26edo: Difference between revisions

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'''26edo''', or 26 equal divisions of the octave (sometimes called '''26-TET''' or '''26-tone equal temperament'''), is the equal tuning featuring steps of (1200/26) ~= 46.15 [[cent]]s, 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]].


==Theory==
26edo has a [[perfect fifth]], 692.3{{c}}, which is tuned even flatter than that of [[19edo]]. Its [[5L 2s|diatonic]] scale is thus very [[soft]] ([[homoioheptatonic]]). Its thirds can still be taken, if inaccurately, to approximate [[6/5]] and [[5/4]], supporting [[Meantone]]. In terms of [[7-limit]] properties, 26edo is notably the smallest EDO to distinguish all of [[9/8]], [[8/7]], [[7/6]], 6/5, and 5/4 (although "9/8" in particular is far closer to [[10/9]]), and does so [[consistent]]ly.


===JI approximation===
Where 26edo truly shines, however, is in higher limits. We can observe it closely approximates both the [[7/4|7th]] and [[11/8|11th]] harmonics (to within half a cent for the former and 3 cents for the latter). Structurally, 26edo's fifth spans 15 edosteps, which means that it can be split into 3 parts and into 5. Both splits result in almost perfectly just intervals: 8/7 serves as 1/3 (5 edosteps), resulting in [[Slendric]] temperament, and [[13/12]] as 1/5 of the fifth (3 edosteps). {{adv|These intervals are tuned particularly well as a result of 26edo approximating the [[natave|natural]] fifth of ''e''<sup>2/5</sup>.}} Anchored by 8/7, 11/8, and 13/12, 26edo consistently represents the 13-[[odd-limit]], beating [[22edo]]'s consistency record of 11. Prime 17 can also be included in the mix, as it is tuned similarly to 13 and 3, although intervals of 15 are inconsistently mapped.
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]].


26edo overall, despite ostensibly supporting familiar harmonic organization in the form of Meantone, presents that organization vastly differently from [[12edo]] due to its flat tuning. Furthermore, 26edo includes [[13edo]] as a subset, and with it, the [[oneirotonic]] scale. And as a claimant for the smallest EDO to merit consideration as a [[17-limit]] system, and with primes 7 and 11 tuned far more accurately than 3 and 5, 26edo is quite capable of supporting harmonic systems that rely at most minimally on the diatonic scale or [[5-limit]] harmony at all.
== Theory ==
=== JI approximation ===
26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.
With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo that [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.
26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7,  which are both tuned extremely well by 26edo.
{{Harmonics in ED|26|31|0}}
{{Harmonics in ED|26|31|0}}


===Edostep interpretations===
=== Edostep interpretations ===
 
26edo's edostep has the following interpretations in the 11-limit:
26edo's edostep has the following 13-limit interpretations:
* 25/24 (the difference between 6/5 and 5/4)
* 25/24 (the difference between 5/4 and 6/5)
* 33/32 (the difference between 4/3 and 11/8)
* 33/32 (the difference between 4/3 and 11/8)
* 36/35 (the difference between 5/4 and 9/7)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 49/48 (the difference between 8/7 and 7/6)
* 64/63 (the difference between 9/8 and 8/7)
* 80/77 (the difference between 11/10 and 8/7)


===Intervals and notation===
=== 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. JI approximations are given in the 17-limit 21-odd-limit, aside from the single edostep and its complement. Approximations within 3{{c}} are given in [brackets]. Harmonics 3-21 are bolded; inconsistent intervals (involving 15) are italicized.


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 !! 17-limit JI approximation !! ADIN interval category
==Compositional theory==
|-
| 0
| 0
| C
| 1/1
| unison
|-
| 1
| 46.2
| C#
| 25/24, 33/32, [36/35], 49/48, 64/63
| superunison
|-
| 2
| 92.3
| Cx, Dbb
| ''15/14'', '''17/16''', 18/17, 21/20, 22/21
| farminor second
|-
| 3
| 138.5
| Db
| 12/11, [13/12], 14/13, ''16/15''
| (supra)minor second
|-
| 4
| 184.6
| D
| '''9/8''', [10/9], 11/10
| (sub)major second
|-
| 5
| 230.8
| D#
| [8/7], 15/13, 17/15
| farmajor second
|-
| 6
| 276.9
| Dx, Ebb
| 7/6, 13/11, 20/17
| farminor third
|-
| 7
| 323.1
| Eb
| 6/5, 17/14
| (supra)minor third
|-
| 8
| 369.2
| E
| '''5/4''', 11/9, 16/13, 21/17, [26/21]
| (sub)major third
|-
| 9
| 415.4
| E#, Fbb
| 9/7, [14/11]
| farmajor third
|-
| 10
| 461.5
| Ex, Fb
| 13/10, [17/13], '''21/16''', 22/17
| subfourth
|-
| 11
| 507.7
| F
| 4/3, ''15/11''
| perfect fourth
|-
| 12
| 553.8
| F#
| '''[11/8]''', 18/13
| (sub)augmented fourth
|-
| 13
| 600
| Fx, Gbb
| 7/5, 10/7, 17/12, 24/17
| tritone
|-
| 14
| 646.2
| Gb
| 13/9, [16/11]
| (supra)diminished fifth
|-
| 15
| 692.3
| G
| '''3/2''', ''22/15''
| perfect fifth
|-
| 16
| 738.5
| G#
| 17/11, 20/13, [26/17], 32/21
| superfifth
|-
| 17
| 784.6
| Gx, Abb
| [11/7], 14/9
| farminor sixth
|-
| 18
| 830.8
| Ab
| 8/5, '''13/8''', 18/11, [21/13], 34/21
| (supra)minor sixth
|-
| 19
| 876.9
| A
| 5/3, 28/17
| (sub)major sixth
|-
| 20
| 923.1
| A#
| 12/7, 17/10, 22/13
| farmajor sixth
|-
| 21
| 969.2
| Ax, Bbb
| '''[7/4]''', 26/15, 30/17
| farminor seventh
|-
| 22
| 1015.4
| Bb
| [9/5], 16/9, 20/11
| (supra)minor seventh
|-
| 23
| 1061.5
| B
| 11/6, 13/7, '''''15/8''''', [24/13]
| (sub)major seventh
|-
| 24
| 1107.7
| B#, Cbb
| 17/9, 21/11, ''28/15'', 32/17, 40/21
| farmajor seventh
|-
| 25
| 1153.8
| Bx, Cb
| [35/18], 48/25, 63/32, 64/33, 96/49
| suboctave
|-
| 26
| 1200
| C
| 2/1
| octave
|}


== Compositional theory ==
=== Tertian structure ===
{| class="wikitable"
{| class="wikitable"
|+Thirds in 26edo
|+Thirds in 26edo
Line 33: Line 206:
|-
|-
!Cents
!Cents
|277
|276.9
|'''323.1'''
|'''323.1'''
|'''369.2'''
|'''369.2'''
Line 39: Line 212:
|-
|-
!Just interpretation
!Just interpretation
|7/6
|7/6 (+10.0{{c}})
|'''6/5'''
|'''6/5 (+7.4{{c}})'''
|'''5/4, 16/13'''
|'''11/9 (+22.2{{c}}) <br /> 16/13 (+9.8{{c}}) <br /> 5/4 (-17.1{{c}})'''
|14/11, 9/7
|14/11 (-2.1{{c}}) <br /> 9/7 (-19.7{{c}})
|-
|-
!Steps
!Steps
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|}
|}


===Chords===
Diatonic thirds are bolded.


=== Chords ===
{{WIP}}
{{WIP}}


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* write about flattone
* write about flattone


===Scales===
=== Scales ===
 
==== Organization of MOSes ====
26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent [[MOS]] scales organize themselves into two loops of three, each linked by the operation of tripling.
{| class="wikitable"
|+
!
! colspan="3" |Loop 1
|-
!Temperament
|[[Slendric]]
|[[Flattone]]
|[[Superkleismic]]
|-
!Scale (albitonic)
|5-5-5-5-6
|4-4-3-4-4-4-3
|2-5-2-5-2-5-5
|-
!Generator
|5\, 21\
|11\, 15\
|7\, 19\
|-
!
! colspan="3" |Loop 2
|-
!Temperament
|[[Quartonic]]
|[[Bleu]]
|[[Roman]]
|-
!Scale (albitonic)
| - (11-note scale has<br>>10 steps interval)
|3-3-3-3-3-3-3-3-2
|7-1-7-1-1-7-1-1
|-
!Generator
|1\, 25\
|3\, 23\
|9\, 17\
|}
{{WIP}}
{{WIP}}


Line 66: Line 279:
104edo is a strong no-5 [[Parapyth]] tuning.
104edo is a strong no-5 [[Parapyth]] tuning.
{{Harmonics in ED|104|47|0}}
{{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}}
{{Cat|edos}}

Latest revision as of 20:39, 22 May 2026

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

26edo has a perfect fifth, 692.3¢, which is tuned even flatter than that of 19edo. Its diatonic scale is thus very soft (homoioheptatonic). Its thirds can still be taken, if inaccurately, to approximate 6/5 and 5/4, supporting Meantone. In terms of 7-limit properties, 26edo is notably the smallest EDO to distinguish all of 9/8, 8/7, 7/6, 6/5, and 5/4 (although "9/8" in particular is far closer to 10/9), and does so consistently.

Where 26edo truly shines, however, is in higher limits. We can observe it closely approximates both the 7th and 11th harmonics (to within half a cent for the former and 3 cents for the latter). Structurally, 26edo's fifth spans 15 edosteps, which means that it can be split into 3 parts and into 5. Both splits result in almost perfectly just intervals: 8/7 serves as 1/3 (5 edosteps), resulting in Slendric temperament, and 13/12 as 1/5 of the fifth (3 edosteps). These intervals are tuned particularly well as a result of 26edo approximating the natural fifth of e2/5. Anchored by 8/7, 11/8, and 13/12, 26edo consistently represents the 13-odd-limit, beating 22edo's consistency record of 11. Prime 17 can also be included in the mix, as it is tuned similarly to 13 and 3, although intervals of 15 are inconsistently mapped.

26edo overall, despite ostensibly supporting familiar harmonic organization in the form of Meantone, presents that organization vastly differently from 12edo due to its flat tuning. Furthermore, 26edo includes 13edo as a subset, and with it, the oneirotonic scale. And as a claimant for the smallest EDO to merit consideration as a 17-limit system, and with primes 7 and 11 tuned far more accurately than 3 and 5, 26edo is quite capable of supporting harmonic systems that rely at most minimally on the diatonic scale or 5-limit harmony at all.

Theory

JI approximation

26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5¢ JND, and therefore 26edo does so as well.

With the very flat tuning of 5, the 4:5:6 triad has more of a submajor quality, and 25/24, the distinction between the main 5-limit triads, is reduced to the size of a quartertone. Similarly to 22edo, this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary chthonic medials), and, as while 7 is accurate, 5 is flat enough in 26edo that 7/5 and 10/7 are both mapped to the 600¢ half-octave tritone.

26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also 11/10; this is characteristic of Flattone, a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that 11/9 is mapped to the same interval as 5/4, and to this we can add the successive mediants, 16/13 as well as 21/17 and 26/21. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.

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 interpretations in the 11-limit:

  • 25/24 (the difference between 6/5 and 5/4)
  • 33/32 (the difference between 4/3 and 11/8)
  • 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
  • 49/48 (the difference between 8/7 and 7/6)
  • 64/63 (the difference between 9/8 and 8/7)
  • 80/77 (the difference between 11/10 and 8/7)

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. JI approximations are given in the 17-limit 21-odd-limit, aside from the single edostep and its complement. Approximations within 3¢ are given in [brackets]. Harmonics 3-21 are bolded; inconsistent intervals (involving 15) are italicized.

Edostep Cents Notation 17-limit JI approximation ADIN interval category
0 0 C 1/1 unison
1 46.2 C# 25/24, 33/32, [36/35], 49/48, 64/63 superunison
2 92.3 Cx, Dbb 15/14, 17/16, 18/17, 21/20, 22/21 farminor second
3 138.5 Db 12/11, [13/12], 14/13, 16/15 (supra)minor second
4 184.6 D 9/8, [10/9], 11/10 (sub)major second
5 230.8 D# [8/7], 15/13, 17/15 farmajor second
6 276.9 Dx, Ebb 7/6, 13/11, 20/17 farminor third
7 323.1 Eb 6/5, 17/14 (supra)minor third
8 369.2 E 5/4, 11/9, 16/13, 21/17, [26/21] (sub)major third
9 415.4 E#, Fbb 9/7, [14/11] farmajor third
10 461.5 Ex, Fb 13/10, [17/13], 21/16, 22/17 subfourth
11 507.7 F 4/3, 15/11 perfect fourth
12 553.8 F# [11/8], 18/13 (sub)augmented fourth
13 600 Fx, Gbb 7/5, 10/7, 17/12, 24/17 tritone
14 646.2 Gb 13/9, [16/11] (supra)diminished fifth
15 692.3 G 3/2, 22/15 perfect fifth
16 738.5 G# 17/11, 20/13, [26/17], 32/21 superfifth
17 784.6 Gx, Abb [11/7], 14/9 farminor sixth
18 830.8 Ab 8/5, 13/8, 18/11, [21/13], 34/21 (supra)minor sixth
19 876.9 A 5/3, 28/17 (sub)major sixth
20 923.1 A# 12/7, 17/10, 22/13 farmajor sixth
21 969.2 Ax, Bbb [7/4], 26/15, 30/17 farminor seventh
22 1015.4 Bb [9/5], 16/9, 20/11 (supra)minor seventh
23 1061.5 B 11/6, 13/7, 15/8, [24/13] (sub)major seventh
24 1107.7 B#, Cbb 17/9, 21/11, 28/15, 32/17, 40/21 farmajor seventh
25 1153.8 Bx, Cb [35/18], 48/25, 63/32, 64/33, 96/49 suboctave
26 1200 C 2/1 octave

Compositional theory

Tertian structure

Thirds in 26edo
Quality Farminor Supraminor Submajor Farmajor
Cents 276.9 323.1 369.2 415.4
Just interpretation 7/6 (+10.0¢) 6/5 (+7.4¢) 11/9 (+22.2¢)
16/13 (+9.8¢)
5/4 (-17.1¢) 		14/11 (-2.1¢)
9/7 (-19.7¢)
Steps 6 7 8 9

Diatonic thirds are bolded.

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

Organization of MOSes

26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent MOS scales organize themselves into two loops of three, each linked by the operation of tripling.

Loop 1
Temperament Slendric Flattone Superkleismic
Scale (albitonic) 5-5-5-5-6 4-4-3-4-4-4-3 2-5-2-5-2-5-5
Generator 5\, 21\ 11\, 15\ 7\, 19\
Loop 2
Temperament Quartonic Bleu Roman
Scale (albitonic) - (11-note scale has
>10 steps interval) 		3-3-3-3-3-3-3-3-2 7-1-7-1-1-7-1-1
Generator 1\, 25\ 3\, 23\ 9\, 17\
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 • 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 • 77 • 80 • 81
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
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