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Huygens vs meanpop: Difference between revisions

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Copy summary of prime-13 extensions from the respective subsections of Meantone family so that a reader doesn't have to hunt them down
 
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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 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.  
In the [[13-limit]], meanpop extends by [[105/104]] (mapping 13/8 to the double-augmented fifth C–G𝄪, tempering out [[144/143]]); whereas meantone forks into fokkertone (also mapping 13/8 to the double-augmented fifth C–G𝄪); grosstone (mapping 13/8 to the double-diminished seventh C–B♭♭♭); and meridetone (mapping 13/8 to the quadruple-augmented fourth C–F𝄪𝄪).  


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.
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.
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== Selected intervals ==
== Selected intervals ==
{| class="wikitable center-3 center-5"
{| class="wikitable center-3 center-5"
! rowspan="2" | JI Interval
! rowspan="2" | JI interval
! colspan="2" | Huygens mapping
! colspan="2" | Huygens mapping
! colspan="2" | Meanpop mapping
! colspan="2" | Meanpop mapping
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|-
|-
| 33/32
| 33/32
| Doubly augmented seventh minus an octave (C-Bx)
| Doubly augmented seventh minus an octave (C–B𝄪)
| +19
| +19
| Diminished second (C-Dbb)
| Diminished second (C–D𝄫)
| -12
| -12
|-
|-
| 22/21
| 22/21
| Augmented unison (C-C#), same as 25/24
| Augmented unison (C–C♯), same as 25/24
| +7
| +7
| Triply diminished third (C-Ebbbb)
| Triply diminished third (C–E𝄫𝄫)
| -24
| -24
|-
|-
| 12/11
| 12/11
| Doubly diminished third (C-Ebbb)
| Doubly diminished third (C–E𝄫♭)
| -17
| -17
| Doubly augmented unison (C-Cx)
| Doubly augmented unison (C–C𝄪)
| +14
| +14
|-
|-
| 11/10
| 11/10
| Doubly augmented unison (C-Cx)
| Doubly augmented unison (C–C𝄪)
| +14
| +14
| Doubly diminished third (C-Ebbb)
| Doubly diminished third (C–E𝄫♭)
| -17
| -17
|-
|-
| 112/99
| 112/99
| Diminished third (C-Ebb), same as 8/7
| Diminished third (C–E𝄫), same as 8/7
| -10
| -10
| Triply augmented unison (C-C#x)
| Triply augmented unison (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–F𝄫♭)
| -22
| -22
|-
|-
| 27/22, 40/33
| 27/22, 40/33
| Doubly diminished fourth (C-Fbb)
| Doubly diminished fourth (C–F𝄫)
| -15
| -15
| Doubly augmented second (C-Dx)
| Doubly augmented second (C–D𝄪)
| +16
| +16
|-
|-
| 11/9
| 11/9
| Doubly augmented second (C-Dx)
| Doubly augmented second (C–D𝄪)
| +16
| +16
| Doubly diminished fourth (C-Fbb)
| Doubly diminished fourth (C–F𝄫)
| -15
| -15
|-
|-
| 14/11
| 14/11
| Diminished fourth (C-Fb), same as 9/7
| Diminished fourth (C–F♭), same as 9/7
| -8
| -8
| Triply augmented second (C-D#x)
| Triply augmented second (C–D𝄪♯)
| +23
| +23
|-
|-
| 15/11
| 15/11
| Doubly diminished fifth (C-Gbb)
| Doubly diminished fifth (C–G𝄫)
| -13
| -13
| Doubly augmented third (C-Ex)
| Doubly augmented third (C–E𝄪)
| +18
| +18
|-
|-
| 11/8
| 11/8
| Doubly augmented third (C-Ex)
| Doubly augmented third (C–E𝄪)
| +18
| +18
| Doubly diminished fifth (C-Gbb)
| Doubly diminished fifth (C–G𝄫)
| -13
| -13
|-
|-
| 16/11
| 16/11
| Doubly diminished sixth (C-Abbb)
| Doubly diminished sixth (C–A𝄫♭)
| -18
| -18
| Doubly augmented fourth (C-Fx)
| Doubly augmented fourth (C–F𝄪)
| +13
| +13
|-
|-
| 22/15
| 22/15
| Doubly augmented fourth (C-Fx)
| Doubly augmented fourth (C–F𝄪)
| +13
| +13
| Doubly diminished sixth (C-Abbb)
| Doubly diminished sixth (C–A𝄫♭)
| -18
| -18
|-
|-
| 11/7
| 11/7
| Augmented fifth (C-G#), same as 14/9
| Augmented fifth (C–G♯), same as 14/9
| +8
| +8
| Triply diminished seventh (C-Bbbbb)
| Triply diminished seventh (C–B𝄫𝄫)
| -23
| -23
|-
|-
| 18/11
| 18/11
| Doubly diminished seventh (C-Bbbb)
| Doubly diminished seventh (C–B𝄫♭)
| -16
| -16
| Doubly augmented fifth (C-Gx)
| Doubly augmented fifth (C–G𝄪)
| +15
| +15
|-
|-
| 33/20, 44/27
| 33/20, 44/27
| Doubly augmented fifth (C-Gx)
| Doubly augmented fifth (C–G𝄪)
| +15
| +15
| Doubly diminished seventh (C-Bbbb)
| Doubly diminished seventh (C–B𝄫♭)
| -16
| -16
|-
|-
| 56/33
| 56/33
| Diminished seventh (C-Bbb), same as 12/7
| Diminished seventh (C–B𝄫), same as 12/7
| -9
| -9
| Triply augmented fifth (C-G#x)
| Triply augmented fifth (C–G𝄪♯)
|  +22
|  +22
|-
|-
| 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–C𝄫♭)
| -21
| -21
|-
|-
| 20/11
| 20/11
| Doubly diminished octave (C-Cbb)
| Doubly diminished octave (C–C𝄫)
| -14
| -14
| Doubly augmented sixth (C-Ax)
| Doubly augmented sixth (C–A𝄪)
| +17
| +17
|-
|-
| 11/6
| 11/6
| Doubly augmented sixth (C-Ax)
| Doubly augmented sixth (C–A𝄪)
| +17
| +17
| Doubly diminished octave (C-Cbb)
| Doubly diminished octave (C–C𝄫)
| -14
| -14
|-
|-
| 21/11
| 21/11
| Diminished octave (C-Cb), same as 48/25
| Diminished octave (C–C♭), same as 48/25
| -7
| -7
| Triply augmented sixth (C-A#x)
| Triply augmented sixth (C–A𝄪♯)
| +24
| +24
|-
|-
| 64/33
| 64/33
| Doubly diminished ninth (C-Dbbb)
| Doubly diminished ninth (C–D𝄫♭)
| -19
| -19
| Augmented seventh (C-B#)
| Augmented seventh (C–B♯)
| +12
| +12
|}
|}
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