Huygens vs meanpop
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.
Extending meantone from the 5-limit to the 7-limit, there is one obvious mapping 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: 11-limit meantone (12&31) and meanpop (19&31).
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.
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.
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.
| JI Interval | Meantone mapping | Meanpop mapping | ||
|---|---|---|---|---|
| Nominals | Fifth steps | Nominals | Fifth steps | |
| 33/32 | Doubly augmented seventh minus an octave (C-Bx) | +19 | Diminished second (C-Dbb) | -12 |
| 22/21 | Augmented unison (C-C#), same as 25/24 | +7 | Triply diminished third (C-Ebbbb) | -24 |
| 12/11 | Doubly diminished third (C-Ebbb) | -17 | Doubly augmented unison (C-Cx) | +14 |
| 11/10 | Doubly augmented unison (C-Cx) | +14 | Doubly diminished third (C-Ebbb) | -17 |
| 112/99 | Diminished third (C-Ebb), same as 8/7 | -10 | Triply augmented unison (C-C#x) | +21 |
| 33/28 | Augmented second (C-D#), same as 7/6 | +9 | Triply diminished fourth (C-Fbbb) | -22 |
| 27/22, 40/33 | Doubly diminished fourth (C-Fbb) | -15 | Doubly augmented second (C-Dx) | +16 |
| 11/9 | Doubly augmented second (C-Dx) | +16 | Doubly diminished fourth (C-Fbb) | -15 |
| 14/11 | Diminished fourth (C-Fb), same as 9/7 | -8 | Triply augmented second (C-D#x) | +23 |
| 15/11 | Doubly diminished fifth (C-Gbb) | -13 | Doubly augmented third (C-Ex) | +18 |
| 11/8 | Doubly augmented third (C-Ex) | +18 | Doubly diminished fifth (C-Gbb) | -13 |
| 16/11 | Doubly diminished sixth (C-Abbb) | -18 | Doubly augmented fourth (C-Fx) | +13 |
| 22/15 | Doubly augmented fourth (C-Fx) | +13 | Doubly diminished sixth (C-Abbb) | -18 |
| 11/7 | Augmented fifth (C-G#), same as 14/9 | +8 | Triply diminished seventh (C-Bbbbb) | -23 |
| 18/11 | Doubly diminished seventh (C-Bbbb) | -16 | Doubly augmented fifth (C-Gx) | +15 |
| 33/20, 44/27 | Doubly augmented fifth (C-Gx) | +15 | Doubly diminished seventh (C-Bbbb) | -16 |
| 56/33 | Diminished seventh (C-Bbb), same as 12/7 | -9 | Triply augmented fifth (C-G#x) | +22 |
| 99/56 | Augmented sixth (C-A#), same as 7/4 | +10 | Triply diminished octave (C-Cbbb) | -21 |
| 20/11 | Doubly diminished octave (C-Cbb) | -14 | Doubly augmented sixth (C-Ax) | +17 |
| 11/6 | Doubly augmented sixth (C-Ax) | +17 | Doubly diminished octave (C-Cbb) | -14 |
| 21/11 | Diminished octave (C-Cb), same as 48/25 | -7 | Triply augmented sixth (C-A#x) | +24 |
| 64/33 | Doubly diminished ninth (C-Dbbb) | -19 | Augmented seventh (C-B#) | +12 |
Tuning spectra
Undecimal meantone
Gencom: [2 4/3; 81/80 99/98 126/125]
Gencom mapping: [⟨1 2 4 7 11], ⟨0 -1 -4 -10 -18]]
| eigenmonzo (unchanged-interval) |
fifth (¢) |
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 |
Tridecimal meantone
Gencom: [2 4/3; 66/65 81/80 99/98 105/104]
Gencom mapping: [⟨1 2 4 7 11 10], ⟨0 -1 -4 -10 -18 -15]]
| eigenmonzo (unchanged interval) |
fifth (¢) |
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 |
Grosstone
Gencom: [2 4/3; 81/80 99/98 126/125 144/143]
Gencom mapping: [⟨1 2 4 7 11 -3], ⟨0 -1 -4 -10 -18 16]]
| eigenmonzo (unchanged interval) |
fifth (¢) |
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 |
Meridetone
Gencom: [2 4/3; 78/77 81/80 99/98 126/125]
Gencom mapping: [⟨1 2 4 7 11 15], ⟨0 -1 -4 -10 -18 -27]]
| eigenmonzo (unchanged interval) |
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 | 697.465 | 13, 15-odd-limit minimax |
| 13/12 | 697.637 | |
| 16/13 | 697.797 | |
| 15/13 | 697.83 | |
| 39/32 | 697.946 | |
| 13/10 | 698.009 | |
| 14/13 | 698.335 | |
| 33/26 | 698.407 | |
| 13/11 | 698.801 | |
| 4/3 | 701.955 |
Meanpop
Gencom: [2 4/3; 81/80 126/125 385/384]
Gencom mapping: [⟨1 2 4 7 -2], ⟨0 -1 -4 -10 13]]
| eigenmonzo (unchanged interval) |
fifth (¢) |
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 |
Tridecimal meanpop
Gencom: [2 4/3; 81/80 105/104 126/125 144/143]
Gencom mapping: [⟨1 2 4 7 -2 10], ⟨0 -1 -4 -10 13 -15]]
| eigenmonzo (unchanged interval) |
fifth (¢) |
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 | |
| 33/26 | 695.824 | |
| 13/10 | 695.838 | |
| 16/13 | 696.035 | |
| 13/11 | 696.043 | 13 and 15-odd-limit minimax |
| 39/32 | 696.405 | |
| 5/4 | 696.578 | 5, 7, 9 and 11-odd-limit minimax |
| 4/3 | 701.955 |
Meanplop
Gencom: [2 4/3; 65/64 78/77 81/80 91/90]
Gencom mapping: [⟨1 2 4 7 -2 2], ⟨0 -1 -4 -10 13 4]]
| eigenmonzo (unchanged interval) |
fifth (¢) |
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 |