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Huygens vs meanpop: Difference between revisions

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"11-limit meantone" and "meanpop", both discussed at [[Meantone_family|Meantone family]], are two different temperaments in the 11 limit. This page compares and contrasts them in detail.
"11-limit meantone" and "meanpop", both discussed at [[meantone family]], are two different temperaments in the 11-limit. This page compares and contrasts them in detail.


Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.
Extending meantone from the 5 limit to the 7 limit, there is one obvious mapping that is not too complex and adds hardly any additional error (so we're not talking about dominant temperament here). This is called "7-limit meantone" or "septimal meantone" and is an amazingly efficient (and beautiful) temperament. But extending it from the 7 limit to the 11 limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit meantone and meanpop.
Line 9: Line 9:
In 13–limit, they extend by the 105/104 comma. Alternatively meantone extends into grosstone by 144/143.
In 13–limit, they extend by the 105/104 comma. Alternatively meantone extends into grosstone by 144/143.


Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo]]. In 31edo the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.
Can meantone and meanpop be combined into a single temperament? Yes! It works wonderfully and that temperament is [[31edo|31EDO]]. In 31EDO the circle of fifths closes perfectly after 31 fifths, so Ex and Gbb are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, if that makes any sense to you.) This makes everything much simpler and results in 121/120 and 243/242 being tempered out, so that 12/11~11/10 is a "neutral second" (exactly half of a minor third), and 11/9 is a "neutral third" (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either meantone or meanpop.


{| class="wikitable"
{| class="wikitable"
Line 19: Line 19:
! | meanpop fifths
! | meanpop fifths
|-
|-
|22/21
| | 22/21
|Augmented prime (C-C#)
| | Augmented prime (C-C#)
| +7
| | +7
|Triply diminished third (C-Ebbbb)
| | Triply diminished third (C-Ebbbb)
| -24
| | -24
|-
|-
| | 12/11, 88/81
| | 12/11, 88/81
Line 37: Line 37:
| | -17
| | -17
|-
|-
|112/99
| | 112/99
|Diminished third (C-Ebb), same as 8/7
| | Diminished third (C-Ebb), same as 8/7
| -10
| | -10
|Triply augmented prime (C-C*)
| | Triply augmented prime (C-C*)
| +21
| | +21
|-
|-
|33/28
| | 33/28
|augmented second (C-D#), same as 7/6
| | Augmented second (C-D#), same as 7/6
| +9
| | +9
|Triply diminished fourth (C-Fbbb)
| | Triply diminished fourth (C-Fbbb)
| -22
| | -22
|-
|-
|40/33
| | 40/33
|Doubly diminished fourth (C-Fbb)
| | Doubly diminished fourth (C-Fbb)
| -15
| | -15
|Doubly augmented second (C-Dx)
| | Doubly augmented second (C-Dx)
| +16
| | +16
|-
|-
| | 11/9, 27/22
| | 11/9, 27/22
Line 67: Line 67:
| | +23
| | +23
|-
|-
|15/11
| | 15/11
|Triply diminished sixth (C-Abbbb)
| | Triply diminished sixth (C-Abbbb)
| -25
| | -25
|Augmented fourth (C-F#), same as 7/5
| | Augmented fourth (C-F#), same as 7/5
| +6
| | +6
|-
|-
| | 11/8
| | 11/8
Line 103: Line 103:
| | +15
| | +15
|-
|-
|33/20
| | 33/20
|Doubly augmented fifth (C-Gx)
| | Doubly augmented fifth (C-Gx)
| +15
| | +15
|Doubly diminished seventh (C-Bbbb)
| | Doubly diminished seventh (C-Bbbb)
| -16
| | -16
|-
|-
|56/33
| | 56/33
|diminished seventh (C-Bbb), same as 12/7
| | Diminished seventh (C-Bbb), same as 12/7
| -9
| | -9
|Triply augmented fifth (C-G*)
| | Triply augmented fifth (C-G*)
| +16
| | +16
|-
|-
|99/56
| | 99/56
|Augmented sixth (C-A#), same as 7/4
| | Augmented sixth (C-A#), same as 7/4
| +10
| | +10
|Triply diminished octave (C-Cbbb)
| | Triply diminished octave (C-Cbbb)
| -21
| | -21
|-
|-
| | 20/11
| | 20/11
Line 133: Line 133:
| | -14
| | -14
|-
|-
|21/11
| | 21/11
|Diminished octave (C-Cb)
| | Diminished octave (C-Cb)
| -7
| | -7
|Triply augmented sixth (C-A*)
| | Triply augmented sixth (C-A*)
| +24
| | +24
|}
|}


=Tuning Spectra=
= Tuning Spectra =


==Spectrum of Undecimal Meantone Tunings by Eigenmonzos==
== Spectrum of Undecimal Meantone Tunings by Eigenmonzos ==
Gencom: [2 4/3; 81/80 99/98 126/125]
 
Gencom map: [{{val|1 2 4 7 11}}, {{val|0 -1 -4 -10 -18}}]


11-limit commas: 81/80, 99/98, 126/125
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |
|-
|-
| | 6/5
| | 6/5
| | 694.786
| | 694.786
| |
|-
|-
| | 9/7
| | 9/7
| | 695.614
| | 695.614
| |
|-
| | 15/14
| | 696.111
| |
|-
|-
| | 7/6
| | 7/6
| | 696.319
| | 696.319
| |
|-
|-
| | 5/4
| | 5/4
| | 696.578 (5, 7, 9 limit minimax)
| | 696.578
| | 5, 7, 9-odd-limit minimax
|-
|-
| | 11/9
| | 11/9
| | 696.713 (11 limit minimax)
| | 696.713
| | 11-odd-limit minimax
|-
|-
| | 8/7
| | 8/7
| | 696.883
| | 696.883
| |
|-
|-
| | 12/11
| | 12/11
| | 697.021
| | 697.021
| |
|-
|-
| | 7/5
| | 7/5
| | 697.085
| | 697.085
| |
|-
|-
|15/11
| | 15/11
|697.158
| | 697.158
| |
|-
|-
|27/22
| | 27/22
|697.159
| | 697.159
| |
|-
|-
|22/21
| | 22/21
|697.22
| | 697.22
| |
|-
|-
| | 11/8
| | 11/8
| | 697.295
| | 697.295
| |
|-
|-
|[[21/16]]
| | 21/16
|697.344
| | 697.344
| |
|-
|-
| | 11/10
| | 11/10
| | 697.5
| | 697.5
| |
|-
|-
|16/15
| | 16/15
|697.654
| | 697.654
| |
|-
|-
|40/33
| | 40/33
|697.797
| | 697.797
| |
|-
|-
| | 14/11
| | 14/11
| | 697.812
| | 697.812
| |
|-
|-
|33/28
| | 33/28
|698.272
| | 698.272
|-
| |  
|112/99
|698.64
|-
|-
| | ''4/3''
| | 112/99
| | ''701.955''
| | 698.640
| |
|-
|-
|''88/81''
| | 4/3
|''710.4335''
| | 701.955
| |
|}
|}


===Tridecimal meantone===
=== Tridecimal meantone ===
13-limit commas: 66/65, 81/80, 99/98, 105/104
Gencom: [2 4/3; 66/65 81/80 99/98 105/104]
 
Gencom map: [{{val|1 2 4 7 11 10}}, {{val|0 -1 -4 -10 -18 -15}}]


{| class="wikitable"
{| class="wikitable"
Line 224: Line 251:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
| | 10/9
| | 691.202
| |
|-
|-
| | 14/13
| | 14/13
| | 694.34
| | 694.340
| |
|-
|-
| | 18/13
| | 18/13
| | 695.124
| | 695.124
| |
|-
|-
| | 15/13
| | 15/13
| | 695.226
| | 695.226
| |
|-
|-
|39/28
| | 39/28
|695.6095
| | 695.609
| |
|-
|-
| | 13/12
| | 13/12
| | 695.612
| | 695.612
| |
|-
|-
| | 13/10
| | 13/10
| | 695.838
| | 695.838
| |
|-
|-
|39/32
| | 16/13
|696.405
| | 696.035
| |
|-
| | 39/32
| | 696.405
| |
|-
| | 5/4
| | 696.578
| | 5, 7, 9-odd-limit minimax
|-
| | 11/9
| | 696.713
| | 11, 13, 15-odd-limit minimax
|-
|-
| | 16/13
| | 4/3
| | 697.467
| | 701.955
| |
|-
|-
|''33/26''
| | 33/26
|''703.186''
| | 703.186
| |
|-
|-
| | ''13/11''
| | 13/11
| |''703.597''
| | 703.597
| |
|}
|}


===Grosstone===
=== Grosstone ===
13-limit commas: 81/80, 99/98, 126/125, 144/143
Gencom: [2 4/3; 81/80 99/98 126/125 144/143]
 
Gencom map: [{{val|1 2 4 7 11 -3}}, {{val|0 -1 -4 -10 -18 16}}]


{| class="wikitable"
{| class="wikitable"
Line 263: Line 319:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
| | 10/9
| | 691.202
| |
|-
|-
|33/26
| | 33/26
|693.178
| | 693.178
| |
|-
|-
|39/32
| | 5/4
|697.168
| | 696.578
| | 5, 7, 9-odd-limit minimax
|-
| | 11/9
| | 696.713
| | 11-odd-limit minimax
|-
| | 39/32
| | 697.168
| |
|-
|-
| | 14/13
| | 14/13
| | 697.242 (13, 15 limit minimax)
| | 697.242
| | 13, 15-odd-limit minimax
|-
|-
| | 13/10
| | 13/10
| | 697.289
| | 697.289
| |
|-
|-
| | 13/11
| | 13/11
| | 697.376
| | 697.376
| |
|-
|-
| | 16/13
| | 16/13
| | 697.467
| | 697.467
| |
|-
|-
| | 15/13
| | 15/13
| | 697.511
| | 697.511
| |
|-
|-
| | 13/12
| | 13/12
| | 697.731
| | 697.731
| |
|-
|-
| | 18/13
| | 18/13
| | 697.966
| | 697.966
| |
|-
| | 4/3
| | 701.955
| |
|}
|}


===Meridetone===
=== Meridetone ===
13-limit commas: 78/77, 81/80, 99/98, 126/125
Gencom: [2 4/3; 78/77 81/80 99/98 126/125]
 
Gencom map: [{{val|1 2 4 7 11 15}}, {{val|0 -1 -4 -10 -18 -27}}]


{| class="wikitable"
{| class="wikitable"
Line 299: Line 383:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
| | 10/9
| | 691.202
| |
|-
| | 5/4
| | 696.578
| | 5, 7, 9-odd-limit minimax
|-
| | 11/9
| | 696.713
| | 11-odd-limit minimax
|-
|-
| | 18/13
| | 18/13
| | 697.465 (13, 15 limit minimax)
| | 697.465
| | 13, 15-odd-limit minimax
|-
|-
| | 13/12
| | 13/12
| | 697.637
| | 697.637
| |
|-
|-
| | 16/13
| | 16/13
| | 697.797
| | 697.797
| |
|-
|-
| | 15/13
| | 15/13
| | 697.83
| | 697.83
| |
|-
|-
|39/32
| | 39/32
|697.946
| | 697.946
| |
|-
|-
| | 13/10
| | 13/10
| | 698.009
| | 698.009
| |
|-
|-
| | 14/13
| | 14/13
| | 698.335
| | 698.335
| |
|-
|-
|33/26
| | 33/26
|698.407
| | 698.407
| |
|-
|-
| | 13/11
| | 13/11
| | 698.801
| | 698.801
| |
|-
| | 4/3
| | 701.955
| |
|}
|}


==Spectrum of Meanpop Tunings by Eigenmonzos==
== Spectrum of Meanpop Tunings by Eigenmonzos ==
11-limit commas: 81/80, 126/125, 385/384
Gencom: [2 4/3; 81/80 126/125 385/384]
 
Gencom map: [{{val|1 2 4 7 -2}}, {{val|0 -1 -4 -10 13}}]


{| class="wikitable"
{| class="wikitable"
Line 335: Line 447:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |
|-
|-
| | 6/5
| | 6/5
| | 694.786
| | 694.786
| |
|-
|-
| | 9/7
| | 9/7
| | 695.614
| | 695.614
| |
|-
|-
|40/33
| | 40/33
|695.815
| | 695.815
| |
|-
|-
|112/99
| | 112/99
|695.886
| | 695.886
| |
|-
|-
| | 11/8
| | 11/8
| | 696.052
| | 696.052
| |
|-
| | 15/14
| | 696.111
| |
|-
|-
| | 11/10
| | 11/10
| | 696.176
| | 696.176
| |
|-
|-
| | 7/6
| | 7/6
| | 696.319
| | 696.319
| |
|-
|-
|27/22
| | 27/22
|696.3635
| | 696.3635
| |
|-
|-
| | 14/11
| | 14/11
| | 696.413
| | 696.413
| |
|-
|-
| | 12/11
| | 12/11
| | 696.474
| | 696.474
| |
|-
| | 15/11
| | 696.497
| |
|-
|-
| | 5/4
| | 5/4
| | 696.578 (5, 7, 9, 11 limit minimax)
| | 696.578
| | 5, 7, 9, 11-odd-limit minimax
|-
|-
| | 11/9
| | 11/9
| | 696.839
| | 696.839
| |
|-
|-
| | 8/7
| | 8/7
| | 696.883
| | 696.883
| |
|-
|-
| | 7/5
| | 7/5
| | 697.085
| | 697.085
| |
|-
| | 16/15
| | 697.654
| |
|-
|-
| | ''4/3''
| | 4/3
| | ''701.955''
| | 701.955
| |
|-
|-
|''22/21''
| | ''22/21''
|''703.356''
| | ''703.356''
| |
|-
|-
|''88/81''
| | ''88/81''
|''707.946''
| | ''707.946''
| |
|}
|}


===Tridecimal meanpop===
=== Tridecimal meanpop ===
13-limit commas: 81/80, 105/104, 126/125, 144/143
Gencom: [2 4/3; 81/80 105/104 126/125 144/143]
 
Gencom map: [{{val|1 2 4 7 -2 10}}, {{val|0 -1 -4 -10 13 -15}}]


{| class="wikitable"
{| class="wikitable"
Line 398: Line 543:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
| | 10/9
| | 691.202
| |
|-
|-
| | 14/13
| | 14/13
| | 694.34
| | 694.340
| |
|-
|-
| | 18/13
| | 18/13
| | 695.124
| | 695.124
| |
|-
|-
| | 15/13
| | 15/13
| | 695.226
| | 695.226
| |
|-
|-
|39/28
| | 39/28
|695.6095
| | 695.609
| |
|-
|-
| | 13/12
| | 13/12
| | 695.612
| | 695.612
| |
|-
|-
|33/26
| | 33/26
|695.824
| | 695.824
| |
|-
|-
| | 13/10
| | 13/10
| | 695.838
| | 695.838
| |
|-
|-
| | 16/13
| | 16/13
| | 696.035
| | 696.035
| |
|-
|-
| | 13/11
| | 13/11
| | 696.043 (13, 15 limit minimax)
| | 696.043
| | 13 and 15-odd-limit minimax
|-
| | 39/32
| | 696.405
| |
|-
| | 5/4
| | 696.578
| | 5, 7, 9 and 11-odd-limit minimax
|-
|-
|39/32
| | 4/3
|696.405
| | 701.955
| |
|}
|}


===Meanplop===
=== Meanplop ===
13-limit commas: 65/64, 78/77, 81/80, 91/90
Gencom: [2 4/3; 65/64 78/77 81/80 91/90]
 
Gencom map: [{{val|1 2 4 7 -2 2}}, {{val|0 -1 -4 -10 13 4}}]


{| class="wikitable"
{| class="wikitable"
Line 437: Line 607:
! | Eigenmonzo
! | Eigenmonzo
! | Fifth
! | Fifth
! | Comments
|-
|-
|39/32
| | ''39/32''
|685.839
| | ''685.839''
| |
|-
|-
| | 16/13
| | 16/13
| | 689.868
| | 689.868
| |
|-
| | 10/9
| | 691.202
| |
|-
|-
| | 13/12
| | 13/12
| | 692.285
| | 692.285
| |
|-
|-
| | 13/10
| | 13/10
| | 693.223
| | 693.223
| |
|-
|-
| | 18/13
| | 18/13
| | 693.897
| | 693.897
| |
|-
|-
| | 15/13
| | 15/13
| | 694.193
| | 694.193
| |
|-
|-
| | 14/13
| | 14/13
| | 694.878
| | 694.878
| |
|-
| | 11/8
| | 696.052
| | 13 and 15-odd-limit minimax
|-
|-
|33/26
| | 5/4
|698.407
| | 696.578
| | 5, 7, 9 and 11-odd-limit minimax
|-
|-
|39/28
| | 33/26
|698.5365
| | 698.407
| |
|-
|-
| | 13/11
| | 13/11
| | 698.801
| | 698.801
| |
|-
| | 4/3
| | 701.955
| |
|}
|}


[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/Huygens_vs_meanpop"