Meantone vs meanpop

From Xenharmonic Wiki
Jump to navigation Jump to search

"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 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, 88/81 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, 99/80 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, 160/99 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, 81/44 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
88/81 707.946

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
39/32 685.839
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