Well temperament: Difference between revisions

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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, ~~though~~ sometimes one can guide construction of the other.

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 ==

* [[Kirnberger]] – 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]])

* [[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]]

# 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

* [[~~Vallotti~~]] – ~~Vallotti/Young~~ temperament

* [[Werckmeister IV]] – Werckmeister temperament IV

# ~~six~~ tempered fifths (C–G, D–A, E–B, ~~F–C~~, ~~G–D~~ and ~~A–E~~) are flat by 1/6 [[~~Pythagorean comma~~]]

# 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

* [[~~Young2~~]] – Young temperament II

* [[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 (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)

* [[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 ==

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== 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.

~~== Other near-equal scales ==~~

* [[Teleic scales]] – unit step generator, patent tuning alternating *[[EdVII|ed(16/9)]] and *ed(9/8)

* [[Kartvelian scales]] – unit step generator, alternating [[EDF|edf]] and [[EdIV|ed(4/3)]]

* [[Well tempered nonet]]

* [[Daseian scales]] - unit step generator, patent tuning alternating [[EDO|edo]]-edo-*[[EDF|edf]]

== External links ==

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[[Category:Well temperament| ]]

[[Category:Historical]]

[[Category:Historical]][[Category:Regular temperament theory]]

@@ Line 4: / Line 4: @@
 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, though sometimes one can guide construction of the other.
+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 ==
-* [[Kirnberger]] – 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]])
+* [[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]]
 # 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
-* [[Vallotti]] – Vallotti/Young temperament
+* [[Werckmeister IV]] – Werckmeister temperament IV
-# six tempered fifths (C–G, D–A, E–B, F–C, G–D and A–E) are flat by 1/6 [[Pythagorean comma]]
+# 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
-* [[Young2]] – Young temperament II
+* [[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 (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)
+* [[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 ==
@@ Line 53: / Line 83: @@
 == 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.
-== Other near-equal scales ==
-* [[Teleic scales]] – unit step generator, patent tuning alternating *[[EdVII|ed(16/9)]] and *ed(9/8)
-* [[Kartvelian scales]] – unit step generator, alternating [[EDF|edf]] and [[EdIV|ed(4/3)]]
-* [[Well tempered nonet]]
-* [[Daseian scales]] - unit step generator, patent tuning alternating [[EDO|edo]]-edo-*[[EDF|edf]]
 == External links ==
@@ Line 68: / Line 92: @@
 [[Category:Well temperament| ]] <!-- main article -->
-[[Category:Historical]]
+[[Category:Historical]][[Category:Regular temperament theory]]