323edo: Difference between revisions

Line 5:

323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

~~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]].

It [[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.

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

Since 323 factors into {{factorisation|323}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].

=== Prime harmonics ===

Line 29:

| {{monzo| 512 -323 }}

| {{mapping| 323 512 }}

| ~~−0~~.0669

| −0.0669

| 0.0669

| 1.80

Line 36:

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

| {{mapping| 323 512 750 }}

| ~~−0~~.0538

| −0.0538

| 0.0577

| 1.55

Line 43:

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

| {{mapping| 323 512 750 907 }}

| ~~−0~~.1146

| −0.1146

| 0.1165

| 3.14

Line 50:

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

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

| ~~−0~~.0431

| −0.0431

| 0.1770

| 4.76

Line 64:

| 1375/1372, 4375/4374, 5632/5625, 14641/14580

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

| ~~−0~~.0066

| −0.0066

| 0.2399

| 6.46

Line 78:

| 540/539, 4375/4374, 12005/11979, 16384/16335

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

| ~~−0~~.2213

| −0.2213

| 0.2375

| 6.39

Line 85:

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

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

| ~~−0~~.1440

| −0.1440

| 0.2773

| 7.47

Line 149:

| [[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

[[Category:Deuteromere]]

[[Category:Lambeth]]

[[Category:Stockhausenic]]

@@ Line 5: / Line 5: @@
 edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].
-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]].
+It [[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.
-Since {{nowrap|323 {{=}} 17 &times; 19}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].
+Since 323 factors into {{factorisation|323}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].
 === Prime harmonics ===
@@ Line 29: / Line 29: @@
 | {{monzo| 512 -323 }}
 | {{mapping| 323 512 }}
-| &minus;0.0669
+| −0.0669
 | 0.0669
 | 1.80
@@ Line 36: / Line 36: @@
 | {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
 | {{mapping| 323 512 750 }}
-| &minus;0.0538
+| −0.0538
 | 0.0577
 | 1.55
@@ Line 43: / Line 43: @@
 | 4375/4374, 589824/588245, 703125/702464
 | {{mapping| 323 512 750 907 }}
-| &minus;0.1146
+| −0.1146
 | 0.1165
 | 3.14
@@ Line 50: / Line 50: @@
 | 676/675, 4096/4095, 4375/4374, 16848/16807
 | {{mapping| 323 512 750 907 1195 }}
-| &minus;0.0431
+| −0.0431
 | 0.1770
 | 4.76
@@ Line 64: / Line 64: @@
 | 1375/1372, 4375/4374, 5632/5625, 14641/14580
 | {{mapping| 323 512 750 907 1117 }} (323)
-| &minus;0.0066
+| −0.0066
 | 0.2399
 | 6.46
@@ Line 78: / Line 78: @@
 | 540/539, 4375/4374, 12005/11979, 16384/16335
 | {{mapping| 323 512 750 907 1118 }} (323e)
-| &minus;0.2213
+| −0.2213
 | 0.2375
 | 6.39
@@ Line 85: / Line 85: @@
 | 364/363, 540/539, 676/675, 4096/4095, 4375/4374
 | {{mapping| 323 512 750 907 1118 1195 }} (323e)
-| &minus;0.1440
+| −0.1440
 | 0.2773
 | 7.47
@@ Line 149: / Line 149: @@
 | [[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
 [[Category:Deuteromere]]
 [[Category:Lambeth]]
 [[Category:Stockhausenic]]