Meantone 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
22/21	Augmented prime (C-C#)	+7	Triply diminished third (C-Ebbbb)	-24
12/11, 88/81	Doubly diminished third (C-Ebbb)	-17	Doubly augmented prime (C-Cx)	+14
11/10	Doubly augmented prime (C-Cx)	+14	Doubly diminished third (C-Ebbb)	-17
112/99	Diminished third (C-Ebb), same as 8/7	-10	Triply augmented prime (C-C*)	+21
33/28	augmented second (C-D#), same as 7/6	+9	Triply diminished fourth (C-Fbbb)	-22
40/33	Doubly diminished fourth (C-Fbb)	-15	Doubly augmented second (C-Dx)	+16
11/9, 27/22	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
15/11	Triply diminished sixth (C-Abbbb)	-25	Augmented fourth (C-F#), same as 7/5	+6
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	Triply augmented third (C-E*)	+25	Diminished fifth (C-Gb), same as 10/7	-6
11/7	Augmented fifth (C-G#), same as 14/9	+8	Triply diminished seventh (C-Bbbbb)	-23
18/11, 44/27	Doubly diminished seventh (C-Bbbb)	-16	Doubly augmented fifth (C-Gx)	+15
33/20	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*)	+16
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, 81/44	Doubly augmented sixth (C-Ax)	+17	Doubly diminished octave (C-Cbb)	-14
21/11	Diminished octave (C-Cb)	-7	Triply augmented sixth (C-A*)	+24

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
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	698.64
4/3	701.955
88/81	710.4335

Tridecimal meantone

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

Eigenmonzo	Fifth
14/13	694.34
18/13	695.124
15/13	695.226
39/28	695.6095
13/12	695.612
13/10	695.838
39/32	696.405
16/13	697.467
33/26	703.186
13/11	703.597

Grosstone

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

Eigenmonzo	Fifth
33/26	693.178
39/32	697.168
14/13	697.242 (13, 15 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

Meridetone

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

Eigenmonzo	Fifth
18/13	697.465 (13, 15 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

Spectrum of Meanpop Tunings by Eigenmonzos

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

Eigenmonzo	Fifth
10/9	691.202
6/5	694.786
9/7	695.614
40/33	695.815
112/99	695.886
11/8	696.052
11/10	696.176
7/6	696.319
27/22	696.3635
14/11	696.413
12/11	696.474
5/4	696.578 (5, 7, 9, 11 limit minimax)
11/9	696.839
8/7	696.883
7/5	697.085
4/3	701.955
22/21	703.356
88/81	707.946

Tridecimal meanpop

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

Eigenmonzo	Fifth
14/13	694.34
18/13	695.124
15/13	695.226
39/28	695.6095
13/12	695.612
33/26	695.824
13/10	695.838
16/13	696.035
13/11	696.043 (13, 15 limit minimax)
39/32	696.405

Meanplop

13-limit commas: 65/64, 78/77, 81/80, 91/90

Eigenmonzo	Fifth
39/32	685.839
16/13	689.868
13/12	692.285
13/10	693.223
18/13	693.897
15/13	694.193
14/13	694.878
33/26	698.407
39/28	698.5365
13/11	698.801