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44edo: Difference between revisions

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→Regular temperament properties: + some missing temps
 
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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; 29th harmonic is not terrible, so either add ratios of 29 or remove ratios of 31.}}
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
!Approximate JI ratios*
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
Line 36: Line 36:
|-
|-
| 1
| 1
| 27.273
| 27.3
| [[64/63]]
| [[65/64]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 43: Line 43:
|-
|-
| 2
| 2
| 54.545
| 54.5
| [[33/32]], [[32/31]]
| [[28/27]], [[32/31]], [[33/32]], [[34/33]], [[36/35]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 50: Line 50:
|-
|-
| 3
| 3
| 81.818
| 81.8
| [[23/22]]
| [[19/18]], [[20/19]], [[23/22]], [[24/23]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 57: Line 57:
|-
|-
| 4
| 4
| 109.091
| 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
|-
|-
| 5
| 5
| 136.364
| 136.4
| [[13/12]]
| [[13/12]], [[14/13]]
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 71: Line 71:
|-
|-
| 6
| 6
| 163.636
| 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
|-
|-
| 7
| 7
| 190.909
| 190.9
| [[28/25]]
| [[19/17]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 85: Line 85:
|-
|-
| 8
| 8
| 218.182
| 218.2
| [[9/8]]
| [[8/7]], ''[[9/8]]'', [[17/15]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 92: Line 92:
|-
|-
| 9
| 9
| 245.455
| 245.5
| [[8/7]]
| [[15/13]], [[22/19]]
| Upmajor 2nd, Downminor 3rd
| Upmajor 2nd, downminor 3rd
| ^M2, vm3
| ^M2, vm3
| ^E, vF
| ^E, vF
|-
|-
| 10
| 10
| 272.727
| 272.7
| [[7/6]], [[32/27]]
| [[7/6]], [[20/17]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 106: Line 106:
|-
|-
| 11
| 11
| 300.000
| 300.0
| [[19/16]]
| [[13/11]], [[19/16]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 113: Line 113:
|-
|-
| 12
| 12
| 327.273
| 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
|-
|-
| 13
| 13
| 354.545
| 354.5
| [[16/13]]
| [[16/13]], [[26/21]], [[39/32]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 127: Line 127:
|-
|-
| 14
| 14
| 381.818
| 381.8
| [[5/4]]
| [[5/4]]
| Dudmajor 3rd, Upmid 3rd
| Dudmajor 3rd, upmid 3rd
| vvM3, ^~3
| vvM3, ^~3
| vvF#
| vvF#
|-
|-
| 15
| 15
| 409.091
| 409.1
| [[80/63]]
| [[19/15]], [[24/19]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 141: Line 141:
|-
|-
| 16
| 16
| 436.364
| 436.4
| [[81/64]], [[9/7]]
| [[9/7]], ''[[14/11]]'', [[22/17]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 148: Line 148:
|-
|-
| 17
| 17
| 463.636
| 463.6
| [[21/16]]
| [[13/10]], [[17/13]]
| Upmajor 3rd, Down 4th
| Upmajor 3rd, down 4th
| ^M3, v4
| ^M3, v4
| ^F#, vG
| ^F#, vG
|-
|-
| 18
| 18
| 490.909
| 490.9
| [[4/3]]
| [[4/3]]
| Perfect 4th
| Perfect 4th
Line 162: Line 162:
|-
|-
| 19
| 19
| 518.182
| 518.2
| [[19/14]]
| [[19/14]]
| Up 4th
| Up 4th
Line 169: Line 169:
|-
|-
| 20
| 20
| 545.455
| 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
|-
|-
| 21
| 21
| 572.727
| 572.7
| [[7/5]]
| [[18/13]], [[32/23]]
| Mid 4th, Updim 5th
| Mid 4th, Updim 5th
| ~4, ^d5
| ~4, ^d5
Line 183: Line 183:
|-
|-
| 22
| 22
| 600.000
| 600.0
| [[99/70]], [[140/99]]
| [[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
|-
|-
| 23
| 23
| 627.273
| 627.3
| [[10/7]]
| [[13/9]], [[23/16]]
| Downaug 4th, Mid 5th
| Downaug 4th, mid 5th
| vA4, ~5
| vA4, ~5
| vvvA, ^^^Ab
| vvvA, ^^^Ab
|-
|-
| 24
| 24
| 654.545
| 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
|-
|-
| 25
| 25
| 681.818
| 681.8
| [[28/19]]
| [[28/19]]
| Down 5th
| Down 5th
Line 211: Line 211:
|-
|-
| 26
| 26
| 709.091
| 709.1
| [[3/2]]
| [[3/2]]
| Perfect 5th
| Perfect 5th
Line 218: Line 218:
|-
|-
| 27
| 27
| 736.364
| 736.4
| [[32/21]]
| [[20/13]], [[26/17]]
| Up 5th, Downminor 6th
| Up 5th, downminor 6th
| ^5, vm6
| ^5, vm6
| ^A, vBb
| ^A, vBb
|-
|-
| 28
| 28
| 763.636
| 763.6
| [[14/9]]
| [[14/9]], ''[[11/7]]'', [[17/11]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 232: Line 232:
|-
|-
| 29
| 29
| 790.909
| 790.9
| [[128/81]]
| [[19/12]], [[30/19]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 239: Line 239:
|-
|-
| 30
| 30
| 818.182
| 818.2
| [[8/5]]
| [[8/5]]
| Dupminor 6th, Downmid 6th
| Dupminor 6th, downmid 6th
| ^^m6, v~6
| ^^m6, v~6
| ^^Bb
| ^^Bb
|-
|-
| 31
| 31
| 845.455
| 845.5
| [[13/8]]
| [[13/8]], [[21/13]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 253: Line 253:
|-
|-
| 32
| 32
| 872.727
| 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
|-
|-
| 33
| 33
| 900.000
| 900.0
| [[27/16]]
| [[22/13]], [[32/19]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 267: Line 267:
|-
|-
| 34
| 34
| 927.273
| 927.3
| [[12/7]]
| [[12/7]], [[17/10]]
| Major 6th
| Major 6th
| M6
| M6
Line 274: Line 274:
|-
|-
| 35
| 35
| 954.545
| 954.5
| [[7/4]]?
| [[19/11]], [[26/15]]
| Upmajor 6th, Downminor 7th
| Upmajor 6th, downminor 7th
| ^M6, vm7
| ^M6, vm7
| ^B, vC
| ^B, vC
|-
|-
| 36
| 36
| 981.818
| 981.8
| [[7/4]], [[16/9]]
| [[7/4]], ''[[16/9]]'', [[30/17]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 288: Line 288:
|-
|-
| 37
| 37
| 1009.091
| 1009.1
| [[9/5]]
| [[34/19]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 295: Line 295:
|-
|-
| 38
| 38
| 1036.364
| 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
|-
|-
| 39
| 39
| 1063.636
| 1063.6
| [[24/13]]
| [[13/7]], [[24/13]]
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 309: Line 309:
|-
|-
| 40
| 40
| 1090.909
| 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#
|-
|-
| 41
| 41
| 1118.182
| 1118.2
| [[40/21]]
| [[19/10]], [[36/19]], [[44/23]], [[23/12]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 323: Line 323:
|-
|-
| 42
| 42
| 1145.455
| 1145.5
| [[27/14]], [[31/16]]
| [[27/14]], [[31/16]], [[33/17]], [[35/18]], [[64/33]]
| Major 7th
| Major 7th
| M7
| M7
Line 330: Line 330:
|-
|-
| 43
| 43
| 1172.727
| 1172.7
| [[63/32]]
| [[128/65]]
| Upmajor 7th, Down 8ve
| Upmajor 7th, down 8ve
| ^M7, v8
| ^M7, v8
| ^C#, vD
| ^C#, vD
|-
|-
| 44
| 44
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
Line 343: Line 343:
| D
| D
|}
|}
<nowiki>* In 2.3.5.7.11.13.17.19.23.29.31 subgroup</nowiki>
<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 416: Line 421:
| 22/21
| 22/21
| [[Nautilus]] (44d)
| [[Nautilus]] (44d)
|-
| 1
| 5\44
| 136.36
| 14/13
| [[Doublethink]]
|-
|-
| 1
| 1
Line 439: 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 475: 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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