113edo: Difference between revisions

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== Theory ==

113edo is ~~distinctly~~ [[consistent]] in the [[13-odd-limit]] with a flat tendency. As a temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

113edo might be notable as a no-fives system, where it is consistent in the [[29-odd-limit]] and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17.29 extension of slendric harmonies known as [[euslendric]].

=== Prime harmonics ===

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=== Subsets and supersets ===

113edo is the 30th [[prime edo]].

113edo is the 30th [[prime edo]], following [[109edo]] and before [[127edo]].

== Intervals ==

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| -179 113 }}

| [{{~~val~~| 113 179 }}]

| {{mapping| 113 179 }}

| +0.338

| 0.338

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| 2.3.5

| 1600000/1594323, 34171875/33554432

| [{{~~val~~| 113 179 262 }}]

| {{mapping| 113 179 262 }}

| +0.801

| 0.712

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| 2.3.5.7

| 225/224, 1029/1024, 1071875/1062882

| [{{~~val~~| 113 179 262 317 }}]

| {{mapping| 113 179 262 317 }}

| +0.820

| 0.617

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| 2.3.5.7.11

| 225/224, 243/242, 385/384, 980000/970299

| [{{~~val~~| 113 179 262 317 391 }}]

| {{mapping| 113 179 262 317 391 }}

| +0.604

| 0.700

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| 2.3.5.7.11.13

| 225/224, 243/242, 325/324, 385/384, 1875/1859

| [{{~~val~~| 113 179 262 317 391 418 }}]

| {{mapping| 113 179 262 317 391 418 }}

| +0.575

| 0.643

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=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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| [[Gaster temperament|Gaster]]

|}

<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|113}}
+{{ED intro}}
 == Theory ==
-edo is distinctly [[consistent]] in the [[13-odd-limit]] with a flat tendency. As a temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.
+edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.
+edo might be notable as a no-fives system, where it is consistent in the [[29-odd-limit]] and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17.29 extension of slendric harmonies known as [[euslendric]].
 === Prime harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-edo is the 30th [[prime edo]].
+edo is the 30th [[prime edo]], following [[109edo]] and before [[127edo]].
 == Intervals ==
@@ Line 16: / Line 18: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 27: / Line 30: @@
 | 2.3
 | {{monzo| -179 113 }}
-| [{{val| 113 179 }}]
+| {{mapping| 113 179 }}
 | +0.338
 | 0.338
@@ Line 34: / Line 37: @@
 | 2.3.5
 | 1600000/1594323, 34171875/33554432
-| [{{val| 113 179 262 }}]
+| {{mapping| 113 179 262 }}
 | +0.801
 | 0.712
@@ Line 41: / Line 44: @@
 | 2.3.5.7
 | 225/224, 1029/1024, 1071875/1062882
-| [{{val| 113 179 262 317 }}]
+| {{mapping| 113 179 262 317 }}
 | +0.820
 | 0.617
@@ Line 48: / Line 51: @@
 | 2.3.5.7.11
 | 225/224, 243/242, 385/384, 980000/970299
-| [{{val| 113 179 262 317 391 }}]
+| {{mapping| 113 179 262 317 391 }}
 | +0.604
 | 0.700
@@ Line 55: / Line 58: @@
 | 2.3.5.7.11.13
 | 225/224, 243/242, 325/324, 385/384, 1875/1859
-| [{{val| 113 179 262 317 391 418 }}]
+| {{mapping| 113 179 262 317 391 418 }}
 | +0.575
 | 0.643
@@ Line 63: / Line 66: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
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 | [[Gaster temperament|Gaster]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct