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The following is a list of intervals of extended [[meantone temperament]]. These intervals constitute the standard vocabulary of intervals for the Western [[common practice]] era. Here 12-EDO refers to the size of the interval in [[equal temperament|12 equal divisions of the octave temperament]], which is the most common meantone temperament of the modern era, 19-EDO to [[19 equal temperament]], 31-EDO to [[31 equal temperament]], and 50-EDO to 50 equal temperament. Note that several of the intervals for 31-EDO and 50-EDO are absent from the table.
The following is a list of intervals of extended [[meantone temperament]]. These intervals constitute the standard vocabulary of intervals for the Western [[common practice]] era. Here 12 EDO refers to the size of the interval in the temperament with [[equal temperament|12 equal divisions of the octave]], which is the most common meantone temperament in the modern era, 19 EDO to [[19 equal temperament]], 31 EDO to [[31 equal temperament]], and 50 EDO to 50 equal temperament. Note that for brevity, several of the intervals for 31 EDO and 50 EDO are omitted from the table.


[[Ross W. Duffin|R.W. Duffin]] writes:
In ''How Equal Temperament Ruined Harmony (and Why You Should Care)'', pp. 91–92, [[Ross W. Duffin]] states: "specifying that the major semitone should be 3/2 the minor semitone [a 3:2 ratio] creates a 31-note division of the octave, which, in turn, closely corresponds to extended-[[quarter-comma meantone]]... the 5:4 ratio [whose] extended-sixth-comma meantone corresponds to the 55-division... extended-fifth-comma meantone [corresponds to] the 43-division of the octave [in which the] ratio of the major to minor semitone is 4:3." The other meantone correspondencies: a 1:1 ratio producing a 12-division (1/11-comma meantone)... "2:1 [which] results in a 19-division (1/3-comma meantone)... 5:3, which results in a 50-division" (2/7-comma meantone) are derived from these statements. [Brackets added for readability.]
: "Specifying that the major semitone should be {{sfrac|3| 2 }} the minor semitone [a 3:2 ratio] creates a 31 note division of the octave, which, in turn, closely corresponds to extended-[[quarter-comma meantone]] ... the 5:4 ratio [whose] extended-sixth-comma meantone corresponds to the 55 division ... extended-fifth-comma meantone [corresponds to] the 43 division of the octave [in which the] ratio of the major to minor semitone is 4:3."
The other meantone correspondencies: a 1:1 ratio produces a 12 division
: ({{sfrac|1| 11 }} comma meantone)... "2:1 [which] results in a 19 division
: ({{sfrac|1| 3 }} comma meantone) ... 5:3, which results in a 50 division"
: ({{sfrac|2| 7 }}&nbsp;comma meantone) are derived from these statements.<ref>{{cite book |author-link=Ross W. Duffin |first=R.W. |last=Duffin |title=How Equal Temperament Ruined Harmony (and why you should care) |pages=91–92}}</ref>{{full citation|date=January 2024}}
[Brackets added for readability.]


The column of ratios gives a ratio or ratios approximated by the interval in [[septimal meantone temperament]]. An augmented interval is increased by a [[chromatic semitone]], and a diminished interval decreased.
The column of ratios gives a ratio or ratios approximated by the interval in [[septimal meantone temperament]]. An augmented interval is increased by a [[chromatic semitone]], and a diminished interval decreased.


{| class="wikitable"
{| class="wikitable"
|- style="text-align:center;background:lightgrey;"
! colspan=2|12-EDO (≈1/11c) || Quarter-<br>comma || colspan=2|19-EDO (≈1/3c) || colspan=2|31-EDO (≈1/4c) || colspan=2|50-EDO (≈2/7c) || rowspan=2|Note<br>(from C) || rowspan=2|Roman<br>No. || rowspan=2|Name || rowspan=2|Classic<br>ratios || rowspan=2|Septimal<br>ratios
|colspan=2| [[12 equal temperament|'''12&nbsp;EDO''']]<br/>(≈{{sfrac|1| 11 }} c)
! Quarter-<br/>comma
|colspan=2| [[19 equal temperament|'''19&nbsp;EDO''']]<br/>(≈{{sfrac|1| 3 }}c)
|colspan=2| [[31 equal temperament|'''31&nbsp;EDO''']]<br/>(≈{{sfrac|1| 4 }}c)
|colspan=2| [[equal temperament|'''50&nbsp;EDO''']]<br/>(≈{{sfrac|2| 7 }}c)
!rowspan=2| Note<br/>(from C)
!rowspan=2| [[musical interval|Roman<br>numeral]]
!rowspan=2| Name
!rowspan=2| Classic<br>ratios
!rowspan=2| Septimal<br/>ratios
|- style="text-align:center;background:lightgrey;"
⚫
| steps || cents || cents || steps || cents || steps || cents || steps || cents
|-
|-
|rowspan=2| {{center|0}}
⚫
! Steps || Cents || Cents || Steps || Cents || Steps || Cents || Steps || Cents
|rowspan=2| {{center|0}}
| {{center|0.00}}
| {{center|0}}
| {{center|0.00}}
| {{center|0}}
| {{center|0.00}}
| {{center|0}}
| {{center|0}}
| {{center|C}}
| {{center|[[Tonic (music)|I]]}}
| [[Unison]]
| 1:1
|-
|-
| {{center|41.06}}
| rowspan=2|{{center|0}} || rowspan=2|{{center|0}} || {{center|0.00}} || {{center|0}} || {{center|0.00}} || {{center|0}} || {{center|0.00}} || {{center|0}} || {{center|0}} || {{center|C}} || {{center|[[Tonic (music)|I]]}} || [[Unison]] || 1:1
|rowspan=2| {{center|1}}
|rowspan=2| {{center|63.16}}
| {{center|1}}
| {{center|38.71}}
| {{center|2}}
| {{center|48}}
| {{center|D{{music|bb}}}}
| {{center|{{music|bb}}II}}
| [[Diminished second]]
| 128:125
| [[Septimal quarter tone|36:35]]
|-
|-
|rowspan=2| {{center|1}}
| {{center|41.06}} || rowspan=2|{{center|1}} || rowspan=2|{{center|63.16}} || {{center|1}} || {{center|38.71}} || {{center|2}} || {{center|48}} || {{center|D{{music|bb}}}} || {{center|{{music|bb}}II}} || [[Diminished second]] || 128:125 || [[Septimal quarter tone|36:35]]
| rowspan=2|{{center|100}}
| {{center|76.05}}
| {{center|2}}
| {{center|77.42}}
| {{center|3}}
| {{center|72}}
| {{center|C{{music|#}}}}
| {{center|{{music|#}}I}}
| [[Chromatic semitone]]
| 25:24
| [[Septimal chromatic semitone|21:20]]
|-
|-
| {{center|117.11}}
| rowspan=2|{{center|1}} || rowspan=2|{{center|100}} || {{center|76.05}} || {{center|2}} || {{center|77.42}} || {{center|3}} || {{center|72}} || {{center|C{{music|#}}}} || {{center|{{music|#}}I}} || [[Chromatic semitone]] || 25:24 || [[Septimal chromatic semitone|21:20]]
| {{center|2}}
|-
| {{center|126.32}}
| {{center|117.11}} || {{center|2}} || {{center|126.32}} || {{center|3}} || {{center|116.13}} || {{center|5}} || {{center|120}} || {{center|D{{music|b}}}} || {{center|{{music|b}}II}} || [[Minor second]] || 16:15, 27:25 || [[Septimal diatonic semitone|15:14]]
| {{center|3}}
| {{center|116.13}}
| {{center|5}}
| {{center|120}}
| {{center|D{{music|b}}}}
| {{center|{{music|b}}II}}
| [[Minor second]]
| 16:15, 27:25
| [[Septimal diatonic semitone|15:14]]
|-
|-
| rowspan=2|{{center|2}} || rowspan=2|{{center|200}} || {{center|193.16}} || {{center|3}} || {{center|189.47}} || {{center|5}} || {{center|193.55}} || {{center|8}} || {{center|192}} || {{center|D}} || {{center|II}} || [[Whole tone]] || 9:8, 10:9
| rowspan=2|{{center|2}} || rowspan=2|{{center|200}} || {{center|193.16}} || {{center|3}} || {{center|189.47}} || {{center|5}} || {{center|193.55}} || {{center|8}} || {{center|192}} || {{center|D}} || {{center|II}} || [[Whole tone]] || 9:8, 10:9

Revision as of 08:25, 10 January 2024

The following is a list of intervals of extended meantone temperament. These intervals constitute the standard vocabulary of intervals for the Western common practice era. Here 12 EDO refers to the size of the interval in the temperament with 12 equal divisions of the octave, which is the most common meantone temperament in the modern era, 19 EDO to 19 equal temperament, 31 EDO to 31 equal temperament, and 50 EDO to 50 equal temperament. Note that for brevity, several of the intervals for 31 EDO and 50 EDO are omitted from the table.

R.W. Duffin writes:

"Specifying that the major semitone should be ⁠3/ 2 ⁠ the minor semitone [a 3:2 ratio] creates a 31 note division of the octave, which, in turn, closely corresponds to extended-quarter-comma meantone ... the 5:4 ratio [whose] extended-sixth-comma meantone corresponds to the 55 division ... extended-fifth-comma meantone [corresponds to] the 43 division of the octave [in which the] ratio of the major to minor semitone is 4:3."

The other meantone correspondencies: a 1:1 ratio produces a 12 division

(⁠1/ 11 ⁠ comma meantone)... "2:1 [which] results in a 19 division
(⁠1/ 3 ⁠ comma meantone) ... 5:3, which results in a 50 division"
(⁠2/ 7 ⁠ comma meantone) are derived from these statements.[1][full citation needed]

[Brackets added for readability.]

The column of ratios gives a ratio or ratios approximated by the interval in septimal meantone temperament. An augmented interval is increased by a chromatic semitone, and a diminished interval decreased.

12 EDO
(≈⁠1/ 11 ⁠ c) 		Quarter-
comma 		19 EDO
(≈⁠1/ 3 ⁠c) 		31 EDO
(≈⁠1/ 4 ⁠c) 		50 EDO
(≈⁠2/ 7 ⁠c) 		Note
(from C) 		Roman
numeral 		Name Classic
ratios 		Septimal
ratios
steps cents cents steps cents steps cents steps cents
0
0
0.00
0
0.00
0
0.00
0
0
C
I
 Unison 1:1
41.06
1
63.16
1
38.71
2
48
Ddouble flat
double flatII
 Diminished second 128:125 36:35
1
100
76.05
2
77.42
3
72
C♯
♯I
 Chromatic semitone 25:24 21:20
117.11
2
126.32
3
116.13
5
120
D♭
♭II
 Minor second 16:15, 27:25 15:14
2
200
193.16
3
189.47
5
193.55
8
192
D
II
 Whole tone 9:8, 10:9
234.22
4
252.63
6
232.26
10
240
Edouble flat
double flatIII
 Diminished third 144:125 8:7
3
300
269.21
7
270.97
11
264
D♯
♯II
 Augmented second 75:64, 125:108 7:6
310.26
5
315.79
8
309.68
13
312
E♭
♭III
 Minor third 6:5, 32:27
4
400
386.31
6
378.95
10
387.10
16
384
E
III
 Major third 5:4
427.37
7
442.11
11
425.81
18
432
F♭
♭IV
 Diminished fourth 32:25 9:7
5
500
462.36
12
464.52
19
456
E♯
♯III
 Augmented third 125:96 21:16
503.42
8
505.26
13
503.23
21
504
F
IV
 Perfect fourth 4:3, 27:20
6
600
579.47
9
568.42
15
580.65
24
576
F♯
♯IV
 Augmented fourth 25:18, 45:32 7:5
620.53
10
631.58
16
619.35
26
624
G♭
♭V
 Diminished fifth 36:25, 64:45 10:7
7
700
696.58
11
694.74
18
696.77
29
696
G
V
 Perfect fifth 3:2, 40:27
737.64
12
757.89
19
735.48
31
744
Adouble flat
double flatVI
 Diminished sixth 192:125 32:21
8
800
772.63
20
774.19
32
768
G♯
♯V
 Augmented fifth 25:16 14:9
813.69
13
821.05
21
812.90
34
816
A♭
♭VI
 Minor sixth 8:5
9
900
889.74
14
884.21
23
890.32
37
888
A
VI
 Major sixth 5:3, 27:16
930.79
15
947.37
24
929.03
39
936
Bdouble flat
double flatVII
 Diminished seventh 128:75, 216:125 12:7
10
1000
965.78
25
967.74
40
960
A♯
♯VI
 Augmented sixth 125:72 7:4
1006.84
16
1010.53
26
1006.45
42
1008
B♭
♭VII
 Minor seventh 9:5, 16:9
11
1100
1082.89
17
1073.68
28
1083.87
45
1080
B
VII
 Major seventh 15:8, 50:27 28:15
1123.95
18
1136.84
29
1122.58
47
1128
C♭
♭VIII
 Diminished octave 48:25 40:21
12
1200
1158.94
30
1161.29
48
1152
B♯
♯VII
 Augmented seventh 125:64 35:18
1200.00
19
1200.00
31
1200.00
50
1200
C
VIII
 Octave 2:1

See also

  1. ^ Duffin, R.W. How Equal Temperament Ruined Harmony (and why you should care). pp. 91–92.
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