Huygens vs meanpop

"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.

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.

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	meantone fifths	meanpop mapping	meanpop fifths
12/11	Doubly diminished third (A-Cbb)	-17	Doubly augmented prime (C-Cx)	+14
11/10	Doubly augmented prime (C-Cx)	+14	Doubly diminished third (A-Cbb)	-17
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###)	+23
11/8	Doubly augmented third (C-Ex)	+18	Doubly diminished fifth (C-Gbb)	-13
16/11	Doubly diminished sixth (A-Fbb)	-18	Doubly augmented fourth (C-Fx)	+13
11/7	Augmented fifth (C-G#), same as 14/9	+8	Triply diminished seventh (A-Gbbb)	-23
18/11	Doubly diminished seventh (A-Gbb)	-16	Doubly augmented fifth (C-G##)	+15
20/11	Doubly diminished octave (C-Cbb)	-14	Doubly augmented sixth (C-A##)	+17
11/6	Doubly augmented sixth (C-A##)	+17	Double diminished octave (C-Cbb)	-14

Tuning Spectra

Spectrum of Undecimal Meantone Tunings by Eigenmonzos

11-limit commas: 81/80, 99/98, 126/125

Eigenmonzo	Fifth
10/9	691.202
6/5	694.786
9/7	695.614
7/6	696.319
5/4	696.578 (5, 7, 9 limit minimax)
11/9	696.713 (11 limit minimax)
8/7	696.883
12/11	697.021
7/5	697.085
11/8	697.295
11/10	697.500
14/11	697.812
4/3	701.955

Tridecimal meantone

13-limit commas: 66/65, 81/80, 99/98, 105/104

Eigenmonzo	Fifth
10/9	691.202
14/13	694.340
6/5	694.786
18/13	695.124
15/13	695.226
13/12	695.612
9/7	695.614
13/10	695.838
16/13	696.035
15/14	696.111
7/6	696.319
5/4	696.578 (5, 7, 9 limit minimax)
11/9	696.713 (11, 13, 15 limit minimax)
8/7	696.883
12/11	697.021
7/5	697.085
15/11	697.158
11/8	697.295
11/10	697.500
16/15	697.654
14/11	697.812
4/3	701.955
13/11	703.597

Grosstone

13-limit commas: 81/80, 99/98, 126/125, 144/143

Eigenmonzo	Fifth
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 limit minimax)
11/9	696.713 (11 limit minimax)
8/7	696.883
12/11	697.021
7/5	697.085
15/11	697.158
14/13	697.242 (13, 15 limit minimax)
13/10	697.289
11/8	697.295
13/11	697.376
16/13	697.467
11/10	697.500
15/13	697.511
16/15	697.654
13/12	697.731
14/11	697.812
18/13	697.966
4/3	701.955

Meridetone

13-limit commas: 78/77, 81/80, 99/98, 126/125

Eigenmonzo	Fifth
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 limit minimax)
11/9	696.713 (11 limit minimax)
8/7	696.883
12/11	697.021
7/5	697.085
15/11	697.158
11/8	697.295
18/13	697.465 (13, 15 limit minimax)
11/10	697.500
13/12	697.637
16/15	697.654
16/13	697.797
14/11	697.812
15/13	697.830
13/10	698.009
14/13	698.335
13/11	698.801
4/3	701.955

Spectrum of Meanpop Tunings by Eigenmonzos

11-limit commas: 81/80, 126/125, 385/384

13-limit commas: 81/80, 105/104, 126/125, 144/143

Eigenmonzo	Fifth
10/9	691.202
14/13	694.340
6/5	694.786
18/13	695.124
15/13	695.226
13/12	695.612
9/7	695.614
13/10	695.838
16/13	696.035
13/11	696.043 (13, 15 limit minimax)
11/8	696.052
15/14	696.111
11/10	696.176
7/6	696.319
14/11	696.413
12/11	696.474
15/11	696.497
5/4	696.578 (5, 7, 9, 11 limit minimax)
11/9	696.839
8/7	696.883
7/5	697.085
16/15	697.654
4/3	701.955