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 interleaved 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; the last step in arriving at meantone is to add an octave to this generator and split it in half to get 47\81, which is the closest approximation to the perfect fifth in 81edo. This produces an interleaved and rotated 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 interleaved 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; the last step in arriving at meantone is to add an octave to this generator and split it in half to get 47\81, which is the closest approximation to the perfect fifth in 81edo. This produces an interleaved and rotated 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 ==