Meantone: Difference between revisions

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== Theory and classification ==

Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few [[cents]] narrower than pure. The rationale for narrowing the fifth is to temper out the [[syntonic comma]]. This means that stacking four fifths (such as ~~C-G-D-A-E~~) results in a major third (~~C-E~~) that is close to just.

Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few [[cents]] narrower than pure. The rationale for narrowing the fifth is to temper out the [[syntonic comma]]. This means that stacking four fifths (such as C–G–D–A–E) results in a major third (C–E) that is close to just.

[[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] {{val| 7 11 16 }}), with a modifier {…"double diminished", "diminished", "minor", "major", "augmented", "double augmented"…} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.

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=== Septimal meantone ===

'''Septimal meantone''' or '''7-limit meantone''' is a natural extension of meantone which also addresses septimal intervals including but not limited to [[7/4]], [[7/5]], and [[7/6]]. By extending the [[circle of fifths]], consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord.

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==== Dominant ====

Dominant is an alternative extension of meantone, which 7/4 is represented by a minor seventh rather than augmented sixth. This results equating 6/5 with 7/6 and 5/4 with 9/7, or tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas comma).

Dominant is an alternative extension of meantone, which 7/4 is represented by a minor seventh rather than augmented sixth. This results equating 6/5 with 7/6 and 5/4 with 9/7, or tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas' comma).

==== Sharptone ====

Sharptone is a low-accuracy temperament which 7/4 is represented by a major sixth. This results equating 7/6 with 9/8 and 4/3 with 7/5, or tempering out [[21/20]] and [[28/27]].

@@ Line 11: / Line 11: @@
 == Theory and classification ==
-Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few [[cents]] narrower than pure. The rationale for narrowing the fifth is to temper out the [[syntonic comma]]. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.
+Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few [[cents]] narrower than pure. The rationale for narrowing the fifth is to temper out the [[syntonic comma]]. This means that stacking four fifths (such as C–G–D–A–E) results in a major third (C–E) that is close to just.
 [[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] {{val| 7 11 16 }}), with a modifier {…"double diminished", "diminished", "minor", "major", "augmented", "double augmented"…} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.
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 === Septimal meantone ===
 {{Wikipedia| Septimal meantone temperament }}
-{{See also|Meantone family #Septimal meantone}}
+{{See also| Meantone family #Septimal meantone }}
 '''Septimal meantone''' or '''7-limit meantone''' is a natural extension of meantone which also addresses septimal intervals including but not limited to [[7/4]], [[7/5]], and [[7/6]]. By extending the [[circle of fifths]], consonant septimal intervals start to appear. For example, 7/4 is represented by an augmented sixth and is notably present in the augmented sixth chord.
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 ==== Dominant ====
-{{See also|Meantone family #Dominant}}
+{{Main| Dominant (temperament) }}
-Dominant is an alternative extension of meantone, which 7/4 is represented by a minor seventh rather than augmented sixth. This results equating 6/5 with 7/6 and 5/4 with 9/7, or tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas comma).
+Dominant is an alternative extension of meantone, which 7/4 is represented by a minor seventh rather than augmented sixth. This results equating 6/5 with 7/6 and 5/4 with 9/7, or tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas' comma).
 ==== Sharptone ====
-{{See also|Meantone family #Sharptone}}
+{{See also| Meantone family #Sharptone }}
 Sharptone is a low-accuracy temperament which 7/4 is represented by a major sixth. This results equating 7/6 with 9/8 and 4/3 with 7/5, or tempering out [[21/20]] and [[28/27]].