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20edo: Difference between revisions

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→Notations: added the modern (as opposed to the classical) version of the ups and downs accidentals
Line 24: Line 24:


== Intervals ==
== Intervals ==
=== Selected 19-limit just intervals ===
{| class="wikitable center-all"
|+[[Direct approximation]] (even if [[inconsistent]])
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[16/13]] '''[[13/8]]'''
| 0.5276
|-
| [[15/14]] [[28/15]]
| 0.5571
|-
| [[10/9]] [[9/5]]
| 2.4037
|-
| '''[[19/16]]''' [[32/19]]
| 2.4869
|-
| [[14/11]] [[11/7]]
| 2.4920
|-
| [[19/13]] [[26/19]]
| 3.0146
|-
| [[15/11]] [[22/15]]
| 3.0492
|-
| [[15/13]] [[26/15]]
| 7.7410
|-
| [[16/15]] [[15/8]]
| 8.2687
|-
| [[14/13]] [[13/7]]
| 8.2982
|-
| [[8/7]] '''[[7/4]]'''
| 8.8259
|-
| [[13/11]] [[22/13]]
| 10.7902
|-
| '''[[11/8]]''' [[16/11]]
| 11.3179
|-
| [[11/9]] [[18/11]]
| 12.5920
|-
| [[11/10]] [[20/11]]
| 14.9957
|-
| '''[[17/16]]''' [[32/17]]
| 15.0445
|-
| [[9/7]] [[14/9]]
| 15.0840
|-
| [[6/5]] [[5/3]]
| 15.6412
|-
| [[7/5]] [[10/7]]
| 17.4878
|-
| '''[[3/2]]''' [[4/3]]
| 18.0449
|-
| [[13/12]] [[24/13]]
| 18.5726
|-
| [[9/8]] [[16/9]]
| 23.9100
|-
| '''[[5/4]]''' [[8/5]]
| 26.3137
|-
| [[7/6]] [[12/7]]
| 26.8709
|-
| [[12/11]] [[11/6]]
| 29.36294
|}
== Notation ==
=== Ups and downs notation ===
Like [[15edo]], every note has many names. D is also C# and Eb. The major 3rd is also a perfect 4th and a dim 5th.
{| class="wikitable center-all right-2"
{| class="wikitable center-all right-2"
|-
|-
! [[Degree]]
![[Degree]]
! [[Cents]]
![[Cents]]
! Approximate ratios
! Approximate ratios
! colspan="3" | [[Ups and downs notation]]
! colspan="3" |[[Ups and downs notation]]
! [[2L 7s|Balzano]] Notation
![[2L 7s|Balzano]]<br>Notation
! [[6L 1s|Archeotonic]] notation
![[6L 1s|Archeotonic]]<br>notation
! Nearest harmonic
! Nearest<br>harmonic
|-
|-
| 0
| 0
Line 143: Line 56:
| 2
| 2
| 120
| 120
| [[15/14]], 14/13
|[[15/14]], 14/13
| dup unison, mid 2nd
| dup unison, mid 2nd
| ^^1, ~2
| ^^1, ~2
Line 183: Line 96:
| 6
| 6
| 360
| 360
| [[16/13]], 5/4
|[[16/13]], 5/4
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 193: Line 106:
| 7
| 7
| 420
| 420
| [[14/11]], [[51/40]]
|[[14/11]], [[51/40]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 213: Line 126:
| 9
| 9
| 540
| 540
| [[15/11]], 11/8
|[[15/11]], 11/8
| up-fourth
| up-fourth
| ^4
| ^4
Line 233: Line 146:
| 11
| 11
| 660
| 660
| [[22/15]], 16/11
|[[22/15]], 16/11
| down-fifth
| down-fifth
| v5
| v5
Line 263: Line 176:
| 14
| 14
| 840
| 840
| [[13/8]], 8/5
|[[13/8]], 8/5
| mid 6th
| mid 6th
| ~6
| ~6
Line 303: Line 216:
| 18
| 18
| 1080
| 1080
| [[28/15]], 15/8, 13/7
|[[28/15]], 15/8, 13/7
| mid 7th
| mid 7th
| ~7
| ~7
Line 331: Line 244:
| 2
| 2
|}
|}
=== Selected 19-limit just intervals ===
{| class="wikitable center-all"
|+[[Direct approximation]] (even if [[inconsistent]])
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[16/13]] '''[[13/8]]'''
| 0.5276
|-
| [[15/14]] [[28/15]]
| 0.5571
|-
| [[10/9]] [[9/5]]
| 2.4037
|-
| '''[[19/16]]''' [[32/19]]
| 2.4869
|-
| [[14/11]] [[11/7]]
| 2.4920
|-
| [[19/13]] [[26/19]]
| 3.0146
|-
| [[15/11]] [[22/15]]
| 3.0492
|-
| [[15/13]] [[26/15]]
| 7.7410
|-
| [[16/15]] [[15/8]]
| 8.2687
|-
| [[14/13]] [[13/7]]
| 8.2982
|-
| [[8/7]] '''[[7/4]]'''
| 8.8259
|-
| [[13/11]] [[22/13]]
| 10.7902
|-
| '''[[11/8]]''' [[16/11]]
| 11.3179
|-
| [[11/9]] [[18/11]]
| 12.5920
|-
| [[11/10]] [[20/11]]
| 14.9957
|-
| '''[[17/16]]''' [[32/17]]
| 15.0445
|-
| [[9/7]] [[14/9]]
| 15.0840
|-
| [[6/5]] [[5/3]]
| 15.6412
|-
| [[7/5]] [[10/7]]
| 17.4878
|-
| '''[[3/2]]''' [[4/3]]
| 18.0449
|-
| [[13/12]] [[24/13]]
| 18.5726
|-
| [[9/8]] [[16/9]]
| 23.9100
|-
| '''[[5/4]]''' [[8/5]]
| 26.3137
|-
| [[7/6]] [[12/7]]
| 26.8709
|-
| [[12/11]] [[11/6]]
| 29.36294
|}
== Notation ==
=== Ups and downs notation ===
Like [[15edo]], every note has many names. D is also C# and Eb. The major 3rd is also a perfect 4th and a dim 5th. 20edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}


=== Sagittal notation ===
=== Sagittal notation ===
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