171edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
Things |
||
| Line 7: | Line 7: | ||
Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -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 [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|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)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -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 [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|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)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | ||
171edo is notably [[Consistency#Consistency to distance d|consistent to distance '''6''']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance '''8''' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit 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 comma|glacier]], and is generally a | 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 comma|glacier]], and is generally a good [[2.3.5.7.13 subgroup|2.3.5.7.13]] and better 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. | 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. | ||
| Line 18: | Line 18: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
171 factors into primes as {{nowrap| 3<sup>2</sup> × 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. | 171 factors into primes as {{nowrap| 3<sup>2</sup> × 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 and is one of the most accurate [[11-limit]] temperaments, with unmatched relative error up until [[1848edo]]. | |||
[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency. | |||
== Intervals == | == Intervals == | ||
Revision as of 08:10, 10 April 2026
| ← 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⟩.
171edo is notably consistent to distance 6 in the 7-prime 15-odd-limit, and to distance 8 in the 9-odd-limit. No other edo is so consistent until 3125edo. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit 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 good 2.3.5.7.13 and better 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 and is one of the most accurate 11-limit temperaments, with unmatched relative error up until 1848edo.
684edo, which quadruples it, achieves 17-odd-limit consistency.
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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
|
 |
|
|
  |
 |
|
|
|
|
|
|
|
 |
|
|
|
|
|
|
|
 |
|
|
|
|
|
|
|
 |
|
|
|
|
| Flat symbol | |
|
|
 |
 |
|
|
|
|
|
|
|
 |
|
|
|
|
|
|
|
 |
|
|
|
|
|
|
|
 |
|
|
|
|
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)