Meantone: Difference between revisions
m Added Wikipedia box, misc. edits |
Improve for future additions of septimal interpretations |
||
| Line 3: | Line 3: | ||
== History == | == History == | ||
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[ | Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[well temperament]]s and eventually [[12edo|12-tone equal temperament]] won in popularity. | ||
== Theory and classification == | == 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 [[Regular Temperaments|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. | Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|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. | ||
[[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] {{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. | [[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] {{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. | ||
| Line 392: | Line 392: | ||
* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings], by [[Kyle Gann]] | * [http://www.kylegann.com/histune.html An Introduction to Historical Tunings], by [[Kyle Gann]] | ||
{{IoT}} | |||
[[Category:Regular temperament theory]] | |||
[[Category:Meantone| ]] <!-- main article --> | [[Category:Meantone| ]] <!-- main article --> | ||
[[Category: | [[Category:Meantone family]] | ||
[[Category: | [[Category:Starling temperaments]] | ||
[[Category:Marvel temperaments]] | |||
<!-- interwiki --> | <!-- interwiki --> | ||
[[de:Mitteltönig]] | [[de:Mitteltönig]] | ||
Revision as of 09:03, 20 September 2021
Meantone is a familiar 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
- 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
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 |
| 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 | |
| f4 = 2f + 2 | 695.630 | Wilson fifth |
| (40\69) | 695.652 | |
| 25/24 | 695.810 | 2/7 comma |
| 81/80 | 695.869 | |
| 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 | |
| (31\53) | 701.887 | |
| 3/2 | 701.955 | Pythagorean tuning |
| 64/63 | 702.272 | |
| 256/189 | 702.301 |