26edo

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26edo

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Prime factorization

2 × 13

Step size

46.1538¢

Fifth

15\26 (692.308¢)

Semitones (A1:m2)

1:3 (46.15¢ : 138.5¢)

Consistency limit

13

Distinct consistency limit

5

26 equal divisions of the octave (abbreviated 26edo or 26ed2), also called 26-tone equal temperament (26tet) or 26 equal temperament (26et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 26 equal parts of about 46.2 ¢ each. Each step represents a frequency ratio of 2^1/26, or the 26th root of 2.

Theory

26edo tempers out 81/80 in the 5-limit, making it a meantone tuning with a very flat fifth (0.088957¢ flat of the 4/9-comma meantone fifth).

In the 7-limit, it tempers out 50/49, 525/512 and 875/864, and supports temperaments like injera, flattone, lemba and doublewide. 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), as it is the denominator of a convergent to log₂7.

26edo's minor sixth (1.6158) is very close to φ ≈ 1.6180 (i.e. the golden ratio).

With a fifth of 15 steps, it can be equally divided into 3 or 5, supporting slendric temperament and bleu temperament respectively.

The structure of 26edo is an interesting beast, with various approaches relating it to various rank-2 temperaments.

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 second of approximately 10/9 instead of 9/8).
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.
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 the 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.
We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give 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.

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest harmonic entropy possible and thus is, in theory, most dissonant, assuming the relatively common values of a = 2 and s = 1.01. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Thanks to its sevenths, 26edo is an ideal tuning for its size for metallic harmony.

Odd harmonics

Approximation of odd harmonics in 26edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-9.6	-17.1	+0.4	-19.3	+2.5	-9.8	+19.4	-12.6	-20.6	-9.2	+17.9
Error	Relative (%)	-20.9	-37.0	+0.9	-41.8	+5.5	-21.1	+42.1	-27.4	-44.6	-20.0	+38.7
Steps (reduced)		41 (15)	60 (8)	73 (21)	82 (4)	90 (12)	96 (18)	102 (24)	106 (2)	110 (6)	114 (10)	118 (14)

Intervals

Degrees	Cents	Approximate Ratios*	Interval Name	Example in D	SKULO Interval Name	Example in D	Solfeges
0	0.00	1/1	P1	D	P1	D	da	do
1	46.15	33/32, 49/48, 36/35, 25/24	A1	D#	A1, S1	D#, SD	du	di
2	92.31	21/20, 22/21, 26/25	d2	Ebb	sm2	sEb	fro	rih
3	138.46	12/11, 13/12, 14/13, 16/15	m2	Eb	m2	Eb	fra	ru
4	184.62	9/8, 10/9, 11/10	M2	E	M2	E	ra	re
5	230.77	8/7, 15/13	A2	E#	SM2	SE	ru	ri
6	276.92	7/6, 13/11, 33/28	d3	Fb	sm3	sF	no	ma
7	323.08	135/112, 6/5	m3	F	m3	F	na	me
8	369.23	5/4, 11/9, 16/13	M3	F#	M3	F#	ma	muh/mi
9	415.38	9/7, 14/11, 33/26	A3	Fx	SM3	SF#	mu	maa
10	461.54	21/16, 13/10	d4	Gb	s4	sG	fo	fe
11	507.69	75/56, 4/3	P4	G	P4	G	fa	fa
12	553.85	11/8, 18/13	A4	G#	A4	G#	fu/pa	fu
13	600.00	7/5, 10/7	AA4, dd5	Gx, Abb	SA4, sd5	SG#, sAb	pu/sho	fi/se
14	646.15	16/11, 13/9	d5	Ab	d5	Ab	sha/so	su
15	692.31	112/75, 3/2	P5	A	P5	A	sa	sol
16	738.46	32/21, 20/13	A5	A#	S5	SA	su	si
17	784.62	11/7, 14/9	d6	Bbb	sm6	sBb	flo	leh
18	830.77	13/8, 8/5	m6	Bb	m6	Bb	fla	le/lu
19	876.92	5/3, 224/135	M6	B	M6	B	la	la
20	923.08	12/7, 22/13	A6	B#	SM6	SB	lu	li
21	969.23	7/4, 26/15	d7	Cb	sm7	sC	tho	ta
22	1015.38	9/5, 16/9, 20/11	m7	C	m7	C	tha	te
23	1061.54	11/6, 13/7, 15/8, 24/13	M7	C#	M7	C#	ta	tu/ti
24	1107.69	21/11, 25/13, 40/21	A7	Cx	SM7	SC#	tu	to
25	1153.85	64/33, 96/49, 35/18, 48/25	d8	Db	d8, s8	Db, sD	do	da
26	1200.00	2/1	P8	D	P8	D	da	do

based on treating 26edo as a 13-limit temperament; other approaches are possible.

Interval quality and chord names in color notation

Using 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 Edoteps	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	14:18:21	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.

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 26edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 26edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/12, 24/13	0.111	0.2
7/4, 8/7	0.405	0.9
11/7, 14/11	2.123	4.6
9/5, 10/9	2.212	4.8
11/8, 16/11	2.528	5.5
13/10, 20/13	7.325	15.9
5/3, 6/5	7.436	16.1
13/9, 18/13	9.536	20.7
3/2, 4/3	9.647	20.9
13/8, 16/13	9.758	21.1
7/6, 12/7	10.052	21.8
13/7, 14/13	10.163	22.0
11/6, 12/11	12.176	26.4
13/11, 22/13	12.287	26.6
15/11, 22/15	16.895	36.6
15/13, 26/15	16.972	36.8
5/4, 8/5	17.083	37.0
7/5, 10/7	17.488	37.9
15/14, 28/15	19.019	41.2
9/8, 16/9	19.295	41.8
15/8, 16/15	19.424	42.1
11/10, 20/11	19.611	42.5
9/7, 14/9	19.699	42.7
11/9, 18/11	21.823	47.3

15-odd-limit intervals in 26edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/12, 24/13	0.111	0.2
7/4, 8/7	0.405	0.9
11/7, 14/11	2.123	4.6
9/5, 10/9	2.212	4.8
11/8, 16/11	2.528	5.5
13/10, 20/13	7.325	15.9
5/3, 6/5	7.436	16.1
13/9, 18/13	9.536	20.7
3/2, 4/3	9.647	20.9
13/8, 16/13	9.758	21.1
7/6, 12/7	10.052	21.8
13/7, 14/13	10.163	22.0
11/6, 12/11	12.176	26.4
13/11, 22/13	12.287	26.6
15/13, 26/15	16.972	36.8
5/4, 8/5	17.083	37.0
7/5, 10/7	17.488	37.9
9/8, 16/9	19.295	41.8
11/10, 20/11	19.611	42.5
9/7, 14/9	19.699	42.7
11/9, 18/11	21.823	47.3
15/8, 16/15	26.730	57.9
15/14, 28/15	27.135	58.8
15/11, 22/15	29.258	63.4

Approximation to irrational intervals

After 13edo, the weird coïncidences continue: acoustic π/2 (17\26) is just in between the ϕ intervals provided by 13edo (16\26 for logarithmic ϕ/2, and 18\26 for acoustic ϕ).

Not until 1076edo do we find a better edo in terms of relative error on these intervals (which is not a very relevant edo for logarithmic ϕ, since 1076 does not belong to the Fibonacci sequence).

However, it should be noted that from an acoustic perspective, acoustic π and acoustic ϕ are both better represented on 23edo.

Direct approximation
Interval	Error (abs, ¢)
2^ϕ / ϕ	0.858
ϕ	2.321
π	2.820
2^ϕ	3.179
π/ϕ	5.141
2^ϕ / π	5.999

Regular temperament properties

Subgroup	Comma basis	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma basis	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-41 26⟩	[⟨26 41]]	+3.043	3.05	6.61
2.3.5	81/80, 78125/73728	[⟨26 41 60]]	+4.489	3.22	6.98
2.3.5.7	50/49, 81/80, 405/392	[⟨26 41 60 73]]	+3.324	3.44	7.45
2.3.5.7.11	45/44, 50/49, 81/80, 99/98	[⟨26 41 60 73 90]]	+2.509	3.48	7.53
2.3.5.7.11.13	45/44, 50/49, 65/64, 78/77, 81/80	[⟨26 41 60 73 90 96]]	+2.531	3.17	6.87
2.3.5.7.11.13.17	45/44, 50/49, 65/64 78/77, 81/80, 85/84	[⟨26 41 60 73 90 96 106]]	+2.613	2.94	6.38
2.3.5.7.11.13.17.19	45/44, 50/49, 57/56, 65/64, 78/77, 81/80, 85/84	[⟨26 41 60 73 90 96 106 110]]	+2.894	2.85	6.18

26et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-, and 29-limit (using the 26i val for the 23- and 29-limit). The next equal temperaments performing better in those subgroups are 27eg, 27eg, 29g, and 46, respectively.

Rank-2 Temperaments

Important MOSes include (in addition to ones found in 13edo):

diatonic (flattone) 4443443 (15\26, 1\1)
chromatic (flattone) 313131331313 (15\26, 1\1)
enharmonic (flattone) 2112112112121121121 (15\26, 1\1)
orgone 5525252 (7\26, 1\1)
orgone 32322322322 (7\26, 1\1)
orgone 212212221222122 (7\26, 1\1)
lemba 553553 (5\26, 1\2)
lemba 3232332323 (5\26, 1\2)
lemba 2122122121221221 (5\26, 1\2)

Periods per 8ve	Generator	Temperaments
1	1\26	Quartonic / quarto
1	3\26	Glacier / bleu / jerome / secund
1	5\26	Cynder / mothra
1	7\26	Orgone / superkleismic
1	9\26	Wesley / roman
1	11\26	Flattone / flattertone
2	1\26	Elvis
2	2\26	Injera
2	3\26	Fifive / crepuscular
2	4\26	Dubbla Unidec / hendec
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].)

Prime limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
5	81/80	[-4 4 -1⟩	21.51	Gu	Syntonic comma, Didymos comma, meantone comma
5	(60 digits)	[-17 62 -35⟩	0.23	Quadla-sepquingu	Senior comma
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma, Avicenna's enharmonic diesis
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Tritonic diesis, Jubilisma
7	875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
7	4000/3969	[5 -4 3 -2⟩	13.47	Sarurutriyo	Octagar comma
7	1728/1715	[6 3 -1 -3⟩	13.07	Triru-agu	Orwellisma
7	1029/1024	[-10 1 0 3⟩	8.43	Latrizo	Gamelisma
7	(12 digits)	[-9 8 -4 2⟩	8.04	Labizogugu	Varunisma
7	(18 digits)	[-26 -1 1 9⟩	3.79	Latritrizo-ayo	Wadisma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
11	65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Orgonisma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	3025/3024	[-4 -3 2 -1 2⟩	0.57	Loloruyoyo	Lehmerisma
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Bilorugu	Kalisma, Gauss' comma
13	105/104	[-3 1 1 1 0 -1⟩	16.57	Thuzoyo	Animist comma

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Scales

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.

Additional scalar bases available

Since the perfect 5th in 26edo 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