22edo: Difference between revisions
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22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22. | 22edo is melodically similar to [[24edo]] as both contain quarter-tones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In [[Sagittal notation]], 11 can be notated as every other note of 22. | ||
==Regular temperament properties== | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" |Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error| Absolute]] (¢) | ! [[TE error| Absolute]] (¢) | ||
| Line 1,001: | Line 1,001: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 35 -22 }} | | {{monzo| 35 -22 }} | ||
| | | {{mapping| 22 35 }} | ||
| −2.25 | | −2.25 | ||
| 2.25 | | 2.25 | ||
| Line 1,008: | Line 1,008: | ||
| 2.3.5 | | 2.3.5 | ||
| 250/243, 2048/2025 | | 250/243, 2048/2025 | ||
| | | {{mapping| 22 35 51 }} | ||
| −0.86 | | −0.86 | ||
| 2.70 | | 2.70 | ||
| Line 1,015: | Line 1,015: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 50/49, 64/63, 245/243 | | 50/49, 64/63, 245/243 | ||
| | | {{mapping| 22 35 51 62 }} | ||
| −1.80 | | −1.80 | ||
| 2.85 | | 2.85 | ||
| Line 1,022: | Line 1,022: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 50/49, 55/54, 64/63, 99/98 | | 50/49, 55/54, 64/63, 99/98 | ||
| | | {{mapping| 22 35 51 62 76 }} | ||
| −1.11 | | −1.11 | ||
| 2.90 | | 2.90 | ||
| Line 1,029: | Line 1,029: | ||
| 2.3.5.7.11.17 | | 2.3.5.7.11.17 | ||
| 50/49, 55/54, 64/63, 85/84, 99/98 | | 50/49, 55/54, 64/63, 85/84, 99/98 | ||
| | | {{mapping| 22 35 51 62 76 90 }} | ||
| −1.09 | | −1.09 | ||
| 2.65 | | 2.65 | ||
| 4.87 | | 4.87 | ||
|} | |} | ||
* 22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. | |||
* 22et does best in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]]. | |||
=== Uniform maps === | |||
===Uniform maps === | |||
{{Uniform map|13|21.5|22.5}} | {{Uniform map|13|21.5|22.5}} | ||
===Commas=== | === Commas === | ||
22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.) | 22et [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.) | ||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
|- | |- | ||
! [[Harmonic limit| Prime <br>limit]] | ! [[Harmonic limit|Prime<br>limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
| Line 1,298: | Line 1,298: | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|- | |- | ||
! Periods <br> per | ! Periods<br>per 8ve | ||
! Generator | ! Generator | ||
! Temperaments | ! Temperaments | ||