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 and Abbreviation | Cents | Approximate Ratios* | Error (abs, ¢) |
|---|---|---|---|---|
| 0 | Natural Unison, 1 | 0.000 | 1/1 | 0.000 |
| 1 | 17.391 | |||
| 2 | 34.783 | |||
| 3 | 52.174 | 20/19 | -1.844 | |
| 4 | 69.565 | |||
| 5 | 86.957 | |||
| 6 | 104.348 | 17/16 | -0.608 | |
| 7 | 121.739 | 15/14 | 2.296 | |
| 8 | 139.130 | 13/12 | 0.558 | |
| 9 | 156.522 | |||
| 10 | 173.913 | |||
| 11 | 191.304 | 19/17 | -1.253 | |
| 12 | 208.696 | 9/8 | 4.786 | |
| 13 | 226.087 | 8/7 | -5.087 | |
| 14 | 243.478 | 23/20 | 1.518 | |
| 15 | 260.870 | 7/6, 29/25 | -6.001, 3.920 | |
| 16 | 278.261 | 27/23 | 0.670 | |
| 17 | 295.652 | 32/27 | 1.517 | |
| 18 | 313.043 | 6/5 | -2.598 | |
| 19 | 330.435 | 23/19 | -0.327 | |
| 20 | 347.826 | 11/9 | 0.418 | |
| 21 | 365.217 | 21/17 | -0.608 | |
| 22 | 382.609 | 5/4 | -3.705 | |
| 23 | 400.000 | |||
| 24 | 417.391 | 14/11 | -0.117 | |
| 25 | 434.783 | 9/7 | -0.301 | |
| 26 | 452.174 | 13/10 | -2.040 | |
| 27 | 469.565 | 21/16 | -1.216 | |
| 28 | 486.957 | |||
| 29 | 504.348 | 4/3 | 6.303 | |
| 30 | 521.739 | 23/17 | -1.580 | |
| 31 | 539.130 | 15/11 | 2.180 | |
| 32 | 556.522 | 11/8, 29/21 | 5.204, -2.275 | |
| 33 | 573.913 | 7/5, 25/18 | -8.600, 5.196 | |
| 34 | 591.304 | 31/22 | -2.413 | |
| 35 | 608.696 | 10/7, 27/19 | -8.792, 0.344 | |
| 36 | 626.087 | 33/23 | 1.088 | |
| 37 | 643.478 | 29/20 | 0.215 | |
| 38 | 660.870 | 19/13, 22/15 | 3.884, -2.180 | |
| 39 | 678.261 | 34/23, 37/25 | 1.580, -0.456 | |
| 40 | 695.652 | 3/2 | -6.303 | |
| 41 | 713.043 | |||
| 42 | 730.435 | 32/21, 29/19 | 1.216, -1.630 | |
| 43 | 747.826 | 17/11 | -5.811 | |
| 44 | 765.217 | 14/9 | 0.301 | |
| 45 | 782.609 | 11/7 | 0.117 | |
| 46 | 800.000 | 27/17 | ||
| 47 | 817.391 | 8/5 | 3.705 | |
| 48 | 834.783 | 34/21 | ||
| 49 | 852.174 | |||
| 50 | 869.565 | |||
| 51 | 886.957 | 5/3 | 2.598 | |
| 52 | 904.348 | |||
| 53 | 921.739 | |||
| 54 | 939.130 | |||
| 55 | 956.522 | |||
| 56 | 973.913 | |||
| 57 | 991.304 | |||
| 58 | 1008.696 | |||
| 59 | 1026.087 | |||
| 60 | 1043.478 | |||
| 61 | 1060.870 | |||
| 62 | 1078.261 | 28/15 | -2.296 | |
| 63 | 1095.652 | |||
| 64 | 1113.043 | |||
| 65 | 1130.435 | |||
| 66 | 1147.826 | |||
| 67 | 1165.217 | |||
| 68 | 1182.609 | |||
| 69 | Octave, 8 | 1200.000 | 2/1 | 0.000 |
*some simpler ratios listed