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'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the perfect octave 2/1. It does not approximate many small prime harmonics well at all but approximates 10/9, 11/8, 17/13, 20/17, and 21/13 well. But the approximations do not fit very well in a temperament (they fit better in a [[Glossary#Neji|neji]]), so [[delta-rational chord|DR]]-based interpretations may be preferred among 13edo users.
'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, so [[Delta-rational chord|DR]]-based interpretations may be preferred among 13edo users.


13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an oneirotonic (5L3s) MOS which is a compressed diatonic. A functional system for 13edo oneirotonic is provided below.
13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an [[oneirotonic]] (5L3s, LLsLLsLs) MOS which is a compressed diatonic. For Jaimbee and Inthar's functional harmony and method, see the [[oneirotonic]] page.


== One functional system for 13edo ==
== Tuning theory ==
The following system has been developed by Jaimbee and Inthar.
=== Intervals ===
{{Proposed}}
Note: The logic of [[User:Ground|ground]]'s notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX
{| class="wikitable"
|+
!Edostep
!Cents
!Interval region name
!ADIN name (Oneirotonic extension)
!Oneirotonic [https://en.xen.wiki/w/TAMNAMS TAMNAMS] name
!Oneirotonic KISS notation
!Ground's notation (on A = 440 Hz)
!26edo subset notation (on A = 440 Hz)
|-
|0
|0
|Unison
|Unison
|Perfect 0-(oneiro)step (P0oneis)
|1
|A
|A
|-
|1
|92.3
|Minor 2nd
|Minor second
|Minor 1-(oneiro)step (m1oneis)
|1# / 2b
|A# / Cb
|Ax / Bbb
|-
|2
|184.6
|Major 2nd
|Major second
|Major 1-(oneiro)step (M1oneis)
|2
|C
|B
|-
|3
|276.9
|(Sub)minor 3rd
|Minor third
|Minor 2-(oneiro)step (m2oneis)
|3
|B
|Bx / Cb
|-
|4
|369.2
|(Sub)major 3rd
|Major third
|Major 2-(oneiro)step (M2oneis)<br/>Diminished 3-(oneiro)step (d3oneis)
|3# / 4b
|B# / Db
|C#
|-
|5
|461.5
|Subfourth
|Fourth
|Perfect 3-(oneiro)step (P3oneis)
|4
|D
|Db
|-
|6
|553.8
|Ultrafourth / Infratritone
|Minor tritone
|Minor 4-(oneiro)step (m4oneis)
|5b
|Fb
|D#
|-
|7
|647.2
|Ultratritone / Infrafifth
|Major tritone
|Major 4-(oneiro)step (M4oneis)
|5
|F
|Eb
|-
|8
|738.5
|Superfifth
|Fifth
|Perfect 5-(oneiro)step (P5oneis)
|6
|E
|E# / Fbb
|-
|9
|830.8
|(Super)minor 6th
|Minor sixth
|Augmented 5-(oneiro)step (A5oneis)<br/>Minor 6-(oneiro)step (m6oneis)
|6# / 7b
|E# / Gb
|F
|-
|10
|923.1
|(Super)major 6th
|Major sixth
|Major 6-(oneiro)step (M6oneis)
|7
|G
|Fx / Gbb
|-
|11
|1015.4
|Minor 7th
|Minor seventh
|Minor 7-(oneiro)step (m7oneis)
|8b
|Xb
|G
|-
|12
|1107.7
|Major 7th
|Major seventh
|Major 7-(oneiro)step (M7oneis)
|8
|X
|Gx / Abb
|-
|13
|1200
|Octave
|Octave
|Perfect 8-(oneiro)step (P8oneis)
|1
|A
|A
|}


13edo's melodically strongest scale is the oneirotonic MOS (preserving the diatonic property of having at least 2 semitones), so it behooves us to find harmonies that work for it.
=== Prime harmonic approximations ===
{{Harmonics in ED|13|23|0}}


The most basic chords in this functional harmony system are:
=== Edostep interpretations ===
* Major +1+1 triad 0-4-7\13: A compressed major triad that sounds desaturated and somewhat bittersweet. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get a major triad. The major triad has the following important tetrad supersets:
13edo's edostep functions in the 2.9.5.21.11.13.17.19 subgroup as:
** 0-2-4-7\13: Reinforces the quasi-DR effect  with an extra tone; approximately +1+1+2.
 
** 0-4-7-10\13: A compressed dominant tetrad; approximately +1+1+1.
* 17/16
** 0-4-7-12\13
* 18/17
* Minor +1+2 triad 0-3-8\13: A bright and brooding if somewhat hollow-sounding minor triad. The important supersets are:
* 19/18
** 0-3-8-10: Approximately +1+2+1
* 20/19
** 0-3-8-12
* 21/20 (the interval between 10/9 and 7/6)
** 0-3-8-15
* 22/21
** 0-3-8-12-15: A concatenation of the minor +1+2 and major +1+1 triads.
* 26/25 (the interval between 5/4 and 13/10)
* 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
* 128/121 (the interval between 11/8 and 16/11)
 
=== Harmonic series approximations ===
13edo approximates the following harmonic series chord fairly well (x indicates notes that are harder to approximate):
 
34:36:38:40:42:x:47:x:52:55:58:61:x:68
 
This can be derived as follows:
# the quasi-13edo isoharmonic chord 5:9:13:17:21 => 17:18:x:20:21:x:x:x:26:x:x:x:x:34
# the simic sixth chord 17:20:26:29 (+1+2+1) => 17:18:x:20:21:x:x:x:26:x:29:x:x:34
# place 11/8 on harmonic 20 => 34:36:x:40:42:x:x:x:52:55:58:x:x:68
# use halfway harmonics 19 and 47 => 34:36:38:40:42:x:47:x:52:55:58:x:x:68
# 61/52 is .6c off from 3\13 => 34:36:38:40:42:x:47:x:52:55:58:61:x:68
 
Making an over-17 13edo neji thus requires you to choose those three notes:
* The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
* The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
* A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.
 
== Multiples ==
 
=== 26edo ===
:''Main article: [[26edo]]''
 
=== 39edo ===
39edo is a Supra (2.3.7.11[17 & 22]) diatonic tuning which has good 11/8 and 9/7 approximations in the mosdiatonic scale, though the 39d val (using the sharp approximation of 7/4) is required. One may favor 39edo over harder Archy tunings for the larger diatonic semitone size.
{{Harmonics in ED|39|31|0}}
 
=== 65edo ===
65edo is notable as the intersection of [[Schismic]] and [[Wurschmidt]]. It is a strong 2.3.5.11.19.23.31.47.49 system.
{{Harmonics in ED|65|47|0}}
 
===104edo===
''See [[26edo#104edo]].''
 
===130edo===
''See [[26edo#130edo]].''
 
== See also ==
* The [[Well-Tempered Triskaidecatonic Clavier]] (WT13C) project
{{Navbox EDO}}
{{Cat|Edos}}

Latest revision as of 18:34, 12 May 2026

13edo, or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, so DR-based interpretations may be preferred among 13edo users.

13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an oneirotonic (5L3s, LLsLLsLs) MOS which is a compressed diatonic. For Jaimbee and Inthar's functional harmony and method, see the oneirotonic page.

Tuning theory

Intervals

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

Note: The logic of ground's notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX

Edostep Cents Interval region name ADIN name (Oneirotonic extension) Oneirotonic TAMNAMS name Oneirotonic KISS notation Ground's notation (on A = 440 Hz) 26edo subset notation (on A = 440 Hz)
0 0 Unison Unison Perfect 0-(oneiro)step (P0oneis) 1 A A
1 92.3 Minor 2nd Minor second Minor 1-(oneiro)step (m1oneis) 1# / 2b A# / Cb Ax / Bbb
2 184.6 Major 2nd Major second Major 1-(oneiro)step (M1oneis) 2 C B
3 276.9 (Sub)minor 3rd Minor third Minor 2-(oneiro)step (m2oneis) 3 B Bx / Cb
4 369.2 (Sub)major 3rd Major third Major 2-(oneiro)step (M2oneis)
Diminished 3-(oneiro)step (d3oneis) 		3# / 4b B# / Db C#
5 461.5 Subfourth Fourth Perfect 3-(oneiro)step (P3oneis) 4 D Db
6 553.8 Ultrafourth / Infratritone Minor tritone Minor 4-(oneiro)step (m4oneis) 5b Fb D#
7 647.2 Ultratritone / Infrafifth Major tritone Major 4-(oneiro)step (M4oneis) 5 F Eb
8 738.5 Superfifth Fifth Perfect 5-(oneiro)step (P5oneis) 6 E E# / Fbb
9 830.8 (Super)minor 6th Minor sixth Augmented 5-(oneiro)step (A5oneis)
Minor 6-(oneiro)step (m6oneis) 		6# / 7b E# / Gb F
10 923.1 (Super)major 6th Major sixth Major 6-(oneiro)step (M6oneis) 7 G Fx / Gbb
11 1015.4 Minor 7th Minor seventh Minor 7-(oneiro)step (m7oneis) 8b Xb G
12 1107.7 Major 7th Major seventh Major 7-(oneiro)step (M7oneis) 8 X Gx / Abb
13 1200 Octave Octave Perfect 8-(oneiro)step (P8oneis) 1 A A

Prime harmonic approximations

Approximation of prime harmonics in 13edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) 0.0 +36.5 -17.1 -45.7 +2.5 -9.8 -12.6 -20.6 +17.9
Relative (%) 0.0 +39.5 -18.5 -49.6 +2.7 -10.6 -13.7 -22.3 +19.4
Steps
(reduced) 		13
(0) 		21
(8) 		30
(4) 		36
(10) 		45
(6) 		48
(9) 		53
(1) 		55
(3) 		59
(7)

Edostep interpretations

13edo's edostep functions in the 2.9.5.21.11.13.17.19 subgroup as:

  • 17/16
  • 18/17
  • 19/18
  • 20/19
  • 21/20 (the interval between 10/9 and 7/6)
  • 22/21
  • 26/25 (the interval between 5/4 and 13/10)
  • 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
  • 128/121 (the interval between 11/8 and 16/11)

Harmonic series approximations

13edo approximates the following harmonic series chord fairly well (x indicates notes that are harder to approximate):

34:36:38:40:42:x:47:x:52:55:58:61:x:68

This can be derived as follows:

  1. the quasi-13edo isoharmonic chord 5:9:13:17:21 => 17:18:x:20:21:x:x:x:26:x:x:x:x:34
  2. the simic sixth chord 17:20:26:29 (+1+2+1) => 17:18:x:20:21:x:x:x:26:x:29:x:x:34
  3. place 11/8 on harmonic 20 => 34:36:x:40:42:x:x:x:52:55:58:x:x:68
  4. use halfway harmonics 19 and 47 => 34:36:38:40:42:x:47:x:52:55:58:x:x:68
  5. 61/52 is .6c off from 3\13 => 34:36:38:40:42:x:47:x:52:55:58:61:x:68

Making an over-17 13edo neji thus requires you to choose those three notes:

  • The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
  • The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
  • A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.

Multiples

26edo

Main article: 26edo

39edo

39edo is a Supra (2.3.7.11[17 & 22]) diatonic tuning which has good 11/8 and 9/7 approximations in the mosdiatonic scale, though the 39d val (using the sharp approximation of 7/4) is required. One may favor 39edo over harder Archy tunings for the larger diatonic semitone size.

Approximation of prime harmonics in 39edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) 0.0 +5.7 +13.7 -15.0 +2.5 -9.8 -12.6 +10.2 -12.9 -14.2 -6.6
Relative (%) 0.0 +18.6 +44.5 -48.7 +8.2 -31.7 -41.1 +33.1 -41.9 -46.1 -21.4
Steps
(reduced) 		39
(0) 		62
(23) 		91
(13) 		109
(31) 		135
(18) 		144
(27) 		159
(3) 		166
(10) 		176
(20) 		189
(33) 		193
(37)

65edo

65edo is notable as the intersection of Schismic and Wurschmidt. It is a strong 2.3.5.11.19.23.31.47.49 system.

Approximation of prime harmonics in 65edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) 0.0 -0.4 +1.4 -8.8 +2.5 +8.7 +5.8 -2.1 -0.6 +4.3 -0.4 +7.1 -4.4 +5.4 -0.9
Relative (%) 0.0 -2.3 +7.5 -47.8 +13.7 +47.1 +31.5 -11.5 -3.2 +23.1 -2.3 +38.6 -24.1 +29.3 -4.8
Steps
(reduced) 		65
(0) 		103
(38) 		151
(21) 		182
(52) 		225
(30) 		241
(46) 		266
(6) 		276
(16) 		294
(34) 		316
(56) 		322
(62) 		339
(14) 		348
(23) 		353
(28) 		361
(36)

104edo

See 26edo#104edo.

130edo

See 26edo#130edo.

See also


View • Talk • EditEqual temperaments
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Macrotonal 5 • 6 • 7 • 8 • 9 • 10 • 11
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