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== Intervals ==
== Intervals ==
{{main| 31edo/Individual degrees }}


{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
Line 94: Line 95:
! colspan="3" | Extended pythagorean notation
! colspan="3" | Extended pythagorean notation
|-
|-
|0
| 0
|0.00
| 0.00
|1/1
| 1/1
|P1
| P1
|perfect unison
| perfect unison
|D
| D
|P1
| P1
|perfect unison
| perfect unison
|D
| D
|-
|-
| 1
| 1
| 38.71
| 38.71
| 45/44, 49/48, 46/45, 128/125, 36/35
| 45/44, 49/48, 46/45, 128/125, 36/35
|^1, d2
| ^1, d2
|up-unison, dim 2nd
| up-unison, dim 2nd
|^D, Ebb
| ^D, Ebb
|d2
| d2
|dim 2nd
| dim 2nd
|Ebb
| Ebb
|-
|-
| 2
| 2
| 77.42
| 77.42
|25/24, 21/20, 22/21, 23/22
| 25/24, 21/20, 22/21, 23/22
|A1, vm2
| A1, vm2
|aug 1sn, downminor 2nd
| aug 1sn, downminor 2nd
|D#, vEb
| D#, vEb
|A1
| A1
|aug 1sn
| aug 1sn
|D#
| D#
|-
|-
| 3
| 3
| 116.13
| 116.13
|15/14, 16/15
| 15/14, 16/15
|m2
| m2
|minor 2nd
| minor 2nd
|Eb
| Eb
|m2
| m2
|minor 2nd
| minor 2nd
|Eb
| Eb
|-
|-
| 4
| 4
| 154.84
| 154.84
|12/11, 11/10, 35/32
| 12/11, 11/10, 35/32
|~2
| ~2
|mid 2nd
| mid 2nd
|vE
| vE
|AA1, dd3
| AA1, dd3
|double-aug 1sn, double-dim 3rd
| double-aug 1sn, double-dim 3rd
|Dx, Fbb
| Dx, Fbb
|-
|-
| 5
| 5
| 193.55
| 193.55
|9/8, 10/9, 19/17, 28/25
| 9/8, 10/9, 19/17, 28/25
|M2
| M2
|major 2nd
| major 2nd
|E
| E
|M2
| M2
|major 2nd
| major 2nd
|E
| E
|-
|-
| 6
| 6
| 232.26
| 232.26
|8/7, 144/125
| 8/7, 144/125
|^M2
| ^M2
|upmajor 2nd
| upmajor 2nd
|^E
| ^E
|d3
| d3
|dim 3rd
| dim 3rd
|Fb
| Fb
|-
|-
| 7
| 7
| 270.97
| 270.97
|7/6, 75/64
| 7/6, 75/64
|vm3
| vm3
|downminor 3rd
| downminor 3rd
|vF
| vF
|A2
| A2
|aug 2nd
| aug 2nd
|E#
| E#
|-
|-
| 8
| 8
| 309.68
| 309.68
|6/5, 25/21
| 6/5, 25/21
|m3
| m3
|minor 3rd
| minor 3rd
|F
| F
|m3
| m3
|minor 3rd
| minor 3rd
|F
| F
|-
|-
| 9
| 9
| 348.39
| 348.39
|11/9, 27/22, 16/13, 60/49, 49/40
| 11/9, 27/22, 16/13, 60/49, 49/40
|~3
| ~3
|mid 3rd
| mid 3rd
|^F
| ^F
|AA2, dd4
| AA2, dd4
|double-aug 2nd, double-dim 4th
| double-aug 2nd, double-dim 4th
|Ex, Gbb
| Ex, Gbb
|-
|-
| 10
| 10
| 387.10
| 387.10
|5/4
| 5/4
|M3
| M3
|major 3rd
| major 3rd
|F#
| F#
|M3
| M3
|major 3rd
| major 3rd
|F#
| F#
|-
|-
| 11
| 11
| 425.81
| 425.81
|9/7, 14/11, 23/18, 32/25
| 9/7, 14/11, 23/18, 32/25
|^M3
| ^M3
|upmajor 3rd
| upmajor 3rd
|^F#
| ^F#
|d4
| d4
|dim 4th
| dim 4th
|Gb
| Gb
|-
|-
| 12
| 12
| 464.52
| 464.52
|21/16, 13/10, 17/13, 125/96
| 21/16, 13/10, 17/13, 125/96
|v4
| v4
|down-4th
| down-4th
|vG
| vG
|A3
| A3
|aug 3rd
| aug 3rd
|Fx
| Fx
|-
|-
| 13
| 13
| 503.23
| 503.23
|4/3
| 4/3
|P4
| P4
|perfect 4th
| perfect 4th
|G
| G
|P4
| P4
|perfect 4th
| perfect 4th
|G
| G
|-
|-
| 14
| 14
| 541.94
| 541.94
|11/8, 15/11, 26/19
| 11/8, 15/11, 26/19
|^4, ~4
| ^4, ~4
|up-4th, mid 4th
| up-4th, mid 4th
|^G
| ^G
|AA3, dd5
| AA3, dd5
|double-aug 3rd, double-dim 5th
| double-aug 3rd, double-dim 5th
|Fx#, Abb
| Fx#, Abb
|-
|-
| 15
| 15
| 580.65
| 580.65
|7/5, 45/32, 25/18
| 7/5, 45/32, 25/18
|A4, vd5
| A4, vd5
|aug 4th, downdim 5th
| aug 4th, downdim 5th
|G#, vAb
| G#, vAb
|A4
| A4
|aug 4th
| aug 4th
|G#
| G#
|-
|-
|16
| 16
|619.35
| 619.35
|10/7, 64/45, 36/25
| 10/7, 64/45, 36/25
|^A4, d5
| ^A4, d5
|upaug 4th, dim 5th
| upaug 4th, dim 5th
|^G#, Ab
| ^G#, Ab
|d5
| d5
|dim 5th
| dim 5th
|Ab
| Ab
|-
|-
|17
| 17
|658.06
| 658.06
|16/11, 22/15, 19/13
| 16/11, 22/15, 19/13
|v5, ~5
| v5, ~5
|down-5th, mid 5th
| down-5th, mid 5th
|vA
| vA
|AA4, dd6
| AA4, dd6
|double-aug 4th, double-dim 6th
| double-aug 4th, double-dim 6th
|Gx, Bbbb
| Gx, Bbbb
|-
|-
|18
| 18
|696.77
| 696.77
|3/2
| 3/2
|P5
| P5
|perfect 5th
| perfect 5th
|A
| A
|P5
| P5
|perfect 5th
| perfect 5th
|A
| A
|-
|-
|19
| 19
|735.48
| 735.48
|32/21, 20/13, 26/17, 192/125
| 32/21, 20/13, 26/17, 192/125
|^5
| ^5
|up-5th
| up-5th
|^A
| ^A
|d6
| d6
|dim 6th
| dim 6th
|Bbb
| Bbb
|-
|-
|20
| 20
|774.19
| 774.19
|14/9, 11/7, 25/16
| 14/9, 11/7, 25/16
|vm6
| vm6
|downminor 6th
| downminor 6th
|vBb
| vBb
|A5
| A5
|aug 5th
| aug 5th
|A#
| A#
|-
|-
|21
| 21
|812.90
| 812.90
|8/5
| 8/5
|m6
| m6
|minor 6th
| minor 6th
|Bb
| Bb
|m6
| m6
|minor 6th
| minor 6th
|Bb
| Bb
|-
|-
|22
| 22
|851.61
| 851.61
|18/11, 44/27, 13/8, 49/30, 80/49
| 18/11, 44/27, 13/8, 49/30, 80/49
|~6
| ~6
|mid 6th
| mid 6th
|vB
| vB
|AA5, dd7
| AA5, dd7
|double-aug 5th, double-dim 7th
| double-aug 5th, double-dim 7th
|Ax, Cbb
| Ax, Cbb
|-
|-
|23
| 23
|890.32
| 890.32
|5/3, 42/25
| 5/3, 42/25
|M6
| M6
|major 6th
| major 6th
|B
| B
|M6
| M6
|major 6th
| major 6th
|B
| B
|-
|-
|24
| 24
|929.03
| 929.03
|12/7, 128/75
| 12/7, 128/75
|^M6
| ^M6
|upmajor 6th
| upmajor 6th
|^B
| ^B
|d7
| d7
|dim 7th
| dim 7th
|Cb
| Cb
|-
|-
|25
| 25
|967.74
| 967.74
|7/4, 125/72
| 7/4, 125/72
|vm7
| vm7
|downminor 7th
| downminor 7th
|vC
| vC
|A6
| A6
|aug 6th
| aug 6th
|B#
| B#
|-
|-
|26
| 26
|1006.45
| 1006.45
|16/9, 9/5, 34/19, 25/14
| 16/9, 9/5, 34/19, 25/14
|m7
| m7
|minor 7th
| minor 7th
|C
| C
|m7
| m7
|minor 7th
| minor 7th
|C
| C
|-
|-
|27
| 27
|1045.16
| 1045.16
|11/6, 20/11, 64/35
| 11/6, 20/11, 64/35
|~7
| ~7
|mid 7th
| mid 7th
|^C
| ^C
|AA6, dd8
| AA6, dd8
|double-aug 6th, double-dim 8ve
| double-aug 6th, double-dim 8ve
|Bx, Dbb
| Bx, Dbb
|-
|-
|28
| 28
|1083.87
| 1083.87
|28/15, 15/8
| 28/15, 15/8
|M7
| M7
|major 7th
| major 7th
|C#
| C#
|M7
| M7
|major 7th
| major 7th
|C#
| C#
|-
|-
|29
| 29
|1122.58
| 1122.58
|48/25, 40/21, 21/11, 44/23
| 48/25, 40/21, 21/11, 44/23
|^M7
| ^M7
|upmajor 7th
| upmajor 7th
|^C#
| ^C#
|d8
| d8
|dim 8ve
| dim 8ve
|Db
| Db
|-
|-
|30
| 30
|1161.29
| 1161.29
|88/45, 96/49, 45/23, 125/64, 35/18
| 88/45, 96/49, 45/23, 125/64, 35/18
|v8
| v8
|down-8ve
| down-8ve
|vD
| vD
|A7
| A7
|aug 7th
| aug 7th
|Cx
| Cx
|-
|-
|31
| 31
|1200.00
| 1200.00
|2/1
| 2/1
|P8
| P8
|perfect 8ve
| perfect 8ve
|D
| D
|P8
| P8
|perfect 8ve
| perfect 8ve
|D
| D
|}
|}


Line 510: Line 511:
| C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list of chords, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list of chords, see [[Ups and Downs Notation #Chords and Chord Progressions]].


== Notations ==
== Notations ==


From the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:


[[File:31edo Sagittal.png|800px]]
[[File:31edo Sagittal.png|800px]]


== Just approximation ==
== Just approximation ==
=== Selected just intervals ===
=== Selected just intervals ===
==== 15-odd-limit interval mappings ====
==== 15-odd-limit interval mappings ====


Line 1,012: Line 1,011:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
[[List of 31et rank two temperaments by badness]]
* [[List of 31et rank two temperaments by badness]]
 
* [[List of edo-distinct 31et rank two temperaments]]
[[List of edo-distinct 31et rank two temperaments]]


{| class="wikitable"
{| class="wikitable"
Line 1,106: Line 1,104:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Overtones in "Mode 8":
| Overtones in "Mode 8":
| | 8
| 8
| | 9
| 9
| | 10
| 10
| | 11
| 11
| | 12
| 12
| | 13
| 13
| | 14
| 14
| | 15
| 15
| | 16
| 16
|-
|-
| | ...as JI Ratio from 1/1:
| …as JI Ratio from 1/1:
| | 1/1
| 1/1
| | 9/8
| 9/8
| | 5/4
| 5/4
| | 11/8
| 11/8
| | 3/2
| 3/2
| | 13/8
| 13/8
| | 7/4
| 7/4
| | 15/8
| 15/8
| | 2/1
| 2/1
|-
|-
| | ...in cents:
| …in cents:
| | 0
| 0
| | 203.9
| 203.9
| | 386.3
| 386.3
| | 551.3
| 551.3
| | 702.0
| 702.0
| | 840.5
| 840.5
| | 968.8
| 968.8
| | 1088.3
| 1088.3
| | 1200.0
| 1200.0
|-
|-
| | Nearest degree of 31edo:
| Nearest degree of 31edo:
| | 0
| 0
| | 5
| 5
| | 10
| 10
| | 14
| 14
| | 18
| 18
| | 22
| 22
| | 25
| 25
| | 28
| 28
| | 31
| 31
|-
|-
| | ...in cents:
| …in cents:
| | 0
| 0
| | 193.5
| 193.5
| | 387.1
| 387.1
| | 541.9
| 541.9
| | 696.8
| 696.8
| | 851.6
| 851.6
| | 967.7
| 967.7
| | 1083.9
| 1083.9
| | 1200.0
| 1200.0
|}
|}


Line 1,172: Line 1,170:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Odd overtones in "Mode 16":
| Odd overtones in "Mode 16":
| | 17
| 17
| | 19
| 19
| | 21
| 21
| | 23
| 23
| | 25
| 25
| | 27
| 27
| | 29
| 29
| | 31
| 31
|-
|-
| | ...as JI Ratio from 1/1:
| …as JI Ratio from 1/1:
| | 17/16
| 17/16
| | 19/16
| 19/16
| | 21/16
| 21/16
| | 23/16
| 23/16
| | 25/16
| 25/16
| | 27/16
| 27/16
| | 29/16
| 29/16
| | 31/16
| 31/16
|-
|-
| | ...in cents:
| …in cents:
| | 105.0
| 105.0
| | 297.5
| 297.5
| | 470.8
| 470.8
| | 628.3
| 628.3
| | 772.6
| 772.6
| | 905.9
| 905.9
| | 1029.6
| 1029.6
| | 1145.0
| 1145.0
|-
|-
| | Nearest degree of 31edo:
| Nearest degree of 31edo:
| | 3
| 3
| | 8
| 8
| | 12
| 12
| | 16
| 16
| | 20
| 20
| | 23
| 23
| | 27
| 27
| | 30
| 30
|-
|-
| | ...in cents:
| …in cents:
| | 116.1
| 116.1
| | 309.7
| 309.7
| | 464.5
| 464.5
| | 619.4
| 619.4
| | 774.2
| 774.2
| | 890.3
| 890.3
| | 1045.1
| 1045.1
| | 1161.3
| 1161.3
|}
|}


Line 1,275: Line 1,273:


== See Also  ==
== See Also  ==
=== Pedagogy ===
=== Pedagogy ===


Retrieved from "https://en.xen.wiki/w/31edo"