1323edo: Difference between revisions

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== Theory ==

1323edo is the smallest ~~uniquely~~ consistent ~~EDO~~ in the 29-odd-limit.

1323edo is the smallest edo [[consistency|distinctly consistent]] in the [[29-odd-limit]]. It is [[enfactoring|enfactored]] in the 7-limit, sharing the same excellent 7-limit approximation with [[441edo]], but it makes for a great higher-limit system by splitting each step of 441edo into three.

It provides the optimal patent val for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave~~. In additoin, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.~~

It provides the [[optimal patent val]] for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave.

1323's divisors are {{EDOs|1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441}}, of which 441EDO is a member of the zeta edos. 1323edo shares the 7-limit mapping with 441edo. As such, it can be interpreted as an improvement for 441edo into the 29-limit by splitting each step of 441edo into three.

=== Prime harmonics ===

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

=== Subsets and supersets ===

Since 1323 factors into {{factorization|1323}}, 1323edo has subset edos {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 7, 27, and 441edo are members of the [[zeta edo]]s.

== Regular temperament properties ==

=== Rank-2 temperaments ===

Note: 7-limit temperaments supported by 441et are not included.

~~=== Rank-2 temperaments ===~~

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

~~!Periods per 8ve~~

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

~~!Generator (reduced)~~

~~!Cents (reduced)~~

~~!Associated ratio~~

~~!Temperaments~~

|-

|3

! Periods per 8ve

~~|177\1323~~

! Generator*

~~|160.544~~

! Cents*

~~|154478651796875~~/~~140737488355328~~

! Associated ratio*

~~|[[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]]~~

! Temperaments

|-

|27

| 27

|299\1323 (5\1323)

| 299\1323 (5\1323)

|271.201 (4.535)

| 271.201 (4.535)

|1375/1176 (?)

| 1375/1176 (?)

|[[Trinealimmal]]

| [[Trinealimmal]]

|}

|}

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1323}}
+{{ED intro}}
 == Theory ==
-edo is the smallest uniquely consistent EDO in the 29-odd-limit.
+edo is the smallest edo [[consistency|distinctly consistent]] in the [[29-odd-limit]]. It is [[enfactoring|enfactored]] in the 7-limit, sharing the same excellent 7-limit approximation with [[441edo]], but it makes for a great higher-limit system by splitting each step of 441edo into three.
-It provides the optimal patent val for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave. In additoin, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.
+It provides the [[optimal patent val]] for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave.
-'s divisors are {{EDOs|1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441}}, of which 441EDO is a member of the zeta edos. 1323edo shares the 7-limit mapping with 441edo. As such, it can be interpreted as an improvement for 441edo into the 29-limit by splitting each step of 441edo into three.
 === Prime harmonics ===
 {{Harmonics in equal|1323}}
-[[Category:Equal divisions of the octave|####]]
+=== Subsets and supersets ===
+Since 1323 factors into {{factorization|1323}}, 1323edo has subset edos {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 7, 27, and 441edo are members of the [[zeta edo]]s.
 == Regular temperament properties ==
+=== Rank-2 temperaments ===
+Note: 7-limit temperaments supported by 441et are not included.
-=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-!Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-!Generator<br>(reduced)
-!Cents<br>(reduced)
-!Associated<br>ratio
-!Temperaments
 |-
-|3
+! Periods<br />per 8ve
-|177\1323
+! Generator*
-|160.544
+! Cents*
-|154478651796875/140737488355328
+! Associated<br />ratio*
-|[[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]]
+! Temperaments
 |-
-|27
+| 27
-|299\1323<br>(5\1323)
+| 299\1323<br />(5\1323)
-|271.201<br>(4.535)
+| 271.201<br />(4.535)
-|1375/1176<br>(?)
+| 1375/1176<br />(?)
-|[[Trinealimmal]]
+| [[Trinealimmal]]
-|}<!-- 4-digit number -->
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct