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{{Problematic}}
'''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''', 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, and the JI approximations it does have do not fit very well in a temperament accessed by a particular scale like oneirotonic (they fit better in a [[Glossary#Neji|neji]]), 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, LLsLLsLs) 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.


== Tuning theory ==
== Tuning theory ==
Line 15: Line 14:
!ADIN name (Oneirotonic extension)
!ADIN name (Oneirotonic extension)
!Oneirotonic [https://en.xen.wiki/w/TAMNAMS TAMNAMS] name
!Oneirotonic [https://en.xen.wiki/w/TAMNAMS TAMNAMS] name
!Fox-Raven notation (N = 261.63 Hz)
!Oneirotonic KISS notation
!Ground's notation (on A = 440 Hz)
!Ground's notation (on A = 440 Hz)
!26edo subset notation (on A = 440 Hz)
!26edo subset notation (on A = 440 Hz)
Line 24: Line 23:
|Unison
|Unison
|Perfect 0-(oneiro)step (P0oneis)
|Perfect 0-(oneiro)step (P0oneis)
|J
|1
|A
|A
|A
|A
Line 33: Line 32:
|Minor second
|Minor second
|Minor 1-(oneiro)step (m1oneis)
|Minor 1-(oneiro)step (m1oneis)
|J# / Kb
|1# / 2b
|A# / Cb
|A# / Cb
|Ax / Bbb
|Ax / Bbb
Line 42: Line 41:
|Major second
|Major second
|Major 1-(oneiro)step (M1oneis)
|Major 1-(oneiro)step (M1oneis)
|K
|2
|C
|C
|B
|B
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|Minor third
|Minor third
|Minor 2-(oneiro)step (m2oneis)
|Minor 2-(oneiro)step (m2oneis)
|L
|3
|B
|B
|Bx / Cb
|Bx / Cb
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|Major third
|Major third
|Major 2-(oneiro)step (M2oneis)<br/>Diminished 3-(oneiro)step (d3oneis)
|Major 2-(oneiro)step (M2oneis)<br/>Diminished 3-(oneiro)step (d3oneis)
|L# / Mb
|3# / 4b
|B# / Db
|B# / Db
|C#
|C#
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|Fourth
|Fourth
|Perfect 3-(oneiro)step (P3oneis)
|Perfect 3-(oneiro)step (P3oneis)
|M
|4
|D
|D
|Db
|Db
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|Minor tritone
|Minor tritone
|Minor 4-(oneiro)step (m4oneis)
|Minor 4-(oneiro)step (m4oneis)
|Nb
|5b
|Fb
|Fb
|D#
|D#
Line 87: Line 86:
|Major tritone
|Major tritone
|Major 4-(oneiro)step (M4oneis)
|Major 4-(oneiro)step (M4oneis)
|N
|5
|F
|F
|Eb
|Eb
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|Fifth
|Fifth
|Perfect 5-(oneiro)step (P5oneis)
|Perfect 5-(oneiro)step (P5oneis)
|O
|6
|E
|E
|E# / Fbb
|E# / Fbb
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|Minor sixth
|Minor sixth
|Augmented 5-(oneiro)step (A5oneis)<br/>Minor 6-(oneiro)step (m6oneis)
|Augmented 5-(oneiro)step (A5oneis)<br/>Minor 6-(oneiro)step (m6oneis)
|O# / Pb
|6# / 7b
|E# / Gb
|E# / Gb
|F
|F
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|Major sixth
|Major sixth
|Major 6-(oneiro)step (M6oneis)
|Major 6-(oneiro)step (M6oneis)
|P
|7
|G
|G
|Fx / Gbb
|Fx / Gbb
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|Minor seventh
|Minor seventh
|Minor 7-(oneiro)step (m7oneis)
|Minor 7-(oneiro)step (m7oneis)
|Qb
|8b
|Xb
|Xb
|G
|G
Line 132: Line 131:
|Major seventh
|Major seventh
|Major 7-(oneiro)step (M7oneis)
|Major 7-(oneiro)step (M7oneis)
|Q
|8
|X
|X
|Gx / Abb
|Gx / Abb
Line 141: Line 140:
|Octave
|Octave
|Perfect 8-(oneiro)step (P8oneis)
|Perfect 8-(oneiro)step (P8oneis)
|J
|1
|A
|A
|A
|A
Line 150: Line 149:


=== Edostep interpretations ===
=== Edostep interpretations ===
13edo's edostep functions in the 2.5.11.13 subgroup as:
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)
* 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)
* 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
Line 157: Line 162:


=== Harmonic series approximations ===
=== Harmonic series approximations ===
13edo approximates the following harmonic series chord well (x indicates notes that are harder to approximate):
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
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:
Making an over-17 13edo neji thus requires you to choose those three notes:
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* 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.
* 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.


== Jaimbee and Inthar's functional system for 13edo ==
== Multiples ==
{{Proposed}}
The following system has been developed by Jaimbee and Inthar.
 
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. Since there are certain similarities of oneirotonic to diatonic, we can build off of these similarities to assign functions to oneirotonic degrees.
 
For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.
=== Basic chords ===
The most basic chords in this functional harmony system are:
* Major triad 0-4-7\13: A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. 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:
** 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.
** 0-4-7-12\13
* Minor +1+2 triad 0-3-8\13: A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26. The important supersets are:
** 0-3-8-10: Approximately +1+2+1.
** 0-3-8-12: Approximately +1+2+2.
** 0-3-8-11: Something like a minor 7th tetrad.
** 0-3-8-15
** 0-3-8-12-15: A concatenation of the minor +1+2 and major +1+1 triads.
* 0-5-9\13: A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
** 0-5-7-9: Approximately +2+1+1.
** 0-5-9-12: A compressed major triad on top of a subfourth.
** 0-5-9-12-15
** 0-5-7-9-12-15-17
* 0-5-7\13: Compressed sus4. Approximately +2+1.
* 0-4-8\13: "Submajor augmented" triad.
* 0-3-6\13: The most diminished-like triad.
 
=== Functional patterns ===
13edo oneiro enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("delta-rational major triad" or just "major"/"Maj") and 0-277-738-923 ("delta-rational minor tetrad" or just "minor"/"min"). One of these chords are on the root in the 6 brightest modes of oneirotonic. In the two darkest modes, I think 0-277-738-1015 or 0-738-1015-277 works well. The chord 0-277-738 will be called "minor triad" or "mintri", and 0-277-646-923 will be called "minor diminished" or "mindim".
 
[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br179hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_7_8_10_12_13%0A%5B%600_%608_3_8_10_'0%5D-%0A%5B%602_5_10_12_'2%5D-%0A%5B%603_5_7_10_'0_'3%5D-%0A%5B%605_8_13_'2_'5%5D-%0A%5B%607_10_13_'3_'7%5D-%0A%5B%608_10_12_'2_'8%5D-%0A%5B%6010_'0_'5_'7_'10%5D-%0A%5B%6012_'2_'5_'8_'12%5D-%0A%5B0_'3_8_10_'13%5D- A progression on the ascending Celephaïsian scale]
 
[https://luphoria.com/xenpaper/#(bpm%3A60)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br274.988hz%7D%0A(env%3A1811)%0A%23_0_2_3_5_6_8_10_12_13%0A%5B%600_%608_0_3_10_%270%5D-%0A%5B%602_%6010_2_5_10%5D12%0A%5B%603_0_3_6_13_%273%5D-%0A%5B%605_2_8_13_15_%278%5D%275%0A%5B%606_0_3_8_10_%276%5D-%0A%5B%608_2_5_10_%275%5D-%0A%5B%6010_0_6_8_%273%5D-%0A%5B%6012_5_8_12_%272%5D-%0A%5B0_3_8_10_12_13%5D- A progression on the ascending Melodic Mnarian scale]
 
Adding 923 and 1108 to chords works well, and for jazzy extensions one can add 185, 461, and 646 to the upper octave.
 
[https://luphoria.com/xenpaper/#(bpm%3A240)(osc%3Asawtooth2)%7B13edo%7D%0A%7Br360hz%7D%0A(env%3A1847)%0A%23_0_2_3_5_6_8_10_11~12_13%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.......13%0A%5B%60%608_%608_11%5D_%6011_2_3---6_3-2-3-5-%0A%5B%605_5_8_11_13_8%5D-_11_13----_13_'2------%0A%5B%600_%608_0%5D_3_6_8-----........%0A%5B%600_%608_0%5D_3_8_10-----.....'3'2'0%0A%5B%603_3_6_8_0_11%5D-13_%5B%602_5_8%5D-_11-_8_%5B0_10%5D-_%5B%6011_8%5D_%5B%6010_6%5D-%5B%6011_8%5D_%5B%6010_6%5D_%5B%608_5%5D%0A%5B%603_%606_%6011_3%5D-_%5B%602_5%5D-_%5B%600_3%5D-.._%0A%5B%608_%608_%6010_%6012_2%5D_%608_%6010_%6012---- A Mnarian loop with an &8 leading tone at the end]
 
[https://luphoria.com/xenpaper/#(bpm%3A30)(osc%3Asawtooth2)%0A%7Br200Hz%7D%7B13edo%7D%0A(env%3A1846)%0A%5B0_4_7_10_%272_%276_%279_%2712%5D-.._%23_Dylydian%0A%5B0_5_9_%270_%272_%274_%277_%2710_%2712%5D-.._%23_Dylathian%0A%5B0_5_8_10_%270_%272_%274_%277_%2712%5D-.._%23_Illarnekian%0A%5B0_3_8_10_%272_%275_%277_%2712%5D-.._%23_Celepha%C3%AFsian%0A%5B0_5_9_%273_%277_%2710_%2712_%27%272%5D-.._%23_Celdorian%0A%5B0_3_6_8_%2711_%272_%275_10%5D-.._%23_Mnarian%0A%5B0_3_6_8_10_%272_%275_%2712%5D-.._%23_Mnionian Some motherchords of oneiro modes]
==== Functional chords on each degree ====
Celephaisian
* 0d: minor
* 1d: minor
* 2d: major
* 3d: minor, mindim
* 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 5d: major
* 6d: minor
* 7d: minor triad, minor 4ms, minor 6ms, minor 7ms
 
==== Progressions ====
kd(maj) means the triad 0 369 646 on kd, kd(min) means the tetrad 0 277 738 923 on kd
 
Common motions:
* 0d(maj or min) → M1d(min)
* 0d(maj or min) → 4d(min) (when ending on 0d this sounds like diatonic V to I)
* 0d(maj or min) → m6d(min)
* 0d(maj) → 5d(maj), 0d(min) → 5d(maj or min) (when ending on 0d this is a "dominant to tonic" motion)
* 0d(maj) → M3d(maj)
* 2d(min) → m1d(maj) → 0d(maj) (pseudo tritone sub)
 
==== Modal harmony ====
Modes can be grouped by their functional properties.
* Dual-fifth: Illarnekian, Celephaïsian, Ultharian; Illarmixian, Ulphrygian
* Dual-fourth: Mnarian, Kadathian, Hlanithian; Mnionian, Mnaeolian
* Major pJI chord on root: Dylathian, Illarnekian; Dylydian, Illarmixian
* Minor pJI chord on root: Celephaïsian, Ultharian, Mnarian, Kadathian; Mnionian, Sardorian, Ulphrygian
* Lower leading tone: Dylathian, Illarnekian, Celephaïsian; Dylydian, Mnionian
* "Neoclassical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian; Mnionian, Mnaeolian(?)
* Upper leading tone: Kadathian, Hlanithian, Sarnathian, Sardorian, Sarlocrian
* Minor 6-mosstep: Hlanithian, Sarnathian, Mnaeolian, Sarlocrian
* 0 462 831 delta-rational chord on root: Dylathian, Dylydian, Hlanithian, Mnaeolian
* "Dorian-like", i.e. no leading tone, 5d is minor, and m7d is major: Ultharian, Mnarian
* 7d is minor: Kadathian, Hlanithian
We'll call degrees that don't have the major delta-rational or the minor delta-rational chord ''dissonant degrees'' (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).
===== Celephaïsian =====
Functional chords on each degree:
* 0d: min
* 1d: min
* 2d: Maj
* 3d: min
* 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 5d: Maj
* 6d: min
* 7d: minor triad, minor 4ms, minor 6ms, minor 7ms
The main resolving degrees (analogues to dominant in diatonic) are 3d and 5d because of their leading tones.
 
Motherchord: 0d-m2d-P5d-M6d-(M7d)-M1d-P3d-M5d-(M7d)
 
Progressions:
* 0d(min) 1d(min) 3d(min or mindim) 0d(min)
* 0d(min) 1d(min) 6d(min) 3d(min or mindim) 0d(min)
* 0d(min) 6d(min) 3d(min or mindim) 0d(min)
* 0d(min) 2d(Maj) 3d(min or mindim) 0d(min)
* 0d(min) 4d(min7) 5d(Maj)/3d(min or mindim) 0d(min)
* 0d(min) 6d(min) 5d(Maj) 0d(min)


Secondary modes:
=== 26edo ===
* 3d Ultharian
:''Main article: [[26edo]]''
* 2d Dylathian
* 5d Illarnekian


===== Ultharian =====
=== 39edo ===
* 0d: min
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.
* 1d: minor triad, minor 4ms, minor 6ms, minor 7ms
{{Harmonics in ED|39|31|0}}
* 2d: Maj
* 3d: min
* 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 5d: min
* 6d: min
* 7d: Maj


Ultharian and Mnarian often behave like Dorian because they lack a leading tone. Resolving degrees: 3d(min), 5d(min), 7d(Maj)
=== 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}}


===== Mnarian =====
===104edo===
* 0d: min
''See [[26edo#104edo]].''
* 1d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 2d: min
* 3d: min
* 4d: Maj
* 5d: min
* 6d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 7d: Maj
Resolving degrees: 3d(min), 5d(min), 7d(Maj)


===== Kadathian =====
===130edo===
* 0d: min
''See [[26edo#130edo]].''
* 1d: Maj
* 2d: min
* 3d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 4d: Maj
* 5d: min
* 6d: minor triad, minor 4ms, minor 6ms, minor 7ms
* 7d: min
Resolving degrees: 2d(min)?, 4d(Maj), 5d(min), 7d(min)
===== Hlanithian and Sarn =====
Main tonic chord is 0-3-8-11\13 (min7), works well with m6d(maj) and m7d(min7).
[[Category:13edo]]
[[Category:Method]]
[[Category:Approaches to tuning systems]]


== See also ==
* The [[Well-Tempered Triskaidecatonic Clavier]] (WT13C) project
{{Navbox EDO}}
{{Cat|Edos}}
{{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
EDOs
Macrotonal 5 • 6 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 74 • 77 • 80 • 81 • 82
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other 88cet Family
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