Meantone
Meantone is a familiar historical temperament based on a chain of fifths (or fourths). The more technical part is discussed in meantone family in the context of the associated family of temperaments.
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.
Septimal meantone
Septimal meantone or 7-limit meantone is a natural extension of meantone which also addresses septimal intervals including but not limited to 7/4, 7/5, and 7/6. By extending the circle of fifths, consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord.
See meantone vs meanpop for a comparison of undecimal (11-limit) extensions.
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 #Tuning spectrum. These different tunings are referred to as "temperaments" in traditional terms.
- Eigenmonzo tunings
- 1/2 syntonic comma meantone – with eigenmonzo 10/9
- 1/3 syntonic comma meantone – with eigenmonzo 5/3
- 2/7 syntonic comma meantone – with eigenmonzo 25/24
- 1/4 syntonic comma meantone – with eigenmonzo 5/4
- 1/5 syntonic comma meantone – with eigenmonzo 15/8
- 1/6 syntonic comma meantone – with eigenmonzo 45/32
- Edo tunings
- POTE tunings
- ~3/2 = 696.239¢ – 5-limit meantone
- ~3/2 = 696.495¢ – 7-limit meantone
- Other optimized tunings
Scales
- Edo tunings
- Meantone5 – pentatonic scale in 31edo
- Meantone7 – diatonic scale in 19edo and 31edo
- Meantone12 – chromatic scale in 31edo
- Eigenmonzo tunings
- Others
Tuning spectrum
| Edo Generator |
Eigenmonzo (unchanged interval) |
Generator (¢) |
Comments |
|---|---|---|---|
| 567/512 | 688.323 | 1/2 septimal comma | |
| [16 -10⟩ | 690.225 | 1/2 Pythagorean comma, Pythagorean dilimma | |
| 51/38 | 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 | ||
| [math]\displaystyle{ f^4 = 2f + 2 }[/math] | 695.630 | Wilson fifth | |
| 40\69 | 695.652 | ||
| 25/24 | 695.810 | 2/7 comma | |
| 81/80 | 695.869 | [clarification needed] | |
| 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-, and 9-odd-limit 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 | [clarification needed] | |
| 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 | ||
| 99\170 | 698.824 | ||
| 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 | [clarification needed] | |
| 31\53 | 701.887 | ||
| 3/2 | 701.955 | Pythagorean tuning | |
| 64/63 | 702.272 | ||
| 256/189 | 702.301 |