69edo

The 69 equal division or 69-EDO, which divides the octave into 69 equal parts of 17.391 cents each. Nice.

Theory

69edo has been called "the love-child of 23edo and quarter-comma meantone". 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.

Odd harmonics

Approximation of odd harmonics in 69edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-6.30	-3.71	+5.09	+4.79	+5.20	-5.75	+7.38	-0.61	-1.86	-1.22	-2.19
	Relative (%)	-36.2	-21.3	+29.3	+27.5	+29.9	-33.0	+42.5	-3.5	-10.7	-7.0	-12.6
Steps (reduced)		109 (40)	160 (22)	194 (56)	219 (12)	239 (32)	255 (48)	270 (63)	282 (6)	293 (17)	303 (27)	312 (36)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)
2.3	[-109, 69⟩	[⟨69 109]]	1.99
2.3.5	81/80, [-41, 1, 17⟩, [-37, -3, 18⟩, [-33, -7, 19⟩	[⟨69 109 160]]	1.86
2.3.5.7	81/80, 3125/3087, 5625/5488, 6144/6125, 17280/16807, 17496/16807	[⟨69 109 160 194]]	0.94
2.3.5.7.11	81/80, 99/98, 441/440, 625/616, 864/847, 1344/1331, 2420/2401, 2560/2541	[⟨69 109 160 194 239]]	0.44

Table of intervals

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

Scales

• Mavka^{[clarification needed]}[11], 3L 8s – 66676667667
• Meantone[7], 5L 2s (gen = 40\69) – 11 11 7 11 11 11 7
• Meantone[12], 7L 5s (gen = 40\69) – 747474774747

Music

• Hypergiant Sakura by Eliora Ben-Gurion