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44edo: Difference between revisions

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=== 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 = Consistency check in column of approximated JI ratios in the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios.}}
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-1 right-2 center-5 center-6"
 
{| class="wikitable center-all right-2 left-3"
|-
|-
! #
! #
Line 38: Line 37:
| 1
| 1
| 27.3
| 27.3
|  
| [[65/64]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 45: Line 44:
| 2
| 2
| 54.5
| 54.5
| [[32/31]], [[33/32]]
| [[28/27]], [[32/31]], [[33/32]], [[34/33]], [[36/35]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 52: Line 51:
| 3
| 3
| 81.8
| 81.8
| [[23/22]]
| [[19/18]], [[20/19]], [[23/22]], [[24/23]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 59: Line 58:
| 4
| 4
| 109.1
| 109.1
| [[16/15]]
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| Dupminor 2nd, Downmid 2nd
| Dupminor 2nd, downmid 2nd
| ^^m2, v~2
| ^^m2, v~2
| ^^Eb
| ^^Eb
Line 66: Line 65:
| 5
| 5
| 136.4
| 136.4
| [[13/12]]
| [[13/12]], [[14/13]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 73: Line 72:
| 6
| 6
| 163.6
| 163.6
| [[11/10]], [[32/29]]
| ''[[10/9]]'', [[11/10]], [[12/11]], [[32/29]]
| Dudmajor 2nd, Upmid 2nd
| Dudmajor 2nd, upmid 2nd
| vvM2, ^~2
| vvM2, ^~2
| vvE
| vvE
Line 80: Line 79:
| 7
| 7
| 190.9
| 190.9
| [[28/25]]
| [[19/17]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 87: Line 86:
| 8
| 8
| 218.2
| 218.2
| [[9/8]]
| [[8/7]], ''[[9/8]]'', [[17/15]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 94: Line 93:
| 9
| 9
| 245.5
| 245.5
|  
| [[15/13]], [[22/19]]
| Upmajor 2nd, Downminor 3rd
| Upmajor 2nd, downminor 3rd
| ^M2, vm3
| ^M2, vm3
| ^E, vF
| ^E, vF
Line 101: Line 100:
| 10
| 10
| 272.7
| 272.7
| [[32/27]]
| [[7/6]], [[20/17]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 108: Line 107:
| 11
| 11
| 300.0
| 300.0
| [[19/16]]
| [[13/11]], [[19/16]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 115: Line 114:
| 12
| 12
| 327.3
| 327.3
| [[11/9]], [[29/24]]
| [[6/5]], ''[[11/9]]'', [[17/14]], [[29/24]]
| Dupminor 3rd, Downmid 3rd
| Dupminor 3rd, downmid 3rd
| ^^m3, v~3
| ^^m3, v~3
| ^^F
| ^^F
Line 122: Line 121:
| 13
| 13
| 354.5
| 354.5
| [[16/13]]
| [[16/13]], [[26/21]], [[39/32]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 130: Line 129:
| 381.8
| 381.8
| [[5/4]]
| [[5/4]]
| Dudmajor 3rd, Upmid 3rd
| Dudmajor 3rd, upmid 3rd
| vvM3, ^~3
| vvM3, ^~3
| vvF#
| vvF#
Line 136: Line 135:
| 15
| 15
| 409.1
| 409.1
|  
| [[19/15]], [[24/19]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 143: Line 142:
| 16
| 16
| 436.4
| 436.4
| [[81/64]]
| [[9/7]], ''[[14/11]]'', [[22/17]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 150: Line 149:
| 17
| 17
| 463.6
| 463.6
|  
| [[13/10]], [[17/13]]
| Upmajor 3rd, Down 4th
| Upmajor 3rd, down 4th
| ^M3, v4
| ^M3, v4
| ^F#, vG
| ^F#, vG
Line 164: Line 163:
| 19
| 19
| 518.2
| 518.2
|  
| [[19/14]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 171: Line 170:
| 20
| 20
| 545.5
| 545.5
| [[11/8]]
| [[11/8]], [[15/11]], [[26/19]]
| Dup 4th, Downmid 4th, Dim 5th
| Dup 4th, downmid 4th, dim 5th
| ^^4, v~4, d5
| ^^4, v~4, d5
| Ab, ^^G
| Ab, ^^G
Line 178: Line 177:
| 21
| 21
| 572.7
| 572.7
|  
| [[18/13]], [[32/23]]
| Mid 4th, Updim 5th
| Mid 4th, Updim 5th
| ~4, ^d5
| ~4, ^d5
Line 185: Line 184:
| 22
| 22
| 600.0
| 600.0
|  
| [[17/12]], [[24/17]], ''[[7/5]]'', ''[[10/7]]''
| Upmid 4th, Downmid 5th
| Upmid 4th, downmid 5th
| ^~4, v~5
| ^~4, v~5
| vvG#, ^^Ab
| vvG#, ^^Ab
Line 192: Line 191:
| 23
| 23
| 627.3
| 627.3
|  
| [[13/9]], [[23/16]]
| Downaug 4th, Mid 5th
| Downaug 4th, mid 5th
| vA4, ~5
| vA4, ~5
| vvvA, ^^^Ab
| vvvA, ^^^Ab
Line 199: Line 198:
| 24
| 24
| 654.5
| 654.5
| [[16/11]]
| [[16/11]], [[19/13]], [[22/15]]
| Aug 4th, Upmid 5th, Dud 5th
| Aug 4th, upmid 5th, dud 5th
| A4, ^~5, vv5
| A4, ^~5, vv5
| G#, vvA
| G#, vvA
Line 206: Line 205:
| 25
| 25
| 681.8
| 681.8
|  
| [[28/19]]
| Down 5th
| Down 5th
| v5
| v5
Line 220: Line 219:
| 27
| 27
| 736.4
| 736.4
|  
| [[20/13]], [[26/17]]
| Up 5th, Downminor 6th
| Up 5th, downminor 6th
| ^5, vm6
| ^5, vm6
| ^A, vBb
| ^A, vBb
Line 227: Line 226:
| 28
| 28
| 763.6
| 763.6
|  
| [[14/9]], ''[[11/7]]'', [[17/11]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 234: Line 233:
| 29
| 29
| 790.9
| 790.9
| [[128/81]]
| [[19/12]], [[30/19]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 242: Line 241:
| 818.2
| 818.2
| [[8/5]]
| [[8/5]]
| Dupminor 6th, Downmid 6th
| Dupminor 6th, downmid 6th
| ^^m6, v~6
| ^^m6, v~6
| ^^Bb
| ^^Bb
Line 248: Line 247:
| 31
| 31
| 845.5
| 845.5
| [[13/8]]
| [[13/8]], [[21/13]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 255: Line 254:
| 32
| 32
| 872.7
| 872.7
| [[5/3]], [[48/29]]
| [[5/3]], ''[[18/11]]'', [[28/17]], [[48/29]]
| Dudmajor 6th, Upmid 6th
| Dudmajor 6th, upmid 6th
| vvM6, ^~6
| vvM6, ^~6
| vvB
| vvB
Line 262: Line 261:
| 33
| 33
| 900.0
| 900.0
| [[27/16]]
| [[22/13]], [[32/19]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 269: Line 268:
| 34
| 34
| 927.3
| 927.3
|  
| [[12/7]], [[17/10]]
| Major 6th
| Major 6th
| M6
| M6
Line 276: Line 275:
| 35
| 35
| 954.5
| 954.5
|  
| [[19/11]], [[26/15]]
| Upmajor 6th, Downminor 7th
| Upmajor 6th, downminor 7th
| ^M6, vm7
| ^M6, vm7
| ^B, vC
| ^B, vC
Line 283: Line 282:
| 36
| 36
| 981.8
| 981.8
| [[16/9]]
| [[7/4]], ''[[16/9]]'', [[30/17]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 290: Line 289:
| 37
| 37
| 1009.1
| 1009.1
| [[9/5]]
| [[34/19]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 297: Line 296:
| 38
| 38
| 1036.4
| 1036.4
| [[20/11]], [[29/16]]
| ''[[9/5]]'', [[11/6]], [[20/11]], [[29/16]]
| Dupminor 7th, Downmid 7th
| Dupminor 7th, downmid 7th
| ^^m7, v~7
| ^^m7, v~7
| ^^C
| ^^C
Line 304: Line 303:
| 39
| 39
| 1063.6
| 1063.6
| [[24/13]]
| [[13/7]], [[24/13]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 311: Line 310:
| 40
| 40
| 1090.9
| 1090.9
| [[15/8]]
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| Dudmajor 7th, Upmid 7th
| Dudmajor 7th, upmid 7th
| vvM7, ^~7
| vvM7, ^~7
| vvC#
| vvC#
Line 318: Line 317:
| 41
| 41
| 1118.2
| 1118.2
| [[40/21]]
| [[19/10]], [[36/19]], [[44/23]], [[23/12]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 325: Line 324:
| 42
| 42
| 1145.5
| 1145.5
| [[31/16]]
| [[27/14]], [[31/16]], [[33/17]], [[35/18]], [[64/33]]
| Major 7th
| Major 7th
| M7
| M7
Line 332: Line 331:
| 43
| 43
| 1172.7
| 1172.7
|  
| [[128/65]]
| Upmajor 7th, Down 8ve
| Upmajor 7th, down 8ve
| ^M7, v8
| ^M7, v8
| ^C#, vD
| ^C#, vD
Line 344: Line 343:
| D
| D
|}
|}
<nowiki/>* As a 2.3.5.11.13.17.19.23.29.31-subgroup temperament
<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 ===
Line 417: Line 421:
| 22/21
| 22/21
| [[Nautilus]] (44d)
| [[Nautilus]] (44d)
|-
| 1
| 5\44
| 136.36
| 14/13
| [[Doublethink]]
|-
|-
| 1
| 1
Line 440: Line 450:
| 409.09
| 409.09
| 5/4
| 5/4
| [[Hocus]] (44)
| [[Hocus]]
|-
| 1
| 17\44
| 463.64
| 72/55
| [[Borwell]] (44e)
|-
| 1
| 19\44
| 518.18
| 88/65
| [[Undecimation]]
|-
|-
| 2
| 2
Line 476: 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]</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 -->
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