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.
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 ¢. 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 (%) | -36.2 | -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) | |
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 40\69) |
Ups and downs notation (Dual sharp fifth 41\69) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 17.4 | ^D, vvE♭♭ | ^D, vE♭ | |
| 2 | 34.8 | ^^D, vE♭♭ | ^^D, E♭ | |
| 3 | 52.2 | 32/31, 33/32, 34/33, 35/34 | vD♯, E♭♭ | ^3D, ^E♭ |
| 4 | 69.6 | 25/24, 26/25 | D♯, ^E♭♭ | ^4D, ^^E♭ |
| 5 | 87 | 20/19, 21/20 | ^D♯, vvE♭ | ^5D, ^3E♭ |
| 6 | 104.3 | 17/16, 35/33 | ^^D♯, vE♭ | v5D♯, ^4E♭ |
| 7 | 121.7 | vD𝄪, E♭ | v4D♯, ^5E♭ | |
| 8 | 139.1 | 13/12 | D𝄪, ^E♭ | v3D♯, v5E |
| 9 | 156.5 | 23/21, 34/31, 35/32 | ^D𝄪, vvE | vvD♯, v4E |
| 10 | 173.9 | 21/19, 31/28, 32/29 | ^^D𝄪, vE | vD♯, v3E |
| 11 | 191.3 | 19/17, 29/26 | E | D♯, vvE |
| 12 | 208.7 | 26/23, 35/31 | ^E, vvF♭ | ^D♯, vE |
| 13 | 226.1 | 33/29 | ^^E, vF♭ | E |
| 14 | 243.5 | 23/20 | vE♯, F♭ | ^E, vF |
| 15 | 260.9 | E♯, ^F♭ | F | |
| 16 | 278.3 | 20/17, 34/29 | ^E♯, vvF | ^F, vG♭ |
| 17 | 295.7 | 19/16 | ^^E♯, vF | ^^F, G♭ |
| 18 | 313 | 6/5 | F | ^3F, ^G♭ |
| 19 | 330.4 | 23/19, 29/24 | ^F, vvG♭♭ | ^4F, ^^G♭ |
| 20 | 347.8 | ^^F, vG♭♭ | ^5F, ^3G♭ | |
| 21 | 365.2 | 21/17, 37/30 | vF♯, G♭♭ | v5F♯, ^4G♭ |
| 22 | 382.6 | 5/4 | F♯, ^G♭♭ | v4F♯, ^5G♭ |
| 23 | 400 | 29/23 | ^F♯, vvG♭ | v3F♯, v5G |
| 24 | 417.4 | 14/11 | ^^F♯, vG♭ | vvF♯, v4G |
| 25 | 434.8 | vF𝄪, G♭ | vF♯, v3G | |
| 26 | 452.2 | 13/10 | F𝄪, ^G♭ | F♯, vvG |
| 27 | 469.6 | 21/16 | ^F𝄪, vvG | ^F♯, vG |
| 28 | 487 | ^^F𝄪, vG | G | |
| 29 | 504.3 | G | ^G, vA♭ | |
| 30 | 521.7 | 23/17 | ^G, vvA♭♭ | ^^G, A♭ |
| 31 | 539.1 | ^^G, vA♭♭ | ^3G, ^A♭ | |
| 32 | 556.5 | 29/21 | vG♯, A♭♭ | ^4G, ^^A♭ |
| 33 | 573.9 | 32/23 | G♯, ^A♭♭ | ^5G, ^3A♭ |
| 34 | 591.3 | 31/22 | ^G♯, vvA♭ | v5G♯, ^4A♭ |
| 35 | 608.7 | 37/26 | ^^G♯, vA♭ | v4G♯, ^5A♭ |
| 36 | 626.1 | 23/16, 33/23 | vG𝄪, A♭ | v3G♯, v5A |
| 37 | 643.5 | 29/20 | G𝄪, ^A♭ | vvG♯, v4A |
| 38 | 660.9 | ^G𝄪, vvA | vG♯, v3A | |
| 39 | 678.3 | 34/23, 37/25 | ^^G𝄪, vA | G♯, vvA |
| 40 | 695.7 | A | ^G♯, vA | |
| 41 | 713 | ^A, vvB♭♭ | A | |
| 42 | 730.4 | 29/19, 32/21, 35/23 | ^^A, vB♭♭ | ^A, vB♭ |
| 43 | 747.8 | 20/13, 37/24 | vA♯, B♭♭ | ^^A, B♭ |
| 44 | 765.2 | A♯, ^B♭♭ | ^3A, ^B♭ | |
| 45 | 782.6 | 11/7 | ^A♯, vvB♭ | ^4A, ^^B♭ |
| 46 | 800 | 35/22 | ^^A♯, vB♭ | ^5A, ^3B♭ |
| 47 | 817.4 | 8/5 | vA𝄪, B♭ | v5A♯, ^4B♭ |
| 48 | 834.8 | 34/21 | A𝄪, ^B♭ | v4A♯, ^5B♭ |
| 49 | 852.2 | ^A𝄪, vvB | v3A♯, v5B | |
| 50 | 869.6 | 33/20 | ^^A𝄪, vB | vvA♯, v4B |
| 51 | 887 | 5/3 | B | vA♯, v3B |
| 52 | 904.3 | 32/19 | ^B, vvC♭ | A♯, vvB |
| 53 | 921.7 | 17/10, 29/17 | ^^B, vC♭ | ^A♯, vB |
| 54 | 939.1 | vB♯, C♭ | B | |
| 55 | 956.5 | 33/19 | B♯, ^C♭ | ^B, vC |
| 56 | 973.9 | ^B♯, vvC | C | |
| 57 | 991.3 | 23/13 | ^^B♯, vC | ^C, vD♭ |
| 58 | 1008.7 | 34/19 | C | ^^C, D♭ |
| 59 | 1026.1 | 29/16 | ^C, vvD♭♭ | ^3C, ^D♭ |
| 60 | 1043.5 | 31/17 | ^^C, vD♭♭ | ^4C, ^^D♭ |
| 61 | 1060.9 | 24/13, 35/19 | vC♯, D♭♭ | ^5C, ^3D♭ |
| 62 | 1078.3 | C♯, ^D♭♭ | v5C♯, ^4D♭ | |
| 63 | 1095.7 | 32/17 | ^C♯, vvD♭ | v4C♯, ^5D♭ |
| 64 | 1113 | 19/10 | ^^C♯, vD♭ | v3C♯, v5D |
| 65 | 1130.4 | 25/13 | vC𝄪, D♭ | vvC♯, v4D |
| 66 | 1147.8 | 31/16, 33/17 | C𝄪, ^D♭ | vC♯, v3D |
| 67 | 1165.2 | ^C𝄪, vvD | C♯, vvD | |
| 68 | 1182.6 | ^^C𝄪, vD | ^C♯, vD | |
| 69 | 1200 | 2/1 | D | D |
Proposed names
| 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
Notation
Ups and downs notation
Using Helmholtz–Ellis accidentals, 69edo can also be notated using ups and downs notation along with Stein–Zimmerman quarter-tone accidentals:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| Flat symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
Here, a sharp raises by four steps, and a flat lowers by four steps, so arrows can be used to fill in the gap.
Sagittal notation
This notation uses the same sagittal sequence as EDOs 62 and 76.
Evo flavor
Revo flavor
Evo-SZ flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
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 | Temperaments |
|---|---|---|
| 1 | 2\69 | Gammy (69de) |
| 1 | 19\69 | Rarity |
| 1 | 20\69 | Mohaha (69e) |
| 1 | 22\69 | Caleb (69) marveltri (69) |
| 1 | 29\69 | Meantone (69d) |
| 3 | 5\69 | Ogene (69bceef) |
| 3 | 6\69 | August (7-limit, 69cdd) Lithium (69) |
| 3 | 9\69 | Nessafof (69e) |
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
- Lithium[9], 3L 6s – 11 6 6 11 6 6 11 6 6
- Lithium[12], 9L 3s – 5 6 6 6 5 6 6 6 5 6 6 6
Music
- Hypergiant Sakura (2021)
- 69 hours before (2023)