171edo: Difference between revisions

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== 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 edo|zeta gap]].

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 edo|zeta gap]].

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]]). So 171et [[support]]s 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 [[helmholtz|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).

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

| {{mapping| 171 271 397 480 592 }}

| ~~−0~~.093

| −0.093

| 0.401

| 5.71

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| 243/242, 364/363, 441/440, 625/624, 2200/2197

| {{mapping| 171 271 397 480 592 633 }}

| ~~−0~~.149

| −0.149

| 0.386

| 5.50

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| 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197

| {{mapping| 171 271 397 480 592 633 699 }}

| ~~−0~~.138

| −0.138

| 0.358

| 5.11

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

| {{mapping| 171 271 397 480 633 }}

| ~~−0~~.001

| −0.001

| 0.220

| 3.13

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| 625/624, 729/728, 833/832, 1225/1224, 2205/2197

| {{mapping| 171 271 397 480 633 699 }}

| ~~−0~~.013

| −0.013

| 0.202

| 2.88

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| [[Enneadecal]]

|}

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

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

* [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]

* "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 ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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 edo|zeta gap]].
+edo 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 edo|zeta gap]].
 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]]). So 171et [[support]]s 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 [[helmholtz|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
@@ Line 57: / Line 57: @@
 | 243/242, 441/440, 4375/4356, 16384/16335
 | {{mapping| 171 271 397 480 592 }}
-| &minus;0.093
+| −0.093
 | 0.401
 | 5.71
@@ Line 64: / Line 64: @@
 | 243/242, 364/363, 441/440, 625/624, 2200/2197
 | {{mapping| 171 271 397 480 592 633 }}
-| &minus;0.149
+| −0.149
 | 0.386
 | 5.50
@@ Line 71: / Line 71: @@
 | 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197
 | {{mapping| 171 271 397 480 592 633 699 }}
-| &minus;0.138
+| −0.138
 | 0.358
 | 5.11
@@ Line 85: / Line 85: @@
 | 625/624, 729/728, 2205/2197, 2401/2400
 | {{mapping| 171 271 397 480 633 }}
-| &minus;0.001
+| −0.001
 | 0.220
 | 3.13
@@ Line 92: / Line 92: @@
 | 625/624, 729/728, 833/832, 1225/1224, 2205/2197
 | {{mapping| 171 271 397 480 633 699 }}
-| &minus;0.013
+| −0.013
 | 0.202
 | 2.88
@@ Line 258: / Line 258: @@
 | [[Enneadecal]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 267: / Line 267: @@
 === 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) &ndash; 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) – rendered by [[レケム]] (2022)
 === 21st century ===
 ; [[birdshite stalactite]]
-* "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]
+* "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 ==