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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== 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"

|-

! Degrees

! Cents

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

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

|-

| 0

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== Notation ==

=== Ups and downs notation ===

44edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Half-sharps and half-flats can be used to avoid triple arrows:

[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[23edo#Second-best fifth notation|23b]], [[30edo#Sagittal notation|30]], and [[37edo#Sagittal notation|37]], and is a superset of the notations for EDOs [[22edo#Sagittal notation|22]] and [[11edo#Sagittal notation|11]].

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In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

== Regular temperament properties ==

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br>per 8ve

! Generator*

! Cents*

! Associated<br>ratio*

! Temperaments

|-

| 1

| 3\44

| 81.82

| 22/21

| [[Nautilus]]

|-

| 1

| 7\44

| 190.91

| 9/8

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

|-

| 1

| 9\44

| 245.46

| 15/13

| [[Immunity]]

|-

| 1

| 13\44

| 354.55

| 11/9

| [[Ringo]]

|-

| 1

| 15\44

| 409.09

| 5/4

| [[Hocus]]

|-

| 2

| 3\44

| 81.82

| 22/21

| [[Harry]]

|-

| 4

| 4\44

| 109.09

| 16/15

| [[Bidia]]

|-

| 11

| 2\44

| 54.55

| 33/32

| [[Hendecatonic]]

|}

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave

== 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

** 7-tone pelog: 4 6 9 6 4 10 5

** 5-tone slendro: 9 9 8 9 9

== Instrument layouts ==

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[[Category:44edo| ]]

[[Category:Equal divisions of the octave|##]]

~~[[Category:Todo:add rank 2 temperaments table]]~~

@@ Line 2: / Line 2: @@
 {{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 13: / Line 17: @@
 == 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"
 |-
 ! Degrees
 ! Cents
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 292: / Line 296: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
+{{Sharpness-sharp6a}}
+Half-sharps and half-flats can be used to avoid triple arrows:
+{{Sharpness-sharp6b}}
+[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp6}}
+If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
+{{Sharpness-sharp6-qt}}
 === Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[23edo#Second-best fifth notation|23b]], [[30edo#Sagittal notation|30]], and [[37edo#Sagittal notation|37]], and is a superset of the notations for EDOs [[22edo#Sagittal notation|22]] and [[11edo#Sagittal notation|11]].
@@ Line 332: / Line 349: @@
 In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
+== Regular temperament properties ==
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 3\44
+| 81.82
+| 22/21
+| [[Nautilus]]
+|-
+| 1
+| 7\44
+| 190.91
+| 9/8
+| [[Spell]]/[[cantrip]]
+|-
+| 1
+| 9\44
+| 245.46
+| 15/13
+| [[Immunity]]
+|-
+| 1
+| 13\44
+| 354.55
+| 11/9
+| [[Ringo]]
+|-
+| 1
+| 15\44
+| 409.09
+| 5/4
+| [[Hocus]]
+|-
+| 2
+| 3\44
+| 81.82
+| 22/21
+| [[Harry]]
+|-
+| 4
+| 4\44
+| 109.09
+| 16/15
+| [[Bidia]]
+|-
+| 11
+| 2\44
+| 54.55
+| 33/32
+| [[Hendecatonic]]
+|}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
+== 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
+** 7-tone pelog: 4 6 9 6 4 10 5
+** 5-tone slendro: 9 9 8 9 9
 == Instrument layouts ==
@@ Line 345: / Line 430: @@
 [[Category:44edo| ]] <!-- main article -->
 [[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
-[[Category:Todo:add rank 2 temperaments table]]