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 (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 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 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.
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 [-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
| # | Cents* | Approximate ratios | ||
|---|---|---|---|---|
| 7-limit | 11-limit extensions | |||
| Meantone | Meanpop | |||
| 0 | 0.0 | 1/1 | ||
| 1 | 696.7 | 3/2 | ||
| 2 | 193.3 | 9/8, 10/9, 28/25 | ||
| 3 | 890.0 | 5/3 | ||
| 4 | 386.6 | 5/4 | ||
| 5 | 1083.3 | 15/8, 28/15 | ||
| 6 | 579.9 | 7/5, 25/18 | ||
| 7 | 76.6 | 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 |
* In 7-limit CWE tuning, octave reduced
Selected intervals
| JI interval | Huygens mapping | 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
| Eigenmonzo (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
| Eigenmonzo (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 |
Grosstone
| Eigenmonzo (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 |
Meridetone
| Eigenmonzo (unchanged-interval) |
Generator (¢) | 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
| Eigenmonzo (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 |
Tridecimal meanpop
| Eigenmonzo (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 | |
| 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 |