171edo: Difference between revisions

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=== Subsets and supersets ===
=== Subsets and supersets ===
171 factors into primes as 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.
171 factors into primes as {{nowrap|3<sup>2</sup> &times; 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 ==
== Intervals ==
Line 21: Line 21:


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 51: Line 43:
| 0.0614
| 0.0614
| 0.87
| 0.87
|-style="border-top: double;"
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 243/242, 441/440, 4375/4356, 16384/16335
| 243/242, 441/440, 4375/4356, 16384/16335
Line 72: Line 64:
| 0.358
| 0.358
| 5.11
| 5.11
|-style="border-top: double;"
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 1331/1323, 1375/1372, 4375/4374
| 385/384, 1331/1323, 1375/1372, 4375/4374
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| 0.418
| 0.418
| 5.96
| 5.96
|-style="border-top: double;"
|- style="border-top: double;"
| 2.3.5.7.13
| 2.3.5.7.13
| 625/624, 729/728, 2205/2197, 2401/2400
| 625/624, 729/728, 2205/2197, 2401/2400
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| 0.202
| 0.202
| 2.88
| 2.88
|}
{{comma basis end}}
* 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.
* 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"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods <br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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| 4/3<br>(225/224)
| 4/3<br>(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==
Line 264: Line 250:
=== Modern renderings ===
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=7IKpuEqHTyk "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)
* [https://www.youtube.com/watch?v=7IKpuEqHTyk "Prelude" from ''Prelude and Fugue in C major, No. 1'', BWV 846, from ''The Well-Tempered Clavier, Book I''] (1722) &ndash; rendered by [[レケム]] (2022)


=== 21st century ===
=== 21st century ===
; [[birdshite stalactite]]
; [[birdshite stalactite]]
* "it's probably gout" from ''clagworks / it's probably gout'' (2024) – [https://open.spotify.com/track/26bfjsdZ8quDTwAkGY5kQF Spotify] | [https://birdshitestalactite.bandcamp.com/track/its-probably-gout Bandcamp] | [https://www.youtube.com/watch?v=uycaqLtws_w YouTube]
* "it's probably gout" from ''clagworks / it's probably gout'' (2024) &ndash; [https://open.spotify.com/track/26bfjsdZ8quDTwAkGY5kQF Spotify] | [https://birdshitestalactite.bandcamp.com/track/its-probably-gout Bandcamp] | [https://www.youtube.com/watch?v=uycaqLtws_w YouTube]


== See also ==
== See also ==
Line 276: Line 262:
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]


[[Category:Enneadecal]]
[[Category:Ennealimmal]]
[[Category:Gammic]]
[[Category:Horwell]]
[[Category:Horwell]]
[[Category:Ennealimmal]]
[[Category:Listen]]
[[Category:Enneadecal]]
[[Category:Mitonic]]
[[Category:Mutt]]
[[Category:Neptune]]
[[Category:Pontiac]]
[[Category:Pontiac]]
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
[[Category:Term (temperament)]]
[[Category:Term (temperament)]]
[[Category:Tertiaseptal]]
[[Category:Tertiaseptal]]
[[Category:Gammic]]
[[Category:Neptune]]
[[Category:Mitonic]]
[[Category:Mutt]]
[[Category:Listen]]

Revision as of 06:19, 16 November 2024

← 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 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

Approximation of prime harmonics in 171edo
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

Main article: 171edo/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

Johann Sebastian Bach

21st century

birdshite stalactite

See also

External links

Retrieved from "https://en.xen.wiki/index.php?title=171edo&oldid=165991"