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Meantone: Difference between revisions

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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&nbsp;3s]], [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[12L&nbsp;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]]; 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 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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{{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 [[huygens 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 ==
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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
|-
|-
! #
! #
! 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
| 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, 28/27
| 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
|-
|-
! #
! #
! 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
| 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
| 27/14, 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
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<nowiki/>* In 7-limit [[CWE]] tuning, octave reduced
<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 ==
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| [[9/5]]
| [[9/5]]
| 691.202
| 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  
| [[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]]
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| [[27/25]]
| [[27/25]]
| 693.352
| 693.352
| [[2/5-comma meantone|2/5 comma]]
| [[2/5-comma meantone|2/5-comma]]
|-
|-
|
|
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| {{nowrap| ''f''<sup>4</sup> − 2''f'' − 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]]
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| [[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
|-
|-
|
|
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|-
|-
|
|
|  
| 3125/2304
| 695.981
| 695.981
| [[5/18-comma meantone|5/18 comma]]
| [[5/18-comma meantone|5/18-comma]]
|-
|-
|
|
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| [[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 − φ)\11 }}
| {{nowrap| (8 − φ)\11 }}
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| 696.319
| 696.319
|
|
|-
|
| [[19/16]]
| 696.340
| As AAAA1
|-
|
| [[17/16]]
| 696.344
| As AAA7
|-
|-
|
|
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|-
|-
|
|
| [[1875/1024]]
| 1875/1024
| 696.895
| 696.895
| [[4/17-comma meantone|4/17 comma]]; 5-limit [[CEE]] tuning
| [[4/17-comma meantone|4/17-comma]]; 5-limit [[CEE]] tuning
|-
|-
|
|
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| [[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'' − 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]]
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|-
|-
|
|
|  
| 1125/1024
| 697.347
| 697.347
| [[3/14-comma meantone|3/14 comma]]
| [[3/14-comma meantone|3/14-comma]]
|-
|-
| {{nowrap|(√(10) − 2)\2}}
| {{nowrap|(√(10) − 2)\2}}
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| [[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]]
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| [[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]]
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|-
|-
|
|
|  
| {{monzo|-23 9 4}}
| 698.514
| 698.514
| [[4/25-comma meantone|4/25 comma]]
| [[4/25-comma meantone|4/25-comma]]
|-
|-
|
|
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| [[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]]
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|
|
| '''700.000'''
| '''700.000'''
| '''Upper bound of 7- and 9-odd-limit diamond monotone''', 1/12 Pythagorean comma, virtually [[1/11-comma meantone|1/11 comma]]†
| '''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
|-
|-
|
|
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