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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.
{{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 does not split the fifth that is not too complex and adds hardly any additional error (so we are not talking about [[dominant (temperament)|dominant]] 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).


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 11-limit huygens, 11/8 is represented by the doubly augmented third, for example C–E𝄪. This is 18 fifths along the [[chain of fifths]]; E𝄪 is 18 fifths up from C. Huygens is tuned best sharp of 31edo, around 697 cents.


{| class="wikitable"
In meanpop, 11/8 is represented by the doubly diminished fifth, for example C–G𝄫. 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 the [[13-limit]], meanpop extends by [[105/104]], whereas meantone forks into fokkertone, grosstone, and meridetone.
 
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 E𝄪 and G𝄫 are the same note. (In other words, the interval of the ''quadruply diminished third'' is tuned to 0 cents, setting a minor third equal to four chromatic semitones. Expressed in tempered fifths and octave-reduced, this interval is the [[31-comma]] {{monzo| -49 31 }}, which is the 3-limit comma tempered out in 31edo.) This makes everything much simpler and results in [[121/120]] and [[243/242]] being tempered out, so that 12/11~11/10 is a true neutral second (exactly half of a minor third), and 11/9 is a true neutral third (exactly half of a perfect fifth). Keep in mind that neither of these things are true in either huygens or meanpop.
 
== Interval chain ==
{| class="wikitable center-1 right-2"
|-
|-
! | JI interval
! rowspan="3" | #
! | Meantone mapping
! rowspan="3" | Cents*
! | Meanpop mapping
! colspan="3" | Approximate ratios
|-
|-
| | 12/11
! rowspan="2" | 7-limit
| | Doubly diminished third (A-Cbb)
! colspan="2" | 11-limit extensions
| | Doubly augmented prime (C-Cx)
|-
|-
| | 11/10
! Meantone
| | Doubly augmented prime (C-Cx)
! Meanpop
| | Doubly diminished third (A-Cbb)
|-
|-
| | 11/9
| 0
| | Doubly augmented second (C-Dx)
| 0.0
| | Doubly diminished fourth (C-Fbb)
| '''1/1'''
|  
|  
|-
|-
| | 14/11
| 1
| | Diminished fourth (C-Fb), same as 9/7
| 696.7
| | Triply augmented second (C-Dx#)
| '''3/2'''
|  
|  
|-
|-
| | 11/8
| 2
| | Doubly augmented third (C-Ex)
| 193.3
| | Doubly diminished fifth (C-Gbb)
| '''9/8''', 10/9, 28/25
|  
|  
|-
|-
| | 16/11
| 3
| | Doubly diminished sixth (A-Fbb)
| 890.0
| | Doubly augmented fourth (C-Fx)
| 5/3
|  
|  
|-
|-
| | 11/7
| 4
| | Augmented fifth (C-G#), same as 14/9
| 386.6
| | Triply diminished seventh (A-Gbbb)
| '''5/4'''
|  
|  
|-
|-
| | 18/11
| 5
| | Doubly diminished seventh (A-Gbb)
| 1083.3
| | Doubly augmented fifth (C-Gx)
| '''15/8''', 28/15
|  
|  
|-
|-
| | 20/11
| 6
| | Doubly diminished octave (C-Cbb)
| 579.9
| | Doubly augmented sixth (C-Ax)
| 7/5, 25/18
|  
|  
|-
|-
| | 11/6
| 7
| | Doubly augmented sixth (C-Ax)
| 76.6
| | Double diminished octave (C-Cbb)
| 21/20, 25/24, 28/27
| 22/21
|
|-
| 8
| 773.2
| 14/9, 25/16
| 11/7
|
|-
| 9
| 269.9
| 7/6
|
|
|-
| 10
| 966.6
| '''7/4'''
|
|
|-
| 11
| 463.2
| 21/16
|
|
|-
| 12
| 1159.9
| 35/18, 49/25, 63/32
| 55/28, 88/45
| 64/33
|-
| 13
| 656.5
| 35/24
| 22/15
| '''16/11'''
|-
| 14
| 153.2
| 35/32
| 11/10
| 12/11
|-
| 15
| 849.8
| 49/30
| 33/20, 44/27
| 18/11
|-
| 16
| 346.5
| 49/40
| 11/9
| 27/22, 40/33
|-
| 17
| 1043.2
| 49/27
| 11/6
| 20/11
|-
| 18
| 539.8
| 49/36
| '''11/8'''
| 15/11
|-
| 19
| 36.5
| 49/48
| 33/32
| 45/44, 56/55
|}
|}
<nowiki/>* In 7-limit [[CWE]] tuning, octave reduced


=Tuning Spectra=
== Selected intervals ==
{| class="wikitable center-3 center-5"
! rowspan="2" | JI interval
! colspan="2" | Huygens mapping
! colspan="2" | Meanpop mapping
|-
! Nominals
! Fifth steps
! Nominals
! Fifth steps
|-
| 33/32
| Doubly augmented seventh minus an octave (C–B𝄪)
| +19
| Diminished second (C–D𝄫)
| -12
|-
| 22/21
| Augmented unison (C–C♯), same as 25/24
| +7
| Triply diminished third (C–E𝄫𝄫)
| -24
|-
| 12/11
| Doubly diminished third (C–E𝄫♭)
| -17
| Doubly augmented unison (C–C𝄪)
| +14
|-
| 11/10
| Doubly augmented unison (C–C𝄪)
| +14
| Doubly diminished third (C–E𝄫♭)
| -17
|-
| 112/99
| Diminished third (C–E𝄫), same as 8/7
| -10
| Triply augmented unison (C–C𝄪♯)
| +21
|-
| 33/28
| Augmented second (C–D♯), same as 7/6
| +9
| Triply diminished fourth (C–F𝄫♭)
| -22
|-
| 27/22, 40/33
| Doubly diminished fourth (C–F𝄫)
| -15
| Doubly augmented second (C–D𝄪)
| +16
|-
| 11/9
| Doubly augmented second (C–D𝄪)
| +16
| Doubly diminished fourth (C–F𝄫)
| -15
|-
| 14/11
| Diminished fourth (C–F♭), same as 9/7
| -8
| Triply augmented second (C–D𝄪♯)
| +23
|-
| 15/11
| Doubly diminished fifth (C–G𝄫)
| -13
| Doubly augmented third (C–E𝄪)
| +18
|-
| 11/8
| Doubly augmented third (C–E𝄪)
| +18
| Doubly diminished fifth (C–G𝄫)
| -13
|-
| 16/11
| Doubly diminished sixth (C–A𝄫♭)
| -18
| Doubly augmented fourth (C–F𝄪)
| +13
|-
| 22/15
| Doubly augmented fourth (C–F𝄪)
| +13
| Doubly diminished sixth (C–A𝄫♭)
| -18
|-
| 11/7
| Augmented fifth (C–G♯), same as 14/9
| +8
| Triply diminished seventh (C–B𝄫𝄫)
| -23
|-
| 18/11
| Doubly diminished seventh (C–B𝄫♭)
| -16
| Doubly augmented fifth (C–G𝄪)
| +15
|-
| 33/20, 44/27
| Doubly augmented fifth (C–G𝄪)
| +15
| Doubly diminished seventh (C–B𝄫♭)
| -16
|-
| 56/33
| Diminished seventh (C–B𝄫), same as 12/7
| -9
| Triply augmented fifth (C–G𝄪♯)
|  +22
|-
| 99/56
| Augmented sixth (C–A♯), same as 7/4
| +10
| Triply diminished octave (C–C𝄫♭)
| -21
|-
| 20/11
| Doubly diminished octave (C–C𝄫)
| -14
| Doubly augmented sixth (C–A𝄪)
| +17
|-
| 11/6
| Doubly augmented sixth (C–A𝄪)
| +17
| Doubly diminished octave (C–C𝄫)
| -14
|-
| 21/11
| Diminished octave (C–C♭), same as 48/25
| -7
| Triply augmented sixth (C–A𝄪♯)
| +24
|-
| 64/33
| Doubly diminished ninth (C–D𝄫♭)
| -19
| Augmented seventh (C–B♯)
| +12
|}
 
== Tuning spectra ==
=== Undecimal meantone ===
{| class="wikitable center-all left-3"
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 6/5
| 694.786
|
|-
| 9/7
| 695.614
|
|-
| 15/14
| 696.111
|
|-
| 7/6
| 696.319
|
|-
| 5/4
| 696.578
| 5, 7, 9-odd-limit minimax
|-
| 11/9
| 696.713
| 11-odd-limit minimax
|-
| 8/7
| 696.883
|
|-
| 12/11
| 697.021
|
|-
| 7/5
| 697.085
|
|-
| 15/11
| 697.158
|
|-
| 27/22
| 697.159
|
|-
| 22/21
| 697.22
|
|-
| 11/8
| 697.295
|
|-
| 21/16
| 697.344
|
|-
| 11/10
| 697.5
|
|-
| 16/15
| 697.654
|
|-
| 40/33
| 697.797
|
|-
| 14/11
| 697.812
|
|-
| 33/28
| 698.272
|
|-
| 112/99
| 698.640
|
|-
| 4/3
| 701.955
|
|}
 
==== Fokkertone ====
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 14/13
| 694.340
|
|-
| 18/13
| 695.124
|
|-
| 15/13
| 695.226
|
|-
| 39/28
| 695.609
|
|-
| 13/12
| 695.612
|
|-
| 13/10
| 695.838
|
|-
| 16/13
| 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
|-
| 4/3
| 701.955
|
|-
| 33/26
| 703.186
|
|-
| 13/11
| 703.597
|
|}


==Spectrum of Undecimal Meantone Tunings by Eigenmonzos==
==== Grosstone ====
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 33/26
| 693.178
|
|-
| 5/4
| 696.578
| 5, 7, 9-odd-limit minimax
|-
| 11/9
| 696.713
| 11-odd-limit minimax
|-
| 39/32
| 697.168
|
|-
| 14/13
| 697.242
| 13, 15-odd-limit minimax
|-
| 13/10
| 697.289
|
|-
| 13/11
| 697.376
|
|-
| 16/13
| 697.467
|
|-
| 15/13
| 697.511
|
|-
| 13/12
| 697.731
|
|-
| 18/13
| 697.966
|
|-
| 4/3
| 701.955
|
|}


{| class="wikitable"
==== Meridetone ====
{| class="wikitable center-all left-3"
|-
|-
! | Eigenmonzo
! Eigenmonzo<br>(unchanged-interval)
! | Fifth
! Generator (¢)
! Comments
|-
|-
| | 10/9
| 10/9
| | 691.202
| 691.202
|
|-
|-
| | 6/5
| 5/4
| | 694.786
| 696.578
| 5, 7, 9-odd-limit minimax
|-
|-
| | 9/7
| 11/9
| | 695.614
| 696.713
| 11-odd-limit minimax
|-
|-
| | 7/6
| 18/13
| | 696.319
| 697.465
| 13, 15-odd-limit minimax
|-
|-
| | 5/4
| 13/12
| | 696.578
| 697.637
|  
|-
|-
| | 11/9
| 16/13
| | 696.713 (minimax tuning)
| 697.797
|  
|-
|-
| | 8/7
| 15/13
| | 696.883
| 697.83
|  
|-
|-
| | 12/11
| 39/32
| | 697.021
| 697.946
|
|-
|-
| | 7/5
| 13/10
| | 697.085
| 698.009
|  
|-
|-
| | 11/8
| 14/13
| | 697.295
| 698.335
|  
|-
|-
| | 11/10
| 33/26
| | 697.500
| 698.407
|  
|-
|-
| | 14/11
| 13/11
| | 697.812
| 698.801
|  
|-
|-
| | 4/3
| 4/3
| | 701.955
| 701.955
|
|}
|}


==Spectrum of Meanpop Tunings by Eigenmonzos==
=== Meanpop ===
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 10/9
| 691.202
|
|-
| 6/5
| 694.786
|
|-
| 9/7
| 695.614
|
|-
| 40/33
| 695.815
|
|-
| 112/99
| 695.886
|
|-
| 11/8
| 696.052
|
|-
| 15/14
| 696.111
|
|-
| 11/10
| 696.176
|
|-
| 7/6
| 696.319
|
|-
| 27/22
| 696.3635
|
|-
| 14/11
| 696.413
|
|-
| 12/11
| 696.474
|
|-
| 15/11
| 696.497
|
|-
| 5/4
| 696.578
| 5, 7, 9, 11-odd-limit minimax
|-
| 11/9
| 696.839
|
|-
| 8/7
| 696.883
|
|-
| 7/5
| 697.085
|
|-
| 16/15
| 697.654
|
|-
| 4/3
| 701.955
|
|-
| 22/21
| 703.356
|
|}


{| class="wikitable"
==== Tridecimal meanpop ====
{| class="wikitable center-all left-3"
|-
|-
! | Eigenmonzo
! Eigenmonzo<br>(unchanged-interval)
! | Fifth
! Generator (¢)
! Comments
|-
|-
| | 10/9
| 10/9
| | 691.202
| 691.202
|
|-
|-
| | 6/5
| 14/13
| | 694.786
| 694.340
|
|-
|-
| | 9/7
| 18/13
| | 695.614
| 695.124
|
|-
|-
| | 11/8
| 15/13
| | 696.052
| 695.226
|  
|-
|-
| | 11/10
| 39/28
| | 696.176
| 695.609
|  
|-
|-
| | 7/6
| 13/12
| | 696.319
| 695.612
|  
|-
|-
| | 14/11
| 33/26
| | 696.413
| 695.824
|  
|-
|-
| | 12/11
| 13/10
| | 696.474
| 695.838
|  
|-
|-
| | 5/4
| 16/13
| | 696.578 (minimax tuning)
| 696.035
|
|-
|-
| | 11/9
| 13/11
| | 696.839
| 696.043
| 13 and 15-odd-limit minimax
|-
|-
| | 8/7
| 39/32
| | 696.883
| 696.405
|
|-
|-
| | 7/5
| 5/4
| | 697.085
| 696.578
| 5, 7, 9 and 11-odd-limit minimax
|-
|-
| | 4/3
| 4/3
| | 701.955
| 701.955
|
|}
|}
<!-- formerly known as meanplop
==== Meanpop variant ====
{| class="wikitable center-all left-3"
|-
! Eigenmonzo<br>(unchanged-interval)
! Generator (¢)
! Comments
|-
| 16/13
| 689.868
|
|-
| 10/9
| 691.202
|
|-
| 13/12
| 692.285
|
|-
| 13/10
| 693.223
|
|-
| 18/13
| 693.897
|
|-
| 15/13
| 694.193
|
|-
| 14/13
| 694.878
|
|-
| 11/8
| 696.052
| 13 and 15-odd-limit minimax
|-
| 5/4
| 696.578
| 5, 7, 9 and 11-odd-limit minimax
|-
| 33/26
| 698.407
|
|-
| 13/11
| 698.801
|
|-
| 4/3
| 701.955
|
|}
-->
[[Category:Meantone]]
[[Category:Temperament extensions]]
Retrieved from "https://en.xen.wiki/w/Huygens_vs_meanpop"