Xenharmonic Wiki

No-threes subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,608 edits
per discussion, pointless temperament even by 1789edo lore standards (and it is not even jacobin)
m Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "
 
(4 intermediate revisions by 2 users not shown)
Line 5: Line 5:
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* For no-fives, see [[#No-threes no-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Insect has a ~55/32 generator, three of which give the ~5/1.
Line 28: Line 28:
[[Comma list]]: [[2100875/2097152]]
[[Comma list]]: [[2100875/2097152]]


{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
{{Mapping|legend=3| 1 2 3 | 0 5 -3 }}


{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
{{Mapping|legend=5| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245
: mapping generators: ~2, ~256/245


Line 52: Line 52:
[[Comma list]]: 128/125
[[Comma list]]: 128/125


{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
{{Mapping|legend=3| 3 7 0 | 0 0 1 }}


{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
: mapping generators: ~5/4, ~7


Line 72: Line 72:
Comma list: 56/55, 128/125
Comma list: 56/55, 128/125


Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}


Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 91: Line 91:
[[Comma list]]: 78125/76832
[[Comma list]]: 78125/76832


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
{{Mapping|legend=3| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


Line 109: Line 109:
Comma list: 245/242, 625/616
Comma list: 245/242, 625/616


Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25
: mapping generators: ~2, ~28/25


Line 130: Line 130:
[[Comma list]]: 1071875/1048576
[[Comma list]]: 1071875/1048576


{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
{{Mapping|legend=3| 1 1 5 | 0 3 -5 }}


{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
{{Mapping|legend=5| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128
: mapping generators: ~2, ~175/128


Line 154: Line 154:
[[Comma list]]: 40960000/40353607
[[Comma list]]: 40960000/40353607


{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
{{Mapping|legend=3| 1 -4 0 | 0 9 4 }}


{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~80/49
: mapping generators: ~2, ~80/49


Line 174: Line 174:
Comma list: 640/637, 10985/10976
Comma list: 640/637, 10985/10976


Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
{{Mapping|legend=2| 1 -4 0 3 | 0 9 4 1 }}


Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8
: mapping generators: ~2, ~13/8


Line 196: Line 196:
Comma list: 170/169, 640/637, 5525/5488
Comma list: 170/169, 640/637, 5525/5488


Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
{{Mapping|legend=2| 1 -4 0 3 9 | 0 9 4 1 -7 }}


Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
{{Mapping|legend=4| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 215: Line 215:
Comma list: 8589934592/8544921875, 30691800524/30517578125
Comma list: 8589934592/8544921875, 30691800524/30517578125


Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773
: mapping generators: ~2, ~5120/3773


Line 231: Line 231:
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
{{Mapping|legend=2| 1 1 20 -49 35 | 0 3 -39 119 -71 }}


Optimal tunings:  
Optimal tunings:  
Line 246: Line 246:
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
{{Mapping|legend=2| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}


Optimal tunings:  
Optimal tunings:  
Line 261: Line 261:
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}


Optimal tunings:  
Optimal tunings:  
Line 280: Line 280:
[[Comma list]]: {{monzo| 1426 -596 -15 }}
[[Comma list]]: {{monzo| 1426 -596 -15 }}


{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
{{Mapping|legend=3| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}


Line 303: Line 303:
[[Comma list]]: 5021863/5000000
[[Comma list]]: 5021863/5000000


{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
{{Mapping|legend=3| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50
: mapping generators: ~2, ~5, ~77/50


Line 321: Line 321:
Comma list: 847/845, 1001/1000
Comma list: 847/845, 1001/1000


Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
{{Mapping|legend=2| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 336: Line 336:
Comma list: 170/169, 221/220, 847/845
Comma list: 170/169, 221/220, 847/845


Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
{{Mapping|legend=2| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 359: Line 359:
[[Comma list]]: 15625/15488
[[Comma list]]: 15625/15488


{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
{{Mapping|legend=3| 2 0 -7 | 0 1 3 }}


{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
{{Mapping|legend=5| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
: mapping generators: ~125/88, ~5/4


Line 379: Line 379:
[[Comma list]]: 33275/32768
[[Comma list]]: 33275/32768


{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
{{Mapping|legend=3| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
: mapping generators, ~2, ~55/32


Line 399: Line 399:
[[Comma list]]: 1331/1280
[[Comma list]]: 1331/1280


{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
{{Mapping|legend=3| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Line 421: Line 421:
[[Comma list]]: 34375/32768
[[Comma list]]: 34375/32768


{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
{{Mapping|legend=3| 1 0 15 | 0 1 -5 }}


{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
{{Mapping|legend=5| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
: mapping generators: ~2, ~5


Line 441: Line 441:
Comma list: 65/64, 6875/6656
Comma list: 65/64, 6875/6656


Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
{{Mapping|legend=2| 1 0 15 6 | 0 1 -5 -1 }}


Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
{{Mapping|legend=4| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 458: Line 458:
Comma list: 65/64, 170/169, 221/220
Comma list: 65/64, 170/169, 221/220


Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
{{Mapping|legend=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 475: Line 475:
[[Comma list]]: 26/25
[[Comma list]]: 26/25


{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
{{Mapping|legend=3| 1 2 3 | 0 1 2 }}
: mapping generators, ~2, ~5/4
: mapping generators, ~2, ~5/4


Line 493: Line 493:
[[Comma list]]: 64000000/62748517
[[Comma list]]: 64000000/62748517


{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
{{Mapping|legend=3| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13
: mapping generators, ~2, ~20/13


Line 523: Line 523:
[[Comma list]]: 5767168/5764801
[[Comma list]]: 5767168/5764801


{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
{{Mapping|legend=3| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
: mapping generators: ~2, ~7


Line 543: Line 543:
[[Comma list]]: 1372/1331
[[Comma list]]: 1372/1331


{{Mapping|legend=2| 3 0 2 | 0 1 1 }}
{{Mapping|legend=3| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7
: mapping generators: ~14/11, ~7


Line 565: Line 565:
[[Comma list]]: 343/338, 847/832
[[Comma list]]: 343/338, 847/832


{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
{{Mapping|legend=3| 1 1 3 1 | 0 4 1 6 }}


{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
{{Mapping|legend=5| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Line 587: Line 587:
[[Comma list]]: 214358881/210827008
[[Comma list]]: 214358881/210827008


{{Mapping|legend=2| 1 0 1 | 0 8 7 }}
{{Mapping|legend=3| 1 0 1 | 0 8 7 }}


{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
: mapping generators: ~2, ~14/11


Line 607: Line 607:
Comma list: 1573/1568, 15488/15379
Comma list: 1573/1568, 15488/15379


Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}


Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
Line 626: Line 626:
[[Comma list]]: 28672/28561
[[Comma list]]: 28672/28561


{{Mapping|legend=2| 1 0 3 | 0 4 1 }}
{{Mapping|legend=3| 1 0 3 | 0 4 1 }}


{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
{{Mapping|legend=5| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~13
: mapping generators: ~2, ~13


Line 646: Line 646:
[[Comma list]]: 4913/4802
[[Comma list]]: 4913/4802


{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
{{Mapping|legend=3| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


Line 664: Line 664:
[[Comma list]]: 2024782584832/2015993900449
[[Comma list]]: 2024782584832/2015993900449


{{Mapping|legend=2| 1 0 1 | 0 10 11 }}
{{Mapping|legend=3| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


Line 682: Line 682:
[[Comma list]]: 1048576/1042139
[[Comma list]]: 1048576/1042139


{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
{{Mapping|legend=3| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
: mapping generators, ~2, ~64/53


Line 704: Line 704:
[[Comma list]]: 1352/1331
[[Comma list]]: 1352/1331


{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
{{Mapping|legend=3| 1 3 3 | 0 2 3 }}


{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
{{Mapping|legend=5| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
: mapping generators, ~2, ~13/11


Line 727: Line 727:
[[Comma list]]: 265837/262144
[[Comma list]]: 265837/262144


{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
{{Mapping|legend=3| 1 0 6 | 0 3 -2 }}


{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
{{Mapping|legend=5| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
: mapping generators, ~2, ~143/128
: mapping generators, ~2, ~143/128


Line 751: Line 751:
[[Comma list]]: 29282/28561
[[Comma list]]: 29282/28561


{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
{{Mapping|legend=3| 4 0 1 | 0 1 1 }}


{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11
: mapping generators, ~13/11, ~11


Line 773: Line 773:
[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
{{Mapping|legend=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
: mapping generators: ~2, ~26/17


Line 791: Line 791:
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432


{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}
{{Mapping|legend=3| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
: mapping generators: ~247/242, ~11


Line 811: Line 811:
[[Comma list]]: 1874161/1874048
[[Comma list]]: 1874161/1874048


{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
{{Mapping|legend=3| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11
: mapping generators: ~44/37, ~11


Line 829: Line 829:
[[Comma list]]: 47525504/47045881
[[Comma list]]: 47525504/47045881


{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
{{Mapping|legend=3| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19
: mapping generators: ~2, ~26/19


Line 847: Line 847:
[[Comma list]]: 209/208, 2057/2048
[[Comma list]]: 209/208, 2057/2048


{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
{{Mapping|legend=3| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13
: mapping generators: ~2, ~11, ~13


Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"