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| Title = Meantone
| Title = Meantone
| Subgroups = 2.3.5, 2.3.5.7
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = 81/80 (2.3.5); <br>81/80, 126/125 (2.3.5.7)
| Comma basis = [[81/80]] (2.3.5); <br>[[81/80]], [[126/125]] (2.3.5.7)
| Generator = 3/2
| Edo join 1 = 12 | Edo join 2 = 19
| Mapping = 1; 1 4 10
| Mapping = 1; 1 4 10
| Generators = 3/2
| Generators tuning = 696.7
| Optimization method = CWE
| MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[12L&nbsp;7s]]
| Pergen = (P8, P5)
| Pergen = (P8, P5)
| Color name = Guti
| Color name = Guti
| Edo join 1 = 12 | Edo join 2 = 19
| Odd limit 1 = 5 | Mistuning 1 = 5.4 | Complexity 1 = 5
| Optimization method = CWE
| Odd limit 2 = 9 | Mistuning 2 = 10.8 | Complexity 2 = 12
| Generator tuning = 696.7
| MOS scales = [[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]]
| Odd limit 1 = 5 | Mistuning 1 = 5.4 | Complexity 1 = 12
| 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. 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]].
'''Meantone''' is a familiar [[Historical temperaments|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.
[[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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== 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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{{Wikipedia| Septimal meantone temperament }}
{{Wikipedia| Septimal meantone temperament }}


'''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 (+10 fifths, C–A♯), and is notably present in the augmented sixth chord; it can also be seen as a diesis-flat minor seventh, as the diesis represents [[36/35]]~[[64/63]]. In septimal meantone, 7/5 is an augmented fourth, 7/6 is an augmented second, and [[9/7]] is a diminished fourth. Notably, septimal meantone equates the interval of a diminished fifth between the third and the seventh of a [[dominant seventh chord]] to [[10/7]], making it a [[9-odd-limit]] [[essentially tempered chord]]. Septimal meantone is best tuned close to [[31edo]] or [[Quarter-comma meantone|1/4-comma]].


See [[meantone vs meanpop]] for a comparison of undecimal (11-limit) extensions.
Extending meantone to the [[11-limit]] is not as simple. For one, there is the factorization of 81/80 as ([[121/120]])*([[243/242]]), and tempering both out leads to [[mohaha]] in the [[2.3.5.11 subgroup]], which splits the perfect fifth into two [[11/9]]~[[27/22]] neutral thirds. Adding back the septimal meantone mapping of 7 (+20 neutral thirds) gives [[migration]], but mohaha has an alternative mapping of [[7/4]] at the semi-diminished seventh (-13 neutral thirds), known as [[mohajira]]. Extensions to prime 11 generated by the perfect fifth are trickier. If 121/120 and 243/242 are not tempered out, then one of them must be mapped positively, and the other negatively. Since 121/120 is the difference between [[11/10]] and [[12/11]], it makes more sense to map it positively, and thus 243/242 negatively, leading 11/9 to be mapped wider than 27/22 and causing inconsistencies. Nonetheless, 31edo supports septimal meantone well while also having a neutral third, and there are two extensions generated by the fifth which map 11/9 to the neutral third. [[Undecimal meantone]] (also known as ''huygens'') maps 11/9 to +16 fifths (C–D𝄪) and 11/8 to +18 fifths (C–E𝄪), tempering out [[99/98]], [[176/175]], and [[441/440]]. Huygens works in the range from 31edo (696.8{{C}}) to 12edo (700{{C}}). The other extension is [[meanpop]], which maps 11/9 to -15 fifths (C–F𝄫) and 11/8 to -13 fifths (C–G𝄫), tempering out [[385/384]] and [[540/539]]. Tunings of meanpop range from 19edo (694.7{{C}}) to 31edo (696.8{{C}}).


=== Other septimal extensions ===
=== Other septimal extensions ===
There are some alternative mappings of the 7-limit meantone, including flattone, dominant and sharptone.
There are some alternative mappings of the 7-limit meantone, including flattone and dominant.


==== Flattone ====
==== Flattone ====
{{Main| Flattone }}
{{Main| Flattone }}


Flattone is an alternative extension of meantone, which represents 7/4 as a diminished seventh rather than an augmented sixth. The fifth interval is adjusted narrower, nine of which give 8/7 and thirteen of which give 10/7 with octave reduction. Alternatively, stacking three minor thirds results in a diminished seventh that is close to 7/4.
Flattone is an alternative extension of meantone, which represents 7/4 as a diminished seventh rather than an augmented sixth. The fifth interval is adjusted narrower, nine of which give 8/7 and thirteen of which give 10/7 with octave reduction. Alternatively, stacking three minor thirds results in a diminished seventh that is close to 7/4. While less accurate than septimal meantone, flattone extends much more easily to the [[11-limit|11-]] and [[13-limit|13-]][[limit]]s, with [[11/8]] being an augmented fourth (+6 fifths, C–F♯) and [[13/8]] being a minor sixth (-4 fifths, C–A♭).


==== Dominant ====
==== Dominant ====
{{Main| Dominant (temperament) }}
{{Main| Dominant (temperament) }}


Dominant is an alternative extension of meantone, which represents 7/4 as a minor seventh rather than an augmented sixth. This equates 6/5 with 7/6 and 5/4 with 9/7, tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas' comma).
Dominant is an alternative extension of meantone, which represents 7/4 as a minor seventh rather than an augmented sixth. This equates 6/5 with 7/6 and 5/4 with 9/7, tempering out [[36/35]] (septimal quarter tone) and [[64/63]] (Archytas' comma). Dominant was named because the [[dominant seventh chord]] of the [[5L 2s|diatonic]] scale represents [[4:5:6:7]] in it.


== 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'''.  


<div><div style="display: inline-grid; margin-right: 25px;">
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|+ style="font-size: 105%;" | Intervals fifthward
|-
|-
! &#35;
! #
! class="unsortable" | Category
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
| P1
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| P5
| 696.7
| 696.7
| '''3/2'''
| '''3/2'''
|-
|-
| 2
| 2
| M2
| 193.3
| 193.3
| '''9/8''', 10/9, 28/25
| '''9/8''', 10/9, 28/25
|-
|-
| 3
| 3
| M6
| 890.0
| 890.0
| 5/3, 42/25
| 5/3
|-
|-
| 4
| 4
| M3
| 386.6
| 386.6
| '''5/4'''
| '''5/4'''
|-
|-
| 5
| 5
| M7
| 1083.3
| 1083.3
| '''15/8''', 28/15
| '''15/8''', 28/15
|-
|-
| 6
| 6
| A4
| 579.9
| 579.9
| 7/5, 25/18
| 7/5, 25/18
|-
|-
| 7
| 7
| A1
| 76.6
| 76.6
| 21/20, 25/24
| 21/20, 25/24, 28/27
|-
|-
| 8
| 8
| A5
| 773.2
| 773.2
| 14/9, 25/16
| 14/9, 25/16
|-
|-
| 9
| 9
| A2
| 269.9
| 269.9
| 7/6
| 7/6
|-
|-
| 10
| 10
| A6
| 966.6
| 966.6
| '''7/4'''
| '''7/4'''
|-
|-
| 11
| 11
| A3
| 463.2
| 463.2
| 21/16
| 21/16
|-
|-
| 12
| 12
| A7
| 1159.9
| 1159.9
| 35/18, 49/25, 63/32
| 35/18, 49/25, 63/32
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</div>
</div>
<div style="display: inline-grid; margin-right: 25px;">
<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|+ style="font-size: 105%;" | Intervals fourthward
|-
|-
! &#35;
! #
! class="unsortable" | Category
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
| P1
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| −1
| −1
| P4
| 503.3
| 503.3
| 4/3
| 4/3
|-
|-
| −2
| −2
| m7
| 1006.7
| 1006.7
| 9/5, 16/9, 25/14
| 9/5, 16/9, 25/14
|-
|-
| −3
| −3
| m3
| 310.0
| 310.0
| 6/5, 25/21
| 6/5
|-
|-
| −4
| −4
| m6
| 813.4
| 813.4
| 8/5
| 8/5
|-
|-
| −5
| −5
| m2
| 116.7
| 116.7
| 15/14, 16/15
| 15/14, 16/15
|-
|-
| −6
| −6
| d5
| 620.1
| 620.1
| 10/7, 36/25
| 10/7, 36/25
|-
|-
| −7
| −7
| d8
| 1123.4
| 1123.4
| 40/21, 48/25
| 27/14, 40/21, 48/25
|-
|-
| −8
| −8
| d4
| 426.8
| 426.8
| 9/7, 32/25
| 9/7, 32/25
|-
|-
| −9
| −9
| d7
| 930.1
| 930.1
| 12/7
| 12/7
|-
|-
| −10
| −10
| d3
| 233.4
| 233.4
| 8/7
| 8/7
|-
|-
| −11
| −11
| d6
| 736.8
| 736.8
| 32/21
| 32/21
|-
|-
| −12
| −12
| d2
| 40.1
| 40.1
| 36/35, 50/49, 64/63
| 36/35, 50/49, 64/63
|}
|}
</div>
</div>
<nowiki/>* In [[CWE]] septimal meantone
<nowiki/>* In 7-limit [[CWE]] tuning, octave reduced


== Chords ==
== Chords and harmony ==
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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== Tunings ==
== Tunings ==
Common meantone tunings can be classified into [[Eigenmonzo|eigenmonzo (unchanged-interval)]] tunings, edo tunings, prime-optimized tunings and others. In eigenmonzo tunings such as the [[quarter-comma meantone]], a certain interval is tuned pure and certain others are equally off. Edo tunings like [[31edo]] have rational size relationship between steps, and happen to send an additional comma to unison. Prime-optimized tunings are optimized for all intervals. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.  
Common meantone tunings can be classified into [[eigenmonzo|eigenmonzo (unchanged-interval)]] tunings, edo tunings, norm-based tunings and others. In eigenmonzo tunings such as the [[quarter-comma meantone]], a certain interval is tuned pure and certain others are equally off. Edo tunings like [[31edo]] have rational size relationship between steps, and happen to send an additional comma to unison. Norm-based tunings are optimized for all intervals. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.  


; Notable eigenmonzo (unchanged-interval) tunings
; Notable eigenmonzo (unchanged-interval) tunings
* [[1/2-comma meantone]] &ndash; with eigenmonzo [[10/9]]
* [[1/2-comma meantone]] – with eigenmonzo [[10/9]]
* [[1/3-comma meantone]] &ndash; with eigenmonzo [[5/3]]
* [[1/3-comma meantone]] – with eigenmonzo [[5/3]]
* [[2/7-comma meantone]] &ndash; with eigenmonzo [[25/24]]
* [[2/7-comma meantone]] – with eigenmonzo [[25/24]]
* [[Quarter-comma meantone|1/4-comma meantone]] &ndash; with eigenmonzo [[5/4]]
* [[Quarter-comma meantone|1/4-comma meantone]] – with eigenmonzo [[5/4]]
* [[1/5-comma meantone]] &ndash; with eigenmonzo [[15/8]]
* [[1/5-comma meantone]] – with eigenmonzo [[15/8]]
* [[1/6-comma meantone]] &ndash; with eigenmonzo [[45/32]]
* [[1/6-comma meantone]] – with eigenmonzo [[45/32]]
* [[Ratwolf|Ratwolf tuning]]
* [[Ratwolf|Ratwolf 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 ===
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~3/2 = 696.895¢<br>(4/17-comma)
| CEE: ~3/2 = 696.8947{{c}}<br>(4/17 comma)
| CSEE: ~3/2 = 696.453¢<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¢
| CTE: ~3/2 = 697.2143{{c}}
| CWE: ~3/2 = 696.651¢
| CWE: ~3/2 = 696.6512{{c}}
| POTE: ~3/2 = 696.2387{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~3/2 = 697.374¢<br>(36/169-comma)
| CBE: ~3/2 = 697.3738{{c}}<br>(36/169 comma)
| CSBE: ~3/2 = 696.787¢<br>(31/129-comma)
| CSBE: ~3/2 = 696.7868{{c}}<br>(31/129 comma)
| POBE: ~3/2 = 696.2984{{c}}
|}
|}


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~3/2 = 696.884¢
| CEE: ~3/2 = 696.8843{{c}}
| CSEE: ~3/2 = 696.725¢
| CSEE: ~3/2 = 696.7248{{c}}
| POEE: ~3/2 = 696.4375{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~3/2 = 696.952¢
| CTE: ~3/2 = 696.9521{{c}}
| CWE: ~3/2 = 696.656¢
| CWE: ~3/2 = 696.6562{{c}}
| POTE: ~3/2 = 696.4949{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~3/2 = 697.015¢
| CBE: ~3/2 = 697.0147{{c}}
| CSBE: ~3/2 = 696.631¢
| 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;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 696.578{{c}}<br>(1/4 comma)
| 5/4
| ~3/2 = 696.165{{c}}<br>(7/26 comma)
| {{Monzo| -13 -2 7 }}
|-
| 7-odd-limit
| ~3/2 = 696.578{{c}}
| 5/4
| ~3/2 = 696.648{{c}}
| {{Monzo| -55 -11 1 25 }}
|-
| 9-odd-limit
| ~3/2 = 696.578{{c}}
| 5/4
| ~3/2 = 696.436{{c}}
| {{Monzo| 19 9 -1 -11 }}
|}
|}


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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator<br />(¢)
! Generator<br>(¢)
! Comments
! Comments
|-
|-
Line 272: Line 343:
| [[27/20]]
| [[27/20]]
| 680.449
| 680.449
| Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)
| Full comma (syntonic comma; from here onwards ''comma'' without an adjective refers to syntonic comma)
|-
|-
| '''[[7edo|4\7]]'''
| '''[[7edo|4\7]]'''
Line 292: Line 363:
| [[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
| [[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
|-
|-
| [[59edo|34\59]]
| [[59edo|34\59]]
Line 312: Line 383:
| [[27/25]]
| [[27/25]]
| 693.352
| 693.352
| [[2/5-comma meantone|2/5 comma]]
| [[2/5-comma meantone|2/5-comma]]
|-
|
| 100/83
| 693.473
| 415&nbsp;Hz A gives 500&nbsp;Hz C
|-
|-
|
|
Line 365: Line 431:
|-
|-
|
|
| {{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
|-
|-
| [[69edo|40\69]]
| [[69edo|40\69]]
Line 377: Line 443:
| [[25/24]]
| [[25/24]]
| 695.810
| 695.810
| [[2/7-comma meantone|2/7 comma]], virtually also [[DR]] 10:12:15
| [[2/7-comma meantone|2/7-comma]], virtually also [[DR]] 10:12:15
|-
|-
|
|
Line 385: Line 451:
|-
|-
|
|
|  
| 3125/2304
| 695.981
| 695.981
| [[5/18-comma meantone|5/18 comma]]
| [[5/18-comma meantone|5/18-comma]]
|-
|-
|
|
Line 402: Line 468:
| [[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
Line 433: Line 499:
| 696.319
| 696.319
|
|
|-
|
| [[19/16]]
| 696.340
| As AAAA1
|-
|
| [[17/16]]
| 696.344
| As AAA7
|-
|-
|
|
Line 448: Line 504:
| 696.399
| 696.399
|
|
|-
|
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 9-odd-limit least squares
|-
|-
|
|
Line 468: Line 519:
| 696.626
| 696.626
|
|
|-
|
| {{monzo| -55 -11 1 25 }}
| 696.648
| [[7-odd-limit]] least squares
|-
|-
| [[31edo|18\31]]
| [[31edo|18\31]]
Line 490: Line 536:
|-
|-
|
|
| [[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
|-
|-
|
|
Line 512: Line 558:
| [[75/64]]
| [[75/64]]
| 697.176
| 697.176
| [[2/9-comma meantone|2/9 comma]]
| [[2/9-comma meantone|2/9-comma]]
|-
|-
|
|
| {{nowrap|''f''<sup>4</sup> + 2''f'' &minus; 8 {{=}} 0}}
| {{nowrap|''f''<sup>4</sup> + 2''f'' − 8 {{=}} 0}}
| 697.278
| 697.278
| 1–3–5 equal-beating tuning ([[DR]] 3:4:5), virtually 5/23 comma
| 1–3–5 equal-beating tuning ([[DR]] 3:4:5), virtually 5/23-comma
|-
|-
| [[74edo|43\74]]
| [[74edo|43\74]]
Line 530: Line 576:
|-
|-
|
|
|  
| 1125/1024
| 697.347
| 697.347
| [[3/14-comma meantone|3/14 comma]]
| [[3/14-comma meantone|3/14-comma]]
|-
|-
| {{nowrap|(√(10) &minus; 2)\2}}
| {{nowrap|(√(10) − 2)\2}}
|
|
| 697.367
| 697.367
Line 547: Line 593:
| [[15/8]]
| [[15/8]]
| 697.654
| 697.654
| [[1/5-comma meantone|1/5 comma]]
| [[1/5-comma meantone|1/5-comma]]
|-
|-
| [[43edo|25\43]]
| [[43edo|25\43]]
Line 597: Line 643:
| [[45/32]]
| [[45/32]]
| 698.371
| 698.371
| [[1/6-comma meantone|1/6 comma]]
| [[1/6-comma meantone|1/6-comma]]
|-
|-
| [[67edo|39\67]]
| [[67edo|39\67]]
Line 605: Line 651:
|-
|-
|
|
|  
| {{monzo|-23 9 4}}
| 698.514
| 698.514
| [[4/25-comma meantone|4/25 comma]]
| [[4/25-comma meantone|4/25-comma]]
|-
|-
|
|
Line 622: Line 668:
| [[135/128]]
| [[135/128]]
| 698.883
| 698.883
| [[1/7-comma meantone|1/7 comma]]
| [[1/7-comma meantone|1/7-comma]]
|-
|-
| [[91edo|53\91]]
| [[91edo|53\91]]
Line 652: Line 698:
|
|
| '''700.000'''
| '''700.000'''
| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone]] (the difference is too small to appear in the digits provided here)
| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone|1/11-comma]]†
|-
|
| [[17/9]]
| 700.209
| As M7
|-
|-
|
|
Line 674: Line 715:
| '''Upper bound of 5-odd-limit diamond monotone'''
| '''Upper bound of 5-odd-limit diamond monotone'''
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave
 
† The difference is too small to appear in the digits provided here


=== Formula for ''n''-comma meantone ===
=== Formula for ''n''-comma meantone ===
The [[generator]] ''g'' of ''n''-comma meantone, where ''n'' is a fraction (like 1/5, 2/9, etc.), can be found by
The [[generator]] ''g'' of ''n''-comma meantone, where ''n'' is a fraction (like 1/5, 2/9, etc.), can be found by


<math>\displaystyle g = g_J - ng_c</math>
$$ g = g_J - ng_c $$


where {{nowrap|''g''<sub>''J''</sub> {{=}} 701.955001}} cents is the size of the just perfect fifth, and ''g''<sub>c</sub> = 21.506290 cents is the size of the syntonic comma.
where {{nowrap|''g''<sub>''J''</sub> {{=}} 701.955001}} cents is the size of the just perfect fifth, and ''g''<sub>c</sub> = 21.506290 cents is the size of the syntonic comma.
Line 685: Line 728:
Conversely, ''n'' can be found by
Conversely, ''n'' can be found by


<math>\displaystyle n = (g_J - g)/g_c</math>
$$ n = (g_J - g)/g_c $$
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~3/2 = 696.353


== Music ==
== Music ==
Line 694: Line 734:


== 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 ==

Latest revision as of 09:42, 22 March 2026

Meantone
Subgroups 2.3.5, 2.3.5.7
Comma basis 81/80 (2.3.5);
81/80, 126/125 (2.3.5.7)
Reduced mapping ⟨1; 1 4 10]
ET join 12 & 19
Generators (CWE) ~3/2 = 696.7 ¢
MOS scales 2L 3s, 5L 2s, 7L 5s, 12L 7s
Ploidacot monocot
Pergen (P8, P5)
Color name Guti
Minimax error 5-odd-limit: 5.4 ¢;
9-odd-limit: 10.8 ¢
Target scale size 5-odd-limit: 5 notes;
9-odd-limit: 12 notes

Meantone is a familiar historical temperament based on a chain of fifths (or fourths), possessing two generating intervals: the octave and the 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 C – G – D – A – E) 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 scales, which are desirable for interval categorization.

Intervals in meantone have standard names based on the number of steps of the diatonic scale they span (this corresponds to the 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.

Technical temperament data is discussed at Meantone family #Meantone in the context of the associated family of temperaments.

English Wikipedia has an article on:

History

See also: Historical temperaments

Meantone tunings with fifths flatter than 700 ¢ were the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various well temperaments 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

Septimal meantone

English Wikipedia has an article on:

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 (+10 fifths, C–A♯), and is notably present in the augmented sixth chord; it can also be seen as a diesis-flat minor seventh, as the diesis represents 36/35~64/63. In septimal meantone, 7/5 is an augmented fourth, 7/6 is an augmented second, and 9/7 is a diminished fourth. Notably, septimal meantone equates the interval of a diminished fifth between the third and the seventh of a dominant seventh chord to 10/7, making it a 9-odd-limit essentially tempered chord. Septimal meantone is best tuned close to 31edo or 1/4-comma.

Extending meantone to the 11-limit is not as simple. For one, there is the factorization of 81/80 as (121/120)*(243/242), and tempering both out leads to mohaha in the 2.3.5.11 subgroup, which splits the perfect fifth into two 11/9~27/22 neutral thirds. Adding back the septimal meantone mapping of 7 (+20 neutral thirds) gives migration, but mohaha has an alternative mapping of 7/4 at the semi-diminished seventh (-13 neutral thirds), known as mohajira. Extensions to prime 11 generated by the perfect fifth are trickier. If 121/120 and 243/242 are not tempered out, then one of them must be mapped positively, and the other negatively. Since 121/120 is the difference between 11/10 and 12/11, it makes more sense to map it positively, and thus 243/242 negatively, leading 11/9 to be mapped wider than 27/22 and causing inconsistencies. Nonetheless, 31edo supports septimal meantone well while also having a neutral third, and there are two extensions generated by the fifth which map 11/9 to the neutral third. Undecimal meantone (also known as huygens) maps 11/9 to +16 fifths (C–D𝄪) and 11/8 to +18 fifths (C–E𝄪), tempering out 99/98, 176/175, and 441/440. Huygens works in the range from 31edo (696.8 ¢) to 12edo (700 ¢). The other extension is meanpop, which maps 11/9 to -15 fifths (C–F𝄫) and 11/8 to -13 fifths (C–G𝄫), tempering out 385/384 and 540/539. Tunings of meanpop range from 19edo (694.7 ¢) to 31edo (696.8 ¢).

Other septimal extensions

There are some alternative mappings of the 7-limit meantone, including flattone and dominant.

Flattone

Main article: Flattone

Flattone is an alternative extension of meantone, which represents 7/4 as a diminished seventh rather than an augmented sixth. The fifth interval is adjusted narrower, nine of which give 8/7 and thirteen of which give 10/7 with octave reduction. Alternatively, stacking three minor thirds results in a diminished seventh that is close to 7/4. While less accurate than septimal meantone, flattone extends much more easily to the 11- and 13-limits, with 11/8 being an augmented fourth (+6 fifths, C–F♯) and 13/8 being a minor sixth (-4 fifths, C–A♭).

Dominant

Main article: Dominant (temperament)

Dominant is an alternative extension of meantone, which represents 7/4 as a minor seventh rather than an augmented sixth. This equates 6/5 with 7/6 and 5/4 with 9/7, tempering out 36/35 (septimal quarter tone) and 64/63 (Archytas' comma). Dominant was named because the dominant seventh chord of the diatonic scale represents 4:5:6:7 in it.

Intervals

Main article: Meantone intervals

In the following tables, odd harmonics 1–15 are labeled in bold.

Intervals fifthward
# Category Cents* Approximate ratios
0 P1 0.0 1/1
1 P5 696.7 3/2
2 M2 193.3 9/8, 10/9, 28/25
3 M6 890.0 5/3
4 M3 386.6 5/4
5 M7 1083.3 15/8, 28/15
6 A4 579.9 7/5, 25/18
7 A1 76.6 21/20, 25/24, 28/27
8 A5 773.2 14/9, 25/16
9 A2 269.9 7/6
10 A6 966.6 7/4
11 A3 463.2 21/16
12 A7 1159.9 35/18, 49/25, 63/32
Intervals fourthward
# Category Cents* Approximate ratios
0 P1 0.0 1/1
−1 P4 503.3 4/3
−2 m7 1006.7 9/5, 16/9, 25/14
−3 m3 310.0 6/5
−4 m6 813.4 8/5
−5 m2 116.7 15/14, 16/15
−6 d5 620.1 10/7, 36/25
−7 d8 1123.4 27/14, 40/21, 48/25
−8 d4 426.8 9/7, 32/25
−9 d7 930.1 12/7
−10 d3 233.4 8/7
−11 d6 736.8 32/21
−12 d2 40.1 36/35, 50/49, 64/63

* In 7-limit CWE tuning, octave reduced

Chords and harmony

Meantone induces didymic chords, the essentially tempered chords 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

Main article: Meantone scales
Edo tunings
Unchanged-interval (eigenmonzo) tunings
  • Meanwoo12 – chromatic scale in 5/4.7-eigenmonzo tuning
  • Meanwoo19 – enharmonic scale in 5/4.7-eigenmonzo tuning
  • Ratwolf – chromatic scale with 20/13 wolf fifth
Others
  • Meaneb471a – chromatic scale in one equal beating tuning of ~3/1 and ~5/1
  • Meaneb471 – chromatic scale in the other equal beating tuning of ~3/1 and ~5/1, also called "metameantone"

Tunings

Common meantone tunings can be classified into eigenmonzo (unchanged-interval) tunings, edo tunings, norm-based tunings and others. In eigenmonzo tunings such as the quarter-comma meantone, a certain interval is tuned pure and certain others are equally off. Edo tunings like 31edo have rational size relationship between steps, and happen to send an additional comma to unison. Norm-based tunings are optimized for all intervals. For a more complete list, see the table below. These different tunings are referred to as "temperaments" in traditional terms.

Notable eigenmonzo (unchanged-interval) tunings
Other optimized tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 696.8947 ¢
(4/17 comma) 		CSEE: ~3/2 = 696.4534 ¢
(11/43 comma) 		POEE: ~3/2 = 695.2311 ¢
Tenney CTE: ~3/2 = 697.2143 ¢ CWE: ~3/2 = 696.6512 ¢ POTE: ~3/2 = 696.2387 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 697.3738 ¢
(36/169 comma) 		CSBE: ~3/2 = 696.7868 ¢
(31/129 comma) 		POBE: ~3/2 = 696.2984 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~3/2 = 696.8843 ¢ CSEE: ~3/2 = 696.7248 ¢ POEE: ~3/2 = 696.4375 ¢
Tenney CTE: ~3/2 = 696.9521 ¢ CWE: ~3/2 = 696.6562 ¢ POTE: ~3/2 = 696.4949 ¢
Benedetti,
Wilson 		CBE: ~3/2 = 697.0147 ¢ CSBE: ~3/2 = 696.6306 ¢ POBE: ~3/2 = 696.4596 ¢

Target tunings

Target tunings
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~3/2 = 696.578 ¢
(1/4 comma) 		5/4 ~3/2 = 696.165 ¢
(7/26 comma) 		[-13 -2 7⟩
7-odd-limit ~3/2 = 696.578 ¢ 5/4 ~3/2 = 696.648 ¢ [-55 -11 1 25⟩
9-odd-limit ~3/2 = 696.578 ¢ 5/4 ~3/2 = 696.436 ¢ [19 9 -1 -11⟩

Tuning spectrum

The below tuning chart assumes septimal meantone and is agnostic to higher-limit extensions.

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator
(¢) 		Comments
27/20 680.449 Full comma (syntonic comma; from here onwards comma without an adjective refers to syntonic comma)
4\7 685.714 Lower bound of 5-odd-limit diamond monotone
51/38 690.603 As P4.
19\33 690.909 33cddd val
9/5 691.202 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
34\59 691.525 59bcddddd val
15\26 692.308 26d val
26\45 693.333 45dd val
27/25 693.352 2/5-comma
45/28 694.651
27/14 694.709
81/70 694.732
11\19 694.737 Lower bound of 7- and 9-odd-limit diamond monotone
5/3 694.786 1/3 comma, lower bound of 5- and 7-odd-limit diamond tradeoff
35/27 695.389
51\88 695.455 88dd val
1\2 + 1\(4π) 695.493 Lucy tuning
9/7 695.614
f4 − 2f − 2 = 0 695.630 1–3–5 equal-beating tuning, Wilson's "metameantone" (DR 4:5:6), virtually 5/17-comma
40\69 695.652 69d val
25/24 695.810 2/7-comma, virtually also DR 10:12:15
36/35 695.936
3125/2304 695.981 5/18-comma
49/27 695.987
29\50 696.000
15/14 696.111
78125/73728 696.165 7/26-comma, 5-odd-limit least squares
(8 − φ)\11 696.214 Golden meantone
49/45 696.245
19/17 696.279 Mediant of 9/8 and 10/9, known as classical meantone
47\81 696.296
7/6 696.319
35/24 696.399
5/4 696.578 1/4 comma, 5-, 7-, and 9-odd-limit minimax
49/48 696.616
49/30 696.626
18\31 696.774
35/32 696.796
7/4 696.883
1875/1024 696.895 4/17-comma; 5-limit CEE tuning
49/40 696.959
7/5 697.085
61\105 697.143
75/64 697.176 2/9-comma
f4 + 2f − 8 = 0 697.278 1–3–5 equal-beating tuning (DR 3:4:5), virtually 5/23-comma
43\74 697.297
21/16 697.344
1125/1024 697.347 3/14-comma
(√(10) − 2)\2 697.367 Tungsten meantone
68\117 697.436 117d val
15/8 697.654 1/5-comma
25\43 697.674
64/63 697.728
21/20 697.781
17/10 697.929 As d7
57\98 697.959 98d val
25/14 698.099
32\55 698.182 55d val
63/40 698.303
17/15 698.331 As d3
45/32 698.371 1/6-comma
39\67 698.507 67d val
[-23 9 4⟩ 698.514 4/25-comma
45/34 698.661 As A3
46\79 698.734 79cdd val
135/128 698.883 1/7-comma
53\91 698.901 91cddd val
17/16 699.009 As m2
25/21 699.384
17/12 699.500 As d5
17/9 699.851 As d8
7\12 700.000 Upper bound of 7- and 9-odd-limit diamond monotone, 1/12 Pythagorean comma, virtually 1/11-comma†
19/16 700.829 As m3
3/2 701.955 Pythagorean tuning, tunings sharper 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. upper bound of 5-, 7-, and 9-odd-limit diamond tradeoff
3\5 720.000 Upper bound of 5-odd-limit diamond monotone

* Besides the octave

† The difference is too small to appear in the digits provided here

Formula for n-comma meantone

The generator g of n-comma meantone, where n is a fraction (like 1/5, 2/9, etc.), can be found by

$$ g = g_J - ng_c $$

where gJ = 701.955001 cents is the size of the just perfect fifth, and gc = 21.506290 cents is the size of the syntonic comma.

Conversely, n can be found by

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

Music

See Quarter-comma meantone #Music.

See also

  • Angel – fifth-equivalent or 5/1-equivalent meantone

External links

Retrieved from "https://en.xen.wiki/index.php?title=Meantone&oldid=226637"