26edo

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	superfourth
13	600	Fx, Gbb	7/5, 10/7	tritone
…	…	…	…	…

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)