1911edo: Difference between revisions

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== Theory ==

1911edo is consistent in the 11-limit and tempers out the [[aluminium comma]] in the 5-limit. It provides the [[optimal patent val]] for the [[protactinium]] temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 subgroup tuning.

1911edo is [[consistent]] in the 11-limit and [[Tempering out|tempers out]] the [[aluminium comma]] in the 5-limit. It provides the [[optimal patent val]] for the [[protactinium]] temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 [[subgroup]] tuning.

=== Odd harmonics ===

=== Subsets and supersets ===

1911edo has subset edos {{EDOs|1, 3, 7, 13, 21, 39, 49, 91, 147, 273, 637}}.

Since 1911 factors into 3 × 72 × 13, 1911edo has subset edos {{EDOs| 3, 7, 13, 21, 39, 49, 91, 147, 273, 637 }}.

== Regular temperament properties ==

=== 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

|-

|13

| 13

|793\1911 (58\1911)

| 793\1911 (58\1911)

|497.959 (36.421)

| 497.959 (36.421)

|4/3 (?)

| 4/3 (?)

|[[Aluminium]]

| [[Aluminium]]

|-

|91

| 91

|793\1911 (16\1911)

| 793\1911 (16\1911)

|497.959 (10.047)

| 497.959 (10.047)

|4/3 (176/175)

| 4/3 (176/175)

|[[Protactinium]]

| [[Protactinium]]

|}

<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 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|1911}}
 == Theory ==
-edo is consistent in the 11-limit and tempers out the [[aluminium comma]] in the 5-limit. It provides the [[optimal patent val]] for the [[protactinium]] temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 subgroup tuning.
+edo is [[consistent]] in the 11-limit and [[Tempering out|tempers out]] the [[aluminium comma]] in the 5-limit. It provides the [[optimal patent val]] for the [[protactinium]] temperament in the 17-limit. However as may stem from consistency only in the 11-limit, the 13th harmonic has a large relative error. As such, 1911edo is best considered as a 2.3.5.7.11.17.19 [[subgroup]] tuning.
 === Odd harmonics ===
 {{Harmonics in equal|1911}}
 === Subsets and supersets ===
-edo has subset edos {{EDOs|1, 3, 7, 13, 21, 39, 49, 91, 147, 273, 637}}.
+Since 1911 factors into 3 × 7<sup>2</sup> × 13, 1911edo has subset edos {{EDOs| 3, 7, 13, 21, 39, 49, 91, 147, 273, 637 }}.
 == Regular temperament properties ==
 === 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
 |-
-|13
+| 13
-|793\1911<br>(58\1911)
+| 793\1911<br>(58\1911)
-|497.959<br>(36.421)
+| 497.959<br>(36.421)
-|4/3<br>(?)
+| 4/3<br>(?)
-|[[Aluminium]]
+| [[Aluminium]]
 |-
-|91
+| 91
-|793\1911<br>(16\1911)
+| 793\1911<br>(16\1911)
-|497.959<br>(10.047)
+| 497.959<br>(10.047)
-|4/3<br>(176/175)
+| 4/3<br>(176/175)
-|[[Protactinium]]
+| [[Protactinium]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct