Lumatone mapping for 20edo: Difference between revisions

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{{Lumatone mapping intro}} You can use the b val, which creates the [[2L 5s]] Balzano mapping.

{{Lumatone mapping intro}} 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.

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 [https://www.youtube.com/shorts/bqYA9flwL_g ''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.

~~This~~ is ~~not particularly optimal~~ for ~~making the best tuned intervals easily playable~~, ~~however. Better options are~~ the [[~~Blackwood~~]] ~~mapping~~.

== 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{{c}} (corresponding to the Blackwood sharp fifth at 720{{c}}).

== 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 [https://www.youtube.com/watch?v=BFUTeFs7Ycg ''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.

Or the [[Tetracot]] ~~mapping~~.

== 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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-{{Lumatone mapping intro}} You can use the b val, which creates the [[2L&nbsp;5s]] Balzano mapping.
+{{Lumatone mapping intro}} You can use the b&nbsp;val, which creates the [[2L&nbsp;5s]] Antidiatonic mapping.
+== Antidiatonic (and Balzano) ==
+Antidiatonic is [[2L&nbsp;5s]], in this case with a 5:2 step ratio, and can be played in this mapping without backtracking. Balzano is [[2L&nbsp;7s]], in this case with a 3:2 step ratio, and can be played in this mapping by slightly backtracking at the large steps.
 {{Lumatone EDO mapping|n=20|start=4|xstep=2|ystep=3}}
+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&nbsp;4s]] (4:1 step ratio) mapping for 20edo that also functions as a [[Blackwood]] mapping, as demonstrated in [https://www.youtube.com/shorts/bqYA9flwL_g ''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.
+{{Lumatone EDO mapping|n=20|start=6|xstep=4|ystep=-3}}
-This is not particularly optimal for making the best tuned intervals easily playable, however. Better options are the [[Blackwood]] mapping.
+== 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{{c}} (corresponding to the Blackwood sharp fifth at 720{{c}}).
 {{Lumatone EDO mapping|n=20|start=12|xstep=4|ystep=-1}}
+== Compressed Skwares (Fibotonic scale) without Blackwood ==
+[[Bryan Deister]] has used a compressed Skwares) [[2L&nbsp;1s]] (7:6 step ratio) mapping for 20edo, as demonstrated in [https://www.youtube.com/watch?v=BFUTeFs7Ycg ''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.
+{{Lumatone EDO mapping|n=20|start=3|xstep=7|ystep=-1}}
-Or the [[Tetracot]] mapping.
+== 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&nbsp;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&nbsp;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.
 {{Lumatone EDO mapping|n=20|start=17|xstep=3|ystep=-1}}
 {{Navbox Lumatone}}