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Meantone: Difference between revisions

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== History ==
== History ==
{{See also|Historical temperaments}}
{{See also| Historical temperaments }}


Meantone with fifths flatter than 700{{cent}} were the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[well temperament]]s and eventually [[12edo|12-tone equal temperament]] won in popularity. However, even today, the vast majority of common-practice Western music theory is based exclusively on meantone, as 12-tone equal temperament is itself a meantone tuning.
Meantone tunings with fifths flatter than 700{{cent}} were the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[well temperament]]s and eventually 12-tone equal temperament won in popularity. However, even today, the vast majority of common-practice Western music theory is based exclusively on meantone, as 12-tone equal temperament is itself a meantone tuning.


== Extensions ==
== Extensions ==
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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.  
'''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 ===
=== Other septimal extensions ===
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== Intervals ==
== Intervals ==
{{main|Meantone intervals}}
{{Main| Meantone intervals }}


In the following tables, odd harmonics 1–15 are labeled in '''bold'''.  
In the following tables, odd harmonics 1–15 are labeled in '''bold'''.  
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{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 right-2"
|-
|-
! #
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
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| 3
| 3
| 890.0
| 890.0
| 5/3, 42/25
| 5/3
|-
|-
| 4
| 4
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| 7
| 7
| 76.6
| 76.6
| 21/20, 25/24
| 21/20, 25/24, 28/27
|-
|-
| 8
| 8
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{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 right-2"
|-
|-
! #
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
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| −3
| −3
| 310.0
| 310.0
| 6/5, 25/21
| 6/5
|-
|-
| −4
| −4
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| −7
| −7
| 1123.4
| 1123.4
| 40/21, 48/25
| 27/14, 40/21, 48/25
|-
|-
| −8
| −8
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|}
|}
</div>
</div>
<nowiki/>* In [[CWE]] septimal meantone
<nowiki/>* In 7-limit [[CWE]] tuning, octave reduced


== Chords ==
== Chords ==
Meantone induces [[didymic chords]], the [[essentially tempered chord]]s and associated progressions which are not found in other temperaments. Notably, the roots of the common chord progression vi-ii-V-I make up such a tetrad. Moreover, the dominant seventh chord and the half-diminished seventh chord can be seen as essentially tempered by septimal meantone.  
Meantone induces [[didymic chords]], the [[essentially tempered chord]]s and associated progressions which are not found in other temperaments. Notably, the roots of the common chord progression vi–ii–V–I make up such a tetrad. Moreover, the dominant seventh chord and the half-diminished seventh chord can be seen as essentially tempered by septimal meantone.  


== Scales ==
== Scales ==
{{Main| Meantone scales }}
{{Main| Meantone scales }}


; EDO tunings
; Edo tunings
* [[Meantone5]] – pentic scale in 31edo
* [[Meantone5]] – pentic scale in 31edo
* [[Meantone7]] – diatonic scale in 31edo
* [[Meantone7]] – diatonic scale in 31edo
* [[Meantone12]] – chromatic 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
* [[Meanwoo12]] – chromatic scale in 5/4.7-eigenmonzo tuning
* [[Meanwoo19]] – enharmonic scale in 5/4.7-eigenmonzo tuning
* [[Meanwoo19]] – enharmonic scale in 5/4.7-eigenmonzo tuning
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* [[Lucy tuning]]
* [[Lucy tuning]]
* Equal beating tunings
* Equal beating tunings
* 5-limit [[DKW theory|DKW]]: ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}


=== Prime-optimized tunings ===
=== Prime-optimized tunings ===
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|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained
! Constrained
! Constrained & skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! 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
! 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
! 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}}
|}
|}


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|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Constrained
! Constrained
! Constrained & skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! 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
! 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
! 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 }}
|}
|}


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|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator<br>(¢)
! Generator<br>(¢)
! Comments
! Comments
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| [[9/5]]
| [[9/5]]
| 691.202
| 691.202
| [[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, a.k.a. lower bound of 9-odd-limit diamond tradeoff
| '''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]]
| [[59edo|34\59]]
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|-
|-
|
|
| {{nowrap|''f''<sup>4</sup> &minus; 2''f'' &minus; 2 {{=}} 0}}
| {{nowrap| ''f''<sup>4</sup> − 2''f'' − 2 {{=}} 0 }}
| 695.630
| 695.630
| 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma
| 1–3–5 equal-beating tuning, Wilson's "metameantone" ([[DR]] 4:5:6), virtually 5/17 comma
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| [[15/14]]
| [[15/14]]
| 696.111
| 696.111
|
|  
|-
|-
|
|  
| [[78125/73728]]
| [[78125/73728]]
| 696.165
| 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
| 696.214
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| 696.399
| 696.399
|
|
|-
|
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 9-odd-limit least squares
|-
|-
|
|
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| 696.626
| 696.626
|
|
|-
|
| {{monzo| -55 -11 1 25 }}
| 696.648
| [[7-odd-limit]] least squares
|-
|-
| [[31edo|18\31]]
| [[31edo|18\31]]
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| [[1875/1024]]
| [[1875/1024]]
| 696.895
| 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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$$ n = (g_J - g)/g_c $$
$$ n = (g_J - g)/g_c $$
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.000{{c}}, ~3/2 = 696.353{{c}}


== Music ==
== Music ==
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== See also ==
== See also ==
* [[Angel]] &ndash; fifth-equivalent or 5/1-equivalent meantone
* [[Angel]] – fifth-equivalent or 5/1-equivalent meantone


== External links ==
== External links ==
Retrieved from "https://en.xen.wiki/w/Meantone"