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44edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{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 [[13/1|13th harmonic]] with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed.  
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.  
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]].
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 ==
{{todo|complete table|inline=1|text=Need to add some ratios to the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error.}}
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-1 right-2 center-5 center-6"
|-
|-
! Degrees
! #
! Cents
! Cents
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 32: Line 36:
|-
|-
| 1
| 1
| 27.273
| 27.3
| [[65/64]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 38: 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 44: Line 50:
|-
|-
| 3
| 3
| 81.818
| 81.8
| [[19/18]], [[20/19]], [[23/22]], [[24/23]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 50: 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 62: 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 74: Line 85:
|-
|-
| 8
| 8
| 218.182
| 218.2
| [[8/7]], ''[[9/8]]'', [[17/15]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 80: 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 92: Line 106:
|-
|-
| 11
| 11
| 300.000
| 300.0
| [[13/11]], [[19/16]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 98: 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 110: 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 122: Line 141:
|-
|-
| 16
| 16
| 436.364
| 436.4
| [[9/7]], ''[[14/11]]'', [[22/17]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 128: 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 140: Line 162:
|-
|-
| 19
| 19
| 518.182
| 518.2
| [[19/14]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 146: 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 158: 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 182: Line 211:
|-
|-
| 26
| 26
| 709.091
| 709.1
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 188: 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 200: Line 232:
|-
|-
| 29
| 29
| 790.909
| 790.9
| [[19/12]], [[30/19]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 206: 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 218: 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 230: Line 267:
|-
|-
| 34
| 34
| 927.273
| 927.3
| [[12/7]], [[17/10]]
| Major 6th
| Major 6th
| M6
| M6
Line 236: 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 248: Line 288:
|-
|-
| 37
| 37
| 1009.091
| 1009.1
| [[34/19]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 254: 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 266: 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 278: 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 284: 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 ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould 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.  
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{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}}
{{Sharpness-sharp6a}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
{{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 ===
=== 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]].
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 ====
==== Evo flavor ====
Line 349: Line 403:
</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.
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 ==
== Regular temperament properties ==
Line 366: Line 420:
| 81.82
| 81.82
| 22/21
| 22/21
| [[Nautilus]]
| [[Nautilus]] (44d)
|-
| 1
| 5\44
| 136.36
| 14/13
| [[Doublethink]]
|-
|-
| 1
| 1
Line 372: Line 432:
| 190.91
| 190.91
| 9/8
| 9/8
| [[Spell]]/[[cantrip]]
| [[Spell]] (44def) / [[cantrip]] (44de)
|-
|-
| 1
| 1
Line 378: Line 438:
| 245.46
| 245.46
| 15/13
| 15/13
| [[Immunity]]
| [[Immunity]] (44cff, 2.3.5.13)
|-
|-
| 1
| 1
Line 384: Line 444:
| 354.55
| 354.55
| 11/9
| 11/9
| [[Ringo]]
| [[Beatles]] / [[ringo]] (44e)
|-
|-
| 1
| 1
Line 391: Line 451:
| 5/4
| 5/4
| [[Hocus]]
| [[Hocus]]
|-
| 1
| 17\44
| 463.64
| 72/55
| [[Borwell]] (44e)
|-
| 1
| 19\44
| 518.18
| 88/65
| [[Undecimation]]
|-
|-
| 2
| 2
Line 396: Line 468:
| 81.82
| 81.82
| 22/21
| 22/21
| [[Harry]]
| [[Harry]] (44ceff)
|-
|-
| 4
| 4
Line 402: Line 474:
| 109.09
| 109.09
| 16/15
| 16/15
| [[Bidia]]
| [[Bidia]] (44d, 7-limit)
|-
| 11
| 2\44
| 54.55
| 33/32
| [[Hendecatonic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


== Scales ==
== Scales ==
Line 420: Line 486:


== 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 ==
Line 433: Line 498:
** [https://www.youtube.com/shorts/Oi3v0c7jbjM ''<nowiki>[short clip]</nowiki>'']
** [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/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 -->
Retrieved from "https://en.xen.wiki/w/44edo"