby producing a rotated and split diatonic layout yielding a range close to 2½ octaves (with some repeated notes to mitigate vertical wraparounds) having a slight downward slant.
+{{Lumatone EDO mapping|n=81|start=44|xstep=4|ystep=5}}
 == Porcupine ==