Well temperament: Difference between revisions

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A '''well temperament''' or a '''circulating temperament''' is a ~~scale with the property that for at least one~~ [[~~interval class~~]] ~~(the interval class~~ is ~~viewed as forming a closed circle within the tuning, hence the name), all of the intervals in the "circle" can be~~ regarded as ~~approximations~~ of ~~some targeted interval, but which is not~~ an [[equal ~~temperament~~]]~~. The targeted (circulating) interval can be a fixed just interval (like a just perfect fifth)~~, ~~or an n-edo interval when~~ the ~~goal is to produce an unequal coloring~~ of ~~n-edo. In~~ the ~~best known examples~~, the ~~interval~~ approximated is a ~~fifth~~ and ~~the scale has twelve notes to an octave~~.

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.

~~== Types ==~~

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.

~~A well~~ temperament ~~may be classified by method as follows:~~

* [[Maximal evenness|Maximally even]] set in ~~a large~~ [[~~EDO|edo]] or another [[equal-step tuning]]~~

* A quasi-equal [[Detempering|detemperament]] ~~(though not all detemperaments of an edo~~ are well ~~temperaments)~~

* [[Neji]]

== Historical well temperaments ==

* [[Kirnberger]] – Kirnberger temperament

~~# the fifths are [[3/2|pure]], except for~~

~~# the C–G, D–A, G–D and A–E fifths are [[quarter comma meantone]]~~

~~# the F#–Db is the wolf fifth, a [[schisma]] flat~~

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

* [[Kirnberger I]] – Kirnberger temperament I

# ~~the fifths are~~ [[~~3/2|pure~~]]~~, except for~~

# one tempered fifth (D–A) is flat by 1 [[syntonic comma]]

# ~~the C–G, D–A, E–B, F–C, G–D and A–E fifths are~~ 1/6 [[~~Pythagorean comma~~]] ~~flat.~~

# one tempered fifth (C#-Ab) is flat by 1 [[schisma]]

# ten [[3/2|pure]] fifths

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

* [[Kirnberger II]] – Kirnberger temperament II

* [[~~Werck3~~]] ~~– Werckmeister temperament III~~

# two tempered fifths (D–A and A–E) are flat by 1/2 syntonic comma (→ [[1/2-comma meantone]])

* [[~~Cauldron~~]]

# one tempered fifth (C#-Ab) is flat by 1 [[schisma]]

* [[~~Bifrost]]~~

# nine [[3/2|pure]] fifths

* [[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)~~]]

~~==Other scales called "well temperaments"==~~

* [[Kirnberger III]] – Kirnberger temperament III

* [[~~Well~~ tempered ~~nonet~~]]

# 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

~~== Articles ==~~

* [[Werck3|Werckmeister III]] – Werckmeister temperament III

# 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

* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings] by [[Kyle Gann]~~] [http://www.webcitation.org/5xe2pcAue Permalink~~]

* [[Werckmeister IV]] – Werckmeister temperament IV

* [http://lumma.org/tuning/gws/circ.html Circulating Temperaments] by [[Gene Ward Smith]~~] [http://www.webcitation.org/5xemAJsWE Permalink~~]

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

* [~~http~~://www.math.uwaterloo.ca/%7Emrubinst/tuning/tuning.html Well v.s. Equal Temperament] by [[Michael Rubinstein~~]] [http://www.webcitation.org/5xemm0tvx Permalink]~~

# two tempered fifths (G#–D# and Eb–Bb) are tuned sharp by 1/3 comma

* [http://www.piano-tuners.org/edfoote/well_tempered_piano.html Six Degrees Of Tonality: The Well Tempered Piano] by [[Edward Foote~~]] [http://www.webcitation.org/5xenGg2uG Permalink]~~

# five [[3/2|pure]] fifths

* [http://www.rollingball.com/~~images/HT5~~.htm ~~Five~~

[[Category:~~Theory~~]]

* [[Werckmeister V]] – Werckmeister temperament V

[[Category:~~Scale theory~~]]

# five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma

[[Category:~~Circulating~~ temperament| ]] ~~~~

# one tempered fifth (G#–D#) is tuned sharp by 1/4 comma

~~{{todo| expand }}~~

# 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 (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 ==

There are several approaches to well temperaments. These are not strictly mutually exclusive, but they provide different frameworks that cater to various goals.

=== Circle of fifths ===

Well temperaments can be structured around the usual uneven distribution of differently-sized fifths, but with a wider palette of fifths, such as [[superpyth]] fifths (approx. 702{{cent}}-720{{cent}}) and [[flattone]] fifths (approx. 691{{cent}}-695{{cent}}). Consequently, major thirds also come in various sizes, sometimes approximating other intervals than the usual [[5/4]], such as [[9/7]] and [[14/11]]. For example: [[Carl Lumma]]'s [[Cauldron]].

The same idea could also be applied to other equal temperaments, using circles of other intervals, possibly with other equaves. For example: [[George Secor]]'s [[secor29htt|29-tone high tolerance temperament]].

=== Detempering or deregularizing ===

Well temperaments can be obtained by [[Detempering|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: [http://lumma.org/tuning/gws/duowell.htm 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 ===

[[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 ==

* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings] by [[Kyle Gann]]

* [http://lumma.org/tuning/gws/circ.html Circulating Temperaments] by [[Gene Ward Smith]]

* [https://www.math.uwaterloo.ca/%7Emrubinst/tuning/tuning.html Well v.s. Equal Temperament] by Michael Rubinstein

* [http://www.piano-tuners.org/edfoote/well_tempered_piano.html Six Degrees Of Tonality: The Well Tempered Piano] by Edward Foote

* [http://www.rollingball.com/TemperamentsFrames.htm Temperaments Visualized] by Jason Kanter

[[Category:Well temperament| ]]

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

@@ Line 1: / Line 1: @@
-A '''well temperament''' or a '''circulating temperament''' is a scale with the property that for at least one [[interval class]] (the interval class is viewed as forming a closed circle within the tuning, hence the name), all of the intervals in the "circle" can be regarded as approximations of some targeted interval, but which is not an [[equal temperament]]. The targeted (circulating) interval can be a fixed just interval (like a just perfect fifth), or an n-edo interval when the goal is to produce an unequal coloring of n-edo. In the best known examples, the interval approximated is a fifth and the scale has twelve notes to an octave.
+{{Wikipedia|Well temperament}}
+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.
-== Types ==
+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.
-A well temperament may be classified by method as follows:
-* [[Maximal evenness|Maximally even]] set in a large [[EDO|edo]] or another [[equal-step tuning]]
-* A quasi-equal [[Detempering|detemperament]] (though not all detemperaments of an edo are well temperaments)
-* [[Neji]]
 == Historical well temperaments ==
-* [[Kirnberger]] – Kirnberger temperament
-# the fifths are [[3/2|pure]], except for
-# the C–G, D–A, G–D and A–E fifths are [[quarter comma meantone]]
-# the F#–Db is the wolf fifth, a [[schisma]] flat
-* [[Vallotti]] – Vallotti/Young temperament
+* [[Kirnberger I]] – Kirnberger temperament I
-# the fifths are [[3/2|pure]], except for
+# one tempered fifth (D–A) is flat by 1 [[syntonic comma]]
-# the C–G, D–A, E–B, F–C, G–D and A–E fifths are 1/6 [[Pythagorean comma]] flat.
+# one tempered fifth (C#-Ab) is flat by 1 [[schisma]]
+# ten [[3/2|pure]] fifths
-* [[Young2]] – Young temperament II
+* [[Kirnberger II]] – Kirnberger temperament II
-* [[Werck3]] – Werckmeister temperament III
+# two tempered fifths (D–A and A–E) are flat by 1/2 syntonic comma (→ [[1/2-comma meantone]])
-* [[Cauldron]]
+# one tempered fifth (C#-Ab) is flat by 1 [[schisma]]
-* [[Bifrost]]
+# nine [[3/2|pure]] fifths
-* [[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)]]
-==Other scales called "well temperaments"==
+* [[Kirnberger III]] – Kirnberger temperament III
-* [[Well tempered nonet]]
+# 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
-== Articles ==
+* [[Werck3|Werckmeister III]] – Werckmeister temperament III
+# 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
-* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings] by [[Kyle Gann]] [http://www.webcitation.org/5xe2pcAue Permalink]
+* [[Werckmeister IV]] – Werckmeister temperament IV
-* [http://lumma.org/tuning/gws/circ.html Circulating Temperaments] by [[Gene Ward Smith]] [http://www.webcitation.org/5xemAJsWE Permalink]
+# five tempered fifths (C–G, D–A, E–B, F#-C# and Bb–F) are tuned flat by 1/3 comma
-* [http://www.math.uwaterloo.ca/%7Emrubinst/tuning/tuning.html Well v.s. Equal Temperament] by [[Michael Rubinstein]] [http://www.webcitation.org/5xemm0tvx Permalink]
+# two tempered fifths (G#–D# and Eb–Bb) are tuned sharp by 1/3 comma
-* [http://www.piano-tuners.org/edfoote/well_tempered_piano.html Six Degrees Of Tonality: The Well Tempered Piano] by [[Edward Foote]] [http://www.webcitation.org/5xenGg2uG Permalink]
+# five [[3/2|pure]] fifths
-* [http://www.rollingball.com/images/HT5.htm Five
-[[Category:Theory]]
+* [[Werckmeister V]] – Werckmeister temperament V
-[[Category:Scale theory]]
+# five tempered fifths (D–A, A-E, F#-C#, C#-G# and F–C) are tuned flat by 1/4 comma
-[[Category:Circulating temperament| ]] <!-- main article -->
+# one tempered fifth (G#–D#) is tuned sharp by 1/4 comma
-{{todo| expand }}
+# 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 (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 ==
+There are several approaches to well temperaments. These are not strictly mutually exclusive, but they provide different frameworks that cater to various goals.
+=== Circle of fifths ===
+Well temperaments can be structured around the usual uneven distribution of differently-sized fifths, but with a wider palette of fifths, such as [[superpyth]] fifths (approx. 702{{cent}}-720{{cent}}) and [[flattone]] fifths (approx. 691{{cent}}-695{{cent}}). Consequently, major thirds also come in various sizes, sometimes approximating other intervals than the usual [[5/4]], such as [[9/7]] and [[14/11]]. For example: [[Carl Lumma]]'s [[Cauldron]].
+The same idea could also be applied to other equal temperaments, using circles of other intervals, possibly with other equaves. For example: [[George Secor]]'s [[secor29htt|29-tone high tolerance temperament]].
+=== Detempering or deregularizing ===
+Well temperaments can be obtained by [[Detempering|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: [http://lumma.org/tuning/gws/duowell.htm 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 ===
+[[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 ==
+* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings] by [[Kyle Gann]]
+* [http://lumma.org/tuning/gws/circ.html Circulating Temperaments] by [[Gene Ward Smith]]
+* [https://www.math.uwaterloo.ca/%7Emrubinst/tuning/tuning.html Well v.s. Equal Temperament] by Michael Rubinstein
+* [http://www.piano-tuners.org/edfoote/well_tempered_piano.html Six Degrees Of Tonality: The Well Tempered Piano] by Edward Foote
+* [http://www.rollingball.com/TemperamentsFrames.htm Temperaments Visualized] by Jason Kanter
+[[Category:Well temperament| ]] <!-- main article -->
+[[Category:Historical]][[Category:Regular temperament theory]]