1323edo: Difference between revisions

(17 intermediate revisions by 8 users not shown)

Line 1:

== 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 ~~trinnealimmal~~ temperament, which has a period of 1\27 octave.

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. 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 ===

=== 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.

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 27

| 299\1323 (5\1323)

| 271.201 (4.535)

| 1375/1176 (?)

| [[Trinealimmal]]

|}

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
-{{EDO intro|1323}}
+{{Infobox ET}}
+{{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 trinnealimmal temperament, which has a period of 1\27 octave.
+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. 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}}
+=== 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.
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 27
+| 299\1323<br />(5\1323)
+| 271.201<br />(4.535)
+| 1375/1176<br />(?)
+| [[Trinealimmal]]
+|}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct