323edo: Difference between revisions

Line 3:

== Theory ==

~~323et~~ [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

323 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].

323edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the 31-limit.

323edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the [[31-limit]].

=== Prime harmonics ===

Line 25:

| 2.3

| {{monzo| 512 -323 }}

| [{{~~val~~| 323 512 }}]

| {{mapping| 323 512 }}

| -0.0669

| 0.0669

Line 32:

| 2.3.5

| {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}

| [{{~~val~~| 323 512 750 }}]

| {{mapping| 323 512 750 }}

| -0.0538

| 0.0577

Line 39:

| 2.3.5.7

| 4375/4374, 589824/588245, 703125/702464

| [{{~~val~~| 323 512 750 907 }}]

| {{mapping| 323 512 750 907 }}

| -0.1146

| 0.1165

Line 46:

| 2.3.5.7.13

| 676/675, 4096/4095, 4375/4374, 16848/16807

| [{{~~val~~| 323 512 750 907 1195 }}]

| {{mapping| 323 512 750 907 1195 }}

| -0.0431

| 0.1770

Line 53:

| 2.3.5.7.13.17

| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374

| [{{~~val~~| 323 512 750 907 1195 1320 }}]

| {{mapping| 323 512 750 907 1195 1320 }}

| +0.0020

| 0.1905

Line 60:

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

| style="border-top: double;" | 540/539, 4375/4374, 12005/11979, 16384/16335

| style="border-top: double;" | [{{~~val~~| 323 512 750 907 1118 }}] (323e)

| style="border-top: double;" | {{mapping| 323 512 750 907 1118 }} (323e)

| style="border-top: double;" | -0.2213

| style="border-top: double;" | 0.2375

Line 67:

| 2.3.5.7.11.13

| 364/363, 540/539, 676/675, 4096/4095, 4375/4374

| [{{~~val~~| 323 512 750 907 1118 1195 }}] (323e)

| {{mapping| 323 512 750 907 1118 1195 }} (323e)

| -0.1440

| 0.2773

Line 74:

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

| style="border-top: double;" | 1375/1372, 4375/4374, 5632/5625, 14641/14580

| style="border-top: double;" | [{{~~val~~| 323 512 750 907 1117 }}] (323)

| style="border-top: double;" | {{mapping| 323 512 750 907 1117 }} (323)

| style="border-top: double;" | -0.0066

| style="border-top: double;" | 0.2399

Line 81:

| 2.3.5.7.11.13

| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374

| [{{~~val~~| 323 512 750 907 1117 1195 }}] (323)

| {{mapping| 323 512 750 907 1117 1195 }} (323)

| +0.0350

| 0.2380

Line 91:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

Line 144:

| [[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

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

[[Category:Deuteromere]]

[[Category:Stockhausenic]]

[[Category:Lambeth]]

@@ Line 3: / Line 3: @@
 == Theory ==
-et [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.
+The equal temperament [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], 589824/588245 and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]]. In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, 5632/5625, [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.
 = 17 × 19, and shares the excellent approximations of [[25/24]] in [[17edo]] and of the [[28/27]] and the [[6/5]] in [[19edo]].
-edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the 31-limit.
+edo is an excellent tuning when considered in the no-11 subgroup, with errors of 25% or less all the way into the [[31-limit]].
 === Prime harmonics ===
@@ Line 25: / Line 25: @@
 | 2.3
 | {{monzo| 512 -323 }}
-| [{{val| 323 512 }}]
+| {{mapping| 323 512 }}
 | -0.0669
 | 0.0669
@@ Line 32: / Line 32: @@
 | 2.3.5
 | {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
-| [{{val| 323 512 750 }}]
+| {{mapping| 323 512 750 }}
 | -0.0538
 | 0.0577
@@ Line 39: / Line 39: @@
 | 2.3.5.7
 | 4375/4374, 589824/588245, 703125/702464
-| [{{val| 323 512 750 907 }}]
+| {{mapping| 323 512 750 907 }}
 | -0.1146
 | 0.1165
@@ Line 46: / Line 46: @@
 | 2.3.5.7.13
 | 676/675, 4096/4095, 4375/4374, 16848/16807
-| [{{val| 323 512 750 907 1195 }}]
+| {{mapping| 323 512 750 907 1195 }}
 | -0.0431
 | 0.1770
@@ Line 53: / Line 53: @@
 | 2.3.5.7.13.17
 | 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
-| [{{val| 323 512 750 907 1195 1320 }}]
+| {{mapping| 323 512 750 907 1195 1320 }}
 | +0.0020
 | 0.1905
@@ Line 60: / Line 60: @@
 | style="border-top: double;" | 2.3.5.7.11
 | style="border-top: double;" | 540/539, 4375/4374, 12005/11979, 16384/16335
-| style="border-top: double;" | [{{val| 323 512 750 907 1118 }}] (323e)
+| style="border-top: double;" | {{mapping| 323 512 750 907 1118 }} (323e)
 | style="border-top: double;" | -0.2213
 | style="border-top: double;" | 0.2375
@@ Line 67: / Line 67: @@
 | 2.3.5.7.11.13
 | 364/363, 540/539, 676/675, 4096/4095, 4375/4374
-| [{{val| 323 512 750 907 1118 1195 }}] (323e)
+| {{mapping| 323 512 750 907 1118 1195 }} (323e)
 | -0.1440
 | 0.2773
@@ Line 74: / Line 74: @@
 | style="border-top: double;" | 2.3.5.7.11
 | style="border-top: double;" | 1375/1372, 4375/4374, 5632/5625, 14641/14580
-| style="border-top: double;" | [{{val| 323 512 750 907 1117 }}] (323)
+| style="border-top: double;" | {{mapping| 323 512 750 907 1117 }} (323)
 | style="border-top: double;" | -0.0066
 | style="border-top: double;" | 0.2399
@@ Line 81: / Line 81: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
-| [{{val| 323 512 750 907 1117 1195 }}] (323)
+| {{mapping| 323 512 750 907 1117 1195 }} (323)
 | +0.0350
 | 0.2380
@@ Line 91: / Line 91: @@
 |+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 144: / Line 144: @@
 | [[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
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Deuteromere]]
 [[Category:Stockhausenic]]
 [[Category:Lambeth]]