ウェル・テンペラメント

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ウェル・テンペラメント

ウェル・テンペラメント（英: well temperament、circulating temperament）は、平均律を近似した音律と考えられる。それは同じ等価音程を持ち、その平均律と同じ移調可能性を有しているが、その平均律と全く同じではない。歴史上のウェル・テンペラメントの多くは純正な 3/2 とわずかにフラットな5度を組み合わせて五度圏を12音で閉じたものと説明される。それは12個の大体等しいステップとなる。

これらのチューニングの利点はそのステップが完全に等しくはないことにあり、個々のコードまたは調がわずかずつ違う性格を持っている。

「テンペラメント」という単語を共有しているにもかかわらず、ウェル・テンペラメントとレギュラーテンペラメントは別々のコンセプトである。実際、ウェル・テンペラメントはイレギュラー（不規則）なテンペラメントである。だが（だからこそ）互いに比較することでそれぞれの構成の理解への助けになるかもしれない。

歴史上のウェル・テンペラメント

• Kirnberger I – Kirnberger temperament I

1. one tempered fifth (D–A) is flat by 1 syntonic comma
2. one tempered fifth (C#-Ab) is flat by 1 schisma
3. ten pure fifths

• Kirnberger II – Kirnberger temperament II

1. two tempered fifths (D–A and A–E) are flat by 1/2 syntonic comma (→ 1/2-comma meantone)
2. one tempered fifth (C#-Ab) is flat by 1 schisma
3. nine pure fifths

• Kirnberger III – Kirnberger temperament III

1. four tempered fifths (C–G, D–A, G–D and A–E) are flat by 1/4 syntonic comma (→ quarter comma meantone)
2. one tempered fifth (F#–Db) is flat by a schisma
3. seven pure fifths

• Werckmeister III – Werckmeister temperament III

1. 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 should be used, so either is acceptable)
2. eight pure fifths

• Werckmeister IV – Werckmeister temperament IV

1. five tempered fifths (C–G, D–A, E–B, F#-C# and Bb–F) are tuned flat by 1/3 comma
2. two tempered fifths (G#–D# and Eb–Bb) are tuned sharp by 1/3 comma
3. five pure fifths

• Werckmeister V – Werckmeister temperament V

1. five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma
2. one tempered fifth (G#–D#) is tuned sharp by 1/4 comma
3. six pure fifths

• Septenarius – Septenarius temperament (Werckmeister VI)

1. 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
2. one tempered fifth (G#–D#) is tuned sharp based on division of string length
3. five pure fifths

• Young I – Young temperament I

1. four tempered fifths (C–G, D–A, G–D and A–E) are tuned flat by 3/16 syntonic comma
2. 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
3. four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb)

• Vallotti – Vallotti/Young temperament II

1. six tempered fifths (C–G, D–A, E–B, F–C, G–D and A–E) are flat by 1/6 Pythagorean comma
2. six pure fifths

• Galilei's tuning

1. eleven 18/17 (~99 ¢) semitones
2. one (2/1)/(18/17)¹¹ (~111.5 ¢) semitone (B-C)

アプローチの分類

ウェル・テンペラメントには複数のアプローチがある。それらは完全に互いに相いれないというわけでもないが、様々なゴールに至るそれぞれの枠組みを表している。

五度圏

ウェル・テンペラメントは複数種類の大きさの5度を偏らせて配置することに基づく。よくあるのは純正完全5度やミーントーンの5度だが、より広い選択肢（スーパーパイスの702 ¢～720 ¢やフラットトーンの691 ¢～695 ¢など）を考えてもよい。結果として長3度も様々なサイズになり、5/4というよりも9/7や14/11に近くなったりする。例: Carl Lumma (en) のCauldron (en) 。

同じ発想を12平均律以外に適用してもよい。異なるジェネレーターによる圏（circle）や非オクターブでもありうる。例: George Secor (en) のsecor29htt (en) 。

Detempering or deregularizing

Well temperaments can be obtained by detempering or deregularizing an equal tuning. This implies going from a rank-1 temperament to a multirank temperament by adding one (or more) extra generator(s) – a common choice is to add a pure octave –, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.

If the main generator is a fifth, then there is only one wolf fifth that closes the circle of fifths, a feature which is often associated to tunings such as quarter-comma meantone. However, these tunings are not always considered as well temperaments because they may not preserve transposability due to their higher mistunings.

If the main generator is different from a fifth, then there are multiple wolf fifths which are evenly distributed along the circle of fifths. Each wolf fifth is typically more in tune than the single wolf fifths of the fifth-generated cases, since the total mistuning is spread out over multiple intervals, but that also means that wolf fifths are more likely to be used frequently in such well temperaments.

Well temperaments based on rank-2 temperaments can be designed to follow the structure of a moment of symmetry (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth.

For examples: Duowell, a well-tuning of Duodene

A similar process is to pick a mos scale with the desired number of tones and a step ratio close to 1. If the step ratio is superparticular, then it is also a maximally even scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually comma-sized. If the superset of the particular detemperament or deregularization is a fine enough equal tuning, it can have sisters with other superpartient step ratios.

Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.

Neji

Nejis 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

• An Introduction to Historical Tunings by Kyle Gann
• Circulating Temperaments by Gene Ward Smith
• Well v.s. Equal Temperament by Michael Rubinstein
• Six Degrees Of Tonality: The Well Tempered Piano by Edward Foote
• Temperaments Visualized by Jason Kanter