171edo: Difference between revisions

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=== 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.

171 factors into primes as {{nowrap|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 ==

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== Regular temperament properties ==

{~~| class="wikitable center-4 center-5 center-6"~~

~~! rowspan="2" | [[Subgroup]]~~

~~! rowspan="2" | [[Comma list|Comma List]]~~

~~! rowspan="2" | [[Mapping]]~~

~~! rowspan="2" | Optimal 8ve Stretch (¢)~~

~~! colspan="2" | Tuning Error~~

|-

~~! [[TE error|Absolute]] (¢)~~

~~! [[TE simple badness|Relative]] (%)~~

|-

| 2.3

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| 0.0614

| 0.87

|-style="border-top: double;"

|- style="border-top: double;"

| 2.3.5.7.11

| 243/242, 441/440, 4375/4356, 16384/16335

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| 0.358

| 5.11

|-style="border-top: double;"

|- style="border-top: double;"

| 2.3.5.7.11

| 385/384, 1331/1323, 1375/1372, 4375/4374

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| 0.418

| 5.96

|-style="border-top: double;"

|- style="border-top: double;"

| 2.3.5.7.13

| 625/624, 729/728, 2205/2197, 2401/2400

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| 0.202

| 2.88

|}

* 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 ===

{~~| class="wikitable center-all left-5"~~

~~|+Table of~~ rank-2 ~~temperaments by generator~~

~~! Periods per 8ve~~

! Generator*

! Cents*

! Associated Ratio*

~~! Temperaments~~

|-

| 1

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| 4/3 (225/224)

| [[Enneadecal]]

|}

~~<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct~~

== Scales ==

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=== Modern renderings ===

; {{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)

=== 21st century ===

; [[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]

== See also ==

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* [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:~~Ennealimmal~~]]

[[Category:Listen]]

[[Category:~~Enneadecal~~]]

[[Category:Mitonic]]

[[Category:Mutt]]

[[Category:Neptune]]

[[Category:Pontiac]]

[[Category:Sesquiquartififths]]

[[Category:Term (temperament)]]

[[Category:Tertiaseptal]]

~~[[Category:Gammic]]~~

~~[[Category:Neptune]]~~

~~[[Category:Mitonic]]~~

~~[[Category:Mutt]]~~

~~[[Category:Listen]]~~

@@ Line 15: / Line 15: @@
 === Subsets and supersets ===
-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.
+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 ==
@@ Line 21: / Line 21: @@
 == 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
@@ Line 51: / Line 43: @@
 | 0.0614
 | 0.87
-|-style="border-top: double;"
+|- style="border-top: double;"
 | 2.3.5.7.11
 | 243/242, 441/440, 4375/4356, 16384/16335
@@ Line 72: / Line 64: @@
 | 0.358
 | 5.11
-|-style="border-top: double;"
+|- style="border-top: double;"
 | 2.3.5.7.11
 | 385/384, 1331/1323, 1375/1372, 4375/4374
@@ Line 79: / Line 71: @@
 | 0.418
 | 5.96
-|-style="border-top: double;"
+|- style="border-top: double;"
 | 2.3.5.7.13
 | 625/624, 729/728, 2205/2197, 2401/2400
@@ Line 93: / Line 85: @@
 | 0.202
 | 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.
 === 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
@@ Line 254: / Line 240: @@
 | 4/3<br>(225/224)
 | [[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 ==
@@ Line 264: / Line 250: @@
 === Modern renderings ===
 ; {{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 ===
 ; [[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 ==
@@ Line 276: / Line 262: @@
 * [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:Ennealimmal]]
+[[Category:Listen]]
-[[Category:Enneadecal]]
+[[Category:Mitonic]]
+[[Category:Mutt]]
+[[Category:Neptune]]
 [[Category:Pontiac]]
 [[Category:Sesquiquartififths]]
 [[Category:Term (temperament)]]
 [[Category:Tertiaseptal]]
-[[Category:Gammic]]
-[[Category:Neptune]]
-[[Category:Mitonic]]
-[[Category:Mutt]]
-[[Category:Listen]]