44edo: Difference between revisions
Added rank 2 temperaments table, resolved todo |
m Text replacement - "Ups and Downs Notation" to "Ups and downs notation" |
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{{ED intro}} | {{ED intro}} | ||
== Theory == | == Theory == | ||
44edo is a double of 22edo, to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the 13th harmonic with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. In the 13-limit it supplies the optimal patent val for [[vigin]] temperament. The [[k*N_subgroups|2*44]] subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for [[88edo]]. | 44edo is a double of [[22edo]], to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the [[13/1|13th harmonic]] with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. | ||
It is on the [[optimal ET sequence]] for 7-, 11- and 13-limit [[nautilus]] temperament, for 11-limit [[spell]] temperament, and for 13-limit [[cantrip]] temperament. In the [[13-limit]] it supplies the optimal [[patent val]] for [[vigin]] temperament. | |||
The [[k*N_subgroups|2*44]] subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for [[88edo]]. | |||
=== Harmonics === | === Harmonics === | ||
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== Intervals == | == Intervals == | ||
In 44edo, sharps and flats alter pitch by 6 EDOsteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo). | In 44edo, sharps and flats alter pitch by 6 EDOsteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo). {{todo|complete table|text=add a column of JI approximations}} | ||
{| class="wikitable center-all right-2 left-3" | {| class="wikitable center-all right-2 left-3" | ||
|- | |- | ||
! Degrees | ! Degrees | ||
! Cents | ! Cents | ||
! colspan="3" | [[Ups and | ! colspan="3" | [[Ups and downs notation]] | ||
|- | |- | ||
| 0 | | 0 | ||
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| 190.91 | | 190.91 | ||
| 9/8 | | 9/8 | ||
| [[Spell]] | | [[Spell]]/[[cantrip]] | ||
|- | |- | ||
| 1 | | 1 | ||
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| [[Bidia]] | | [[Bidia]] | ||
|- | |- | ||
| | | 11 | ||
| | | 2\44 | ||
| | | 54.55 | ||
| | | 33/32 | ||
| [[ | | [[Hendecatonic]] | ||
|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave | ||
== Scales == | == Scales == | ||
* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 5 13 8 12 6 | |||
* Approximations of [[gamelan]] scales: | * Approximations of [[gamelan]] scales: | ||
** 5-tone pelog: 4 6 15 4 15 | ** 5-tone pelog: 4 6 15 4 15 | ||