69ed7
69 equal divisions of the 7th harmonic (abbreviated 69ed7), is the tuning system that divides the 7th harmonic into 69 equal parts of about 48.82356 ¢ each. Each step represents a frequency ratio of [math]\displaystyle{ 7^{\frac{1}{69}} }[/math], or the 69th root of 7.
Theory
69ed7 is a nice and strong 3.5.7.11.13 system.
All ratios in the 3.5.7.11.13 subgroup and 21-integer-limit are approximated in 69ed7 with less than 7.7 ¢ error.
69ed7 is close to the different methods for approaching the 3.5.7.11.13 subgroup in the Triple Bohlen-Piece aera:
- The finite Euler product maxima of the 3.5.7.11.13 subgroup (sigma = 1 gives 48.82085 ¢, sigma = 1/2 gives 48.82100 ¢).
- The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.
With a size of 48.82356 ¢, 69ed7 gives a better approximation than 39edt to Triple Bohlen-Pierce just intonation, yet still remaining a simple division.
Intervals and approximation to JI
| 69ed7 degree |
Cents | Ratios in the 3.5.7.11.13 subgroup |
Error (abs, ¢) | Error (rel, %) |
|---|---|---|---|---|
| 5 | 244.1 | 15/13 | -3.623 | -7.4 |
| 6 | 292.9 | 13/11 | 3.732 | 7.6 |
| 7 | 341.8 | 11/9 | -5.643 | -11.6 |
| 9 | 439.4 | 9/7 | 4.328 | 8.9 |
| 11 | 537.1 | 15/11 | 0.108 | 0.2 |
| 12 | 585.9 | 7/5 | 3.371 | 6.9 |
| 13 | 634.7 | 13/9 | -1.911 | -3.9 |
| 16 | 781.2 | 11/7 | -1.315 | -2.7 |
| 17 | 830.0 | 21/13 | -0.253 | -0.5 |
| 18 | 878.8 | 5/3 | -5.535 | -11.3 |
| 21 | 1025.3 | 9/5 | 7.699 | 15.8 |
| 22 | 1074.1 | 13/7 | 2.417 | 4.9 |
| 23 | 1122.9 | 21/11 | 3.479 | 7.1 |
| 27 | 1318.2 | 15/7 | -1.207 | -2.5 |
| 28 | 1367.1 | 11/5 | 2.056 | 4.2 |
| 30 | 1464.7 | 7/3 | -2.164 | -4.4 |
| 34 | 1660.0 | 13/5 | 5.787 | 11.9 |
| 39 | 1904.1 | 3/1 | 2.164 | 4.4 |
| 46 | 2245.9 | 11/3 | -3.479 | -7.1 |
| 51 | 2490.0 | 21/5 | 5.535 | 11.3 |
| 52 | 2538.8 | 13/3 | 0.253 | 0.5 |
| 57 | 2782.9 | 5/1 | -3.371 | -6.9 |
| 69 | 3368.8 | 7/1 | 0.0 | 0.0 |
| 78 | 3808.2 | 9/1 | 4.328 | 8.9 |
| 85 | 4150.0 | 11/1 | -1.315 | -2.7 |
| 91 | 4442.9 | 13/1 | 2.417 | 4.9 |
| 96 | 4687.1 | 15/1 | -1.207 | -2.5 |
| 108 | 5272.9 | 21/1 | 2.164 | 4.4 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | |
|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +20.6 | +2.2 | -3.4 | +0.0 | -1.3 | +2.4 |
| Relative (%) | +42.2 | +4.4 | -6.9 | +0.0 | -2.7 | +4.9 | |
| Steps (reduced) |
25 (25) |
39 (39) |
57 (57) |
69 (0) |
85 (16) |
91 (22) | |
Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7
Commas with numerator < 1000000:
245/243, 275/273, 847/845, 1331/1323, 1575/1573, 1625/1617, 1875/1859, 2197/2187, 2205/2197, 3125/3087, 4459/4455, 6655/6561, 6655/6591, 8125/8019, 9295/9261, 9375/9317, 9625/9477, 11011/10935, 12005/11979, 14641/14625, 15625/15309, 15625/15379, 16807/16731, 16875/16807, 26411/26325, 28875/28561, 29575/29403, 41503/41067, 42875/42471, 46475/45927, 60025/59049, 60025/59319, 75625/74529, 78125/77077, 91125/91091, 107811/107653, 109375/107811, 117975/117649, 153125/150579, 161051/159705, 196625/194481, 200475/199927, 218491/216513, 219615/218491, 366025/361179, 378125/369603, 373527/371293, 378125/371293, 390625/382239, 398125/395307, 408375/405769, 456533/455625, 538265/531441, 539539/531441, 546875/531441, 546875/533871, 717409/710775, 714025/713097, 717409/714025, 759375/753571, 823543/820125, 823875/823543, 831875/823543, 859375/842751, 983125/964467