36edo: Difference between revisions

36edo's patent val, like 12, tempers out 81/80, 128/125, and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675 and 1029/1000, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242, and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank-3 temperament [[melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77 and 91/90, in the 17-limit 51/50, and in the 19-limit 76/75 and 96/95.

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=== Colored notes ===

One way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, {{colored note|blue|A}} is 33{{frac|3}}{{c}} below {{colored note|A}} and {{colored note|red|A}} is 33{{frac|3}} cents above {{colored note|A}}. Or, the colors could be written out (red A, blue C♯, etc.) or abbreviated as rA, bC♯, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo).

=== Ups and downs notation ===

{{sharpness-sharp3a|36}}

Alternatively, one can use sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

 

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.

This notation uses the same sagittal sequence as [[43edo#Sagittal notation|43-EDO]], is a subset of the notation for [[72edo#Sagittal notation|72-EDO]], and is a superset of the notations for EDOs [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], and [[6edo#Sagittal notation|6]].

{{Q-odd-limit intervals|35.98|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36e val mapping}}

{{clear}}

| Schismina

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

 

 

If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.

 

 

 

Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within 9.7{{c}}. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings.

|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings

* 12-tone 4&9edo polymicrotonal scale: 4 4 1 3 4 2 2 4 3 1 4 4

* 12-tone 6&9edo polymicrotonal scale: 4 2 2 4 4 2 2 4 4 2 2 4

* 12-tone 9&12edo polymicrotonal scale: 4 2 2 4 3 3 3 3 3 3 2 4

* 12-tone 12&18edo polymicrotonal scale: 4 4 2 2 3 3 3 3 2 4 2 4

* 18-tone 9&12edo polymicrotonal scale: 3 1 2 2 1 3 3 1 2 2 1 3 3 1 2 2 1 3

* 24-tone 12&18edo polymicrotonal scale: 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2

|+[[Catnip]][24] subsets

|Bright catnip[24] MOS: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1  

• 12edo plus 1 extra min7 note: 3 3 3 3 3 3 3 3 3 2 1 3 3

 

• 12edo with 7/4 replacing 9/5: 3 3 3 3 3 3 3 3 3 2 4 3

 

• 12edo with 7/4 replacing 9/5 & 7/6 replacing 6/5 3 3 2 4 3 3 3 3 3 2 4 3

 

• 12-tone chord 30:34:35:36:37:38:40:35:47:52:53:56 approximated from [[30afdo]]^: 6 2 1 2 1 3 6 2 6 1 2 4  

 

　　　　•  Rotated [[6afdo]]: 6 6 9 8 7

 

　　　　• Flattened Ionian pentatonic: 11 4 6 11 4

 

　　　　• Flattened blues Aeolian pentatonic I: 8 7 6 2 13

 

　　　　• Flattened cosmic: 15 6 2 7 6

 

　　　　• Catnip moonbeam: 6 3 12 11 4  

 

• 12-tone chord 24:25:27:28:30:32:33:36:38:39:42:45 approximated from [[24afdo]]: 2 4 2 4 3 2 4 3 2 3 4 3

 

• 12-tone chord 18:19:20:22:23:24:26:27:29:31:32:35 approximated from [[18afdo]]: 3 3 4 3 2 4 2 4 3 2 4 2

 

 

Dark catnip[24] MOS: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2  

 

• 12edo plus 1 extra maj7 note: 3 3 3 3 3 3 3 3 3 3 1 2 3

 

• 12edo plus 1 extra maj2 note: 3 3 1 2 3 3 3 3 3 3 3 3 3

 

• 12edo but 6/5, 8/5 & 9/5 are sharp not flat: 3 3 4 2 3 3 3 4 2 4 2 3

 

• 12edo but 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 3 3 4 2 3 4 2 4 2 4 2 3

 

• 12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat: 4 2 4 2 3 3 3 4 2 4 2 3

 

• 12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 4 2 4 2 3 4 2 4 2 4 2 3

 

• 12-tone, approximates the chord 42:45:47:48:51:56:59:63:68:71:76:78 from [[42afdo]]^: 4 2 1 3 5 3 3 4 2 4 2 3

 

　　　　• Sharpened minor: 7 3 5 6 4 6 5

 

　　　　• Sharpened minor pentatonic: 10 5 6 10 5

 

　　　　• Sharpened minor harmonic pentatonic I: 7 3 11 12 3

 

　　　　• Sharpened Phyrgian pentatonic: 4 6 11 4 11

 

　　　　• Sharpened blues Aeolian pentatonic I: 10 5 6 4 11

 

　　　　• Sharpened blues Aeolian hexatonic: 10 5 3 3 4 11

 

　　　　• Sharpened blues Dorian hexatonic: 10 5 6 6 4 5

 

　　　　• Sharpened blues pentachordal I: 6 4 5 3 3 15

 

　　　　• Sharpened akebono I: 6 4 11 6 9

 

　　　　• Sharpened hirajoshi: 6 4 11 4 11

 

　　　　• Extra sharpened hirajoshi: 7 3 11 4 11

 

　　　　• Catnip Deja Vu: 10 11 4 6 5

 

　　　　• Catnip underpass: 10 11 6 4 5  

 

• 12-tone chord 18:19:20:21:22:24:25:27:28:30:32:34 approximated from [[18afdo]]: 3 3 2 3 4 2 4 2 3 4 3 3

 

 

36edo can be played on the [[Lumatone]] (see [[Lumatone mapping for 36edo]]) and using three instruments tuned to 12edo with different root notes (that is, a sixth-tone apart).