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.

== 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.
 {{Lumatone EDO mapping|n=81|start=76|xstep=8|ystep=-3}}
+== 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}}
+== 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}}