Lumatone mapping for 81edo: Difference between revisions

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{{Lumatone mapping intro}} (Note that the 81b sharp fifth is shared with [[27edo]], thereby making three mutually exclusive rings of fifths.)

== Diatonic ==

== Meantone ==

The [[patent val]] diatonic mapping would be a perfectly respectable [[Meantone]] mapping on a hypothetical XL-sized Lumatone having at least 405 keys in the space of five octaves.

=== Diatonic ===

The [[patent val]] diatonic mapping would be a perfectly respectable [[Meantone]] mapping on a hypothetical XL-sized Lumatone having at least 405 keys in the space of five octaves. Since meantone is a highly efficient temperament in general, and 81 is the optimal patent val for [[golden meantone]], the full gamut is not necessary to play the best chords and a pedal that shifts the pitch by a single edostep would be an effective way to enable you to play in all keys.

== ~~Meantone-related rank-3 mapping~~ ==

=== 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. The range is just over three octaves, with no missed notes, and the octaves slant down moderately.

[[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 ==

The [[7L 1s]] scale covers 4/5ths of the notes and makes the 5th & 11th harmonics very easy to play, although it uses the second best approximation to the 3rd and 7th harmonics.

== Lithium ==

The Lithium scale divides the octave in three to take advantage of the efficiency of meantone in a different way, giving you a near equal scale with the [[3L 6s]] mos covering 8/9ths of the gamut and the [[9L 3s]] covering all of it with the same efficiency as the chromatic one, with octaves closer to horizontal.

=== 3L 6s ===

=== 9L 3s ===

== Other Mappings ==

If you want maximum range with no concern for good temperaments, the [[3L 7s]] mos generated by a submajor third of 25/81 is the most efficient one that can hit all the notes in the middle octaves.

@@ Line 1: / Line 1: @@
 {{Lumatone mapping intro}} (Note that the 81b sharp fifth is shared with [[27edo]], thereby making three mutually exclusive rings of fifths.)
-== Diatonic ==
+== Meantone ==
-The [[patent val]] diatonic mapping would be a perfectly respectable [[Meantone]] mapping on a hypothetical XL-sized Lumatone having at least 405 keys in the space of five octaves.
+=== Diatonic ===
+The [[patent val]] diatonic mapping would be a perfectly respectable [[Meantone]] mapping on a hypothetical XL-sized Lumatone having at least 405 keys in the space of five octaves. Since meantone is a highly efficient temperament in general, and 81 is the optimal patent val for [[golden meantone]], the full gamut is not necessary to play the best chords and a pedal that shifts the pitch by a single edostep would be an effective way to enable you to play in all keys.
 {{Lumatone EDO mapping|n=81|start=75|xstep=13|ystep=-5}}
-== Meantone-related rank-3 mapping ==
+=== 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. The range is just over three octaves, with no missed notes, and the octaves slant down moderately.
+[[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}}
+== 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 ==
+The [[7L&nbsp;1s]] scale covers 4/5ths of the notes and makes the 5th & 11th harmonics very easy to play, although it uses the second best approximation to the 3rd and 7th harmonics.
+{{Lumatone EDO mapping|n=81|start=9|xstep=11|ystep=-7}}
-{{Lumatone EDO mapping|n=81|start=76|xstep=8|ystep=-3}}
+== Lithium ==
+The Lithium scale divides the octave in three to take advantage of the efficiency of meantone in a different way, giving you a near equal scale with the [[3L&nbsp;6s]] mos covering 8/9ths of the gamut and the [[9L&nbsp;3s]] covering all of it with the same efficiency as the chromatic one, with octaves closer to horizontal.
+=== 3L 6s ===
+{{Lumatone EDO mapping|n=81|start=62|xstep=7|ystep=6}}
+=== 9L 3s ===
+{{Lumatone EDO mapping|n=81|start=66|xstep=7|ystep=-1}}
+== Other Mappings ==
+If you want maximum range with no concern for good temperaments, the [[3L&nbsp;7s]] mos generated by a submajor third of 25/81 is the most efficient one that can hit all the notes in the middle octaves.
+{{Lumatone EDO mapping|n=81|start=41|xstep=6|ystep=7}}
 {{Navbox Lumatone}}