94th-octave temperaments: Difference between revisions
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| " |
explain what makes it also so cool on top of just perfect fifth. |
||
| Line 1: | Line 1: | ||
{{Technical data page}} | {{Technical data page}} | ||
{{Infobox fractional-octave|94}} | {{Infobox fractional-octave|94}} | ||
[[94edo]] is a "wheel" for some [[fractional-octave temperaments]] because of its excellent [[3/2|perfect fifth]]. | [[94edo]] is a "wheel" for some [[fractional-octave temperaments]] because of its excellent [[3/2|perfect fifth]], which is a semiconvergent. Hence a natural option is to temper out the [[94-comma]] while leaving other generators as independent. | ||
Not only 94edo is notable for its semiconvergent [[3/2|perfect fifth]], it is also the first ET [[consistent]] in the [[23-odd-limit]]. On top of that, [[282edo]], which is 94 × 3, is also consistent equally as far (on top of being the first distinctly consistent in it), thus providing natural octave-merger option. | |||
Hence considered below is garistearn. | |||
== Garistearn == | == Garistearn == | ||
Latest revision as of 19:27, 2 October 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
94edo is a "wheel" for some fractional-octave temperaments because of its excellent perfect fifth, which is a semiconvergent. Hence a natural option is to temper out the 94-comma while leaving other generators as independent.
Not only 94edo is notable for its semiconvergent perfect fifth, it is also the first ET consistent in the 23-odd-limit. On top of that, 282edo, which is 94 × 3, is also consistent equally as far (on top of being the first distinctly consistent in it), thus providing natural octave-merger option.
Hence considered below is garistearn.
Garistearn
Named by Xenllium in 2021, the garistearn temperament (94 & 282) tempers out the same commas as 94edo in the 2.3.7 subgroup while adding an independent generator for prime 5.
Subgroup: 2.3.5.7
Comma list: 118098/117649, 33554432/33480783
Mapping: [⟨94 149 0 264], ⟨0 0 1 0]]
- mapping generators: ~1029/1024, ~5
- WE: ~1029/1024 = 12.7638 ¢, ~5/4 = 386.7156 ¢ (~5120/5103 = 3.8011 ¢)
- error map: ⟨-0.201 -0.146 -0.000 +0.822]
- CWE: ~1029/1024 = 12.7660 ¢, ~5/4 = 386.6637 ¢ (~5120/5103 = 3.6850 ¢)
- error map: ⟨0.000 +0.173 +0.350 +1.387]
Optimal ET sequence: 94, 188, 282, 658d, 940dd
Badness (Sintel): 7.77
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 33554432/33480783
Mapping: [⟨94 149 0 264 107], ⟨0 0 1 0 1]]
Optimal tunings:
- WE: ~1029/1024 = 12.7637 ¢, ~5/4 = 386.5270 ¢ (~5120/5103 = 3.6174 ¢)
- CWE: ~1029/1024 = 12.7660 ¢, ~5/4 = 386.4545 ¢ (~5120/5103 = 3.4758 ¢)
Optimal ET sequence: 94, 188e, 282, 376, 658de
Badness (Sintel): 2.72
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 28672/28561
Mapping: [⟨94 149 0 264 107 348], ⟨0 0 1 0 1 0]]
Optimal tunings:
- WE: ~169/168 = 12.7628 ¢, ~5/4 = 386.7181 ¢ (~352/351 = 3.8344 ¢)
- CWE: ~169/168 = 12.7660 ¢, ~5/4 = 386.6570 ¢ (~352/351 = 3.6783 ¢)
Optimal ET sequence: 94, 188e, 282, 658deff, 940ddeefff
Badness (Sintel): 1.90
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873
Mapping: [⟨94 149 0 264 107 348 166], ⟨0 0 1 0 1 0 1]]
Optimal tunings:
- WE: ~169/168 = 12.7628 ¢, ~5/4 = 386.7471 ¢ (~352/351 = 3.8623 ¢)
- CWE: ~169/168 = 12.7660 ¢, ~5/4 = 386.6751 ¢ (~352/351 = 3.6963 ¢)
Optimal ET sequence: 94, 188eg, 282, 658deff, 940ddeefffg
Badness (Sintel): 1.41