44edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|44}}
{{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 ===
{{harmonics in equal|44}}
{{Harmonics in equal|44|columns=11}}
{{Harmonics in equal|44|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 44edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
44edo has subsets {{EDOs|2, 4, 11, 22}}.  
44edo has subsets {{EDOs| 2, 4, 11, 22 }}.  


One step of 44edo is very close (only 0.0086 cents sharp) to [[64/63]] (the septimal comma). [[Ruthenium]] temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example [[1012edo]], [[1848edo]], and [[2684edo]].
One step of 44edo is very close (only 0.0086 cents sharp) to [[64/63]] (the septimal comma). [[Ruthenium]] temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example [[1012edo]], [[1848edo]], and [[2684edo]]. The aforementioned 88edo, which doubles it, is a [[meantone]] tuning that corrects the 7th harmonic to near-just, although at the expense of increasing relative error of the 13th and 19th harmonics; alternatively, if it is treated as directly approximating the 9th harmonic, then it also corrects the 9th harmonic to near-just.


== 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).  
{| class="wikitable center-all right-2 left-3"
 
{| class="wikitable center-1 right-2 center-5 center-6"
|-
|-
! Degrees
! #
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation]]
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 27: Line 36:
|-
|-
| 1
| 1
| 27.273
| 27.3
| [[65/64]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 33: Line 43:
|-
|-
| 2
| 2
| 54.545
| 54.5
| [[28/27]], [[32/31]], [[33/32]], [[34/33]], [[36/35]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 39: Line 50:
|-
|-
| 3
| 3
| 81.818
| 81.8
| [[19/18]], [[20/19]], [[23/22]], [[24/23]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 45: Line 57:
|-
|-
| 4
| 4
| 109.091
| 109.1
| Dupminor 2nd, Downmid 2nd
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| Dupminor 2nd, downmid 2nd
| ^^m2, v~2
| ^^m2, v~2
| ^^Eb
| ^^Eb
|-
|-
| 5
| 5
| 136.364
| 136.4
| [[13/12]], [[14/13]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 57: Line 71:
|-
|-
| 6
| 6
| 163.636
| 163.6
| Dudmajor 2nd, Upmid 2nd
| ''[[10/9]]'', [[11/10]], [[12/11]], [[32/29]]
| Dudmajor 2nd, upmid 2nd
| vvM2, ^~2
| vvM2, ^~2
| vvE
| vvE
|-
|-
| 7
| 7
| 190.909
| 190.9
| [[19/17]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 69: Line 85:
|-
|-
| 8
| 8
| 218.182
| 218.2
| [[8/7]], ''[[9/8]]'', [[17/15]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 75: Line 92:
|-
|-
| 9
| 9
| 245.455
| 245.5
| Upmajor 2nd, Downminor 3rd
| [[15/13]], [[22/19]]
| Upmajor 2nd, downminor 3rd
| ^M2, vm3
| ^M2, vm3
| ^E, vF
| ^E, vF
|-
|-
| 10
| 10
| 272.727
| 272.7
| [[7/6]], [[20/17]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 87: Line 106:
|-
|-
| 11
| 11
| 300.000
| 300.0
| [[13/11]], [[19/16]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 93: Line 113:
|-
|-
| 12
| 12
| 327.273
| 327.3
| Dupminor 3rd, Downmid 3rd
| [[6/5]], ''[[11/9]]'', [[17/14]], [[29/24]]
| Dupminor 3rd, downmid 3rd
| ^^m3, v~3
| ^^m3, v~3
| ^^F
| ^^F
|-
|-
| 13
| 13
| 354.545
| 354.5
| [[16/13]], [[26/21]], [[39/32]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 105: Line 127:
|-
|-
| 14
| 14
| 381.818
| 381.8
| Dudmajor 3rd, Upmid 3rd
| [[5/4]]
| Dudmajor 3rd, upmid 3rd
| vvM3, ^~3
| vvM3, ^~3
| vvF#
| vvF#
|-
|-
| 15
| 15
| 409.091
| 409.1
| [[19/15]], [[24/19]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 117: Line 141:
|-
|-
| 16
| 16
| 436.364
| 436.4
| [[9/7]], ''[[14/11]]'', [[22/17]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 123: Line 148:
|-
|-
| 17
| 17
| 463.636
| 463.6
| Upmajor 3rd, Down 4th
| [[13/10]], [[17/13]]
| Upmajor 3rd, down 4th
| ^M3, v4
| ^M3, v4
| ^F#, vG
| ^F#, vG
|-
|-
| 18
| 18
| 490.909
| 490.9
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 135: Line 162:
|-
|-
| 19
| 19
| 518.182
| 518.2
| [[19/14]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 141: Line 169:
|-
|-
| 20
| 20
| 545.455
| 545.5
| Dup 4th, Downmid 4th, Dim 5th
| [[11/8]], [[15/11]], [[26/19]]
| Dup 4th, downmid 4th, dim 5th
| ^^4, v~4, d5
| ^^4, v~4, d5
| Ab, ^^G
| Ab, ^^G
|-
|-
| 21
| 21
| 572.727
| 572.7
| [[18/13]], [[32/23]]
| Mid 4th, Updim 5th
| Mid 4th, Updim 5th
| ~4, ^d5
| ~4, ^d5
Line 153: Line 183:
|-
|-
| 22
| 22
| 600.000
| 600.0
| Upmid 4th, Downmid 5th
| [[17/12]], [[24/17]], ''[[7/5]]'', ''[[10/7]]''
| Upmid 4th, downmid 5th
| ^~4, v~5
| ^~4, v~5
| vvG#, ^^Ab
| vvG#, ^^Ab
|-
|-
| 23
| 23
| 627.273
| 627.3
| Downaug 4th, Mid 5th
| [[13/9]], [[23/16]]
| Downaug 4th, mid 5th
| vA4, ~5
| vA4, ~5
| vvvA, ^^^Ab
| vvvA, ^^^Ab
|-
|-
| 24
| 24
| 654.545
| 654.5
| Aug 4th, Upmid 5th, Dud 5th
| [[16/11]], [[19/13]], [[22/15]]
| Aug 4th, upmid 5th, dud 5th
| A4, ^~5, vv5
| A4, ^~5, vv5
| G#, vvA
| G#, vvA
|-
|-
| 25
| 25
| 681.818
| 681.8
| [[28/19]]
| Down 5th
| Down 5th
| v5
| v5
Line 177: Line 211:
|-
|-
| 26
| 26
| 709.091
| 709.1
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 183: Line 218:
|-
|-
| 27
| 27
| 736.364
| 736.4
| Up 5th, Downminor 6th
| [[20/13]], [[26/17]]
| Up 5th, downminor 6th
| ^5, vm6
| ^5, vm6
| ^A, vBb
| ^A, vBb
|-
|-
| 28
| 28
| 763.636
| 763.6
| [[14/9]], ''[[11/7]]'', [[17/11]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 195: Line 232:
|-
|-
| 29
| 29
| 790.909
| 790.9
| [[19/12]], [[30/19]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 201: Line 239:
|-
|-
| 30
| 30
| 818.182
| 818.2
| Dupminor 6th, Downmid 6th
| [[8/5]]
| Dupminor 6th, downmid 6th
| ^^m6, v~6
| ^^m6, v~6
| ^^Bb
| ^^Bb
|-
|-
| 31
| 31
| 845.455
| 845.5
| [[13/8]], [[21/13]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 213: Line 253:
|-
|-
| 32
| 32
| 872.727
| 872.7
| Dudmajor 6th, Upmid 6th
| [[5/3]], ''[[18/11]]'', [[28/17]], [[48/29]]
| Dudmajor 6th, upmid 6th
| vvM6, ^~6
| vvM6, ^~6
| vvB
| vvB
|-
|-
| 33
| 33
| 900.000
| 900.0
| [[22/13]], [[32/19]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 225: Line 267:
|-
|-
| 34
| 34
| 927.273
| 927.3
| [[12/7]], [[17/10]]
| Major 6th
| Major 6th
| M6
| M6
Line 231: Line 274:
|-
|-
| 35
| 35
| 954.545
| 954.5
| Upmajor 6th, Downminor 7th
| [[19/11]], [[26/15]]
| Upmajor 6th, downminor 7th
| ^M6, vm7
| ^M6, vm7
| ^B, vC
| ^B, vC
|-
|-
| 36
| 36
| 981.818
| 981.8
| [[7/4]], ''[[16/9]]'', [[30/17]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 243: Line 288:
|-
|-
| 37
| 37
| 1009.091
| 1009.1
| [[34/19]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 249: Line 295:
|-
|-
| 38
| 38
| 1036.364
| 1036.4
| Dupminor 7th, Downmid 7th
| ''[[9/5]]'', [[11/6]], [[20/11]], [[29/16]]
| Dupminor 7th, downmid 7th
| ^^m7, v~7
| ^^m7, v~7
| ^^C
| ^^C
|-
|-
| 39
| 39
| 1063.636
| 1063.6
| [[13/7]], [[24/13]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 261: Line 309:
|-
|-
| 40
| 40
| 1090.909
| 1090.9
| Dudmajor 7th, Upmid 7th
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| Dudmajor 7th, upmid 7th
| vvM7, ^~7
| vvM7, ^~7
| vvC#
| vvC#
|-
|-
| 41
| 41
| 1118.182
| 1118.2
| [[19/10]], [[36/19]], [[44/23]], [[23/12]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 273: Line 323:
|-
|-
| 42
| 42
| 1145.455
| 1145.5
| [[27/14]], [[31/16]], [[33/17]], [[35/18]], [[64/33]]
| Major 7th
| Major 7th
| M7
| M7
Line 279: Line 330:
|-
|-
| 43
| 43
| 1172.727
| 1172.7
| Upmajor 7th, Down 8ve
| [[128/65]]
| Upmajor 7th, down 8ve
| ^M7, v8
| ^M7, v8
| ^C#, vD
| ^C#, vD
|-
|-
| 44
| 44
| 1200.000
| 1200.0
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki/>* As a 19-limit temperament, with additional ratios of 23, 29, and 31
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|44}}


== Notation ==
== Notation ==
===Sagittal notation===
=== Stein–Zimmermann–Gould notation ===
In the following diagrams, 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.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}


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]].
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
====Evo flavor====
{{Sharpness-sharp6-qt-szg}}


=== Kite's ups and downs notation ===
44edo can also be notated with [[Kite's ups and downs notation|Kite's 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}}
=== 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]].
==== Evo flavor ====
<imagemap>
<imagemap>
File:44-EDO_Evo_Sagittal.svg
File:44-EDO_Evo_Sagittal.svg
Line 309: Line 379:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:44-EDO_Revo_Sagittal.svg
File:44-EDO_Revo_Sagittal.svg
Line 322: Line 391:
</imagemap>
</imagemap>


====Evo-SZ flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:44-EDO_Evo-SZ_Sagittal.svg
File:44-EDO_Evo-SZ_Sagittal.svg
Line 334: Line 402:
default [[File:44-EDO_Evo-SZ_Sagittal.svg]]
default [[File:44-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
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]] (44d)
|-
| 1
| 5\44
| 136.36
| 14/13
| [[Doublethink]]
|-
| 1
| 7\44
| 190.91
| 9/8
| [[Spell]] (44def) / [[cantrip]] (44de)
|-
| 1
| 9\44
| 245.46
| 15/13
| [[Immunity]] (44cff, 2.3.5.13)
|-
| 1
| 13\44
| 354.55
| 11/9
| [[Beatles]] / [[ringo]] (44e)
|-
| 1
| 15\44
| 409.09
| 5/4
| [[Hocus]]
|-
| 1
| 17\44
| 463.64
| 72/55
| [[Borwell]] (44e)
|-
| 1
| 19\44
| 518.18
| 88/65
| [[Undecimation]]
|-
| 2
| 3\44
| 81.82
| 22/21
| [[Harry]] (44ceff)
|-
| 4
| 4\44
| 109.09
| 16/15
| [[Bidia]] (44d, 7-limit)
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|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 ==
== Instrument layouts ==
[[Lumatone mapping for 44edo]]
* [[Lumatone mapping for 44edo]]
 
* [[Skip fretting system 44 2 11]]
[[Skip fretting system 44 2 11]]


== Music ==
== Music ==
* ''[https://m.youtube.com/watch?v=ljaSpsQP2qc Improvisation in 44edo]'' - composed and played by [[Bryan Deister]] (May 2023), transcribed by [[Stephen Weigel]] (Sept 2024)
; [[Bryan Deister]]
* ''[https://m.youtube.com/watch?v=U_7nziRy91g Leaning Dream - Pizza Tower]'' - adapted into 44edo by [[Bryan Deister]] (July 2024)
* [https://m.youtube.com/watch?v=ljaSpsQP2qc ''Improvisation in 44edo''] (composed and played by Bryan Deister in May 2023, transcribed by [[Stephen Weigel]] in Sept 2024)
* ''[https://youtu.be/8iS_yIYYS9Q 44edo improvisation]'' - [[Bryan Deister]] (Oct 2024)
* [https://m.youtube.com/watch?v=U_7nziRy91g ''Leaning Dream - Pizza Tower''] (adapted into 44edo by Bryan Deister in July 2024)
* [https://youtu.be/8iS_yIYYS9Q ''44edo improvisation''] (Oct 2024)
* [https://www.youtube.com/shorts/_GoQNEW24fQ ''44edo improv''] (Oct 2025)
* ''Buried Treasure - 44edo'' (2026)
** [https://www.youtube.com/shorts/Oi3v0c7jbjM ''<nowiki>[short clip]</nowiki>'']
** [https://www.youtube.com/shorts/ZOoiGuUA-9Y ''<nowiki>[short 2]</nowiki>'']
** [https://www.youtube.com/watch?v=lclipVgCvf4 ''<nowiki>[complete song original release]</nowiki>'']
** [https://www.youtube.com/watch?v=JUVotVQwpiY ''<nowiki>[complete song Orpheum release]</nowiki>'']


[[Category:44edo| ]] <!-- main article -->
[[Category:44edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Todo:add rank 2 temperaments table]]

Latest revision as of 18:58, 14 May 2026

← 43edo 44edo 45edo →
Prime factorization 22 × 11
Step size 27.2727 ¢ 
Fifth 26\44 (709.091 ¢) (→ 13\22)
Semitones (A1:m2) 6:2 (163.6 ¢ : 54.55 ¢)
Consistency limit 5
Distinct consistency limit 5

44 equal divisions of the octave (abbreviated 44edo or 44ed2), also called 44-tone equal temperament (44tet) or 44 equal temperament (44et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 44 equal parts of about 27.3 ¢ each. Each step represents a frequency ratio of 21/44, or the 44th root of 2.

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.

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 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

Approximation of odd harmonics in 44edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +7.1 -4.5 +13.0 -13.0 -5.9 +4.9 +2.6 +4.1 +2.5 -7.1 -1.0
Relative (%) +26.2 -16.5 +47.6 -47.7 -21.5 +18.1 +9.7 +15.2 +9.1 -26.2 -3.7
Steps
(reduced) 		70
(26) 		102
(14) 		124
(36) 		139
(7) 		152
(20) 		163
(31) 		172
(40) 		180
(4) 		187
(11) 		193
(17) 		199
(23)
Approximation of odd harmonics in 44edo (continued)
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) -9.0 -5.9 +6.8 +0.4 +1.3 +8.5 -5.9 +12.1 +7.3 +6.7 +9.8
Relative (%) -33.0 -21.5 +24.9 +1.5 +4.7 +31.2 -21.6 +44.2 +26.8 +24.4 +35.8
Steps
(reduced) 		204
(28) 		209
(33) 		214
(38) 		218
(42) 		222
(2) 		226
(6) 		229
(9) 		233
(13) 		236
(16) 		239
(19) 		242
(22)

Subsets and supersets

44edo has subsets 2, 4, 11, 22.

One step of 44edo is very close (only 0.0086 cents sharp) to 64/63 (the septimal comma). Ruthenium temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example 1012edo, 1848edo, and 2684edo. The aforementioned 88edo, which doubles it, is a meantone tuning that corrects the 7th harmonic to near-just, although at the expense of increasing relative error of the 13th and 19th harmonics; alternatively, if it is treated as directly approximating the 9th harmonic, then it also corrects the 9th harmonic to near-just.

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).

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 Perfect 1sn P1 D
1 27.3 65/64 Up 1sn ^1 ^D
2 54.5 28/27, 32/31, 33/32, 34/33, 36/35 Minor 2nd m2 Eb
3 81.8 19/18, 20/19, 23/22, 24/23 Upminor 2nd ^m2 ^Eb
4 109.1 15/14, 16/15, 17/16, 18/17 Dupminor 2nd, downmid 2nd ^^m2, v~2 ^^Eb
5 136.4 13/12, 14/13 Mid 2nd ~2 vvvE, ^^^Eb
6 163.6 10/9, 11/10, 12/11, 32/29 Dudmajor 2nd, upmid 2nd vvM2, ^~2 vvE
7 190.9 19/17 Downmajor 2nd vM2 vE
8 218.2 8/7, 9/8, 17/15 Major 2nd M2 E
9 245.5 15/13, 22/19 Upmajor 2nd, downminor 3rd ^M2, vm3 ^E, vF
10 272.7 7/6, 20/17 Minor 3rd m3 F
11 300.0 13/11, 19/16 Upminor 3rd ^m3 ^F
12 327.3 6/5, 11/9, 17/14, 29/24 Dupminor 3rd, downmid 3rd ^^m3, v~3 ^^F
13 354.5 16/13, 26/21, 39/32 Mid 3rd ~3 ^^^F, vvvF#
14 381.8 5/4 Dudmajor 3rd, upmid 3rd vvM3, ^~3 vvF#
15 409.1 19/15, 24/19 Downmajor 3rd vM3 vF#
16 436.4 9/7, 14/11, 22/17 Major 3rd M3 F#
17 463.6 13/10, 17/13 Upmajor 3rd, down 4th ^M3, v4 ^F#, vG
18 490.9 4/3 Perfect 4th P4 G
19 518.2 19/14 Up 4th ^4 ^G
20 545.5 11/8, 15/11, 26/19 Dup 4th, downmid 4th, dim 5th ^^4, v~4, d5 Ab, ^^G
21 572.7 18/13, 32/23 Mid 4th, Updim 5th ~4, ^d5 ^^^G, vvvG#
22 600.0 17/12, 24/17, 7/5, 10/7 Upmid 4th, downmid 5th ^~4, v~5 vvG#, ^^Ab
23 627.3 13/9, 23/16 Downaug 4th, mid 5th vA4, ~5 vvvA, ^^^Ab
24 654.5 16/11, 19/13, 22/15 Aug 4th, upmid 5th, dud 5th A4, ^~5, vv5 G#, vvA
25 681.8 28/19 Down 5th v5 vA
26 709.1 3/2 Perfect 5th P5 A
27 736.4 20/13, 26/17 Up 5th, downminor 6th ^5, vm6 ^A, vBb
28 763.6 14/9, 11/7, 17/11 Minor 6th m6 Bb
29 790.9 19/12, 30/19 Upminor 6th ^m6 ^Bb
30 818.2 8/5 Dupminor 6th, downmid 6th ^^m6, v~6 ^^Bb
31 845.5 13/8, 21/13 Mid 6th ~6 ^^^Bb, vvvB
32 872.7 5/3, 18/11, 28/17, 48/29 Dudmajor 6th, upmid 6th vvM6, ^~6 vvB
33 900.0 22/13, 32/19 Downmajor 6th vM6 vB
34 927.3 12/7, 17/10 Major 6th M6 B
35 954.5 19/11, 26/15 Upmajor 6th, downminor 7th ^M6, vm7 ^B, vC
36 981.8 7/4, 16/9, 30/17 Minor 7th m7 C
37 1009.1 34/19 Upminor 7th ^m7 ^C
38 1036.4 9/5, 11/6, 20/11, 29/16 Dupminor 7th, downmid 7th ^^m7, v~7 ^^C
39 1063.6 13/7, 24/13 Mid 7th ~7 ^^^C, vvvC#
40 1090.9 15/8, 17/9, 28/15, 32/17 Dudmajor 7th, upmid 7th vvM7, ^~7 vvC#
41 1118.2 19/10, 36/19, 44/23, 23/12 Downmajor 7th vM7 vC#
42 1145.5 27/14, 31/16, 33/17, 35/18, 64/33 Major 7th M7 C#
43 1172.7 128/65 Upmajor 7th, down 8ve ^M7, v8 ^C#, vD
44 1200.0 2/1 Perfect 8ve P8 D

* As a 19-limit temperament, with additional ratios of 23, 29, and 31

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 44edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 44edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 4.7
11/10, 20/11 1.368 5.0
13/12, 24/13 2.209 8.1
15/13, 26/15 2.287 8.4
15/8, 16/15 2.640 9.7
5/4, 8/5 4.496 16.5
13/8, 16/13 4.927 18.1
7/6, 12/7 5.856 21.5
11/8, 16/11 5.863 21.5
3/2, 4/3 7.136 26.2
11/9, 18/11 7.138 26.2
13/7, 14/13 8.065 29.6
11/7, 14/11 8.417 30.9
15/11, 22/15 8.504 31.2
9/5, 10/9 8.505 31.2
13/9, 18/13 9.345 34.3
13/10, 20/13 9.422 34.5
7/5, 10/7 9.785 35.9
15/14, 28/15 10.352 38.0
13/11, 22/13 10.790 39.6
5/3, 6/5 11.631 42.6
7/4, 8/7 12.992 47.6
11/6, 12/11 12.999 47.7
9/8, 16/9 13.001 47.7
15-odd-limit intervals in 44edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 4.7
11/10, 20/11 1.368 5.0
13/12, 24/13 2.209 8.1
15/13, 26/15 2.287 8.4
15/8, 16/15 2.640 9.7
5/4, 8/5 4.496 16.5
13/8, 16/13 4.927 18.1
7/6, 12/7 5.856 21.5
11/8, 16/11 5.863 21.5
3/2, 4/3 7.136 26.2
13/7, 14/13 8.065 29.6
15/11, 22/15 8.504 31.2
13/9, 18/13 9.345 34.3
13/10, 20/13 9.422 34.5
15/14, 28/15 10.352 38.0
13/11, 22/13 10.790 39.6
5/3, 6/5 11.631 42.6
7/4, 8/7 12.992 47.6
11/6, 12/11 12.999 47.7
9/8, 16/9 14.272 52.3
7/5, 10/7 17.488 64.1
9/5, 10/9 18.767 68.8
11/7, 14/11 18.856 69.1
11/9, 18/11 20.135 73.8

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

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

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

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

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

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

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 37, and is a superset of the notations for edos 22 and 11.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8027/26

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 3\44 81.82 22/21 Nautilus (44d)
1 5\44 136.36 14/13 Doublethink
1 7\44 190.91 9/8 Spell (44def) / cantrip (44de)
1 9\44 245.46 15/13 Immunity (44cff, 2.3.5.13)
1 13\44 354.55 11/9 Beatles / ringo (44e)
1 15\44 409.09 5/4 Hocus
1 17\44 463.64 72/55 Borwell (44e)
1 19\44 518.18 88/65 Undecimation
2 3\44 81.82 22/21 Harry (44ceff)
4 4\44 109.09 16/15 Bidia (44d, 7-limit)

* Octave-reduced form, reduced to the first half-octave

Scales

  • Evacuated planet[idiosyncratic term] (approximated from 66afdo): 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

Music

Bryan Deister
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