Meantone: Difference between revisions

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Line 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)

| ~~Generator~~ = 3/2

| Edo join 1 = 12 | Edo join 2 = 19

| Mapping = 1; 1 4 10

| Generators = 3/2

| Generators tuning = 696.7

| Optimization method = CWE

| MOS scales = [[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]]

| Pergen = (P8, P5)

| Color name = Guti

~~| Edo join 1 = 12 | Edo join 2 = 19~~

~~| Optimization method = CWE~~

~~| Generator tuning = 696.7~~

~~| MOS scales = [[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]]~~

| Odd limit 1 = 5 | Mistuning 1 = 5.4 | Complexity 1 = 5

| Odd limit 2 = 9 | Mistuning 2 = 10.8 | Complexity 2 = 12

}}

'''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.

Line 33:

'''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]].

'''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 ===

Line 56:

{| 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*

! class="unsortable" | 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

Line 116:

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</div>

{| 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*

! class="unsortable" | 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

@@ Line 7: / Line 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)
-| Generator = 3/2
+| Edo join 1 = 12 | Edo join 2 = 19
 | 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)
 | Color name = Guti
-| Edo join 1 = 12 | Edo join 2 = 19
-| Optimization method = CWE
-| 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 = 5
 | Odd limit 2 = 9 | Mistuning 2 = 10.8 | Complexity 2 = 12
 }}
-'''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.
@@ Line 33: / Line 33: @@
 {{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 (+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]].
+'''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 ===
@@ Line 56: / Line 56: @@
 <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*
 ! class="unsortable" | 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
@@ Line 116: / Line 131: @@
 </div>
 <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*
 ! class="unsortable" | 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