Meantone: Difference between revisions
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== History == | == History == | ||
Meantone | Meantone with fifths flatter than 700{{c}} were 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. However, even today, the vast majority of common-practice Western music theory is based exclusively on meantone, as 12-tone equal temperament is itself a meantone tuning. | ||
== 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 | 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 {{dash|C, G, D, A, E|hair|med}}) 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. | ||
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; Notable eigenmonzo (unchanged-interval) tunings | ; Notable eigenmonzo (unchanged-interval) tunings | ||
* [[1/2-comma meantone]] | * [[1/2-comma meantone]] – with eigenmonzo [[10/9]] | ||
* [[1/3-comma meantone]] | * [[1/3-comma meantone]] – with eigenmonzo [[5/3]] | ||
* [[2/7-comma meantone]] | * [[2/7-comma meantone]] – with eigenmonzo [[25/24]] | ||
* [[Quarter-comma meantone|1/4-comma meantone]] | * [[Quarter-comma meantone|1/4-comma meantone]] – with eigenmonzo [[5/4]] | ||
* [[1/5-comma meantone]] | * [[1/5-comma meantone]] – with eigenmonzo [[15/8]] | ||
* [[1/6-comma meantone]] | * [[1/6-comma meantone]] – with eigenmonzo [[45/32]] | ||
* [[Ratwolf|Ratwolf tuning]] | * [[Ratwolf|Ratwolf tuning]] | ||
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| Weil || CWE: ~3/2 = 696.6512¢ | | Weil || CWE: ~3/2 = 696.6512¢ | ||
|- | |- | ||
| Equilateral || CEE: ~3/2 = 696.8947¢<br>[[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.1875 (4/17-comma tuning) | | Equilateral || CEE: ~3/2 = 696.8947¢<br />[[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.1875 (4/17-comma tuning) | ||
|- | |- | ||
| Skewed-equilateral || CSEE: ~3/2 = 696.4534¢<br>Eigenmonzo (unchanged-interval) basis: 2.48828125/3 (11/43-comma tuning) | | Skewed-equilateral || CSEE: ~3/2 = 696.4534¢<br />Eigenmonzo (unchanged-interval) basis: 2.48828125/3 (11/43-comma tuning) | ||
|- | |- | ||
| Benedetti/Wilson || CBE: ~3/2 = 697.3738¢<br>Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 25 36 }} (36/169-comma tuning) | | Benedetti/Wilson || CBE: ~3/2 = 697.3738¢<br />Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 25 36 }} (36/169-comma tuning) | ||
|- | |- | ||
| Skewed-Benedetti/Wilson || CSBE: ~3/2 = 696.7868¢<br>Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 5 31 }} (31/129-comma tuning) | | Skewed-Benedetti/Wilson || CSBE: ~3/2 = 696.7868¢<br />Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 5 31 }} (31/129-comma tuning) | ||
|} | |} | ||
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| Weil || CWE: ~3/2 = 696.6562¢ | | Weil || CWE: ~3/2 = 696.6562¢ | ||
|- | |- | ||
| Equilateral || CEE: ~3/2 = 696.8843¢<br>Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 1 4 10 }} | | Equilateral || CEE: ~3/2 = 696.8843¢<br />Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 1 4 10 }} | ||
|- | |- | ||
| Skewed-equilateral || CSEE: ~3/2 = 696.7248¢<br>Eigenmonzo (unchanged-interval) basis: 2.4117715/9 | | Skewed-equilateral || CSEE: ~3/2 = 696.7248¢<br />Eigenmonzo (unchanged-interval) basis: 2.4117715/9 | ||
|- | |- | ||
| Benedetti/Wilson || CBE: ~3/2 = 697.0147¢<br>Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 1225 1764 2250 }} | | Benedetti/Wilson || CBE: ~3/2 = 697.0147¢<br />Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 1225 1764 2250 }} | ||
|- | |- | ||
| Skewed-Benedetti/Wilson || CSBE: ~3/2 = 696.6306¢<br>Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 -3290 3171 7215 }} | | Skewed-Benedetti/Wilson || CSBE: ~3/2 = 696.6306¢<br />Eigenmonzo (unchanged-interval) basis: 2.{{monzo| 0 -3290 3171 7215 }} | ||
|} | |} | ||
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{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
! Edo<br>Generator | ! Edo<br />Generator | ||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | ! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]] | ||
! Generator<br>(¢) | ! Generator<br />(¢) | ||
! Comments | ! Comments | ||
|- | |- | ||