Xenharmonic Wiki

Huygens vs meanpop: Difference between revisions

← Older edit
VisualWikitext
Cleanup and some really important corrections
made it so undecimal meantone/huygens isn't just referred to as "meantone".
 
(13 intermediate revisions by 6 users not shown)
Line 1: Line 1:
"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.
{{Breadcrumb|Meantone}}


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.
{{Wikipedia| Septimal meantone temperament#11-limit meantone }}


In 11-limit meantone, 11/8 is represented by the doubly augmented third, for example C-Ex (where "x" represents the standard double sharp symbol, equivalent in meaning to "##"). This is 18 fifths along the circle of fifths; Ex is 18 fifths up from C.
'''Undecimal meantone''' (also known as '''huygens''') and '''meanpop''', both discussed at [[meantone family]], are two different temperaments in the 11-limit. This page compares and contrasts them in detail.


In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down.
Extending meantone from the 5-limit to the 7-limit, there is one obvious mapping (for standard meantone tunings) which doesn't split the fifth 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: huygens (12&31) and meanpop (19&31).


In 13–limit, they extend by the 105/104 comma. Alternatively meantone extends into grosstone by 144/143.
In 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C-Ex (where "x" represents the standard double sharp symbol, equivalent in meaning to "##"). This is 18 fifths along the circle of fifths; Ex is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.


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.
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C-Gbb. This is in the opposite direction along the circle of fifths - 13 fifths down. Meanpop is tuned best flat of 31edo, around 696 cents.
 
In 13-limit, they extend by the 105/104 comma. Alternatively huygens extends into grosstone by 144/143.
 
Can huygens 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 huygens or meanpop.


{| class="wikitable center-3 center-5"
{| class="wikitable center-3 center-5"
! rowspan="2" | JI Interval
! rowspan="2" | JI Interval
! colspan="2" | Meantone mapping
! colspan="2" | Huygens mapping
! colspan="2" | Meanpop mapping
! colspan="2" | Meanpop mapping
|-
|-
! Nominals
! Nominals
! Fifth Steps
! Fifth steps
! Nominals
! Nominals
! Fifth Steps
! Fifth steps
|-
| 33/32
| Doubly augmented seventh minus an octave (C-Bx)
| +19
| Diminished second (C-Dbb)
| -12
|-
|-
| 22/21
| 22/21
Line 33: Line 43:
| +14
| +14
|-
|-
| 11/10, 88/81
| 11/10
| Doubly augmented unison (C-Cx)
| Doubly augmented unison (C-Cx)
| +14
| +14
Line 57: Line 67:
| +16
| +16
|-
|-
| 11/9, 99/80
| 11/9
| Doubly augmented second (C-Dx)
| Doubly augmented second (C-Dx)
| +16
| +16
Line 99: Line 109:
| -23
| -23
|-
|-
| 18/11, 160/99
| 18/11
| Doubly diminished seventh (C-Bbbb)
| Doubly diminished seventh (C-Bbbb)
| -16
| -16
Line 115: Line 125:
| -9
| -9
| Triply augmented fifth (C-G#x)
| Triply augmented fifth (C-G#x)
| +16
| +22
|-
|-
| 99/56
| 99/56
Line 123: Line 133:
| -21
| -21
|-
|-
| 20/11, 81/44
| 20/11
| Doubly diminished octave (C-Cbb)
| Doubly diminished octave (C-Cbb)
| -14
| -14
Line 140: Line 150:
| Triply augmented sixth (C-A#x)
| Triply augmented sixth (C-A#x)
| +24
| +24
|-
| 64/33
| Doubly diminished ninth (C-Dbbb)
| -19
| Augmented seventh (C-B#)
| +12
|}
|}


= Tuning Spectra =
== Tuning spectra ==
 
=== Undecimal meantone ===
== Spectrum of Undecimal Meantone Tunings by Eigenmonzos ==
Gencom: [2 4/3; 81/80 99/98 126/125]
Gencom: [2 4/3; 81/80 99/98 126/125]


Gencom map: [{{val|1 2 4 7 11}}, {{val|0 -1 -4 -10 -18}}]
Gencom mapping: [{{val|1 2 4 7 11}}, {{val|0 -1 -4 -10 -18}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! | Fifth
! | fifth<br>(¢)
! | Comments
! | 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
| | 15/14
| | 696.111
| | 696.111
| |  
| |
|-
|-
| | 7/6
| | 7/6
| | 696.319
| | 696.319
| |  
| |
|-
|-
| | 5/4
| | 5/4
Line 185: Line 200:
| | 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
| | 112/99
| | 698.640
| | 698.640
| |  
| |
|-
|-
| | 4/3
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|}
|}


=== Tridecimal meantone ===
==== Tridecimal meantone ====
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 11 10}}, {{val|0 -1 -4 -10 -18 -15}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |  
| |
|-
|-
| | 14/13
| | 14/13
| | 694.340
| | 694.340
| |  
| |
|-
|-
| | 18/13
| | 18/13
| | 695.124
| | 695.124
| |  
| |
|-
|-
| | 15/13
| | 15/13
| | 695.226
| | 695.226
| |  
| |
|-
|-
| | 39/28
| | 39/28
| | 695.609
| | 695.609
| |  
| |
|-
|-
| | 13/12
| | 13/12
| | 695.612
| | 695.612
| |  
| |
|-
|-
| | 13/10
| | 13/10
| | 695.838
| | 695.838
| |  
| |
|-
|-
| | 16/13
| | 16/13
| | 696.035
| | 696.035
| |  
| |
|-
|-
| | 39/32
| | 39/32
| | 696.405
| | 696.405
| |  
| |
|-
|-
| | 5/4
| | 5/4
Line 301: Line 316:
| | 4/3
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|-
|-
| | 33/26
| | 33/26
| | 703.186
| | 703.186
| |  
| |
|-
|-
| | 13/11
| | 13/11
| | 703.597
| | 703.597
| |  
| |
|}
|}


=== Grosstone ===
==== Grosstone ====
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 11 -3}}, {{val|0 -1 -4 -10 -18 16}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |  
| |
|-
|-
| | 33/26
| | 33/26
| | 693.178
| | 693.178
| |  
| |
|-
|-
| | 5/4
| | 5/4
Line 341: Line 356:
| | 39/32
| | 39/32
| | 697.168
| | 697.168
| |  
| |
|-
|-
| | 14/13
| | 14/13
Line 349: Line 364:
| | 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
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|}
|}


=== Meridetone ===
==== Meridetone ====
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 11 15}}, {{val|0 -1 -4 -10 -18 -27}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |  
| |
|-
|-
| | 5/4
| | 5/4
Line 405: Line 420:
| | 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
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|}
|}


== Spectrum of Meanpop Tunings by Eigenmonzos ==
=== Meanpop ===
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 -2}}, {{val|0 -1 -4 -10 13}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | 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
| | 15/14
| | 696.111
| | 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
| | 15/11
| | 696.497
| | 696.497
| |  
| |
|-
|-
| | 5/4
| | 5/4
Line 509: Line 524:
| | 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
| | 16/15
| | 697.654
| | 697.654
| |  
| |
|-
|-
| | 4/3
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|-
|-
| | ''22/21''
| | 22/21
| | ''703.356''
| | 703.356
| |
| |
|-
| | ''88/81''
| | ''707.946''
| |  
|}
|}


=== Tridecimal meanpop ===
==== Tridecimal meanpop ====
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 -2 10}}, {{val|0 -1 -4 -10 13 -15}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | comments
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 691.202
| |  
| |
|-
|-
| | 14/13
| | 14/13
| | 694.340
| | 694.340
| |  
| |
|-
|-
| | 18/13
| | 18/13
| | 695.124
| | 695.124
| |  
| |
|-
|-
| | 15/13
| | 15/13
| | 695.226
| | 695.226
| |  
| |
|-
|-
| | 39/28
| | 39/28
| | 695.609
| | 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
Line 597: Line 608:
| | 4/3
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|}
|}


=== Meanplop ===
==== Meanplop ====
Gencom: [2 4/3; 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}}]
Gencom mapping: [{{val|1 2 4 7 -2 2}}, {{val|0 -1 -4 -10 13 4}}]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Eigenmonzo
! | eigenmonzo<br>(unchanged interval)
! | Fifth
! | fifth<br>(¢)
! | Comments
! | comments
|-
| | ''39/32''
| | ''685.839''
| |
|-
|-
| | 16/13
| | 16/13
| | 689.868
| | 689.868
| |  
| |
|-
|-
| | 10/9
| | 10/9
| | 691.202
| | 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
| | 11/8
Line 653: Line 660:
| | 33/26
| | 33/26
| | 698.407
| | 698.407
| |  
| |
|-
|-
| | 13/11
| | 13/11
| | 698.801
| | 698.801
| |  
| |
|-
|-
| | 4/3
| | 4/3
| | 701.955
| | 701.955
| |  
| |
|}
|}


[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/Huygens_vs_meanpop"