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 7-limit microtemperament, 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. It is also almost consistent in the 17-odd-limit, only missing 15/11 and 22/15.
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). Therefore, 171et supports a number of notable 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 helmholtz temperament, tempering out 32805/32768, and the no-fives temperament tempering out [-59 39 0 -1⟩.
All intervals with an odd limit below 10,000,000 in the 7-limit are mapped consistently by 171edo. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify commas by size. For example, one step represents 225/224, two steps 126/125, three steps 81/80, and four steps 64/63.
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. In the 2.3.13 subgroup, it provides the optimal patent val for glacier, and is generally a great 2.3.5.7.13 and 2.3.5.7.13.17 subgroup temperament.
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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | +3.07 | +1.58 | +0.31 | -2.78 | +3.30 | +2.00 | -1.18 |
| Relative (%) | +0.0 | -2.9 | -5.0 | -5.8 | +43.7 | +22.5 | +4.4 | -39.6 | +47.1 | +28.5 | -16.8 | |
| Steps (reduced) |
171 (0) |
271 (100) |
397 (55) |
480 (138) |
592 (79) |
633 (120) |
699 (15) |
726 (42) |
774 (90) |
831 (147) |
847 (163) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.29 | -0.99 | +0.76 | +1.16 | -3.33 | +0.48 | -1.10 | -2.11 | +2.76 | -3.23 | +0.38 |
| Relative (%) | +18.3 | -14.1 | +10.9 | +16.5 | -47.4 | +6.8 | -15.6 | -30.1 | +39.3 | -46.0 | +5.3 | |
| Steps (reduced) |
891 (36) |
916 (61) |
928 (73) |
950 (95) |
979 (124) |
1006 (151) |
1014 (159) |
1037 (11) |
1052 (26) |
1058 (32) |
1078 (52) | |
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. 342edo, which doubles 171, provides an excellent correction for the 11th harmonic.
Intervals
Notation
Ups and downs notation
171edo can be notated using ups and downs with quarter-tone accidentals:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 |
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Approximation to JI
15-odd-limit intervals
The following tables show how 15-odd-limit intervals are represented in 171edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 9/7, 14/9 | 0.004 | 0.1 |
| 9/5, 10/9 | 0.052 | 0.7 |
| 7/5, 10/7 | 0.056 | 0.8 |
| 15/14, 28/15 | 0.145 | 2.1 |
| 5/3, 6/5 | 0.148 | 2.1 |
| 3/2, 4/3 | 0.201 | 2.9 |
| 7/6, 12/7 | 0.204 | 2.9 |
| 5/4, 8/5 | 0.349 | 5.0 |
| 9/8, 16/9 | 0.401 | 5.7 |
| 7/4, 8/7 | 0.405 | 5.8 |
| 15/8, 16/15 | 0.549 | 7.8 |
| 13/11, 22/13 | 1.490 | 21.2 |
| 13/8, 16/13 | 1.578 | 22.5 |
| 13/12, 24/13 | 1.778 | 25.3 |
| 13/10, 20/13 | 1.926 | 27.5 |
| 13/9, 18/13 | 1.979 | 28.2 |
| 13/7, 14/13 | 1.982 | 28.2 |
| 15/13, 26/15 | 2.127 | 30.3 |
| 11/8, 16/11 | 3.068 | 43.7 |
| 11/6, 12/11 | 3.269 | 46.6 |
| 15/11, 22/15 | 3.400 | 48.5 |
| 11/10, 20/11 | 3.417 | 48.7 |
| 11/9, 18/11 | 3.469 | 49.4 |
| 11/7, 14/11 | 3.473 | 49.5 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 9/7, 14/9 | 0.004 | 0.1 |
| 9/5, 10/9 | 0.052 | 0.7 |
| 7/5, 10/7 | 0.056 | 0.8 |
| 15/14, 28/15 | 0.145 | 2.1 |
| 5/3, 6/5 | 0.148 | 2.1 |
| 3/2, 4/3 | 0.201 | 2.9 |
| 7/6, 12/7 | 0.204 | 2.9 |
| 5/4, 8/5 | 0.349 | 5.0 |
| 9/8, 16/9 | 0.401 | 5.7 |
| 7/4, 8/7 | 0.405 | 5.8 |
| 15/8, 16/15 | 0.549 | 7.8 |
| 13/11, 22/13 | 1.490 | 21.2 |
| 13/8, 16/13 | 1.578 | 22.5 |
| 13/12, 24/13 | 1.778 | 25.3 |
| 13/10, 20/13 | 1.926 | 27.5 |
| 13/9, 18/13 | 1.979 | 28.2 |
| 13/7, 14/13 | 1.982 | 28.2 |
| 15/13, 26/15 | 2.127 | 30.3 |
| 11/8, 16/11 | 3.068 | 43.7 |
| 11/6, 12/11 | 3.269 | 46.6 |
| 11/10, 20/11 | 3.417 | 48.7 |
| 11/9, 18/11 | 3.469 | 49.4 |
| 11/7, 14/11 | 3.473 | 49.5 |
| 15/11, 22/15 | 3.617 | 51.5 |
Consistent circles
171edo contains consistent circles of 7/6, 6/5, and 9/7, each with 9, 19, and 171 notes respectively.
| Note count |
Interval | Closing error |
Consistency | Associated edostep |
|---|---|---|---|---|
| 9 | 7/6 | -26.2% | Normal | 2\9 = 38\171 |
| 19 | 6/5 | +40.1% | Normal | 5\19 = 45\171 |
| 171 | 9/7 | +8.8% | Strong | 62\171 |
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 and in the 9-odd-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* |
Temperament |
|---|---|---|---|---|
| 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 distinct
Scales
- Nestoria7
- Nestoria12
- Diamond9: 26 3 4 5 7 10 7 9 12 5 12 9 7 10 7 5 4 3 26
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)