# 26edo

(Redirected from 26-edo)

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" (1.6158) is very close to φ ≈ 1.6180 (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 Approximate Ratios* Solfege Interval
Name
Example
in D
cents pions 7mus
0 0.00 1/1 do P1 D
1 46.15 48.92 59.08 (3B.13816) 33/32, 49/48, 36/35, 25/24 di A1 D#
2 92.31 97.85 118.15 (76.27816) 21/20, 22/21, 26/25 rih d2 Ebb
3 138.46 146.77 177.23 (B1.3B16) 12/11, 13/12, 14/13, 16/15 ru m2 Eb
4 184.615 195.69 236.31 (EC.4F16) 9/8, 10/9, 11/10 re M2 E
5 230.77 243.615 295.385 (127.62816) 8/7, 15/13 ri A2 E#
6 276.92 293.54 354.46 (162.7616) 7/6, 13/11, 33/28 ma d3 Fb
7 323.08 342.46 413.54 (19D.8A16) 6/5 me m3 F
8 369.23 391.385 472.615 (1D8.9D816) 5/4, 11/9, 16/13 muh/mi M3 F#
9 415.385 440.31 531.69 (213.B116) 9/7, 14/11, 33/26 maa A3 Fx
10 461.54 489.23 590.77 (24E.C516) 21/16, 13/10 fe d4 Gb
11 507.69 538.15 649.85 (289.D8816) 4/3 fa P4 G
12 553.85 587.08 708.92 (2C4.EC816) 11/8, 18/13 fu A4 G#
13 600 636 768 (30016) 7/5, 10/7 fi/se AA4, dd5 Gx, Abb
14 646.15 684.92 827.08 (33B.13816) 16/11, 13/9 su d5 Ab
15 692.31 733.85 886.15 (376.27816) 3/2 sol P5 A
16 738.46 782.77 945.23 (3B1.3B16) 32/21, 20/13 si A5 A#
17 784.615 831.69 1004.31 (3EC.4F16) 11/7, 14/9 leh d6 Bbb
18 830.77 880.615 1063.385 (427.62816) 13/8, 8/5 le/lu m6 Bb
19 876.92 929.54 1122.46 (462.7616) 5/3 la M6 B
20 923.08 978.46 1181.54 (49D.8A16) 12/7, 22/13 li A6 B#
21 969.23 1027.315 1240.615 (4D8.9D816) 7/4 ta d7 Cb
22 1015.385 1076.31 1299.69 (513.B116) 9/5, 16/9, 20/11 te m7 C
23 1061.54 1125.23 1358.77 (54E.C516) 11/6, 13/7, 15/8, 24/13 tu/ti M7 C#
24 1107.69 1174.15 1417.85 (589.D8816) 21/11, 25/13, 40/21 to A7 Cx
25 1153.85 1223.08 1476.92 (5C4.EC816)) 64/33, 96/49, 35/18, 48/25 da d8 Db
26 1200 1272 1536 (60016) 2/1 do P8 D
• based on treating 26-EDO as a 13-limit temperament; other approaches are possible.

Using Kite's color notation, qualities can be loosely associated with colors:

Quality Color Monzo format Examples
diminished zo {a, b, 0, 1} 7/6, 7/4
minor fourthward wa {a, b}, b < -1 32/27, 16/9
" gu {a, b, -1} 6/5, 9/5
major yo {a, b, 1} 5/4, 5/3
" fifthward wa {a, b}, b > 1 9/8, 27/16
augmented ru {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 zo, gu, yo and ru triads:

color of the 3rd JI chord Notes as EDO steps Notes of C chord Written name Spoken name
zo 6:7:9 0-6-15 C Ebb G C(b3) or C(d3) C flat-three or C dim-three
gu 10:12:15 0-7-15 C Eb G Cm C minor
yo 4:5:6 0-8-15 C E G C C major or C
ru 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).

### Best direct mapping, even if inconsistent

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

### Patent val mapping

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
15/13, 26/15 16.972
5/4, 8/5 17.083
7/5, 10/7 17.488
9/8, 16/9 19.295
11/10, 20/11 19.611
9/7, 14/9 19.699
11/9, 18/11 21.823
16/15, 15/8 26.730
15/14, 28/15 27.135
15/11, 22/15 29.258

## 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.

## 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

## Compositions

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)

Public Rituals « Jim Aikin's Oblong Blob “The Triumphal Procession of Nebuchadnezzar“