Meantone: Difference between revisions

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Make tunings shown by types
→Spectrum of meantone tunings by eigenmonzos: remove 11- and 13-limit eigenmonzos for (1) they're unreadable (2) they're present in meantone vs meanpop
Line 61: Line 61:
| 688.323
| 688.323
| 1/2 septimal comma
| 1/2 septimal comma
|-
| 16/13
| 689.868
| Meanplop
|-
|-
| {{monzo| 16 -10 }}
| {{monzo| 16 -10 }}
Line 81: Line 77:
| 691.202
| 691.202
| 1/2 comma
| 1/2 comma
|-
| 13/12
| 692.285
| Meanplop
|-
|-
| [[26edo|(15\26)]]
| [[26edo|(15\26)]]
Line 97: Line 89:
| 692.867
| 692.867
| 1/3 septimal comma
| 1/3 septimal comma
|-
| 33/26
| 693.178
| Grosstone
|-
| 13/10
| 693.223
| Meanplop
|-
|-
| [[45edo|(26\45)]]
| [[45edo|(26\45)]]
Line 113: Line 97:
| 693.352
| 693.352
| 2/5 comma
| 2/5 comma
|-
| 18/13
| 693.897
| Meanplop
|-
|-
| 19683/16384
| 19683/16384
Line 125: Line 105:
| 694.165
| 694.165
| 2/7 septimal comma
| 2/7 septimal comma
|-
| 15/13
| 694.193
| Meanplop
|-
| [[14/13]]
| 694.340
| Tridecimal meantone
|-
|-
| [[56/45]]
| [[56/45]]
Line 153: Line 125:
| 694.786
| 694.786
| 1/3 comma
| 1/3 comma
|-
| 14/13
| 694.878
| Meanplop
|-
| [[18/13]]
| 695.124
| Tridecimal meantone
|-
|-
| 5103/4095
| 5103/4095
| 695.139
| 695.139
| 1/4 septimal comma
| 1/4 septimal comma
|-
| [[15/13]]
| 695.226
| Tridecimal Meantone
|-
|-
| {{monzo| 27 -17 }}
| {{monzo| 27 -17 }}
Line 185: Line 145:
| 695.493
| 695.493
| Lucy Tuning
| Lucy Tuning
|-
| 39/28
| 695.6095
| Tridecimal meantone, tridecimal meanpop
|-
| [[13/12]]
| 695.612
| Tridecimal meantone
|-
|-
| [[9/7]]
| [[9/7]]
Line 209: Line 161:
| 695.810
| 695.810
| 2/7 comma
| 2/7 comma
|-
| 40/33
| 695.815
| Meanpop
|-
| [[13/10]]
| 695.838
| Ratwolf fifth, tridecimal meantone and meanpop eigenmonzo
|-
|-
| [[81/80]]
| [[81/80]]
Line 237: Line 181:
| 696.000
| 696.000
|
|
|-
| 16/13
| 696.035
| Tridecimal meantone, tridecimal meanpop
|-
| 13/11
| 696.043
| 13- and 15-odd-limit minimax (tridecimal meanpop)
|-
| 11/8
| 696.052
| Meanpop
|-
|-
| 8192/6561
| 8192/6561
| 696.09
| 696.090
| 1/4 Pythagorean comma, Pythagorean diminished fourth
| 1/4 Pythagorean comma, Pythagorean diminished fourth
|-
|-
Line 261: Line 193:
| 696.165
| 696.165
| [[5-odd-limit]] least squares
| [[5-odd-limit]] least squares
|-
| 11/10
| 696.176
| Meanpop
|-
|-
| (8 - φ)\11
| (8 - φ)\11
Line 285: Line 213:
| 696.319
| 696.319
|
|
|-
| 27/22
| 696.3635
| Meanpop
|-
|-
| [[48/35]]
| [[48/35]]
| 696.399
| 696.399
|
|
|-
| 39/32
| 696.405
| Tridecimal meantone, tridecimal meanpop
|-
| 14/11
| 696.413
| Meanpop
|-
|-
| {{Monzo| 19 9 -1 -11 }}
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 696.436
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
| 12/11
| 696.474
| Meanpop
|-
|-
| 16384/15309
| 16384/15309
Line 329: Line 241:
| 696.648
| 696.648
| [[7-odd-limit]] least squares
| [[7-odd-limit]] least squares
|-
| [[11/9]]
| 696.713
| 11-, 13- and 15-odd-limit (tridecimal meantone) minimax
|-
|-
| [[31edo|(18\31)]]
| [[31edo|(18\31)]]
Line 341: Line 249:
| 696.796
| 696.796
|
|
|-
| 11/9
| 696.839
| Meanpop
|-
|-
| [[8/7]]
| [[8/7]]
Line 353: Line 257:
| 696.959
| 696.959
|
|
|-
| [[12/11]]
| 697.021
| Undecimal meantone
|-
|-
| [[7/5]]
| [[7/5]]
| 697.085
| 697.085
|
|
|-
| [[15/11]]
| 697.158
| rowspan="2" | Undecimal meantone
|-
| [[27/22]]
| 697.159
|-
| 39/32
| 697.168
| Grosstone
|-
|-
| [[75/64]]
| [[75/64]]
| 697.176
| 697.176
|2/9 comma
|2/9 comma
|-
| 22/21
| 697.220
| Undecimal meantone
|-
| 14/13
| 697.242
| 13- and 15-odd-limit minimax (grosstone)
|-
|-
| [[2187/2048]]
| [[2187/2048]]
| 697.263
| 697.263
| 1/5 Pythagorean comma, Pythagorean apotome
| 1/5 Pythagorean comma, Pythagorean apotome
|-
| 13/10
| 697.289
| Grosstone
|-
| [[11/8]]
| 697.295
| Undecimal meantone
|-
|-
| [[74edo|(43\74)]]
| [[74edo|(43\74)]]
Line 404: Line 277:
| 697.344
| 697.344
|
|
|-
| [[13/11]]
| 697.376
| Meridetone
|-
|-
| 45927/32768
| 45927/32768
| 697.411
| 697.411
| 1/6 septimal comma
| 1/6 septimal comma
|-
| 18/13
| 697.465
| 13-, 15-odd-limit minimax (meridetone)
|-
| [[16/13]]
| 696.467
| Grosstone
|-
| [[11/10]]
| 697.500
| Undecimal meantone
|-
| 15/13
| 697.511
| Grosstone
|-
| 13/12
| 697.637
| Meridetone
|-
|-
| [[16/15]]
| [[16/15]]
Line 448: Line 297:
| 697.781
| 697.781
|
|
|-
| 40/33, 16/13
| 697.797
| Undecimal meantone, meridetone
|-
| [[14/11]]
| 697.812
| Undecimal meantone
|-
| 15/13
| 697.830
| Meridetone
|-
| [[18/13]]
| 697.966
| Grosstone
|-
| 13/10
| 698.009
| Meridetone
|-
|-
| [[1024/729]]
| [[1024/729]]
Line 484: Line 313:
| 698.182
| 698.182
|
|
|-
| 33/28
| 698.272
| Undecimal meantone
|-
|-
| [[80/63]]
| [[80/63]]
Line 500: Line 325:
| 698.371
| 698.371
| 1/6 comma
| 1/6 comma
|-
| 33/26
| 698.407
| Meanplop, meridetone
|-
|-
| [[67edo|(39\67)]]
| [[67edo|(39\67)]]
Line 512: Line 333:
| 698.604
| 698.604
| 1/7 Pythagorean comma, Pythagorean limma
| 1/7 Pythagorean comma, Pythagorean limma
|-
| 112/99
| 698.640
| Undecimal meantone
|-
|-
| 45/34
| 45/34
Line 524: Line 341:
| 698.734
| 698.734
|
|
|-
| 13/11
| 698.801
| Meridetone, meanplop
|-
|-
| [[135/128]]
| [[135/128]]
Line 572: Line 385:
| ''702.301''
| ''702.301''
|
|
|-
| ''33/26''
| ''703.186''
| ''Tridecimal meantone''
|-
| ''22/21''
| ''703.356''
| ''Meanpop''
|-
| ''13/11''
| ''703.597''
| ''Tridecimal meantone''
|-
| ''88/81''
| ''707.946''
| ''Meanpop''
|-
| ''88/81''
| ''710.4335''
| ''Undecimal meantone''
|}
|}


Revision as of 05:49, 27 March 2021

Meantone is a familar historical temperament based on a chain of fifths (or fourths), which is discussed in meantone family in the context of the associated family of temperaments, and in meantone vs meanpop in terms of 11-limit extensions.

History

Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various Well Temperaments and eventually 12-tone equal temperament won in popularity.

Theory and classification

Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a rank-2 temperament. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.

Intervals in meantone have standard names based on the number of steps of the diatonic scale they span (this corresponds to the val ⟨7 11 16]), with a modifier {…"double diminished", "diminished", "minor", "major", "augmented", "double augmented"…} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.

Tunings

Common meantone tunings include various eigenmonzo tunings such as the quarter-comma meantone, edo tunings like 31edo, and otherwise optimized tunings like the POTE tuning, shown below. For a more complete list, see #Spectrum of meantone tunings by eigenmonzos. These different tunings are referred to as "temperaments" in traditional terms.

Eigenmonzo tunings
Edo tunings
POTE tunings
  • ~3/2 = 696.239¢ – 5-limit meantone
  • ~3/2 = 696.495¢ – 7-limit meantone
Other optimized tunings

Scales

[5/4 7] eigenmonzos:

Spectrum of meantone tunings by eigenmonzos

Eigenmonzo Fifth size Comments
567/512 688.323 1/2 septimal comma
[16 -10⟩ 690.225 1/2 Pythagorean comma, Pythagorean dilimma
76/51 690.603
[-19 9 0 2⟩ 691.049 2/5 septimal comma
10/9 691.202 1/2 comma
(15\26) 692.308
[31 -19⟩ 692.571 2/5 Pythagorean comma
2048/1701 692.867 1/3 septimal comma
(26\45) 693.333
27/25 693.352 2/5 comma
19683/16384 694.135 1/3 Pythagorean comma, Pythagorean augmented second
[-23 11 0 2⟩ 694.165 2/7 septimal comma
56/45 694.651
28/27 694.709
81/70 694.732
(11\19) 694.737
6/5, 25/18 694.786 1/3 comma
5103/4095 695.139 1/4 septimal comma
[27 -17⟩ 695.252 2/7 Pythagorean comma, 17-comma
35/27 695.389
(51\88) 695.455
1\2 + 1\(4π) 695.493 Lucy Tuning
9/7 695.614
f4 = 2f + 2 695.630 Wilson fifth
(40\69) 695.652
25/24 695.810 2/7 comma
81/80 695.869
112/99 695.886 Meanpop
36/35 695.936
54/49 695.987
(29\50) 696.000
8192/6561 696.090 1/4 Pythagorean comma, Pythagorean diminished fourth
15/14 696.111
78125/73728 696.165 5-odd-limit least squares
(8 - φ)\11 696.214 Golden meantone
49/45 696.245
19/17 696.279 Classical meantone
(47\81) 696.296
7/6 696.319
48/35 696.399
[19 9 -1 -11⟩ 696.436 9-odd-limit least squares
16384/15309 696.502 1/5 septimal comma
5/4 696.578 5-, 7-, 9- and 11-odd-limit (meanpop) minimax, 1/4 comma
49/48 696.616
60/49 696.626
[-55 -11 1 25⟩ 696.648 7-odd-limit least squares
(18\31) 696.774
35/32 696.796
8/7 696.883
49/40 696.959
7/5 697.085
75/64 697.176 2/9 comma
2187/2048 697.263 1/5 Pythagorean comma, Pythagorean apotome
(43\74) 697.297
21/16 697.344
45927/32768 697.411 1/6 septimal comma
16/15 697.654 1/5 comma
(25\43) 697.674
64/63 697.728
21/20 697.781
1024/729 698.045 1/6 Pythagorean comma, lesser Pythagorean tritone
[- 17 9 0 1⟩ 698.060 1/7 septimal comma
28/25 698.099
(32\55) 698.182
80/63 698.303
17/15 698.331
45/32 698.371 1/6 comma
(39\67) 698.507
256/243 698.604 1/7 Pythagorean comma, Pythagorean limma
45/34 698.661 [clarification needed]
(46\79) 698.734
135/128 698.883 1/7 comma
17/16 699.009 [clarification needed]
25/21 699.384
(7\12) 700.000
18/17 700.209 [clarification needed]
19/16 700.829 [clarification needed]
81/80 701.792
(31\53) 701.887
3/2 701.955 Pythagorean tuning
64/63 702.272
256/189 702.301
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