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 [[~~The Riemann zeta function and tuning #Zeta EDO lists|~~zeta integral edo]] but not enough to make it a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~|gap ~~edo~~]].

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]].

Remarkable 5-limit commas 171et 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 comma]]). 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 [[~~Schismatic family~~|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).

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 comma]]). 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]]).

~~171 factors into primes~~ as 3~~2<~~/~~sup> × 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 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.

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 ===

=== 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 ==

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

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

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

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

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

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

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| 2.3

| {{monzo| -271 171 }}

| [{{~~val~~| 171 271 }}]

| {{mapping| 171 271 }}

| +0.063

| 0.0633

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| 2.3.5

| 32805/32768, {{monzo| 1 -27 18 }}

| [{{~~val~~| 171 271 397 }}]

| {{mapping| 171 271 397 }}

| +0.092

| 0.0660

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| 2.3.5.7

| 2401/2400, 4375/4374, 32805/32768

| [{{~~val~~| 171 271 397 480 }}]

| {{mapping| 171 271 397 480 }}

| +0.105

| 0.0614

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| style="border-top: double;" | 2.3.5.7.11

| style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335

| style="border-top: double;" | [{{~~val~~| 171 271 397 480 592 }}]

| style="border-top: double;" | {{mapping| 171 271 397 480 592 }}

| style="border-top: double;" | −0.093

| style="border-top: double;" | 0.401

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| 2.3.5.7.11.13

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

| [{{~~val~~| 171 271 397 480 592 633 }}]

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

| −0.149

| 0.386

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| 2.3.5.7.11.13.17

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

| [{{~~val~~| 171 271 397 480 592 633 699 }}]

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

| −0.138

| 0.358

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| style="border-top: double;" | 2.3.5.7.11

| style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374

| style="border-top: double;" | [{{~~val~~| 171 271 397 480 591 }}] (171e)

| style="border-top: double;" | {{mapping| 171 271 397 480 591 }} (171e)

| style="border-top: double;" | +0.312

| style="border-top: double;" | 0.418

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| style="border-top: double;" | 2.3.5.7.13

| style="border-top: double;" | 625/624, 729/728, 2205/2197, 2401/2400

| style="border-top: double;" | [{{~~val~~| 171 271 397 480 633 }}]

| style="border-top: double;" | {{mapping| 171 271 397 480 633 }}

| style="border-top: double;" | −0.001

| style="border-top: double;" | 0.220

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| 2.3.5.7.13.17

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

| [{{~~val~~| 171 271 397 480 633 699 }}]

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

| −0.013

| 0.202

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator ~~ (Reduced)~~

! Generator*

! Cents ~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| 182.46

| 10/9

| [[~~Chromatic pairs #Terrain|~~Terrain]] / [[domain]]

| [[Terrain]] / [[domain]]

|-

| 3

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| 498.25 (98.25)

| 4/3 (200/189)

| [[~~Schismatic_family#~~Term|Term]] / terminal / terminator

| [[Term (temperament)|Term]] / terminal / terminator

|-

| 9

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== Music ==

; [[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]

; [[レケム]]

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

~~[[Category:171edo| ]] ~~

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Horwell]]

[[Category:Ennealimmal]]

@@ 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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] but not enough to make it a [[The Riemann zeta function and tuning #Zeta EDO lists|gap edo]].
+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]].
-Remarkable 5-limit commas 171et 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 comma]]). 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 [[Schismatic family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
+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 comma]]). 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]]).
-factors into primes as 3<sup>2</sup> × 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.
+edo 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]].
-edo 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]].
+edo 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.
-edo 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 ===
 {{Harmonics in equal|171|columns=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.
 == Intervals ==
@@ Line 22: / Line 23: @@
 {| class="wikitable center-4 center-5 center-6"
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
 ! rowspan="2" | Optimal<br>8ve Stretch (¢)
@@ Line 32: / Line 33: @@
 | 2.3
 | {{monzo| -271 171 }}
-| [{{val| 171 271 }}]
+| {{mapping| 171 271 }}
 | +0.063
 | 0.0633
@@ Line 39: / Line 40: @@
 | 2.3.5
 | 32805/32768, {{monzo| 1 -27 18 }}
-| [{{val| 171 271 397 }}]
+| {{mapping| 171 271 397 }}
 | +0.092
 | 0.0660
@@ Line 46: / Line 47: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, 32805/32768
-| [{{val| 171 271 397 480 }}]
+| {{mapping| 171 271 397 480 }}
 | +0.105
 | 0.0614
@@ Line 53: / Line 54: @@
 | style="border-top: double;" | 2.3.5.7.11
 | style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335
-| style="border-top: double;" | [{{val| 171 271 397 480 592 }}]
+| style="border-top: double;" | {{mapping| 171 271 397 480 592 }}
 | style="border-top: double;" | &minus;0.093
 | style="border-top: double;" | 0.401
@@ Line 60: / Line 61: @@
 | 2.3.5.7.11.13
 | 243/242, 364/363, 441/440, 625/624, 2200/2197
-| [{{val| 171 271 397 480 592 633 }}]
+| {{mapping| 171 271 397 480 592 633 }}
 | &minus;0.149
 | 0.386
@@ Line 67: / Line 68: @@
 | 2.3.5.7.11.13.17
 | 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197
-| [{{val| 171 271 397 480 592 633 699 }}]
+| {{mapping| 171 271 397 480 592 633 699 }}
 | &minus;0.138
 | 0.358
@@ Line 74: / Line 75: @@
 | style="border-top: double;" | 2.3.5.7.11
 | style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374
-| style="border-top: double;" | [{{val| 171 271 397 480 591 }}] (171e)
+| style="border-top: double;" | {{mapping| 171 271 397 480 591 }} (171e)
 | style="border-top: double;" | +0.312
 | style="border-top: double;" | 0.418
@@ Line 81: / Line 82: @@
 | style="border-top: double;" | 2.3.5.7.13
 | style="border-top: double;" | 625/624, 729/728, 2205/2197, 2401/2400
-| style="border-top: double;" | [{{val| 171 271 397 480 633 }}]
+| style="border-top: double;" | {{mapping| 171 271 397 480 633 }}
 | style="border-top: double;" | &minus;0.001
 | style="border-top: double;" | 0.220
@@ Line 88: / Line 89: @@
 | 2.3.5.7.13.17
 | 625/624, 729/728, 833/832, 1225/1224, 2205/2197
-| [{{val| 171 271 397 480 633 699 }}]
+| {{mapping| 171 271 397 480 633 699 }}
 | &minus;0.013
 | 0.202
@@ Line 99: / Line 100: @@
 |+Table of rank-2 temperaments by generator
 ! Periods <br>per 8ve
-! Generator <br>(Reduced)
+! Generator*
-! Cents <br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 222: / Line 223: @@
 | 182.46
 | 10/9
-| [[Chromatic pairs #Terrain|Terrain]] / [[domain]]
+| [[Terrain]] / [[domain]]
 |-
 | 3
@@ Line 234: / Line 235: @@
 | 498.25<br>(98.25)
 | 4/3<br>(200/189)
-| [[Schismatic_family#Term|Term]] / terminal / terminator
+| [[Term (temperament)|Term]] / terminal / terminator
 |-
 | 9
@@ Line 261: / Line 262: @@
 == Music ==
 ; [[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]
 ; [[レケム]]
@@ Line 272: / Line 273: @@
 * [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
-[[Category:171edo| ]]  <!-- main article -->
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Horwell]]
 [[Category:Ennealimmal]]