View source for List of just intonation intervals


This is a list of just intervals, similar to the list of [[EDO|EDOs]] and the [[list of regular temperaments]]. It exists to compile information on a number of just intonation intervals.

'''Commas should not redirect here!''' They should instead redirect to their corresponding entry on the temperament page, or if present, a page for the temperament itself.

{{Adv|The formula for "good edos" is the edos that satisfy, for interval x in cents and edo n: 
<br> abs(round(xn/1200)-xn/1200)*sqrt(12/n)<(1/16)  for n < or equal to 31
<br> abs(round(xn/1200)-xn/1200)*((n^2)/1024)*sqrt(12/31)<(1/16)  for n > 31
<br><br> EDOs below 5 are excluded.}}

== 3-limit ==
{| class="wikitable"
|+(MOS) diatonic intervals
!Fifthspan
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
| -7
|4096/2187
|{{Cents from ratio|ratio=4096/2187}}
|10, 11, 21, 22, 31, 32, 42, 53, 74, 95, 190
|Diatonic diminished octave
|
|-
| -6
|1024/729
|{{Cents from ratio|ratio=1024/729}}
|47, 49, 51, 53, 102
|Diatonic diminished fifth
|Found in a diminished triad. A stack of two diatonic minor thirds
|-
| -5
|256/243
|{{Cents from ratio|ratio=256/243}}
|13, 14, 26, 27, 40, 53, 80, 93, 133, 306
|[[Diatonic semitone]]
|Small step of the diatonic scale.
|-
| -4
|128/81
|{{Cents from ratio|ratio=128/81}}
|6, 44, 47, 50, 53, 56, 103
|Diatonic minor sixth
|
|-
| -3
|32/27
|{{Cents from ratio|ratio=32/27}}
|8, 12, 33, 37, 41, 45, 49, 53, 57, 102
|Diatonic minor third
|Middle interval in a Pythagorean minor chord
|-
| -2
|16/9
|{{Cents from ratio|ratio=16/9}}
|6, 12, 18, 24, 29, 30, 35, 41, 47, 53, 59, 100, 153
|Diatonic minor seventh
|
|-
| -1
|4/3
|{{Cents from ratio|ratio=4/3}}
|12, 17, 24, 29, 36, 41, 53, 94, 200
|[[Perfect fourth]]
|
|-
|0
|1/1
|{{Cents from ratio|ratio=1/1}}
|(All)
|Perfect unison
|Represents a multiplication by 1, i.e. no change in pitch
|-
|'''+1'''
|'''3/2'''
|'''{{Cents from ratio|ratio=3/2}}'''
|'''12, 17, 24, 29, 36, 41, 53, 94, 200'''
|[[Perfect fifth|'''Perfect fifth''']]
|'''Generator of Pythagorean tuning, most consonant interval within the octave after 2/1 itself'''
|-
| +2
|9/8
|{{Cents from ratio|ratio=9/8}}
|6, 12, 18, 24, 29, 30, 35, 41, 47, 53, 59, 100, 153
|[[Diatonic major second]]
|Large step of the diatonic scale.
|-
| +3
|27/16
|{{Cents from ratio|ratio=27/16}}
|8, 12, 33, 37, 41, 45, 49, 53, 57, 102
|Diatonic major sixth
|
|-
| +4
|81/64
|{{Cents from ratio|ratio=81/64}}
|6, 44, 47, 50, 53, 56, 103
|Diatonic major third
|Middle interval in a Pythagorean major chord
|-
| +5
|243/128
|{{Cents from ratio|ratio=243/128}}
|13, 14, 26, 27, 40, 53, 80, 93, 133, 306
|Diatonic major seventh
|
|-
| +6
|729/512
|{{Cents from ratio|ratio=729/512}}
|47, 49, 51, 53, 102
|Diatonic augmented fourth
|Stack of 3 tones (tritone)
|-
| +7
|2187/2048
|{{Cents from ratio|ratio=2187/2048}}
|10, 11, 21, 22, 31, 32, 42, 53, 74, 95, 190
|[[Chromatic semitone]]
|Chroma of the diatonic scale.
|}

== 5-limit ==
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|128/125
|41.1
|27, 28, 29, 30, 31, 32, 58, 59, 117, 146, 292
|augmented diesis
|
|-
|25/24
|70.7
|16, 17, 18, 33, 34, 35, 51, 68, 85, 102
|
|
|-
|135/128
|92.2
|13, 25, 26, 27, 39, 52, 65, 78, 91, 104
|major chroma
|
|-
|16/15
|111.7
|11, 21, 22, 32, 33, 43, 54, 86
|classical diatonic semitone
|
|-
|27/25
|133.2
|9, 18, 27, 36, 45, 54, 63, 72, 81, 90
|
|
|-
|10/9
|182.4
|13, 20, 26, 33, 46, 79, 125, 250
|grave whole tone
|
|-
|75/64
|274.6
|13, 22, 26, 31, 35, 48, 70, 83, 118
|classical subminor third
|Also an augmented second.
|-
|6/5
|315.6
|15, 19, 23, 34, 38, 42, 57, 76
|classical minor third
|
|-
|'''5/4'''
|'''386.3'''
|'''22, 25, 28, 31, 34, 56, 59, 87, 146, 292'''
|'''classical major third'''
|
|-
|32/25
|427.4
|14, 17, 28, 31, 42, 45, 59, 73, 87, 146, 219, 292
|classical supermajor third
|Also a diminished fourth.
|-
|27/20
|519.6
|7, 14, 16, 23, 30, 37, 44, 60, 67, 97, 194
|acute fourth
|
|-
|45/32
|590.2
|59, 61, 63, 122
|classical narrow tritone
|
|-
|64/45
|609.8
|59, 61, 63, 122
|classical wide tritone
|
|-
|40/27
|680.4
|7, 14, 16, 23, 30, 37, 44, 60, 67, 97, 194
|grave fifth
|
|-
|25/16
|772.6
|14, 17, 28, 31, 42, 45, 59, 73, 87, 146, 219, 292
|classical subminor sixth
|
|-
|8/5
|813.7
|22, 25, 28, 31, 34, 56, 59, 87, 146, 292
|classical minor sixth
|
|-
|5/3
|884.4
|15, 19, 23, 34, 38, 42, 57, 76
|classical major sixth
|
|-
|128/75
|925.4
|13, 22, 26, 31, 35, 48, 70, 83, 118
|classical supermajor sixth
|
|-
|9/5
|1017.6
|13, 20, 26, 33, 46, 79, 125, 250
|
|
|}

== 7-limit ==

=== 2.3.7 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|49/48
|35.7
|31, 32, 33, 34, 35, 36, 67, 101, 168
|interseptimal diesis
|
|-
|28/27
|63
|18, 19, 20, 37, 38, 39, 57, 76
|
|
|-
|8/7
|231.2
|5, 21, 26, 31, 36, 52, 57, 83, 109, 218
|septimal supermajor second
|
|-
|7/6
|266.9
|9, 18, 27, 36, 45, 54, 63, 72
|septimal subminor third
|
|-
|9/7
|435.1
|11, 22, 25, 33, 36, 44, 47, 58, 69, 80, 91, 171
|septimal supermajor third
|
|-
|21/16
|470.8
|5, 18, 23, 28, 33, 46, 51, 56, 79, 130, 209
|septimal subfourth
|
|-
|32/21
|729.2
|5, 18, 23, 28, 33, 46, 51, 56, 79, 130, 209
|septimal superfifth
|
|-
|14/9
|764.9
|11, 22, 25, 33, 36, 44, 47, 58, 69, 80, 91, 171
|septimal subminor sixth
|
|-
|12/7
|933.1
|9, 18, 27, 36, 45, 54, 63, 72
|septimal supermajor sixth
|
|-
|'''7/4'''
|'''968.8'''
|'''5, 21, 26, 31, 36, 52, 57, 83, 109, 218'''
|'''[[harmonic seventh]]'''
|
|-
|27/14
|1137
|18, 19, 20, 37, 38, 39, 57, 76
|
|
|}

=== 2.3.5.7 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|50/49
|35
|31, 32, 33, 34, 35, 36, 68, 69, 103, 137, 240
|jubilisma
|
|-
|36/35
|48.8
|23, 24, 25, 26, 49, 50, 74, 123
|
|
|-
|21/20
|84.5
|14, 15, 28, 29, 42, 43, 57, 71, 142, 213
|
|
|-
|15/14
|119.4
|10, 20, 30, 31, 40, 50, 201
|septimal major semitone
|
|-
|35/32
|155.1
|8, 15, 16, 23, 31, 39, 54, 62, 85, 116, 147, 294
|septimal neutral second
|
|-
|28/25
|196.2
|6, 12, 18, 24, 25, 30, 31, 37, 43, 49, 55, 61, 104, 159, 263
|
|
|-
|60/49
|350.6
|7, 17, 24, 31, 34, 41, 48, 65, 89
|septimal artoneutral third
|
|-
|49/40
|351.3
|17, 24, 31, 34, 41, 58, 82
|septimal tendoneutral third
|
|-
|63/50
|400.1
|6, 9, 12, 15, 18, ... 99, 102, 105
|
|
|-
|35/27
|449.3
|8, 16, 24, 32, 40, 48, 56, 211
|septimal ultramajor third
|
|-
|64/49
|462.3
|13, 18, 26, 31, 39, 44, 52, 109, 122, 244
|
|
|-
|48/35
|546.8
|11, 22, 24, 33, 35, 44, 46, 57, 68, 79, 90
|septimal neutral fourth
|
|-
|7/5
|582.5
|29, 31, 33, 35, 37, 68, 70, 103
|septimal narrow tritone
|
|-
|10/7
|617.5
|29, 31, 33, 35, 37, 68, 70, 103
|septimal wide tritone
|
|-
|35/24
|653.2
|11, 22, 24, 33, 35, 44, 46, 57, 68, 79, 90
|septimal neutral fifth
|
|-
|49/32
|737.7
|13, 18, 26, 31, 39, 44, 52, 109, 122, 244
|
|
|-
|54/35
|750.7
|8, 16, 24, 32, 40, 48, 56, 211
|septimal inframinor sixth
|
|-
|100/63
|799.9
|6, 9, 12, 15, 18, ... 99, 102, 105
|
|
|-
|25/14
|1003.8
|6, 12, 18, 24, 25, 30, 31, 37, 43, 49, 55, 61, 104, 159, 263
|
|
|-
|64/35
|1044.9
|8, 15, 16, 23, 31, 39, 54, 62, 85, 116, 147, 294
|septimal neutral seventh
|
|-
|35/18
|1151.2
|23, 24, 25, 26, 49, 50, 74, 123
|
|
|}

== 11-limit ==

=== 2.3.5.11 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|11/8
|551.3
|11, 13, 24, 26, 35, 37, 50, 61, 74, 111
|
|
|-
|11/9
|347.4
|7, 14, 21, 24, 31, 38, 45, 76, 114, 152
|
|
|-
|11/10
|165
|7, 15, 22, 29, 36, 44, 51, 58, 80, 160, 240
|
|
|-
|11/6
|1049.4
|8, 16, 24, 32, 40, 48, 56, 247
|
|
|-
|12/11
|150.6
|8, 16, 24, 32, 40, 48, 56, 247
|
|
|-
|16/11
|648.7
|11, 13, 24, 26, 35, 37, 50, 61, 74, 111
|
|
|-
|18/11
|852.6
|7, 14, 21, 24, 31, 38, 45, 76, 114, 152
|
|
|-
|20/11
|1035
|7, 15, 22, 29, 36, 44, 51, 58, 80, 160, 240
|
|
|-
|27/22
|354.5
|10, 17, 27, 34, 37, 44, 61, 88, 132
|
|
|-
|22/15
|663
|9, 18, 20, 27, 29, 38, 47, 67, 76, 181
|
|
|-
|15/11
|537
|9, 18, 20, 27, 29, 38, 47, 67, 76, 181
|
|
|-
|33/32
|53.3
|21, 22, 23, 24, 44, 45, 46, 68, 90, 135
|
|
|-
|33/25
|480.6
|5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55
|
|
|-
|33/20
|867
|11, 18, 25, 29, 36, 47, 54, 191
|
|
|-
|44/27
|845.5
|10, 17, 27, 34, 37, 44, 61, 88, 132
|
|
|-
|25/22
|221.3
|11, 16, 22, 27, 32, 33, 38, 49, 65, 76, 103, 141
|
|
|-
|44/25
|978.7
|11, 16, 22, 27, 32, 33, 38, 49, 65, 76, 103, 141
|
|
|-
|55/54
|31.8
|35, 36, 37, 38, 39, 40, 75, 76, 113, 151
|
|
|}

=== 2.3.5.7.11 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|44/35
|396.2
|6, 9, 12, 15, 18, 21, 24, 27, 30, 103, 106
|
|
|-
|66/35
|1098.1
|12, 23, 24, 35, 36, 47, 59, 106
|
|
|-
|35/33
|101.9
|12, 23, 24, 35, 36, 47, 59, 106
|
|
|-
|33/28
|284.4
|17, 21, 25, 34, 38, 42, 59, 76, 135, 173
|
|
|-
|21/11
|1119.5
|14, 15, 29, 30, 31, 44, 45, 60, 149, 164
|
|
|-
|22/21
|80.5
|14, 15, 29, 30, 31, 44, 45, 60, 149, 164
|
|
|-
|14/11
|417.5
|20, 23, 26, 29, 43, 46, 69, 92
|
|
|-
|11/7
|782.5
|20, 23, 26, 29, 43, 46, 69, 92
|
|
|}

== 13-limit ==

=== 2.3.5.7.13 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|
|
|
|
|
|}

=== 2.3.5.7.11.13 ===
{| class="wikitable"
!Ratio
!Cents
!Good edos
!Name
!Notes
|-
|
|
|
|
|
|}

== Higher limits ==