26edo: Difference between revisions

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{{problematic}}
'''26edo''', or 26 equal divisions of the octave, is the equal tuning featuring steps of (1200/26) ~= 46.15 cents, 26 of which stack to the perfect octave [[2/1]].
'''26edo''', or 26 equal divisions of the octave, is the equal tuning featuring steps of (1200/26) ~= 46.15 cents, 26 of which stack to the perfect octave [[2/1]].


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Similar to [[19edo]], 26edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb.
Similar to [[19edo]], 26edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb.


{| class="wikitable"
|+
! Edostep !! Cents !! Notation !! 13-limit JI approximation !! ADIN interval category
|-
| 0
| 0
| C
| 1/1
| unison
|-
| 1
| 46.2
| C#
| 25/24, 33/32, 36/35, 49/48
| superunison
|-
| 2
| 92.3
| Cx, Dbb
| 21/20, 22/21, 26/25
| farminor second
|-
| 3
| 138.5
| Db
| 12/11, 13/12, 14/13
| supraminor second
|-
| 4
| 184.6
| D
| 9/8, 10/9, 11/10
| submajor second
|-
| 5
| 230.8
| D#
| 8/7
| supermajor second
|-
| 6
| 276.9
| Dx, Ebb
| 7/6, 13/11
| farminor third
|-
| 7
| 323.1
| Eb
| 6/5
| supraminor third
|-
| 8
| 369.2
| E
| 5/4, 16/13, 11/9
| submajor third
|-
| 9
| 415.4
| E#, Fbb
| 14/11, 9/7
| farmajor third
|-
| 10
| 461.5
| Ex, Fb
| 21/16, 13/10
| subfourth
|-
| 11
| 507.7
| F
| 4/3
| perfect fourth
|-
| 12
| 553.8
| F#
| 11/8
| subaugmented fourth
|-
| 13
| 600
| Fx, Gbb
| 7/5, 10/7
| tritone
|-
| 14
| 646.2
| Gb
| 16/11
| supradiminished fifth
|-
| 15
| 692.3
| G
| 3/2
| perfect fifth
|-
| 16
| 738.5
| G#
| 32/21, 20/13
| superfifth
|-
| 17
| 784.6
| Gx, Abb
| 11/7, 14/9
| farminor sixth
|-
| 18
| 830.8
| Ab
| 8/5, 13/8, 18/11
| supraminor sixth
|-
| 19
| 876.9
| A
| 5/3
| submajor sixth
|-
| 20
| 923.1
| A#
| 12/7, 22/13
| farmajor sixth
|-
| 21
| 969.2
| Ax, Bbb
| 7/4
| subminor seventh
|-
| 22
| 1015.4
| Bb
| 9/5, 16/9, 20/11
| supraminor seventh
|-
| 23
| 1061.5
| B
| 11/6, 13/7, 24/13
| submajor seventh
|-
| 24
| 1107.7
| B#, Cbb
| 21/11, 25/13, 40/21
| farmajor seventh
|-
| 25
| 1153.8
| Bx, Cb
| 64/33, 96/49, 35/18, 48/25
| suboctave
|-
| 26
| 1200
| C
| 2/1
| octave
|}


==Compositional theory==
==Compositional theory==
{| class="wikitable"
|+Thirds in 26edo
!Quality
|Farminor
|'''Supraminor'''
|'''Submajor'''
|Farmajor
|-
!Cents
|276.9
|'''323.1'''
|'''369.2'''
|415.4
|-
!Just interpretation
|7/6
|'''6/5'''
|'''5/4, 16/13'''
|14/11, 9/7
|-
!Steps
|6
|'''7'''
|'''8'''
|9
|}


===Chords===
===Chords===
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{{WIP}}
{{WIP}}
== Multiples ==
=== 104edo ===
104edo is a strong no-5 [[Parapyth]] tuning.
{{Harmonics in ED|104|47|0}}
=== 130edo ===
130edo adds 26edo's accurate 7/4 and [[10edo]]'s accurate 13/8 to [[65edo]], resulting in a strong 2.3.5.7.11.13.19.23.31.47 system. It is a good [[Hemiwurschmidt]] tuning. It is also useful as an example for interval categorization.
{{Harmonics in ED|130|47|0}}


{{navbox EDO}}
{{navbox EDO}}
{{Cat|edos}}

Latest revision as of 17:45, 11 March 2026

26edo, or 26 equal divisions of the octave, is the equal tuning featuring steps of (1200/26) ~= 46.15 cents, 26 of which stack to the perfect octave 2/1.

Theory

JI approximation

26edo is characterized by a flat tuning of harmonics 3, 5, and 13 and slightly sharp but accurate tunings of 7 and 11. Although its primes 3, 5, and 13 are damaged, 26edo can be used as a 13-limit temperament as it is consistent to the 13-odd-limit. Additionally, the fact that primes 3 and 13 are flat by about the same amount and 5 is flat by about double that means that intervals such as 13/12 and 10/9 are approximated well. The accurate 7 combined with the flat 5 means that 7/5 and 10/7 are both mapped to the 600¢ half octave tritone, tempering out 50/49. 16/13 and 11/9 are mapped to the same interval as 5/4, tempering out 65/64, 144/143, and 45/44.


Approximation of prime harmonics in 26edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) 0.0 -9.6 -17.1 +0.4 +2.5 -9.8 -12.6 -20.6 +17.9 -14.2 +8.8
Relative (%) 0.0 -20.9 -37.0 +0.9 +5.5 -21.1 -27.4 -44.6 +38.7 -30.8 +19.1
Steps

(reduced)

26

(0)

41

(15)

60

(8)

73

(21)

90

(12)

96

(18)

106

(2)

110

(6)

118

(14)

126

(22)

129

(25)

Edostep interpretations

26edo's edostep has the following 13-limit interpretations:

  • 25/24 (the difference between 5/4 and 6/5)
  • 33/32 (the difference between 4/3 and 11/8)
  • 36/35 (the difference between 5/4 and 9/7)
  • 49/48 (the difference between 8/7 and 7/6)

Intervals and notation

Similar to 19edo, 26edo can be notated entirely with standard diatonic notation, with #/b = 1\26 and x/bb = 2\26. The equivalences are Cx = Dbb, E# = Fbb, and Ex = Fb.

Edostep Cents Notation 13-limit JI approximation ADIN interval category
0 0 C 1/1 unison
1 46.2 C# 25/24, 33/32, 36/35, 49/48 superunison
2 92.3 Cx, Dbb 21/20, 22/21, 26/25 farminor second
3 138.5 Db 12/11, 13/12, 14/13 supraminor second
4 184.6 D 9/8, 10/9, 11/10 submajor second
5 230.8 D# 8/7 supermajor second
6 276.9 Dx, Ebb 7/6, 13/11 farminor third
7 323.1 Eb 6/5 supraminor third
8 369.2 E 5/4, 16/13, 11/9 submajor third
9 415.4 E#, Fbb 14/11, 9/7 farmajor third
10 461.5 Ex, Fb 21/16, 13/10 subfourth
11 507.7 F 4/3 perfect fourth
12 553.8 F# 11/8 subaugmented fourth
13 600 Fx, Gbb 7/5, 10/7 tritone
14 646.2 Gb 16/11 supradiminished fifth
15 692.3 G 3/2 perfect fifth
16 738.5 G# 32/21, 20/13 superfifth
17 784.6 Gx, Abb 11/7, 14/9 farminor sixth
18 830.8 Ab 8/5, 13/8, 18/11 supraminor sixth
19 876.9 A 5/3 submajor sixth
20 923.1 A# 12/7, 22/13 farmajor sixth
21 969.2 Ax, Bbb 7/4 subminor seventh
22 1015.4 Bb 9/5, 16/9, 20/11 supraminor seventh
23 1061.5 B 11/6, 13/7, 24/13 submajor seventh
24 1107.7 B#, Cbb 21/11, 25/13, 40/21 farmajor seventh
25 1153.8 Bx, Cb 64/33, 96/49, 35/18, 48/25 suboctave
26 1200 C 2/1 octave

Compositional theory

Thirds in 26edo
Quality Farminor Supraminor Submajor Farmajor
Cents 276.9 323.1 369.2 415.4
Just interpretation 7/6 6/5 5/4, 16/13 14/11, 9/7
Steps 6 7 8 9

Chords

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

TODO:

  • write about flattone

Scales

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

Multiples

104edo

104edo is a strong no-5 Parapyth tuning.

Approximation of prime harmonics in 104edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) 0.0 +1.9 -5.5 +0.4 +2.5 +1.8 -1.1 +2.5 -5.2 -2.7 -2.7 +2.5 -2.1 -3.8 +3.7
Relative (%) 0.0 +16.4 -48.1 +3.5 +21.9 +15.4 -9.6 +21.6 -45.0 -23.0 -23.6 +21.7 -18.5 -33.2 +32.3
Steps

(reduced)

104

(0)

165

(61)

241

(33)

292

(84)

360

(48)

385

(73)

425

(9)

442

(26)

470

(54)

505

(89)

515

(99)

542

(22)

557

(37)

564

(44)

578

(58)

130edo

130edo adds 26edo's accurate 7/4 and 10edo's accurate 13/8 to 65edo, resulting in a strong 2.3.5.7.11.13.19.23.31.47 system. It is a good Hemiwurschmidt tuning. It is also useful as an example for interval categorization.

Approximation of prime harmonics in 130edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) 0.0 -0.4 +1.4 +0.4 +2.5 -0.5 -3.4 -2.1 -0.6 +4.3 -0.4 -2.1 -4.4 -3.8 -0.9
Relative (%) 0.0 -4.5 +14.9 +4.4 +27.4 -5.7 -37.0 -23.1 -6.3 +46.2 -4.6 -22.9 -48.2 -41.4 -9.7
Steps

(reduced)

130

(0)

206

(76)

302

(42)

365

(105)

450

(60)

481

(91)

531

(11)

552

(32)

588

(68)

632

(112)

644

(124)

677

(27)

696

(46)

705

(55)

722

(72)


View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 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 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 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
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