69edo

The 69 equal division or 69-EDO, which divides the octave into 69 equal parts of 17.391 cents each, has been called "the love-child of 23edo and quarter-comma meantone". Nice. As a meantone system, it is on the flat side, with a fifth of 695.652 cents. Such a fifth is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to that of "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.

In the 7-limit it is a mohajira system, tempering out 6144/6125, but not a septimal meantone system, as 126/125 maps to one step. It also supports the 12&69 temperament tempering out 3125/3087 along with 81/80. In the 11-limit it tempers out 99/98, and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in 31EDO but not in 69.

Degree	Name	Cents	Approximate Ratios*	Error (abs, ¢)
0	Natural Unison, 1	0.000	1/1	0.000
1	Ptolemy's comma	17.391	100/99	-0.008
2	Jubilisma, lesser septimal sixth tone	34.783	50/49, 101/99	-0.193, 0.157
3	lesser septendecimal quartertone, _____	52.174	34/33, 101/98	0.491, -0.028
4	_____	69.565	76/73	-0.158
5	Small undevicesimal semitone	86.957	20/19	-1.844
6	Large septendecimal semitone	104.348	17/16	-0.608
7	Septimal diatonic semitone	121.739	15/14	2.296
8	Tridecimal neutral second	139.130	13/12	0.558
9	Vicesimotertial neutral second	156.522	23/21	-0.972
10	Undevicesimal large neutral second, undevicesimal whole tone	173.913	21/19	0.645
11	Quasi-meantone	191.304	19/17	-1.253
12	Whole tone	208.696	9/8	4.786
13	Septimal whole tone	226.087	8/7	-5.087
14	Vicesimotertial semifourth	243.478	23/20	1.518
15	Subminor third, undetricesimal subminor third	260.870	7/6, 29/25	-6.001, 3.920
16	Vicesimotertial subminor third	278.261	27/23	0.670
17	Pythagorean minor third	295.652	32/27	1.517
18	Classic minor third	313.043	6/5	-2.598
19	Vicesimotertial supraminor third	330.435	23/19	-0.327
20	Undecimal neutral third	347.826	11/9	0.418
21	Septendecimal submajor third	365.217	21/17	-0.608
22	Classic major third	382.609	5/4	-3.705
23	Undetricesimal major third, Septendecimal major third	400.000	29/23, 34/27	-1.303, 0.910
24	Undecimal major third	417.391	14/11	-0.117
25	Supermajor third	434.783	9/7	-0.301
26	Barbados third	452.174	13/10	-2.040
27	Septimal sub-fourth	469.565	21/16	-1.216
28	_____	486.957	53/40	-0.234
29	Just perfect fourth	504.348	4/3	6.303
30	Vicesimotertial acute fourth	521.739	23/17	-1.580
31	Undecimal augmented fourth	539.130	15/11	2.180
32	Undecimal superfourth, undetricesimal superfourth	556.522	11/8, 29/21	5.204, -2.275
33	Narrow tritone, classic augmented fourth	573.913	7/5, 25/18	-8.600, 5.196
34	_____	591.304	31/22	-2.413
35	High tritone, undevicesimal tritone	608.696	10/7, 27/19	-8.792, 0.344
36	_____	626.087	33/23	1.088
37	Undetricesimal tritone	643.478	29/20	0.215
38	Undevicesimal diminished fifth, undecimal diminished fifth	660.870	19/13, 22/15	3.884, -2.180
39	Vicesimotertial grave fifth, _____	678.261	34/23, 37/25	1.580, -0.456
40	Just perfect fifth	695.652	3/2	-6.303
41	_____	713.043	80/53	0.234
42	Super-fifth, undetricesimal super-fifth	730.435	32/21, 29/19	1.216, -1.630
43	Septendecimal subminor sixth	747.826	17/11	-5.811
44	Subminor sixth	765.217	14/9	0.301
45	Undecimal minor sixth	782.609	11/7	0.117
46	Septendecimal subminor sixth	800.000	27/17	-0.910
47	Classic minor sixth	817.391	8/5	3.705
48	Septendecimal supraminor sixth	834.783	34/21	0.608
49	Undecimal neutral sixth	852.174	18/11	-0.418
50	Vicesimotertial submajor sixth	869.565	38/23	0.327
51	Classic major sixth	886.957	5/3	2.598
52	Pythagorean major sixth	904.348	27/16	-1.517
53	Septendecimal major sixth, undetricesimal major sixth	921.739	17/10, 29/17	3.097, -2.883
54	Supermajor sixth, undetricesimal supermajor sixth	939.130	12/7, 50/29	6.001, -3.920
55	Vicesimotertial supermajor sixth	956.522	40/23	-1.518
56	Harmonic seventh	973.913	7/4	5.087
57	Pythagorean minor seventh	991.304	16/9	-4.786
58	Quasi-meantone minor seventh	1008.696	34/19	1.253
59	Minor neutral undevicesimal seventh	1026.087	38/21	-0.645
60	Vicesimotertial neutral seventh	1043.478	42/23	0.972
61	Tridecimal neutral seventh	1060.870	24/13	-0.558
62	Septimal diatonic major seventh	1078.261	28/15	-2.296
63	Small septendecimal major seventh	1095.652	32/17	0.608
64	Small undevicesimal semitone	1113.043	20/19	1.844
65	_____	1130.435	73/38	0.158
66	Septendecimal supermajor seventh	1147.826	33/17	-0.491
67	_____	1165.217	49/25	-0.193
68	_____	1182.609	99/50	0.008
69	Octave, 8	1200.000	2/1	0.000

*some simpler ratios listed