Well temperament: Difference between revisions
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A '''well temperament''' (or '''circulating temperament''') is a [[tuning system]] which is regarded as an approximation of an [[equal tuning]], has the same [[equave]] as that equal tuning and preserves the transposability of the equal tuning, but is not exactly the equal tuning being approximated. Historical well temperaments were formed by stacking a combination of pure [[3/2]] and flattened fifths and had 12 nearly equal steps. | A '''well temperament''' (or '''circulating temperament''') is a [[tuning system]] which is regarded as an approximation of an [[equal tuning]], has the same [[equave]] as that equal tuning and preserves the transposability of the equal tuning, but is not exactly the equal tuning being approximated. Historical well temperaments were formed by stacking a combination of pure [[3/2]] and flattened fifths and had 12 nearly equal steps. | ||
One of the advantages of these tunings is that because they are not quite equal, each chord (or key) has a slightly different character because the interval sizes have changed slightly. | One of the advantages of these tunings is that because they are not quite equal, each chord (or key) has a slightly different character because the interval sizes have changed slightly. | ||
Despite their sharing the word "temperament" in their names, well temperaments and the modern notion of [[regular temperament]]s are two different concepts. In fact, a well temperament is an ''ir''regular temperament. But sometimes one can guide construction of the other. | |||
== Historical well temperaments == | == Historical well temperaments == | ||
* [[Kirnberger]] – Kirnberger temperament III | |||
# four tempered fifths (C–G, D–A, G–D and A–E) are flat by 1/4 | * [[Kirnberger I]] – Kirnberger temperament I | ||
# one tempered fifth (D–A) is flat by 1 [[syntonic comma]] | |||
# one tempered fifth (C#-Ab) is flat by 1 [[schisma]] | |||
# ten [[3/2|pure]] fifths | |||
* [[Kirnberger II]] – Kirnberger temperament II | |||
# two tempered fifths (D–A and A–E) are flat by 1/2 syntonic comma (→ [[1/2-comma meantone]]) | |||
# one tempered fifth (C#-Ab) is flat by 1 [[schisma]] | |||
# nine [[3/2|pure]] fifths | |||
* [[Kirnberger III]] – Kirnberger temperament III | |||
# four tempered fifths (C–G, D–A, G–D and A–E) are flat by 1/4 syntonic comma (→ [[quarter comma meantone]]) | |||
# one tempered fifth (F#–Db) is flat by a [[schisma]] | # one tempered fifth (F#–Db) is flat by a [[schisma]] | ||
# seven [[3/2|pure]] fifths | # seven [[3/2|pure]] fifths | ||
* [[Werck3]] – Werckmeister temperament III | * [[Werck3|Werckmeister III]] – Werckmeister temperament III | ||
# four tempered fifths (C–G, D–A, G–D and B–F#) are tuned flat by 1/4 Pythagorean comma | # four tempered fifths (C–G, D–A, G–D and B–F#) are tuned flat by 1/4 comma (''Werckmeister did not specify whether the syntonic or [[pythagorean comma|Pythagorean]] comma should be used, so either is acceptable'') | ||
# eight [[3/2|pure]] fifths | # eight [[3/2|pure]] fifths | ||
* [[ | * [[Werckmeister IV]] – Werckmeister temperament IV | ||
# | # five tempered fifths (C–G, D–A, E–B, F#-C# and Bb–F) are tuned flat by 1/3 comma | ||
# six pure fifths | # two tempered fifths (G#–D# and Eb–Bb) are tuned sharp by 1/3 comma | ||
# five [[3/2|pure]] fifths | |||
* [[Werckmeister V]] – Werckmeister temperament V | |||
# five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma | |||
# one tempered fifth (G#–D#) is tuned sharp by 1/4 comma | |||
# six [[3/2|pure]] fifths | |||
* [[Septenarius]] – Septenarius temperament (Werckmeister VI) | |||
# six tempered fifths (C-G, G-D, D-A, B-F#, F#-C# and Bb-F) are tuned flat based on division of string length | |||
# one tempered fifth (G#–D#) is tuned sharp based on division of string length | |||
# five [[3/2|pure]] fifths | |||
* [[ | * [[Young I]] – Young temperament I | ||
# four tempered fifths (C–G, D–A, G–D and A–E) are tuned flat by 3/16 syntonic comma | # four tempered fifths (C–G, D–A, G–D and A–E) are tuned flat by 3/16 syntonic comma | ||
# four tempered fifths (E-B, B–F#, Bb–F and F–C) are tuned flat by 1/4 Pythagorean comma less 3/16 syntonic comma | # four tempered fifths (E-B, B–F#, Bb–F and F–C) are tuned flat by 1/4 Pythagorean comma less 3/16 syntonic comma | ||
# four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb) | # four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb) | ||
* [[Vallotti]] – Vallotti/Young temperament II | |||
# six tempered fifths (C–G, D–A, E–B, F–C, G–D and A–E) are flat by 1/6 Pythagorean comma | |||
# six pure fifths | |||
* [[Galilei's tuning]] | |||
# eleven [[18/17]] (~99{{cent}}) semitones | |||
# one (2/1)/(18/17)<sup>11</sup> (~111.5{{cent}}) semitone (B-C) | |||
== Classification by approaches == | == Classification by approaches == | ||
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[[Neji]]s are [[primodal]] scales that more or less roughly approximate the equal tuning with the corresponding number of tones per equave. These scales achieve consonance by ensuring that all intervals share a relatively small common denominator, instead of focusing on a few very simple intervals such as the perfect fifth ([[3/2]]) or the classical major third ([[5/4]]). | [[Neji]]s are [[primodal]] scales that more or less roughly approximate the equal tuning with the corresponding number of tones per equave. These scales achieve consonance by ensuring that all intervals share a relatively small common denominator, instead of focusing on a few very simple intervals such as the perfect fifth ([[3/2]]) or the classical major third ([[5/4]]). | ||
== | == Relation to regular temperaments == | ||
Through the lens of regular temperament theory, a well temperament can be viewed as a result of applying an irregular [[tuning map]] to the abstract intervals of an [[equal temperament]] (i.e. a rank-1 abstract regular temperament), though tuning maps in the technical sense are defined to be regular. However, note that when nejis are considered well temperaments in this sense, the JI ratios the intervals are said to represent and the actual JI ratios of the neji tuning must be distinguished, and the JI ratios that occur in the neji should not be assumed to be consistent with the val. | |||
== External links == | == External links == | ||
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[[Category:Well temperament| ]] <!-- main article --> | [[Category:Well temperament| ]] <!-- main article --> | ||
[[Category:Historical]][[Category:Regular temperament theory]] | |||