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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 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 EDO for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents.  
171edo is an excellent edo 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 ===
=== Prime harmonics ===
{{Primes in edo|171}}
{{Harmonics in equal|171|columns=11}}


== Intervals ==
== Intervals ==
Line 21: Line 21:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 51: Line 51:
| 0.87
| 0.87
|-
|-
| 2.3.5.7.11
| style="border-top: double;" | 2.3.5.7.11
| 243/242, 441/440, 4375/4356, 16384/16335
| style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335
| [{{val| 171 271 397 480 592 }}] (171)
| style="border-top: double;" | [{{val| 171 271 397 480 592 }}] (171)
| -0.093
| style="border-top: double;" | -0.093
| 0.401
| style="border-top: double;" | 0.401
| 5.71
| style="border-top: double;" | 5.71
|-
|-
| 2.3.5.7.11
| style="border-top: double;" | 2.3.5.7.11
| 385/384, 1331/1323, 1375/1372, 4375/4374
| style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374
| [{{val| 171 271 397 480 591 }}] (171e)
| style="border-top: double;" | [{{val| 171 271 397 480 591 }}] (171e)
| +0.312
| style="border-top: double;" | +0.312
| 0.418
| style="border-top: double;" | 0.418
| 5.96
| style="border-top: double;" | 5.96
|}
|}
* 171et is lower in relative error than any previous ETs in the 7-limit. Not until 441 do we find a better ET in terms of absolute error, and not until 3125 do we find one in terms of relative error.
* 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] do we find one in terms of relative error.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-

Revision as of 09:42, 26 November 2022

← 170edo 171edo 172edo →
Prime factorization 32 × 19
Step size 7.01754 ¢ 
Fifth 100\171 (701.754 ¢)
Semitones (A1:m2) 16:13 (112.3 ¢ : 91.23 ¢)
Consistency limit 13
Distinct consistency limit 13

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 gap edo.

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).

171 factors into primes as 32 × 19, and it 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.

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 edo 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

Approximation of prime harmonics in 171edo
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)

Intervals

Main article: 171edo/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]] (171) -0.093 0.401 5.71
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
  • 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		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 71\171
(2\171) 		385.96
(14.04) 		5/4
(126/125) 		Mutt
3 55\171
(2\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)
19 71\171
(1\171) 		498.25
(7.02) 		4/3
(225/224) 		Enneadecal

Scales

See also

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