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→Spectrum of meantone tunings by eigenmonzos: improve and standardize tuning spectra tables
Line 59: Line 59:
* [[Ratwolf]]
* [[Ratwolf]]


== Spectrum of meantone tunings by eigenmonzos ==
== Tuning spectrum ==
{| class="wikitable center-1 right-2"
 
{| class="wikitable center-all"
|-
|-
! [[Eigenmonzo]]
! ET<br>generator
! Fifth size
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! Comments
! fifth<br>(¢)
! comments
|-
|-
|
| 567/512
| 567/512
| 688.323
| 688.323
| 1/2 septimal comma
| 1/2 septimal comma
|-
|-
|
| {{monzo| 16 -10 }}
| {{monzo| 16 -10 }}
| 690.225
| 690.225
| 1/2 Pythagorean comma, Pythagorean dilimma
| 1/2 Pythagorean comma, Pythagorean dilimma
|-
|-
|
| [[51/38]]
| [[51/38]]
| 690.603
| 690.603
|  
|
|-
|-
|
| {{monzo| -19 9 0 2 }}
| {{monzo| -19 9 0 2 }}
| 691.049
| 691.049
| 2/5 septimal comma
| 2/5 septimal comma
|-
|-
|
| [[10/9]]
| [[10/9]]
| 691.202
| 691.202
| 1/2 comma
| 1/2 comma
|-
|-
| [[26edo|(15\26)]]
| [[26edo|15\26]]
|
| 692.308
| 692.308
|  
|
|-
|-
|
| {{monzo| 31 -19 }}
| {{monzo| 31 -19 }}
| 692.571
| 692.571
| 2/5 Pythagorean comma
| 2/5 Pythagorean comma
|-
|-
|
| 2048/1701
| 2048/1701
| 692.867
| 692.867
| 1/3 septimal comma
| 1/3 septimal comma
|-
|-
| [[45edo|(26\45)]]
| [[45edo|26\45]]
|
| 693.333
| 693.333
|
|
|-
|-
|
| [[27/25]]
| [[27/25]]
| 693.352
| 693.352
| 2/5 comma
| 2/5 comma
|-
|-
|
| 19683/16384
| 19683/16384
| 694.135
| 694.135
| 1/3 Pythagorean comma, Pythagorean augmented second
| 1/3 Pythagorean comma, Pythagorean augmented second
|-
|-
|
| {{monzo| -23 11 0 2 }}
| {{monzo| -23 11 0 2 }}
| 694.165
| 694.165
| 2/7 septimal comma
| 2/7 septimal comma
|-
|-
|
| [[56/45]]
| [[56/45]]
| 694.651
| 694.651
|  
|
|-
|-
|
| [[28/27]]
| [[28/27]]
| 694.709
| 694.709
|  
|
|-
|-
|
| [[81/70]]
| [[81/70]]
| 694.732
| 694.732
|  
|
|-
|-
| [[19edo|(11\19)]]
| [[19edo|11\19]]
|
| 694.737
| 694.737
|
|
|-
|-
|
| [[6/5]], [[25/18]]
| [[6/5]], [[25/18]]
| 694.786
| 694.786
| 1/3 comma
| 1/3 comma
|-
|-
|
| 5103/4095
| 5103/4095
| 695.139
| 695.139
| 1/4 septimal comma
| 1/4 septimal comma
|-
|-
|
| {{monzo| 27 -17 }}
| {{monzo| 27 -17 }}
| 695.252
| 695.252
| 2/7 Pythagorean comma, [[17-comma]]
| 2/7 Pythagorean comma, [[17-comma]]
|-
|-
|
| [[35/27]]
| [[35/27]]
| 695.389
| 695.389
|  
|
|-
|-
| [[88edo|(51\88)]]
| [[88edo|51\88]]
|
| 695.455
| 695.455
|  
|
|-
|-
| 1\2 + 1\(4π)
| 1\2 + 1\(4π)
|
| 695.493
| 695.493
| [[Lucy Tuning]]
| [[Lucy Tuning]]
|-
|-
|
| [[9/7]]
| [[9/7]]
| 695.614
| 695.614
|
|
|-
|-
| ''f''<sup>4</sup> = 2''f'' + 2
|
| <math>f^4 = 2f + 2</math>
| 695.630
| 695.630
| Wilson fifth
| Wilson fifth
|-
|-
| [[69edo|(40\69)]]
| [[69edo|40\69]]
|
| 695.652
| 695.652
|
|
|-
|-
|
| [[25/24]]
| [[25/24]]
| 695.810
| 695.810
| 2/7 comma
| 2/7 comma
|-
|-
|
| [[81/80]]
| [[81/80]]
| 695.869
| 695.869
|
|
|-
|-
|
| [[36/35]]
| [[36/35]]
| 695.936
| 695.936
|
|
|-
|-
|
| [[54/49]]
| [[54/49]]
| 695.987
| 695.987
|
|
|-
|-
| [[50edo|(29\50)]]
| [[50edo|29\50]]
|
| 696.000
| 696.000
|
|
|-
|-
|
| 8192/6561
| 8192/6561
| 696.090
| 696.090
| 1/4 Pythagorean comma, Pythagorean diminished fourth
| 1/4 Pythagorean comma, Pythagorean diminished fourth
|-
|-
|
| [[15/14]]
| [[15/14]]
| 696.111
| 696.111
|
|
|-
|-
|
| 78125/73728
| 78125/73728
| 696.165
| 696.165
Line 199: Line 234:
|-
|-
| (8 - φ)\11
| (8 - φ)\11
|
| 696.214
| 696.214
| Golden meantone
| Golden meantone
|-
|-
|
| [[49/45]]
| [[49/45]]
| 696.245
| 696.245
|
|
|-
|-
|
| [[19/17]]
| [[19/17]]
| 696.279
| 696.279
| Classical meantone
| Classical meantone
|-
|-
| [[81edo|(47\81)]]
| [[81edo|47\81]]
|
| 696.296
| 696.296
|
|
|-
|-
|
| [[7/6]]
| [[7/6]]
| 696.319
| 696.319
|
|
|-
|-
|
| [[48/35]]
| [[48/35]]
| 696.399
| 696.399
|
|
|-
|-
|
| {{Monzo| 19 9 -1 -11 }}
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 696.436
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
|-
|
| 16384/15309
| 16384/15309
| 696.502
| 696.502
| 1/5 septimal comma
| 1/5 septimal comma
|-
|-
|
| [[5/4]]
| [[5/4]]
| 696.578
| 696.578
| 5-, 7-, and 9-odd-limit minimax, 1/4 comma
| 5-, 7-, and 9-odd-limit minimax, 1/4 comma
|-
|-
|
| [[49/48]]
| [[49/48]]
| 696.616
| 696.616
|
|
|-
|-
|
| [[60/49]]
| [[60/49]]
| 696.626
| 696.626
|
|
|-
|-
|
| {{monzo| -55 -11 1 25 }}
| {{monzo| -55 -11 1 25 }}
| 696.648
| 696.648
| [[7-odd-limit]] least squares
| [[7-odd-limit]] least squares
|-
|-
| [[31edo|(18\31)]]
| [[31edo|18\31]]
|
| 696.774
| 696.774
|
|
|-
|-
|
| [[35/32]]
| [[35/32]]
| 696.796
| 696.796
|
|
|-
|-
|
| [[8/7]]
| [[8/7]]
| 696.883
| 696.883
|
|
|-
|-
|
| [[49/40]]
| [[49/40]]
| 696.959
| 696.959
|
|
|-
|-
|
| [[7/5]]
| [[7/5]]
| 697.085
| 697.085
|
|
|-
|-
|
| [[75/64]]
| [[75/64]]
| 697.176
| 697.176
| 2/9 comma
| 2/9 comma
|-
|-
|
| [[2187/2048]]
| [[2187/2048]]
| 697.263
| 697.263
| 1/5 Pythagorean comma, Pythagorean apotome
| 1/5 Pythagorean comma, Pythagorean apotome
|-
|-
| [[74edo|(43\74)]]
| [[74edo|43\74]]
|
| 697.297
| 697.297
|
|
|-
|-
|
| [[21/16]]
| [[21/16]]
| 697.344
| 697.344
|
|
|-
|-
|
| 45927/32768
| 45927/32768
| 697.411
| 697.411
| 1/6 septimal comma
| 1/6 septimal comma
|-
|-
|
| [[16/15]]
| [[16/15]]
| 697.654
| 697.654
| 1/5 comma
| 1/5 comma
|-
|-
| [[43edo|(25\43)]]
| [[43edo|25\43]]
|
| 697.674
| 697.674
|
|
|-
|-
|
| [[64/63]]
| [[64/63]]
| 697.728
| 697.728
|
|
|-
|-
|
| [[21/20]]
| [[21/20]]
| 697.781
| 697.781
|
|
|-
|-
|
| [[1024/729]]
| [[1024/729]]
| 698.045
| 698.045
| 1/6 Pythagorean comma, lesser Pythagorean tritone
| 1/6 Pythagorean comma, lesser Pythagorean tritone
|-
|-
|
| {{monzo| -17 9 0 1 }}
| {{monzo| -17 9 0 1 }}
| 698.060
| 698.060
| 1/7 septimal comma
| 1/7 septimal comma
|-
|-
|
| [[28/25]]
| [[28/25]]
| 698.099
| 698.099
|
|
|-
|-
| [[55edo|(32\55)]]
| [[55edo|32\55]]
|
| 698.182
| 698.182
|
|
|-
|-
|
| [[80/63]]
| [[80/63]]
| 698.303
| 698.303
|
|
|-
|-
|
| 17/15
| 17/15
| 698.331
| 698.331
| {{clarify}}
| {{clarify}}
|-
|-
|
| [[45/32]]
| [[45/32]]
| 698.371
| 698.371
| 1/6 comma
| 1/6 comma
|-
|-
| [[67edo|(39\67)]]
| [[67edo|39\67]]
|
| 698.507
| 698.507
|
|
|-
|-
|
| [[256/243]]
| [[256/243]]
| 698.604
| 698.604
| 1/7 Pythagorean comma, Pythagorean limma
| 1/7 Pythagorean comma, Pythagorean limma
|-
|-
|
| 45/34
| 45/34
| 698.661
| 698.661
| {{clarify}}
| {{clarify}}
|-
|-
| [[79edo|(46\79)]]
| [[79edo|46\79]]
|
| 698.734
| 698.734
|
|
|-
|-
| [[170edo|(99\170)]]
| [[170edo|99\170]]
|
| 698.824
| 698.824
|  
|
|-
|-
|
| [[135/128]]
| [[135/128]]
| 698.883
| 698.883
| 1/7 comma
| 1/7 comma
|-
|-
|
| [[17/16]]
| [[17/16]]
| 699.009
| 699.009
| {{clarify}}
| {{clarify}}
|-
|-
|
| [[25/21]]
| [[25/21]]
| 699.384
| 699.384
|
|
|-
|-
| [[12edo|(7\12)]]
| [[12edo|7\12]]
|
| 700.000
| 700.000
|
|
|-
|-
| ''[[18/17]]''
|
| ''700.209''
| [[18/17]]
| 700.209
| {{clarify}}
| {{clarify}}
|-
|-
| ''[[19/16]]''
|
| ''700.829''
| [[19/16]]
| 700.829
| {{clarify}}
| {{clarify}}
|-
|-
| ''[[81/80]]''
|
| ''701.792''
| [[81/80]]
|  
| 701.792
|
|-
|-
| [[53edo|(''31\53'')]]
| [[53edo|31\53]]
| ''701.887''
|
| 701.887
|
|
|-
|-
| ''[[3/2]]''
|
| ''701.955''
| [[3/2]]
| ''[[Pythagorean tuning]]''
| 701.955
| [[Pythagorean tuning]]
|-
|-
| ''[[64/63]]''
|
| ''702.272''
| [[64/63]]
| 702.272
|
|
|-
|-
| ''256/189''
|
| ''702.301''
| 256/189
| 702.301
|
|
|}
|}
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