Well temperament - Revision history

Vector at 21:50, 23 March 2026

2026-03-23T21:50:35Z

← Older revision		Revision as of 21:50, 23 March 2026
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	A '''well temperament''' is a scale (usually an [[irregular temperament]]) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified. As a special case, equal temperaments are also considered well temperaments, and are in fact the intersection of well temperaments and [[Regular temperament\|regular temperaments]].		A '''well temperament''' is a scale (usually an [[irregular temperament]]) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified. As a special case, equal temperaments themselves are also considered well temperaments, and are in fact the intersection of well temperaments and [[Regular temperament\|regular temperaments]].

	Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time.		Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time.
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	== Derivation ==		== Derivation ==
	~~The~~ historical concept of 12-tone well temperament may be derived as the solution to the problem of 'wolf fifths' in the circle of fifths. In the extreme case, this leads to equal temperaments; well temperaments leave all the fifths unequal but close to just to achieve other goals.		A 12-tone well temperament may be expressed by providing the size of each step of the tuning system, or each interval over a given root (such as A). However, due to historical convention, well temperaments are usually described by providing the tuning of each [[perfect fifth]] along the circle.

			This makes more sense given that the concept of 12-tone well temperament may be derived as the solution to the problem of 'wolf fifths' in the circle of fifths. In the extreme case, this leads to equal temperaments; general well temperaments leave all the fifths unequal but close to just to achieve other goals.

	In meantone[12], 11 tempered fifths of 697.5 cents produce a wolf fifth of 727.5 cents. This may be solved by only tuning, for instance, nine of the fifths flat, that way the meantone major third is still accessible in a wide range of keys, but the detuned fifths become only about 709.2 cents: much more acceptable		In meantone[12], 11 tempered fifths of 697.5 cents produce a wolf fifth of 727.5 cents. This may be solved by only tuning, for instance, nine of the fifths flat, that way the meantone major third is still accessible in a wide range of keys, but the detuned fifths become only about 709.2 cents: much more acceptable

Vector at 21:48, 23 March 2026

2026-03-23T21:48:45Z

← Older revision		Revision as of 21:48, 23 March 2026
Line 1:		Line 1:
	A '''well temperament''' is a scale (~~formally~~ an [[irregular temperament]]) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified.		A '''well temperament''' is a scale (usually an [[irregular temperament]]) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified. As a special case, equal temperaments are also considered well temperaments, and are in fact the intersection of well temperaments and [[Regular temperament\|regular temperaments]].

	Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time~~. Additionally, note that 12edo was considered a well temperament~~.		Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time.

	In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 [[detempering]] of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism - in fact, they emerge from the opposite set of constraints as regular temperament theory. While regular temperaments tune each generator to the same size and allow arbitrary numbers of notes, well temperaments fix the number of notes and allow the tuning of each generator to vary, and are thus more useful when tuning fixed-pitch instruments such as keyboards.		In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 [[detempering]] of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism - in fact, they emerge from the opposite set of constraints as regular temperament theory. While regular temperaments tune each generator to the same size and allow arbitrary numbers of notes, well temperaments fix the number of notes and allow the tuning of each generator to vary, and are thus more useful when tuning fixed-pitch instruments such as keyboards.

Vector at 21:47, 23 March 2026

2026-03-23T21:47:20Z

← Older revision		Revision as of 21:47, 23 March 2026
Line 3:		Line 3:
	Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time. Additionally, note that 12edo was considered a well temperament.		Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time. Additionally, note that 12edo was considered a well temperament.

	In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 [[detempering]] of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism. ~~However~~, ~~they do have musical utility~~.		In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 [[detempering]] of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism - in fact, they emerge from the opposite set of constraints as regular temperament theory. While regular temperaments tune each generator to the same size and allow arbitrary numbers of notes, well temperaments fix the number of notes and allow the tuning of each generator to vary, and are thus more useful when tuning fixed-pitch instruments such as keyboards.

			== Derivation ==
			The historical concept of 12-tone well temperament may be derived as the solution to the problem of 'wolf fifths' in the circle of fifths. In the extreme case, this leads to equal temperaments; well temperaments leave all the fifths unequal but close to just to achieve other goals.

	~~''TODO~~: ~~complete page''~~		In meantone[12], 11 tempered fifths of 697.5 cents produce a wolf fifth of 727.5 cents. This may be solved by only tuning, for instance, nine of the fifths flat, that way the meantone major third is still accessible in a wide range of keys, but the detuned fifths become only about 709.2 cents: much more acceptable

			Alternatively, one might choose to vary the tunings of the flat fifths; eight 698.5c fifths, two 695.5c fifths, and two 710.5c fifths produces a large range of this tunings, many of which may be reasonably heard as 5-limit.

			== Examples ==

Vector: Created page with "A '''well temperament''' is a scale (formally an irregular temperament) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified. Historically, the usage of the term "well temperament" was limited t..."

2026-03-23T10:39:19Z

Created page with "A '''well temperament''' is a scale (formally an irregular temperament) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified. Historically, the usage of the term "well temperament" was limited t..."

New page

A '''well temperament''' is a scale (formally an [[irregular temperament]]) approximating an equal tuning, which has the same average step size as said equal temperament (that is, in general no single note in an equal tuning becomes two different ones in a well temperament). Like musical scales in general, they are generally assumed to be periodic (usually octave-equivalent) unless otherwise specified.

Historically, the usage of the term "well temperament" was limited to specifically 12-tone well temperaments, due to the conceptualization of a temperament as a circle of twelve fifths, which could be tuned equally ([[equal temperament]]), as a chain of eleven fifths "broken" by one diminished sixth (a 12-note scale belonging to a [[regular temperament]]), or in a number of other combinations of tunings. On top of this, well temperaments were generally expected to be closer to 12edo than [[meantone]] or [[pythagorean]] tunings, which were considered their own categories of temperament due to the objectives of tempering at the time. Additionally, note that 12edo was considered a well temperament.

In terms of modern xenharmonic theory, a well temperament can be considered a 1-to-1 [[detempering]] of an equal tuning, and this is the sense used on the wiki. As a result, they have historically attracted very little attention from xenharmonic theorists interested in abstract formalism. However, they do have musical utility.

''TODO: complete page''