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[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.
[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.


[[Schismatic]] temperament flattens the fifth very slightly such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. Thus a triad with 5/4 is written as C F♭ G (unless the notation has accidentals for [[81/80]], e.g. C vE G).
[[Schismatic]] temperament flattens the fifth very slightly such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).


== Approximations by edos ==
== Approximations by edos ==
12edo approximates 3/2 to within only 2¢. [[29edo]], [[41edo]] and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 [[telic]] system, 53edo is notably a 3-strong 3-2 telic system.
12edo approximates 3/2 to within only 2¢. [[29edo]], [[41edo]], and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 [[telic]] system, 53edo is notably a 3-strong 3-2 telic system.


The following edos (up to 200) approximate 3/2 to within both 7¢ and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.
The following edos (up to 200) approximate 3/2 to within both 7¢ and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.
Line 47: Line 47:
! Relative
! Relative
Error (%)
Error (%)
! ↕
! ↕
! class="unsortable" | Equally accurate
! class="unsortable" | Equally accurate
multiples
multiples
|-
|-
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 ||↓ || [[24edo|14\24]], [[36edo|21\36]]
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 ||↑ ||  
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || ↑ ||  
|-
|-
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 ||↑ ||  
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || ↑ ||  
|-
|-
|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 ||↑ || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 || ↑ || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
|-
|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 ||↓ || [[106edo|62\106]], [[159edo|93\159]]
|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 || ↓ || [[106edo|62\106]], [[159edo|93\159]]
|-
|-
|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 ||↓ || [[130edo|76\130]], [[195edo|114\195]]
|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 || ↓ || [[130edo|76\130]], [[195edo|114\195]]
|-
|-
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 ||↑ ||  
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || ↑ ||  
|-
|-
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 ||↓ ||  
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || ↓ ||  
|-
|-
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 ||↓ ||  
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || ↓ ||  
|-
|-
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 ||↑ || [[188edo|110\188]]
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || ↑ || [[188edo|110\188]]
|-
|-
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 ||↑ ||  
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || ↑ ||  
|-
|-
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 ||↓ ||  
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || ↓ ||  
|-
|-
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||↑ ||  
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||↑ ||  
|-
|-
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 ||↓ ||  
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || ↓ ||  
|-
|-
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 ||↑ ||  
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || ↑ ||  
|-
|-
| [[171edo|171]] || 100\171 || 0.200 || 2.859 ||↓ ||  
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || ↓ ||  
|-
|-
| [[176edo|176]] || 103\176 || 0.318 || 4.660 ||↑ ||  
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || ↑ ||  
|-
|-
| [[183edo|183]] || 107\183 || 0.316 || 4.814 ||↓ ||  
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || ↓ ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 ||↑ ||  
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || ↑ ||  
|}
|}
Edos can be classified by their approximation of 3/2 as:
Edos can be classified by their approximation of 3/2 as:
*'''Superflat''' edos have fifths narrower than 4\7 = ~686¢
* '''Superflat''' edos have fifths narrower than {{nowrap|4\7 {{=}} ~686{{c}}}}
*'''Perfect''' edos have fifths of exactly 4\7
* '''Perfect''' edos have fifths of exactly 4\7
*'''Diatonic''' edos have fifths between 4\7 and 3\5 = 720¢
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap|3\5 {{=}} 720{{c}}}}
*'''Pentatonic''' have fifths of exactly 3\5
* '''Pentatonic''' have fifths of exactly 3\5
*'''Supersharp''' edos have fifths wider than 3\5
* '''Supersharp''' edos have fifths wider than 3\5
 
{| class="wikitable sortable"
{| class="wikitable sortable"
|+Comparison of the fifths of edos 5 to 31
|+ style="font-size: 105%;" | Comparison of the fifths of edos 5 to 31
|-
! Edo
! Edo
! Degree
! Degree
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! Error (¢)
! Error (¢)
|-
|-
|[[5edo]]
| [[5edo]]
| 3\5
| 3\5
| 720.000
| 720.000
| pentatonic edo
| Pentatonic edo
|  +18.045
|  +18.045
|-
|-
|[[7edo]]
| [[7edo]]
| 4\7
| 4\7
| 685.714
| 685.714
| perfect edo
| perfect edo
| -16.241
| −16.241
|-
|-
|[[8edo]]
| [[8edo]]
| 5\8
| 5\8
| 750.000
| 750.000
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|  +48.045
|  +48.045
|-
|-
|[[9edo]]
| [[9edo]]
| 5\9
| 5\9
| 666.667
| 666.667
| superflat edo
| superflat edo
| -35.288
| −35.288
|-
|-
|[[10edo]]
| [[10edo]]
| 6\10
| 6\10
| 720.000
| 720.000
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|  +18.045
|  +18.045
|-
|-
|[[11edo]]
| [[11edo]]
| 6\11
| 6\11
| 654.545
| 654.545
| superflat edo
| superflat edo
| -47.41
| −47.41
|-
|-
|[[12edo]]
| [[12edo]]
| 7\12
| 7\12
| 700.000
| 700.000
| diatonic edo
| diatonic edo
| -1.955
| −1.955
|-
|-
|[[13edo]]
| [[13edo]]
| 8\13
| 8\13
| 738.462
| 738.462
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|  +36.507
|  +36.507
|-
|-
|[[14edo]]
| [[14edo]]
| 8\14
| 8\14
| 685.714
| 685.714
| perfect edo
| perfect edo
| -16.241
| −16.241
|-
|-
|[[15edo]]
| [[15edo]]
| 9\15
| 9\15
| 720.000
| 720.000
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|  +18.045
|  +18.045
|-
|-
|[[16edo]]
| [[16edo]]
| 9\16
| 9\16
| 675.000
| 675.000
| superflat edo
| superflat edo
| -26.955
| −26.955
|-
|-
|[[17edo]]
| [[17edo]]
| 10\17
| 10\17
| 705.882
| 705.882
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|  +3.927
|  +3.927
|-
|-
|[[18edo]]
| [[18edo]]
| 11\18
| 11\18
| 733.333
| 733.333
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|  +31.378
|  +31.378
|-
|-
|[[19edo]]
| [[19edo]]
| 11\19
| 11\19
| 694.737
| 694.737
| diatonic edo
| diatonic edo
| -7.218
| −7.218
|-
|-
|[[20edo]]
| [[20edo]]
| 12\20
| 12\20
| 720.000
| 720.000
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|  +18.045
|  +18.045
|-
|-
|[[21edo]]
| [[21edo]]
| 12\21
| 12\21
| 685.714
| 685.714
| perfect edo
| perfect edo
| -16.241
| −16.241
|-
|-
|[[22edo]]
| [[22edo]]
| 13\22
| 13\22
| 709.091
| 709.091
Line 205: Line 208:
|  +7.136
|  +7.136
|-
|-
|[[23edo]]
| [[23edo]]
| 13\23
| 13\23
| 678.261
| 678.261
| superflat edo
| superflat edo
| -23.694
| −23.694
|-
|-
|[[24edo]]
| [[24edo]]
| 14\24
| 14\24
| 700.000
| 700.000
| diatonic edo
| diatonic edo
| -1.955
| −1.955
|-
|-
|[[25edo]]
| [[25edo]]
| 15\25
| 15\25
| 720.000
| 720.000
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|  +18.045
|  +18.045
|-
|-
|[[26edo]]
| [[26edo]]
| 15\26
| 15\26
| 692.308
| 692.308
| diatonic edo
| diatonic edo
| -9.647
| −9.647
|-
|-
|[[27edo]]
| [[27edo]]
| 16\27
| 16\27
| 711.111
| 711.111
Line 235: Line 238:
|  +9.156
|  +9.156
|-
|-
|[[28edo]]
| [[28edo]]
| 16\28
| 16\28
| 685.714
| 685.714
| perfect edo
| perfect edo
| -16.241
| −16.241
|-
|-
|[[29edo]]
| [[29edo]]
| 17\29
| 17\29
| 703.448
| 703.448
Line 247: Line 250:
|  +1.493
|  +1.493
|-
|-
|[[30edo]]
| [[30edo]]
| 17\30
| 17\30
| 720.000
| 720.000
Line 253: Line 256:
|  +18.045
|  +18.045
|-
|-
|[[31edo]]
| [[31edo]]
| 18\31
| 18\31
| 696.774
| 696.774
| diatonic edo
| diatonic edo
| -5.181
| −5.181
|}
|}


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