EDO

An equal division of the octave (EDO or edo, /ˈidoʊ/ EE-doh) is a tuning system constructed by dividing the octave into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain MOS will have crudely similar properties.

The notation m\n denotes m steps of n-edo, i.e. the frequency ratio 2^(m/n).

List of edos

Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.

Popular edos are highlighted. Temperaments are capitalized and can be found in the List of regular temperaments.

Edo	Description	First twelve steps (¢)	Fifth (¢)	Edostep interpretation	Example basic (in 2...23) and erac subgroups
1	Equivalent to the 2-limit.	1200	1200		2
					2
2	Just a 12edo tritone.	600, 1200	600		2.11.23?
					2.<3.>>5.>>7?
3	A 12edo augmented triad.	400, 800, 1200	800		2.5.13?
					2.>3.5?
4	A diminished tetrad.	300, 600, 900, 1200	600		2.19.23?
					2.<3.<5.<7?
5	Collapsed diatonic, and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.	240, 480, 720, 960, 1200	720	9/8, 8/7, 7/6	2.3.7
					2.>>3.<7
6	Subset of 12edo. Good approximation of Didacus temperament.	200, 400, 600, 800, 1000, 1200	600, 800		2.9.5.7.23
					2.xx3.5.7.<11.>13.23
7	Equalized diatonic, and the first edo to (very vaguely) support diatonic functional harmony.	171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200	685.7	9/8, 10/9, 16/15	2.3.5.13
					2.<3.<<5.>13
8	Minimal version of Ammonite temperament.	150, 300, 450, 600, 750, 900, 1050, 1200	750		2.19.23
					2.x3.x5.x7.x11.x13
9	The first edo to support the antidiatonic scale and temperaments like Semabila, loosely resembling the pelog scale. It contains approximations to many 7-limit intervals, but not the 7/4 itself (see erac subgroup).	133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200	666.7	9/8, 16/15, 25/24	2.5.11
					2.<<3.>5.<<7
10	The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting mosh.	120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200	720	16/15, 10/9, 81/80, 36/35	2.3.5.7.13
					2.>>3.<7.13
11	Basic smitonic and checkertonic. Good example of Orgone.	109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9	654.5, 763.6		2.9.7.11.15
					2.x3.x5.7.11
12	The basic tuning of diatonic, and consequently the most widespread EDO. Supports the 5-limit decently well.	100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200	700	256/243, chromatic semitone, 16/15, 25/24	2.3.5.17.19
					2.3.>5.>>7.17.19
13	Basic oneirotonic, archeotonic, and gramitonic.	92.3, 184.6, 276.9, 369.2, 461.5, 553.8, 646.2, 738.5, 830.8, 923.1, 1015.4, 1107.7	646.2, 738.5		2.5.11.13.17
14	Basic semiquartal.	85.7, 171.4, 257.1, 342.9, 428.6, 514.3, 600, 685.7, 771.4, 857.1, 942.9, 1028.6	685.7	28/27, 21/20, 15/14	2.3.7.13
15	The basic tuning of Zarlino's intense diatonic, a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.	80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960	720	81/80, 25/24, 16/15, 33/32, 36/35	2.3.5.7.11.23
16	The most popular antidiatonic edo, which supports Trismegistus and Mavila.	75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900	675, 750		2.5.7.13.19
17	Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic.	70.6, 141.2, 211.8, 282.4, 352.9, 423.5, 494.1, 564.7, 635.3, 705.9, 776.5, 847.1	705.9		2.3.13.23
18	Straddle-3 version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.	66.7, 133.3, 200, 266.7, 333.3, 400, 466.7, 533.3, 600, 666.7, 733.3, 800	666.6, 733.3		2.9.5.7.13
					2.xx3.>5.>>7.<11.<13
19	A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.	63.2, 126.3, 189.5, 252.6, 315.8, 378.9, 442.1, 505.3, 568.4, 631.6, 694.7, 757.9	694.7	25/24, diaschisma, 36/35, 28/27	2.3.5.23
20	Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.	60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720	660, 720		2.7.11.13.19
21	Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.	57.1, 114.3, 171.4, 228.6, 285.7, 342.9, 400, 457.1, 514.3, 571.4, 628.6, 685.7	685.7, 742.9		2.3.5.7.23
					2.x>3.x<5.7.x<11.x<13.23
22	Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.	54.5, 109.1, 163.6, 218.2, 272.7, 327.3, 381.8, 436.4, 490.9, 545.5, 600, 654.5	709.1		2.3.5.7.11.17
					2.3.5.7>.11
23	The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.	52.2, 104.3, 156.5, 208.7, 260.9, 313, 365.2, 417.4, 469.6, 521.7, 573.9, 626.1	678.3		2.x3.x5.x7.x11.13.17.23
24	Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.	50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600	700		2.3.11.13.17.19
25	A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus.	48, 96, 144, 192, 240, 288, 336, 384, 432, 480, 528, 576	720		2.5.7.19
26	A simple tuning of Flattone. Has an absurdly accurate 7/4.	46.2, 92.3, 138.5, 184.6, 230.8, 276.9, 323.1, 369.2, 415.4, 461.5, 507.7, 553.8	692.7		2.3.7.11.13
27	A good tuning for Archy and Sensi. It has 3/2 at 16 steps.	44.4, 88.9, 133.3, 177.8, 222.2, 266.7, 311.1, 355.6, 400, 444.4, 488.9, 533.3	711.1		2.3.5.7.13.23
28	A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard oneirotonic scale converging on Buzzard temperament.	42.9, 85.7, 128.6, 171.4, 214.3, 257.1, 300, 342.9, 385.7, 428.6, 471.4, 514.3	685.7, 728.6		2.3.5.7.11
29	Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.	41.4, 82.8, 124.1, 165.5, 206.9, 248.3, 289.7, 331, 372.4, 413.8, 455.2, 496.6	703.4		2.3.19.23
...
31	The definitive Septimal Meantone tuning.	38.7, 77.4, 116.1, 154.8, 193.5, 232.3, 271, 309.7, 348.4, 387.1, 425.8, 464.5	696.8		2.3.5.7.11.23
...
34	An accurate medium-sized non-Meantone 5-limit edo.	35.3, 70.6, 105.9, 141.2, 176.5, 211.8, 247.1, 282.4, 317.6, 352.9, 388.2, 423.5	705.9		2.3.5.13.23
					2.3.5.x7.13.x19.23
...
36	Triple 12edo, which functions as an extremely accurate septal Compton and Slendric system.	33.3, 66.7, 100, 133.3, 166.7, 200, 233.3, 266.7, 300, 333.3, 366.7, 400	700		2.3.7.13.17.19.23
37	An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.	32.4, 64.9, 97.3, 129.7, 162.2, 194.6, 227, 259.5, 291.9, 324.3, 356.8, 389.2	681.1, 713.5		2.9.5.7.11.13.17.19
					2.<x3.5.7.11.13.17.19
...
39	Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.	30.8, 61.5, 92.3, 123.1, 153.8, 184.6, 215.4, 246.2, 276.9, 307.7, 338.5, 369.2	707.7		2.3.11
...
41	The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. One of two viably small tunings of 11-limit penslen.	29.3, 58.5, 87.8, 117.1, 146.3, 175.6, 204.9, 234.1, 263.4, 292.7, 322, 351.2	702.4		2.3.5.7.11.13.19
...
46	The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. One of two viably small tunings of 11-limit penslen.	26.1, 52.2, 78.3, 104.3, 130.4, 156.5, 182.6, 208.7, 234.8, 260.9, 287, 313	704.3		2.3.5.7.11.13.17.23
...
53	Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).	22.6, 45.3, 67.9, 90.6, 113.2, 135.8, 158.5, 181.1, 203.8, 226.4, 249.1, 271.7	701.9		2.3.5.7.13.19
...
58	Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.	20.7, 41.4, 62.1, 82.8, 103.4, 124.1, 144.8, 165.5, 186.2, 206.9, 227.6, 248.3	703.4		2.3.7.17
...
159	Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.	7.5, 15.1, 22.6, 30.2, 37.7, 45.3, 52.8, 60.4, 67.9, 75.5, 83, 90.6	701.9		-
...
311	An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.	3.9, 7.7, 11.6, 15.4, 19.3, 23.2, 27, 30.9, 34.7, 38.6, 42.4, 46.3	702.3		-