1323edo: Difference between revisions

Line 3:

== 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 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. In addition, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.

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 ===

1323's divisors are {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 441edo is a member of the [[zeta edo]]s. 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.

== 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

! Periods per 8ve

!Generator~~ (reduced)~~

! Generator*

!Cents~~ (reduced)~~

! Cents*

!Associated ~~ratio~~

! Associated Ratio

!Temperaments

! Temperaments

<!--

|-

|3

| 3

|177\1323

| 177\1323

|160.544

| 160.544

|154478651796875/140737488355328

| 154478651796875/140737488355328

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

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

-->

|-

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

@@ Line 3: / Line 3: @@
 == 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 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. In addition, it tunes well 441 & 1308 temperament, which is a member of the augmented-cloudy equivalence continuum.
-'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 ===
+'s divisors are {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 441edo is a member of the [[zeta edo]]s. 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.
 == 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
+! Periods<br>per 8ve
-!Generator<br>(reduced)
+! Generator*
-!Cents<br>(reduced)
+! Cents*
-!Associated<br>ratio
+! Associated<br>Ratio
-!Temperaments
+! Temperaments
+<!--
 |-
-|3
+| 3
-|177\1323
+| 177\1323
-|160.544
+| 160.544
-|154478651796875/140737488355328
+| 154478651796875/140737488355328
-|[[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]]
+| [[Augmented-cloudy equivalence continuum#441 & 1308|441 & 1308]]
+-->
 |-
-|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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct