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Lumatone mapping for 20edo

Revision as of 09:11, 13 August 2025 by Lucius Chiaraviglio (talk | contribs) (→Tetracot (+ Diminished + Blackwood + Skwares): --> Skwares/Squares, since Tetracot uses the 5th harmonic)
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There are many conceivable ways to map 20edo onto the onto the Lumatone keyboard. However, it has 4 mutually-exclusive rings of fifths, so the Standard Lumatone mapping for Pythagorean is not one of them. You can use the b val, which creates the 2L 5s Antidiatonic mapping.

Antidiatonic (and Balzano)

Antidiatonic is 2L 5s, in this case with a 5:2 step ratio, and can be played in this mapping without backtracking. Balzano is 2L 7s, in this case with a 3:2 step ratio, and can be played in this mapping by slightly backtracking at the large steps.

 
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The Antidiatonic/Balzano mapping is not particularly optimal for making the best tuned intervals easily playable, however, which the Blackwood and Tetracot mappings solve.

Blackwood + Diminished

Bryan Deister has used a Diminished 4L 4s (4:1 step ratio) mapping for 20edo that also functions as a Blackwood mapping, as demonstrated in 20edo prelude (2025). In this mapping, the octave can be interpreted as being divided into five parts (Blackwood temperament) or 4\20 or four parts of 5\20 (Diminished temperament). Octaves slope, as expected for a Blackwood mapping, but with plenty of repeated vertical notes, and the range is a bit over 5½ octaves.

 
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Blackwood + Skwares

Combining Blackwood with Skwares is another possibility. In the version of Skwares used for this, instead of substantially sharpening the undecimal major third ~14/11 (or even a septimal major third ~9/7) to get a diatonic fourth ~4/3 (by stacking and octave-reducing four generators), the undecimal major third (7\20 as one move right and one move down-right) is near-just, and the fourth is very flat at 480 ¢ (corresponding to the Blackwood sharp fifth at 720 ¢).

 
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Compressed Skwares (Fibotonic scale) without Blackwood

Bryan Deister has used a compressed Skwares) 2L 1s (7:6 step ratio) mapping for 20edo, as demonstrated in 20edo groove (2025), which leaves out the Blackwood division of the octave in favor of maximizing range. Octaves are nearly level (sloping down very gradually), and the range is a bit over 11 octaves, which suggests dividing the keyboard into zones for different instruments.

 
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Tetracot (+ Diminished + Blackwood + Skwares/Squares)

Tetracot tempers together the Ptolmeic whole tone ~10/9 with the undecimal submajor second ~11/10 to make a fourth of the fifth ~3/2; however, in the case of 20edo, this requires using the 20c val to make the Ptolmeic whole tone map to its near-just direct approximation value (instead of its very flat patent val mapping) (and make the undecimal submajor whole tone fairly sharp) to get the sharp Blackwood fifth. Octaves slope up, and the range is 4¾ octaves; notes are abundantly repeated, but unfortunately the upwards slope of the octaves is inconvenient for division into multiple manuals. The Tetracot mapping does have an embedded 4L 4s (3:2 step ratio) Diminished scale available by alternating right and down-right motions; it has embedded Blackwood rings by way of moving two keys at a time down-right; and it even has an embedded 2L 1s (7:6 step ratio) scale for Skwares/Squares), thus making these temperaments also accessible if one needs to switch temperaments frequently, thus gaining considerable versatility in exchange for the lost range.

 
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