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
Template:Comma basis begin |- | 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 |- style="border-top: double;" | 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 |- style="border-top: double;" | 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 |- style="border-top: double;" | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
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)