69edo
| ← 68edo | 69edo | 70edo → |
69 equal divisions of the octave (abbreviated 69edo or 69ed2), also called 69-tone equal temperament (69tet) or 69 equal temperament (69et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 69 equal parts of about 17.4 ¢ each. Each step represents a frequency ratio of 21/69, or the 69th root of 2. 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.
69edo offers two kinds of meantone 12-tone scales. One is the raw meantone scale, which has a 7:4 step ratio, and other is period-3 lithium scale, which has a 6:5 step ratio and stems from a temperament tempering out 3125/3087 along with 81/80. It should be noted that while the lithium scale has a meantone fifth, it produces a tcherepnin scale instead of traditional diatonic.
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. 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.
The concoctic scale for 69edo is 22\69, and the corresponding rank two temperament is 22 & 69, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit.
Odd harmonics
| 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 | -21 | +29 | +28 | +30 | -33 | +42 | -3 | -11 | -7 | -13 | |
| Steps (reduced) |
109 (40) |
160 (22) |
194 (56) |
219 (12) |
239 (32) |
255 (48) |
270 (63) |
282 (6) |
293 (17) |
303 (27) |
312 (36) | |
Table of intervals
| Degree | Carmen's naming system | 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
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-109 69⟩ | [⟨69 109]] | +1.99 | 1.99 | 11.43 |
| 2.3.5 | 81/80, [-41 1 17⟩ | [⟨69 109 160]] | +1.86 | 1.64 | 9.40 |
| 2.3.5.7 | 81/80, 126/125, 4117715/3981312 | [⟨69 109 160 193]] (69d) | +2.49 | 1.79 | 10.28 |
| 2.3.5.7 | 81/80, 3125/3087, 6144/6125 | [⟨69 109 160 194]] (69) | +0.94 | 2.13 | 12.23 |
Rank 2 temperaments
| Periods
per 8ve |
Generator | Names |
|---|---|---|
| 1 | 2\69 | Gammy |
| 1 | 19\69 | Rarity |
| 1 | 20\69 | Mohaha, Tritomere |
| 1 | 22\69 | Caleb, Marveltri |
| 1 | 29\69 | Meantone |
| 3 | 5\69 | Ogene |
| 3 | 6\69 | August, Lithium |
| 3 | 9\69 | Nessafof |
Scales
- Supermajor[11], 3L 8s – 6 6 6 7 6 6 6 7 6 6 7
- Meantone[7], 5L 2s (gen = 40\69) – 11 11 7 11 11 11 7
- Meantone[12], 7L 5s (gen = 40\69) – 7 4 7 4 7 4 7 7 4 7 4 7
- Baroque[12], 4 7 7 4 7 4 7 4 7 7 4 7 (as proposed by Pianoteq plugin)
- Lithium[9], 11 6 6 11 6 6 11 6 6 - 3L 6s
- Lithium[12], 5 6 6 6 5 6 6 6 5 6 6 6 - 9L 3s