Xenial: Difference between revisions
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Created page with "{{Infobox regtemp | Title = Xenial | Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23 | Comma basis = 126/125, 177147/175616 (7-limit); <br>126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255 (23-limit) | Edo join 1 = 19 | Edo join 2 = 70 | Mapping = 1; -9 -17 -33 22 -21 26 27 -3 | Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE | MOS scales = 6L 1s, 6L 7s, 13L 6s, [..." Tags: Mobile edit Mobile web edit |
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| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE | | Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE | ||
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]] | | MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]] | ||
| Pergen = (P8, P11/ | | Pergen = (P8, P11/9) | ||
| Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51 | | Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51 | ||
}} | }} | ||
| Line 15: | Line 15: | ||
== Tunings == | == Tunings == | ||
=== Norm-based tunings === | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~10/9 = 188.8535 {{c}} | |||
| CWE: ~10/9 = 188.8544 {{c}} | |||
| POTE: ~10/9 = 188.8548 {{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~10/9 = 188.7849 {{c}} | |||
| CWE: ~10/9 = 188.7755 {{c}} | |||
| POTE: ~10/9 = 188.7744 {{c}} | |||
|} | |||
=== Tuning spectrum === | === Tuning spectrum === | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
Revision as of 11:03, 3 May 2026
| Xenial |
126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255 (23-limit)
Xenial is a rank-2 temperament that is generated by a sharpened minor whole tone of ~10/9, so that nine generators reach 4/3, 17 reach 8/5, 21 reach 16/13 and 33 reach 8/7 with octave reduction. It is also generated by dividing 11th harmonic into 22 equal parts, 17th harmonic into 26 equal parts, or 19th harmonic into 27 equal parts.
See Starling temperaments #Xenial for more technical data.
Tunings
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 188.8535 ¢ | CWE: ~10/9 = 188.8544 ¢ | POTE: ~10/9 = 188.8548 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~10/9 = 188.7849 ¢ | CWE: ~10/9 = 188.7755 ¢ | POTE: ~10/9 = 188.7744 ¢ |
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 9/5 | 182.404 | ||
| 13/10 | 186.447 | ||
| 5 ⧵ 32 | 187.500 | 32cddefgh val Lower bound of 7-odd-limit diamond monotone | |
| 23/12 | 187.720 | ||
| 13/9 | 187.794 | ||
| 23/13 | 188.208 | ||
| 8 ⧵ 51 | 188.235 | 51cdh val Lower bound of 9-odd-limit diamond monotone | |
| 23/18 | 188.291 | ||
| 13/12 | 188.452 | ||
| 15/14 | 188.492 | ||
| 13/8 | 188.546 | ||
| 11 ⧵ 70 | 188.571 | Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone | |
| 7/5 | 188.593 | ||
| 21/20 | 188.621 | ||
| 23/14 | 188.648 | ||
| 17/16 | 188.652 | ||
| 23/21 | 188.654 | ||
| 3/2 | 188.672 | ||
| 23/15 | 188.6959 | ||
| 11/8 | 188.6963 | ||
| 23/20 | 188.711 | ||
| 14 ⧵ 89 | 188.764 | 19, 21 and 23-odd-limit diamond monotone (singleton) | |
| 21/16 | 188.791 | ||
| 19/16 | 188.797 | ||
| 7/4 | 188.823 | ||
| 7/6 | 188.880 | ||
| 15/8 | 188.913 | ||
| 9/7 | 189.006 | ||
| 21/13 | 189.036 | ||
| 5/4 | 189.040 | ||
| 14/13 | 189.308 | ||
| 5/3 | 189.455 | ||
| 3 ⧵ 19 | 189.473 | Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone | |
| 15/13 | 190.452 | ||
| 23/16 | 190.575 |