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39edo: Difference between revisions

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39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).
39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).
39edo is a reasonable tuning of [[triforce]] beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp.


=== Odd harmonics ===
=== Odd harmonics ===
Line 22: Line 24:


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 right-9 right-10"
{| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"
|-
|-
! rowspan="2" | Steps
! rowspan="2" | Steps
! rowspan="2" | Cents
! rowspan="2" | Cents
! rowspan="2" | Approximate ratios (2.3.5.11)
! rowspan="2" | Ratios of the<br>[[2.3.5.11 subgroup]]
! colspan="2" |Intervals of 7
! colspan="2" | Intervals of 7
! colspan="3" rowspan="2" | [[Ups and downs notation]]
! colspan="3" rowspan="2" | [[Ups and downs notation]]
! colspan="3" rowspan="2" | [[Nearest just interval]] <br>(Ratio, cents, error)
|-
|-
!Patent val
! Patent val
!Superpyth (39d) val
! 39d val
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| colspan=3 | [[1/1]]
|
|
| P1
| P1
| perfect unison
| perfect unison
| D
| D
| 1/1
| 0.00
| None
|-
|-
| 1
| 1
| 30.8
| 30.8
| [[55/54]], [[81/80]]
| [[55/54]], [[81/80]]
|''28/27,'' 64/63
| ''[[28/27]]'', [[64/63]]
|''[[36/35]],'' [[50/49]]'', [[56/55]]''
| ''[[36/35]]'', [[50/49]], ''[[56/55]]''
| ^1, <br>vm2
| ^1, <br>vm2
| up unison, <br>downminor 2nd
| up unison, <br>downminor 2nd
| ^D, <br>vEb
| ^D, <br>vEb
| 57/56
| 30.64
| +0.1271
|-
|-
| 2
| 2
| 61.5
| 61.5
| [[33/32]]
| [[33/32]]
|''21/20,'' 36/35
| ''[[21/20]]'', [[36/35]]
|[[28/27]], ''[[49/48]]''
| [[28/27]], ''[[49/48]]''
| m2
| m2
| minor 2nd
| minor 2nd
| Eb
| Eb
| 29/28
| 60.75
| +0.7872
|-
|-
| 3
| 3
| 92.3
| 92.3
| ''[[16/15]]'', ''[[25/24]]''
| ''[[16/15]]'', ''[[25/24]]''
|''50/49''
| ''[[50/49]]''
|[[21/20]]
| [[21/20]]
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
| 39/37
| 91.14
| +1.1691
|-
|-
| 4
| 4
Line 86: Line 73:
|
|
|
|
|[[15/14]]
| [[15/14]]
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| 44/41
| 122.26
| +0.8214
|-
|-
| 5
| 5
| 153.8
| 153.8
| [[11/10]], [[12/11]]
| [[11/10]], [[12/11]]
|''15/14''
| ''[[15/14]]''
|
|
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
| vvE
| vvE
| 35/32
| 155.14
| -1.2934
|-
|-
| 6
| 6
Line 114: Line 95:
| downmajor 2nd
| downmajor 2nd
| vE
| vE
| 10/9
| 182.40
| +2.2117
|-
|-
| 7
| 7
Line 122: Line 100:
| [[9/8]]
| [[9/8]]
|
|
|''[[8/7]]''
| ''[[8/7]]''
| M2
| M2
| major 2nd
| major 2nd
| E
| E
| 17/15
| 216.69
| -1.3021
|-
|-
| 8
| 8
| 246.2
| 246.2
|
|
|8/7, ''7/6''
| [[8/7]], ''[[7/6]]''
|[[81/70]]
| [[81/70]]
| ^M2, <br>vm3
| ^M2, <br>vm3
| upmajor 2nd, <br>downminor 3rd
| upmajor 2nd, <br>downminor 3rd
| ^E, <br>vF
| ^E, <br>vF
| 15/13
| 247.74
| -1.5873
|-
|-
| 9
| 9
| 276.9
| 276.9
|
|
|''81/70''
| ''[[81/70]]''
|[[7/6]]
| [[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
| 27/23
| 277.59
| -0.6676
|-
|-
| 10
| 10
Line 162: Line 131:
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| 43/36
| 307.61
| +0.0846
|-
|-
| 11
| 11
Line 174: Line 140:
| dupminor 3rd
| dupminor 3rd
| ^^F
| ^^F
| 17/14
| 336.13
| +2.3320
|-
|-
| 12
| 12
Line 186: Line 149:
| dudmajor 3rd
| dudmajor 3rd
| vvF#
| vvF#
| 26/21
| 369.75
| -0.5160
|-
|-
| 13
| 13
| 400.0
| 400.0
| [[5/4]]
| [[5/4]]
|''14/11''
| ''[[14/11]]''
|
|
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| 34/27
| 399.09
| +0.9096
|-
|-
| 14
| 14
| 430.8
| 430.8
|
|
|''35/27''
| ''[[35/27]]''
|[[9/7]], [[14/11]]
| [[9/7]], [[14/11]]
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
| 41/32
| 429.06
| +1.7068
|-
|-
| 15
| 15
| 461.5
| 461.5
|
|
|''9/7''
| ''[[9/7]]''
|[[35/27]]
| [[35/27]]
| v4
| v4
| down 4th
| down 4th
| vG
| vG
| 30/23
| 459.99
| +1.5441
|-
|-
| 16
| 16
Line 234: Line 185:
| perfect 4th
| perfect 4th
| G
| G
| 85/64
| 491.27
| +1.0386
|-
|-
| 17
| 17
Line 246: Line 194:
| up 4th
| up 4th
| ^G
| ^G
| 23/17
| 523.32
| -0.2420
|-
|-
| 18
| 18
| 553.8
| 553.8
| [[11/8]]
| [[11/8]]
|''7/5''
| ''[[7/5]]''
|
|
| ^^4
| ^^4
| dup 4th
| dup 4th
| ^^G
| ^^G
| 11/8
| 551.32
| +2.5283
|-
|-
| 19
| 19
Line 266: Line 208:
|
|
|
|
|[[7/5]]
| [[7/5]]
| vvA4, <br>^d5
| vvA4, <br>^d5
| dudaug 4th, <br>updim 5th
| dudaug 4th, <br>updim 5th
| vvG#, <br>^Ab
| vvG#, <br>^Ab
| 7/5
| 582.51
| +2.1032
|-
|-
| 20
| 20
Line 278: Line 217:
|
|
|
|
|[[10/7]]
| [[10/7]]
| vA4, <br>^^d5
| vA4, <br>^^d5
| downaug 4th, <br>dupdim 5th
| downaug 4th, <br>dupdim 5th
| vG#, <br>^^Ab
| vG#, <br>^^Ab
| 10/7
| 617.49
| -2.1032
|-
|-
| 21
| 21
| 646.2
| 646.2
| [[16/11]]
| [[16/11]]
|''10/7''
| ''[[10/7]]''
|
|
| vv5
| vv5
| dud 5th
| dud 5th
| vvA
| vvA
| 16/11
| 648.68
| -2.5283
|-
|-
| 22
| 22
Line 306: Line 239:
| down 5th
| down 5th
| vA
| vA
| 34/23
| 676.68
| +0.2420
|-
|-
| 23
| 23
Line 318: Line 248:
| perfect 5th
| perfect 5th
| A
| A
| 128/85
| 708.73
| -1.0386
|-
|-
| 24
| 24
| 738.5
| 738.5
|
|
|''14/9''
| ''[[14/9]]''
|[[54/35]]
| [[54/35]]
| ^5
| ^5
| up 5th
| up 5th
| A^
| A^
| 23/15
| 740.01
| -1.5441
|-
|-
| 25
| 25
| 769.2
| 769.2
|
|
|''54/35''
| ''[[54/35]]''
|[[11/7]], [[14/9]]
| [[11/7]], [[14/9]]
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
| 64/41
| 770.94
| -1.7068
|-
|-
| 26
| 26
| 800.0
| 800.0
| [[8/5]]
| [[8/5]]
|''11/7''
| ''[[11/7]]''
|
|
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| 27/17
| 800.91
| -0.9096
|-
|-
| 27
| 27
Line 366: Line 284:
| dupminor 6th
| dupminor 6th
| ^^Bb
| ^^Bb
| 21/13
| 830.25
| +0.5160
|-
|-
| 28
| 28
Line 378: Line 293:
| dudmajor 6th
| dudmajor 6th
| vvB
| vvB
| 28/17
| 863.87
| -2.3320
|-
|-
| 29
| 29
Line 390: Line 302:
| downmajor 6th
| downmajor 6th
| vB
| vB
| 72/43
| 892.39
| -0.0846
|-
|-
| 30
| 30
| 923.1
| 923.1
|
|
|''140/81''
| ''[[140/81]]''
|[[12/7]]
| [[12/7]]
| M6
| M6
| major 6th
| major 6th
| B
| B
| 46/27
| 922.41
| +0.6676
|-
|-
| 31
| 31
| 953.8
| 953.8
|
|
|7/4, ''12/7''
| [[7/4]], ''[[12/7]]''
|[[140/81]]
| [[140/81]]
| ^M6, <br>vm7
| ^M6, <br>vm7
| upmajor 6th, <br>downminor 7th
| upmajor 6th, <br>downminor 7th
| ^B, <br>vC
| ^B, <br>vC
| 26/15
| 952.26
| +1.5873
|-
|-
| 32
| 32
Line 422: Line 325:
| [[16/9]]
| [[16/9]]
|
|
|''[[7/4]]''
| ''[[7/4]]''
| m7
| m7
| minor 7th
| minor 7th
| C
| C
| 30/17
| 983.31
| +1.3021
|-
|-
| 33
| 33
Line 438: Line 338:
| upminor 7th
| upminor 7th
| ^C
| ^C
| 9/5
| 1017.60
| -2.2117
|-
|-
| 34
| 34
| 1046.2
| 1046.2
| [[11/6]], [[20/11]]
| [[11/6]], [[20/11]]
|''28/15''
| ''[[28/15]]''
|
|
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
| ^^C
| ^^C
| 64/35
| 1044.86
| +1.2934
|-
|-
| 35
| 35
Line 458: Line 352:
|
|
|
|
|[[28/15]]
| [[28/15]]
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
| 41/22
| 1077.74
| -0.8214
|-
|-
| 36
| 36
| 1107.7
| 1107.7
| ''[[15/8]]'', ''[[48/25]]''
| ''[[15/8]]'', ''[[48/25]]''
|''49/25''
| ''[[49/25]]''
|[[40/21]]
| [[40/21]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| 74/39
| 1108.86
| -1.1691
|-
|-
| 37
| 37
| 1138.5
| 1138.5
| [[64/33]]
| [[64/33]]
|35/18'', 40/21''
| [[35/18]], ''[[40/21]]''
|[[27/14]], ''[[96/49]]''
| [[27/14]], ''[[96/49]]''
| M7
| M7
| major 7th
| major 7th
| C#
| C#
| 56/29
| 1139.25
| -0.7872
|-
|-
| 38
| 38
| 1169.2
| 1169.2
| [[55/28]], [[108/55]], [[160/81]]
| [[108/55]], [[160/81]]
|63/32, ''27/14''
| [[63/32]], ''[[27/14]]''
|''[[35/18]], [[49/25]]''
| ''[[35/18]]'', [[49/25]]
| ^M7, <br>v8
| ^M7, <br>v8
| upmajor 7th, <br>down 8ve
| upmajor 7th, <br>down 8ve
| ^C#, <br>vD
| ^C#, <br>vD
| 112/57
| 1169.36
| -0.1271
|-
|-
| 39
| 39
| 1200.0
| 1200.0
| [[2/1]]
| colspan=3 | [[2/1]]
|
|
| P8
| P8
| perfect 8ve
| perfect 8ve
| D
| D
| 2/1
| 1200.00
| None
|}
|}
<nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic''


Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].
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