26edo

26edo divides the octave into 26 equal parts of 46.154 cents each. It tempers out 81/80 in the 5-limit, making it a meantone tuning with a very flat fifth. In the 7-limit, it tempers out 50/49, 525/512 and 875/864, and supports injera, flattone, lemba and doublewide temperaments. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the 13 odd limit consistently. 26edo has a very good approximation of the harmonic seventh (7/4).

26edo's "minor sixth" is very close to phi (i. e., the golden ratio).

Structure


The structure of 26edo is an interesting beast, with various approaches relating it to various rank two temperaments.
1. In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately 10/9 or 8/7, but NOT 9/8).
2. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo.
3. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and | -3 0 0 6 -4>. The 65536/65219 comma, the orgonisma, leads to orgone temperament with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The | -3 0 0 6 -4> comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14.
4. We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).
5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and 33edo even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.

Intervals

Degrees
Size in
cents
Approximate Ratios*
Solfege
Interval
Name
Example
in D
0
0.00
1/1
do
P1
D
1
46.15
33/32, 49/48, 36/35, 25/24
di
A1
D#
2
92.31
21/20, 22/21, 26/25
rih
d2
Ebb
3
138.46
12/11, 13/12, 14/13, 16/15
ru
m2
Eb
4
184.62
9/8, 10/9, 11/10
re
M2
E
5
230.77
8/7, 15/13
ri
A2
E#
6
276.92
7/6, 13/11, 33/28
ma
d3
Fb
7
323.08
6/5
me
m3
F
8
369.23
5/4, 11/9, 16/13
muh/mi
M3
F#
9
415.38
9/7, 14/11, 33/26
maa
A3
Fx
10
461.54
21/16, 13/10
fe
d4
Gb
11
507.69
4/3
fa
P4
G
12
553.85
11/8, 18/13
fu
A4
G#
13
600.00
7/5, 10/7
fi/se
AA4, dd5
Gx, Abb
14
646.15
16/11, 13/9
su
d5
Ab
15
692.31
3/2
sol
P5
A
16
738.46
32/21, 20/13
si
A5
A#
17
784.62
11/7, 14/9
leh
d6
Bbb
18
830.77
13/8, 8/5
le/lu
m6
Bb
19
876.92
5/3
la
M6
B
20
923.08
12/7, 22/13
li
A6
B#
21
969.23
7/4
ta
d7
Cb
22
1015.38
9/5, 16/9, 20/11
te
m7
C
23
1061.54
11/6, 13/7, 15/8, 24/13
tu/ti
M7
C#
24
1107.69
21/11, 25/13, 40/21
to
A7
Cx
25
1153.85
64/33, 96/49, 35/18, 48/25
da
d8
Db
26
1200.00
2/1
do
P8
D
*based on treating 26-EDO as a 13-limit temperament; other approaches are possible.

Using color notation, qualities can be loosely associated with colors:
quality
color
monzo format
examples
diminished
blue
{a, b, 0, 1}
7/6, 7/4
minor
fourthward white
{a, b}, b < -1
32/27, 16/9
"
green
{a, b, -1}
6/5, 9/5
major
yellow
{a, b, 1}
5/4, 5/3
"
fifthward white
{a, b}, b > 1
9/8, 27/16
augmented
red
{a, b, 0, -1}
9/7, 12/7
All 26edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Spelling certain chords properly may require triple sharps and flats, especially if the tonic is anything other than the 11 keys in the Eb-C# range. Here are the blue, green, yellow and red triads
color of the 3rd
JI chord
notes as edosteps
notes of C chord
written name
spoken name
blue
6:7:9
0-6-15
C Ebb G
C(b3) or C(d3)
C flat-three or C dim-three
green
10:12:15
0-7-15
C Eb G
Cm
C minor
yellow
4:5:6
0-8-15
C E G
C
C major or C
red
14:18:27
0-9-15
C E# G
C(#3) or C(A3)
C sharp-three or C aug-three
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 26edo (ordered by absolute error).
Interval, complement
Error (abs., in cents)
13/12, 24/13
0.111
8/7, 7/4
0.405
14/11, 11/7
2.123
10/9, 9/5
2.212
11/8, 16/11
2.528
13/10, 20/13
7.325
6/5, 5/3
7.436
18/13, 13/9
9.536
4/3, 3/2
9.647
16/13, 13/8
9.758
7/6, 12/7
10.052
14/13, 13/7
10.163
12/11, 11/6
12.176
13/11, 22/13
12.287
15/11, 22/15
16.895
15/13, 26/15
16.972
5/4, 8/5
17.083
7/5, 10/7
17.488
15/14, 28/15
19.019
9/8, 16/9
19.295
16/15, 15/8
19.424
11/10, 20/11
19.611
9/7, 14/9
19.699
11/9, 18/11
21.823

Rank two temperaments

List of 26et rank two temperaments by badness
List of edo-distinct 26et rank two temperaments
Periods
per octave
Generator
Temperaments
1
1\26
Quartonic/Quarto
1
3\26
Jerome/Bleu/Secund
1
5\26
Cynder/Mothra
1
7\26
Superkleismic/Orgone/Shibboleth
1
9\26
Roman/Wesley
1
11\26
Meantone/Flattone
2
1\26
Elvis
2
2\26
Injera
2
3\26
Fifive/Crepuscular
2
4\26
Unidec/Hendec/Dubbla
2
5\26
Lemba
2
6\26
Doublewide/Cavalier
13
1\26
Triskaidekic

Hendec in 26et

Hendec, the 13-limit 26&46 temperament with generator ~10/9, concentrates the intervals of greatest accuracy in 26et into the lower ranges of complexity. It has a period of half an octave, with 13/12 reachable by four generators, 8/7 by two, 14/11 by one, 10/9 by one, and 11/8 by three. All of these are tuned to within 2.5 cents of accuracy.

Commas

26et tempers out the following commas. (Note: This assumes the val < 26 41 60 73 90 96 |.)
Ratio
Monzo
Size (Cents)
Name 1
Name 2
Name 3
81/80
| -4 4 -1 >
21.51
Syntonic Comma
Didymos Comma
Meantone Comma
5696703/5695946
| -17 62 -35 >
0.23
Senior


525/512
| -9 1 2 1 >
43.41
Avicennma
Avicenna's Enharmonic Diesis

50/49
| 1 0 2 -2 >
34.98
Tritonic Diesis
Jubilisma

875/864
| -5 -3 3 1 >
21.90
Keema


4000/3969
| 5 -4 3 -2 >
13.47
Octagar


1728/1715
| 6 3 -1 -3 >
13.07
Orwellisma
Orwell Comma

1029/1024
| -10 1 0 3 >
8.43
Gamelisma


321489/320000
| -9 8 -4 2 >
8.04
Varunisma


1065875/1063543
| -26 -1 1 9 >
3.79
Wadisma


4375/4374
| -1 -7 4 1 >
0.40
Ragisma


99/98
| -1 2 0 -2 1 >
17.58
Mothwellsma


100/99
| 2 -2 2 0 -1 >
17.40
Ptolemisma


105/104
| -3 1 1 1 0 -1 >
16.57
Animist


65536/65219
| 16 0 0 -2 -3 >
8.39
Orgonisma


385/384
| -7 -1 1 1 1 >
4.50
Keenanisma


441/440
| -3 2 -1 2 -1 >
3.93
Werckisma


3025/3024
| -4 -3 2 -1 2 >
0.57
Lehmerisma


9801/9800
| -3 4 -2 -2 2 >
0.18
Kalisma
Gauss' Comma

Orgone Temperament


Andrew Heathwaite first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:

The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. MOS of type 4L 3s (mish).
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.

The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. MOS of type 4L 7s.
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108 1200.

The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates 16:11 and 3g approximates 7:4 (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.

37edo is another orgone tuning, and 89edo is better even than 26. If we take 11 and 26 to be the edges of the Orgone Spectrum, we may fill in the rest of the spectrum thus:

3\11








19\70



16\59





29\107


13\48






36\133



23\85





33\122

10\37







37\137



27\100





44\163


17\63






41\152



24\89





31\115
7\26





Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.

orgone_heptatonic.jpg


Additional Scalar Bases Available in 26-EDO:

Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (though further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).

-Igs

Literature


Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.

Compositions


Canon 3-in-1 on a ground ‘The tempest’, by Claudi Meneghin
A Time-Yellowed Photo of the Cliffs Hangs on the Wall by Igliashon Jones
Two Pairs of Socks by Igliashon Jones
Etude in 26-tone equal temperament play by Herman Miller
under the heatDome play by Jon Lyle Smith
A New Recording of Organ Study #1 play by Daniel Thompson
Little Fugue in 26 by Cameron Bobro
26 tone fugue (unfinished) by Peter Kosmorsky (based on the melody he was singing in the shower, in orgone, to the presumed confusion of those in earshot)
Microtonal music in 26-EDO by Shaahin Mohajeri
Chiapas Palenque by Zach Curley
Injera Jam by Zach Curley
Guitar Serenade in Q Major by Zach Curley
Public Rituals « Jim Aikin's Oblong Blob “The Triumphal Procession of Nebuchadnezzar“
Melopœia Project – 26 EDO album based on Tolkein’s Silmarillion – Ainulindalë – Unfretted
Morpheous Wing in 26 edo by Chris Vaisvil