171edo
| ← 170edo | 171edo | 172edo → |
The 171 equal divisions of the octave (171edo), or the 171(-tone) equal temperament (171tet, 171et) when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave into 171 parts of about 7.02 cents each, a size close to 225/224, the marvel comma.
Theory
171edo is a remarkable edo which serves as a microtemperament for the 7-limit, approximating the 9-odd-limit tonality diamond within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh zeta integral edo but not enough to make it a zeta gap.
Remarkable 5-limit commas 171et tempers out are 32805/32768 (schisma), [1 -27 18⟩ (ennealimma), [-14 -19 19⟩ (enneadeca), and [-29 -11 20⟩ (gammic comma), and remarkable 7-limit commas 171et tempers out are 2401/2400 (breedsma), 4375/4374 (ragisma), 65625/65536 (horwell comma), 250047/250000 (landscape comma), 420175/419904 (wizma), and 703125/702464 (meter comma). So 171et supports a number of 7-limit rank-2 temperaments: pontiac, sesquiquartififths, term, ennealimmal, tertiaseptal, supermajor, enneadecal, neptune, mitonic, and mutt. It notably provides the optimal patent val for the rank-3 horwell temperament, and is also an excellent tuning for the 5-limit schismatic microtemperament, tempering out 32805/32768, and the no-fives temperament tempering out [-59 39 0 -1⟩ (nanisma).
171edo is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament jove, which tempers out 243/242 (rastma) and 441/440, not to mention 540/539 and 2401/2400. Jove can be extended by adding 364/363 for the 13-limit and 595/594 for the 17-limit, which 171edo also supports. Alternatively, the 171e val can be used, which tempers out 385/384.
171edo is an excellent tuning for the Carlos Gamma scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | +3.07 | +1.58 | +0.31 | -2.78 | +3.30 | +2.00 | -1.18 | +1.29 | -0.99 | +0.76 | +1.16 |
| Relative (%) | +0.0 | -2.9 | -5.0 | -5.8 | +43.7 | +22.5 | +4.4 | -39.6 | +47.1 | +28.5 | -16.8 | +18.3 | -14.1 | +10.9 | +16.5 | |
| Steps (reduced) |
171 (0) |
271 (100) |
397 (55) |
480 (138) |
592 (79) |
633 (120) |
699 (15) |
726 (42) |
774 (90) |
831 (147) |
847 (163) |
891 (36) |
916 (61) |
928 (73) |
950 (95) | |
Subsets and supersets
171 factors into primes as 32 × 19, and 171edo shares the nearly pure 7/6 of 9edo and the nearly pure 6/5 of 19edo, with every 7-limit interval expressible in terms of 2, 6/5, 7/6, and any one of primes 3, 5, or 7.
Intervals
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-271 171⟩ | [⟨171 271]] | +0.063 | 0.0633 | 0.90 |
| 2.3.5 | 32805/32768, [1 -27 18⟩ | [⟨171 271 397]] | +0.092 | 0.0660 | 0.94 |
| 2.3.5.7 | 2401/2400, 4375/4374, 32805/32768 | [⟨171 271 397 480]] | +0.105 | 0.0614 | 0.87 |
| 2.3.5.7.11 | 243/242, 441/440, 4375/4356, 16384/16335 | [⟨171 271 397 480 592]] | −0.093 | 0.401 | 5.71 |
| 2.3.5.7.11.13 | 243/242, 364/363, 441/440, 625/624, 2200/2197 | [⟨171 271 397 480 592 633]] | −0.149 | 0.386 | 5.50 |
| 2.3.5.7.11.13.17 | 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197 | [⟨171 271 397 480 592 633 699]] | −0.138 | 0.358 | 5.11 |
| 2.3.5.7.11 | 385/384, 1331/1323, 1375/1372, 4375/4374 | [⟨171 271 397 480 591]] (171e) | +0.312 | 0.418 | 5.96 |
| 2.3.5.7.13 | 625/624, 729/728, 2205/2197, 2401/2400 | [⟨171 271 397 480 633]] | −0.001 | 0.220 | 3.13 |
| 2.3.5.7.13.17 | 625/624, 729/728, 833/832, 1225/1224, 2205/2197 | [⟨171 271 397 480 633 699]] | −0.013 | 0.202 | 2.88 |
- 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 441 do we find a better equal temperaments in terms of absolute error, and not until 3125 do we find one in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 5\171 | 35.09 | 234375/229376 | Gammic |
| 1 | 11\171 | 77.19 | 256/245 | Tertiaseptal / tertia (171e) |
| 1 | 17\171 | 119.30 | 15/14 | Septidiasemi / sedia |
| 1 | 20\171 | 140.35 | 243/224 | Tsaharuk |
| 1 | 25\171 | 175.44 | 448/405 | Sesquiquartififths / sesquart |
| 1 | 26\171 | 182.46 | 10/9 | Minortone / mitonic / mineral (171) / ore (171e) / goldmine (171ef) |
| 1 | 34\171 | 238.60 | 147/128 | Tokko |
| 1 | 46\171 | 322.81 | 3087/2560 | Senior / seniority |
| 1 | 49\171 | 343.86 | 8000/6561 | Geb |
| 1 | 56\171 | 392.98 | 2744/2187 | Emmthird |
| 1 | 61\171 | 428.07 | 2800/2187 | Osiris |
| 1 | 62\171 | 435.09 | 9/7 | Supermajor |
| 1 | 64\171 | 449.12 | 35/27 | Semidimi |
| 1 | 65\171 | 456.14 | 125/96 | Qak |
| 1 | 70\171 | 491.23 | 3645/2744 | Fifthplus |
| 1 | 71\171 | 498.25 | 4/3 | Helmholtz / pontiac |
| 1 | 83\171 | 582.46 | 7/5 | Neptune |
| 3 | 20\171 | 140.35 | 243/224 | Septichrome |
| 3 | 23\171 | 161.40 | 192/175 | Pnict |
| 3 | 26\171 | 182.46 | 10/9 | Terrain / domain |
| 3 | 55\171 (2\171) |
385.96 (14.04) |
5/4 (126/125) |
Mutt |
| 3 | 71\171 (14\171) |
498.25 (98.25) |
4/3 (200/189) |
Term / terminal / terminator |
| 9 | 45\171 (7\171) |
315.79 (49.12) |
6/5 (36/35) |
Ennealimmal (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171ef) |
| 9 | 10\171 | 70.17 | (336/323) | Enneasoteric (171f) |
| 19 | 71\171 (1\171) |
498.25 (7.02) |
4/3 (225/224) |
Enneadecal |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
Music
Modern renderings
- "Prelude" from Prelude and Fugue in C major, No. 1, BWV 846, from The Well-Tempered Clavier, Book I (1722) – rendered by レケム (2022)