44edo: Difference between revisions

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== 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 ===

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! Degrees

! Cents

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! colspan="3" | [[Ups and downs notation]]

|-

| 0

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| 190.91

| 9/8

| [[Spell]]

| [[Spell]]/[[cantrip]]

|-

| 1

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== Scales ==

* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 5 13 8 12 6

* Approximations of [[gamelan]] scales:

** 5-tone pelog: 4 6 15 4 15

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 {{ED intro}}
 == Theory ==
-edo 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]].
+edo 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 ===
@@ Line 18: / Line 22: @@
 ! Degrees
 ! Cents
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 367: / Line 371: @@
 | 190.91
 | 9/8
-| [[Spell]]
+| [[Spell]]/[[cantrip]]
 |-
 | 1
@@ Line 408: / Line 412: @@
 == Scales ==
+* Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]]): 5 13 8 12 6
 * Approximations of [[gamelan]] scales:
 ** 5-tone pelog: 4 6 15 4 15