Meantone: Difference between revisions

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| Odd limit 2 = 9 | Mistuning 2 = 10.8 | Complexity 2 = 31

}}

'''Meantone''' is a familiar historical [[temperament]] based on a [[chain of fifths]] (or fourths), possessing two [[generator|generating intervals]]: the [[octave]] and the [[3/2|fifth]], from which all pitches are composed. ~~In the common tunings of meantone (such~~ as ~~[[quarter-comma meantone]] and [[12edo]]), all the fifths are tuned the same, qualifying meantone as a [[regular temperament]] and more specifically~~ a [[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]], 81/80, which means that stacking four fifths (such as {{dash|C, G, D, A, E|hair|med}}) results in a major third (C–E) that is close to the just interval [[5/4]] rather than the more complex Pythagorean interval [[81/64]]; good tunings of meantone also lead to soft [[diatonic]] and [[Chromatic scale|chromatic]] scales, which are desirable for interval categorization.

'''Meantone''' is a familiar historical [[temperament]] based on a [[chain of fifths]] (or fourths), possessing two [[generator|generating intervals]]: the [[octave]] and the [[3/2|fifth]], from which all pitches are composed. This qualifies it as a [[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]], 81/80, which means that stacking four fifths (such as {{dash|C, G, D, A, E|hair|med}}) results in a major third (C–E) that is close to the just interval [[5/4]] rather than the more complex Pythagorean interval [[81/64]]; good tunings of meantone also lead to soft [[diatonic]] and [[Chromatic scale|chromatic]] scales, which are desirable for interval categorization.

[[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''' 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; it can also be seen as a diesis-flat minor seventh.

See [[~~meantone~~ vs meanpop]] for a comparison of undecimal (11-limit) extensions.

See [[huygens vs meanpop]] for a comparison of undecimal (11-limit) extensions.

=== Other septimal extensions ===

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* [[Meantone7]] – diatonic scale in 31edo

* [[Meantone12]] – chromatic scale in 31edo

; ~~Eigenmonzo (unchanged~~-interval) tunings

; Unchanged-interval (eigenmonzo) tunings

* [[Meanwoo12]] – chromatic scale in 5/4.7-eigenmonzo tuning

* [[Meanwoo19]] – enharmonic scale in 5/4.7-eigenmonzo tuning

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* [[Lucy tuning]]

* Equal beating tunings

* 5-limit [[DKW theory|DKW]]: ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}

=== Prime-optimized tunings ===

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|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~3/2 = 696.~~895~~{{c}} (4/17-comma)

| CEE: ~3/2 = 696.8947{{c}} (4/17 comma)

| CSEE: ~3/2 = 696.~~453~~{{c}} (11/43-comma)

| CSEE: ~3/2 = 696.4534{{c}} (11/43 comma)

| POEE: ~3/2 = 695.2311{{c}}

|-

! Tenney

| CTE: ~3/2 = 697.~~214~~{{c}}

| CTE: ~3/2 = 697.2143{{c}}

| CWE: ~3/2 = 696.~~651~~{{c}}

| CWE: ~3/2 = 696.6512{{c}}

| POTE: ~3/2 = 696.2387{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 697.~~374~~{{c}} (36/169-comma)

| CBE: ~3/2 = 697.3738{{c}} (36/169 comma)

| CSBE: ~3/2 = 696.~~787~~{{c}} (31/129-comma)

| CSBE: ~3/2 = 696.7868{{c}} (31/129 comma)

| POBE: ~3/2 = 696.2984{{c}}

|}

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|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~3/2 = 696.~~884~~{{c}}

| CEE: ~3/2 = 696.8843{{c}}

| CSEE: ~3/2 = 696.~~725~~{{c}}

| CSEE: ~3/2 = 696.7248{{c}}

| POEE: ~3/2 = 696.4375{{c}}

|-

! Tenney

| CTE: ~3/2 = 696.~~952~~{{c}}

| CTE: ~3/2 = 696.9521{{c}}

| CWE: ~3/2 = 696.~~656~~{{c}}

| CWE: ~3/2 = 696.6562{{c}}

| POTE: ~3/2 = 696.4949{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 697.~~015~~{{c}}

| CBE: ~3/2 = 697.0147{{c}}

| CSBE: ~3/2 = 696.~~631~~{{c}}

| CSBE: ~3/2 = 696.6306{{c}}

| POBE: ~3/2 = 696.4596{{c}}

|}

=== Target tunings ===

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | Minimax tunings

|-

! Target

! Generator

! Eigenmonzo*

|-

| 5-odd-limit

| ~3/2 = 696.578{{c}}

| 5/4

|-

| 7-odd-limit

| ~3/2 = 696.578{{c}}

| 5/4

|-

| 9-odd-limit

| ~3/2 = 696.578{{c}}

| 5/4

|}

{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | Least squares tunings

|-

! Target

! Generator

! Eigenmonzo*

|-

| 5-odd-limit

| ~3/2 = 696.165{{c}} (7/26 comma)

| {{Monzo| -13 -2 7 }}

|-

| 7-odd-limit

| ~3/2 = 696.648{{c}}

| {{Monzo| -55 -11 1 25 }}

|-

| 9-odd-limit

| ~3/2 = 696.436{{c}}

| {{Monzo| 19 9 -1 -11 }}

|}

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|-

! Edo generator

! [[Eigenmonzo|~~Eigenmonzo~~ (~~unchanged-interval~~)]]*

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator (¢)

! Comments

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| [[9/5]]

| 691.202

| '''Lower bound of 9-odd-limit diamond tradeoff''' [[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9

| '''Lower bound of 9-odd-limit diamond tradeoff''' [[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9

|-

| [[59edo|34\59]]

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|-

|

| {{nowrap|''f''4 ~~−~~ 2''f'' ~~−~~ 2 {{=}} 0}}

| {{nowrap| ''f''4 − 2''f'' − 2 {{=}} 0 }}

| 695.630

| 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma

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| [[15/14]]

| 696.111

|

|-

|

| [[78125/73728]]

| 696.165

| [[7/26-comma meantone|7/26 comma]], [[5-odd-limit]] least squares

| [[7/26-comma meantone|7/26 comma]], 5-odd-limit least squares

|-

| {{nowrap|(8 ~~−~~ φ)\11}}

| {{nowrap| (8 − φ)\11 }}

|

| 696.214

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| 696.399

|

|-

|

~~| {{Monzo| 19 9 -1 -11 }}~~

~~| 696.436~~

~~| 9-odd-limit least squares~~

|-

|

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| 696.626

|

|-

|

~~| {{monzo| -55 -11 1 25 }}~~

~~| 696.648~~

~~| [[7-odd-limit]] least squares~~

|-

| [[31edo|18\31]]

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| [[1875/1024]]

| 696.895

| [[4/17-comma meantone|4/17 comma]]; ~~2.3.~~5 [[CEE]] tuning

| [[4/17-comma meantone|4/17 comma]]; 5-limit [[CEE]] tuning

|-

|

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$$ n = (g_J - g)/g_c $$

~~=== Other tunings ===~~

* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}

== Music ==

@@ Line 18: / Line 18: @@
 | Odd limit 2 = 9 | Mistuning 2 = 10.8 | Complexity 2 = 31
 }}
-'''Meantone''' is a familiar historical [[temperament]] based on a [[chain of fifths]] (or fourths), possessing two [[generator|generating intervals]]: the [[octave]] and the [[3/2|fifth]], from which all pitches are composed. In the common tunings of meantone (such as [[quarter-comma meantone]] and [[12edo]]), all the fifths are tuned the same, qualifying meantone as a [[regular temperament]] and more specifically a [[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]], 81/80, which means that stacking four fifths (such as {{dash|C, G, D, A, E|hair|med}}) results in a major third (C–E) that is close to the just interval [[5/4]] rather than the more complex Pythagorean interval [[81/64]]; good tunings of meantone also lead to soft [[diatonic]] and [[Chromatic scale|chromatic]] scales, which are desirable for interval categorization.
+'''Meantone''' is a familiar historical [[temperament]] based on a [[chain of fifths]] (or fourths), possessing two [[generator|generating intervals]]: the [[octave]] and the [[3/2|fifth]], from which all pitches are composed. This qualifies it as a [[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]], 81/80, which means that stacking four fifths (such as {{dash|C, G, D, A, E|hair|med}}) results in a major third (C–E) that is close to the just interval [[5/4]] rather than the more complex Pythagorean interval [[81/64]]; good tunings of meantone also lead to soft [[diatonic]] and [[Chromatic scale|chromatic]] scales, which are desirable for interval categorization.
 [[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.
@@ Line 35: / Line 35: @@
 '''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; it can also be seen as a diesis-flat minor seventh.
-See [[meantone vs meanpop]] for a comparison of undecimal (11-limit) extensions.
+See [[huygens vs meanpop]] for a comparison of undecimal (11-limit) extensions.
 === Other septimal extensions ===
@@ Line 187: / Line 187: @@
 * [[Meantone7]] – diatonic scale in 31edo
 * [[Meantone12]] – chromatic scale in 31edo
-; Eigenmonzo (unchanged-interval) tunings
+; Unchanged-interval (eigenmonzo) tunings
 * [[Meanwoo12]] – chromatic scale in 5/4.7-eigenmonzo tuning
 * [[Meanwoo19]] – enharmonic scale in 5/4.7-eigenmonzo tuning
@@ Line 213: / Line 213: @@
 * [[Lucy tuning]]
 * Equal beating tunings
+* 5-limit [[DKW theory|DKW]]: ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}
 === Prime-optimized tunings ===
@@ Line 219: / Line 220: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
-| CEE: ~3/2 = 696.895{{c}}<br>(4/17-comma)
+| CEE: ~3/2 = 696.8947{{c}}<br>(4/17 comma)
-| CSEE: ~3/2 = 696.453{{c}}<br>(11/43-comma)
+| CSEE: ~3/2 = 696.4534{{c}}<br>(11/43 comma)
+| POEE: ~3/2 = 695.2311{{c}}
 |-
 ! Tenney
-| CTE: ~3/2 = 697.214{{c}}
+| CTE: ~3/2 = 697.2143{{c}}
-| CWE: ~3/2 = 696.651{{c}}
+| CWE: ~3/2 = 696.6512{{c}}
+| POTE: ~3/2 = 696.2387{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 697.374{{c}}<br>(36/169-comma)
+| CBE: ~3/2 = 697.3738{{c}}<br>(36/169 comma)
-| CSBE: ~3/2 = 696.787{{c}}<br>(31/129-comma)
+| CSBE: ~3/2 = 696.7868{{c}}<br>(31/129 comma)
+| POBE: ~3/2 = 696.2984{{c}}
 |}
@@ Line 241: / Line 246: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
-| CEE: ~3/2 = 696.884{{c}}
+| CEE: ~3/2 = 696.8843{{c}}
-| CSEE: ~3/2 = 696.725{{c}}
+| CSEE: ~3/2 = 696.7248{{c}}
+| POEE: ~3/2 = 696.4375{{c}}
 |-
 ! Tenney
-| CTE: ~3/2 = 696.952{{c}}
+| CTE: ~3/2 = 696.9521{{c}}
-| CWE: ~3/2 = 696.656{{c}}
+| CWE: ~3/2 = 696.6562{{c}}
+| POTE: ~3/2 = 696.4949{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 697.015{{c}}
+| CBE: ~3/2 = 697.0147{{c}}
-| CSBE: ~3/2 = 696.631{{c}}
+| CSBE: ~3/2 = 696.6306{{c}}
+| POBE: ~3/2 = 696.4596{{c}}
+|}
+=== Target tunings ===
+{| class="wikitable center-all mw-collapsible mw-collapsed"
+|+ style="white-space: nowrap;" | Minimax tunings
+|-
+! Target
+! Generator
+! Eigenmonzo*
+|-
+| 5-odd-limit
+| ~3/2 = 696.578{{c}}
+| 5/4
+|-
+| 7-odd-limit
+| ~3/2 = 696.578{{c}}
+| 5/4
+|-
+| 9-odd-limit
+| ~3/2 = 696.578{{c}}
+| 5/4
+|}
+{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
+|+ style="white-space: nowrap;" | Least squares tunings
+|-
+! Target
+! Generator
+! Eigenmonzo*
+|-
+| 5-odd-limit
+| ~3/2 = 696.165{{c}}<br>(7/26 comma)
+| {{Monzo| -13 -2 7 }}
+|-
+| 7-odd-limit
+| ~3/2 = 696.648{{c}}
+| {{Monzo| -55 -11 1 25 }}
+|-
+| 9-odd-limit
+| ~3/2 = 696.436{{c}}
+| {{Monzo| 19 9 -1 -11 }}
 |}
@@ Line 265: / Line 315: @@
 |-
 ! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator<br>(¢)
 ! Comments
@@ Line 292: / Line 342: @@
 | [[9/5]]
 | 691.202
-| '''Lower bound of 9-odd-limit diamond tradeoff'''<br />[[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9
+| '''Lower bound of 9-odd-limit diamond tradeoff'''<br>[[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9
 |-
 | [[59edo|34\59]]
@@ Line 360: / Line 410: @@
 |-
 |
-| {{nowrap|''f''<sup>4</sup> &minus; 2''f'' &minus; 2 {{=}} 0}}
+| {{nowrap| ''f''<sup>4</sup> − 2''f'' − 2 {{=}} 0 }}
 | 695.630
 | 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma
@@ Line 397: / Line 447: @@
 | [[15/14]]
 | 696.111
 |
 |-
 |
 | [[78125/73728]]
 | 696.165
-| [[7/26-comma meantone|7/26 comma]], [[5-odd-limit]] least squares
+| [[7/26-comma meantone|7/26 comma]], 5-odd-limit least squares
 |-
-| {{nowrap|(8 &minus; φ)\11}}
+| {{nowrap| (8 − φ)\11 }}
 |
 | 696.214
@@ Line 443: / Line 493: @@
 | 696.399
 |
-|-
-|
-| {{Monzo| 19 9 -1 -11 }}
-| 696.436
-| 9-odd-limit least squares
 |-
 |
@@ Line 463: / Line 508: @@
 | 696.626
 |
-|-
-|
-| {{monzo| -55 -11 1 25 }}
-| 696.648
-| [[7-odd-limit]] least squares
 |-
 | [[31edo|18\31]]
@@ Line 487: / Line 527: @@
 | [[1875/1024]]
 | 696.895
-| [[4/17-comma meantone|4/17 comma]]; 2.3.5 [[CEE]] tuning
+| [[4/17-comma meantone|4/17 comma]]; 5-limit [[CEE]] tuning
 |-
 |
@@ Line 683: / Line 723: @@
 $$ n = (g_J - g)/g_c $$
-=== Other tunings ===
-* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}
 == Music ==