149edo: Difference between revisions

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~~'''149edo''' is the [[equal division of the octave]] into 149 equal parts of 8.054 [[cent]]s each.~~

== Theory ==

149edo is the smallest division which is ~~uniquely~~ [[consistent]] through the [[17-odd-limit]]~~. It provides the [[optimal patent val]] for 7-, 11-, 13-, and 17-limit [[heinz]] temperament and the rank-3 temperament [[Gamelismic family #Ominous|ominous]] in the 13- and 17-limits~~. It has a general flat tendency, with the fifth 1.28 ~~cents~~ flat, but the major third is a quarter of a cent sharp. In the 5-limit it tempers out the [[sensipent comma]], 78732/78125; in the 7-limit, [[1029/1024]], [[3136/3125]] and [[19683/19600]]; in the 11-limit [[385/384]] and [[441/440]]; in the 13-limit [[351/350]] and [[676/675]]; in the 17-limit [[273/272]] and [[561/560]]; in the 19-limit [[286/285]] and [[343/342]].

149edo is the smallest division which is [[consistency|uniquely consistent]] through the [[17-odd-limit]]. It has a general flat tendency, with the fifth 1.28{{c}} flat, but the major third is a quarter of a cent sharp.

~~149edo~~ is the ~~35th~~ [[~~prime EDO~~]].

In the 5-limit it [[tempering out|tempers out]] the [[sensipent comma]], 78732/78125; in the [[7-limit]], [[1029/1024]], [[3136/3125]] and [[19683/19600]]; in the [[11-limit]] [[385/384]] and [[441/440]]; in the [[13-limit]] [[351/350]], [[676/675]] and [[729/728]]; in the [[17-limit]] [[273/272]] and [[561/560]]; in the [[19-limit]] [[286/285]] and [[343/342]]. It provides the [[optimal patent val]] for 7-, 11-, 13-, and 17-limit [[heinz]] temperament and the rank-3 temperament [[gamelismic family #Ominous|ominous]] in the 13- and 17-limit.

It is also usable in the [[23-limit]], only missing [[19/11]], [[21/11]], and their [[octave complement]]s in the [[23-odd-limit]]. In the [[27-odd-limit]], additional inconsistencies include [[25/21]], [[25/22]], [[27/20]], [[27/25]], [[27/19]], and their octave complements.

=== Prime harmonics ===

=== Subsets and supersets ===

149edo is the 35th [[prime edo]]. As such, it does not contain any nontrivial subset edos.

[[894edo]], which slices its step in six, is a notable system for the higher-limit, also consistent to the 17-odd-limit.

== Regular temperament properties ==

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

! rowspan="2" | Subgroup

|-

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

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

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

Line 21:

Line 30:

|-

| 2.3

| {{~~monzo~~| -236 149 }}

| {{Monzo| -236 149 }}

| [{{~~val~~| 149 236 }}]

| {{Mapping| 149 236 }}

| +0.405

| 0.405

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| 2.3.5

| 78732/78125, {{monzo| -34 20 1 }}

| [{{~~val~~| 149 236 346 }}]

| {{Mapping| 149 236 346 }}

| +0.232

| 0.411

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Line 45:

| 2.3.5.7

| 1029/1024, 3136/3125, 19683/19600

| [{{~~val~~| 149 236 346 418 }}]

| {{Mapping| 149 236 346 418 }}

| +0.386

| 0.445

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Line 52:

| 2.3.5.7.11

| 385/384, 441/440, 3136/3125, 19683/19600

| [{{~~val~~| 149 236 346 418 515 }}]

| {{Mapping| 149 236 346 418 515 }}

| +0.521

| 0.481

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| 2.3.5.7.11.13

| 351/350, 385/384, 441/440, 676/675, 847/845

| [{{~~val~~| 149 236 346 418 515 551 }}]

| {{Mapping| 149 236 346 418 515 551 }}

| +0.567

| 0.451

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| 2.3.5.7.11.13.17

| 273/272, 351/350, 385/384, 441/440, 676/675, 847/845

| [{{~~val~~| 149 236 346 418 515 551 609 }}]

| {{Mapping| 149 236 346 418 515 551 609 }}

| +0.495

| 0.453

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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 ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

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| 193.29

| 28/25

| [[~~Luna]] / [[hemithirds~~]]

| [[Hemithirds]]

|-

| 1

Line 138:

Line 148:

| [[Heinz]]

|}

<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

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

~~[[Category:Prime EDO]]~~

[[Category:Heinz]]

@@ Line 1: / Line 1: @@
-'''149edo''' is the [[equal division of the octave]] into 149 equal parts of 8.054 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-edo is the smallest division which is uniquely [[consistent]] through the [[17-odd-limit]]. It provides the [[optimal patent val]] for 7-, 11-, 13-, and 17-limit [[heinz]] temperament and the rank-3 temperament [[Gamelismic family #Ominous|ominous]] in the 13- and 17-limits. It has a general flat tendency, with the fifth 1.28 cents flat, but the major third is a quarter of a cent sharp. In the 5-limit it tempers out the [[sensipent comma]], 78732/78125; in the 7-limit, [[1029/1024]], [[3136/3125]] and [[19683/19600]]; in the 11-limit [[385/384]] and [[441/440]]; in the 13-limit [[351/350]] and [[676/675]]; in the 17-limit [[273/272]] and [[561/560]]; in the 19-limit [[286/285]] and [[343/342]].
+edo is the smallest division which is [[consistency|uniquely consistent]] through the [[17-odd-limit]]. It has a general flat tendency, with the fifth 1.28{{c}} flat, but the major third is a quarter of a cent sharp.
-edo is the 35th [[prime EDO]].
+In the 5-limit it [[tempering out|tempers out]] the [[sensipent comma]], 78732/78125; in the [[7-limit]], [[1029/1024]], [[3136/3125]] and [[19683/19600]]; in the [[11-limit]] [[385/384]] and [[441/440]]; in the [[13-limit]] [[351/350]], [[676/675]] and [[729/728]]; in the [[17-limit]] [[273/272]] and [[561/560]]; in the [[19-limit]] [[286/285]] and [[343/342]]. It provides the [[optimal patent val]] for 7-, 11-, 13-, and 17-limit [[heinz]] temperament and the rank-3 temperament [[gamelismic family #Ominous|ominous]] in the 13- and 17-limit.
+It is also usable in the [[23-limit]], only missing [[19/11]], [[21/11]], and their [[octave complement]]s in the [[23-odd-limit]]. In the [[27-odd-limit]], additional inconsistencies include [[25/21]], [[25/22]], [[27/20]], [[27/25]], [[27/19]], and their octave complements.
 === Prime harmonics ===
-{{Primes in edo|149}}
+{{Harmonics in equal|149}}
+=== Subsets and supersets ===
+edo is the 35th [[prime edo]]. As such, it does not contain any nontrivial subset edos.
+[[894edo]], which slices its step in six, is a notable system for the higher-limit, also consistent to the 17-odd-limit.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
@@ Line 21: / Line 30: @@
 |-
 | 2.3
-| {{monzo| -236 149 }}
+| {{Monzo| -236 149 }}
-| [{{val| 149 236 }}]
+| {{Mapping| 149 236 }}
 | +0.405
 | 0.405
@@ Line 29: / Line 38: @@
 | 2.3.5
 | 78732/78125, {{monzo| -34 20 1 }}
-| [{{val| 149 236 346 }}]
+| {{Mapping| 149 236 346 }}
 | +0.232
 | 0.411
@@ Line 36: / Line 45: @@
 | 2.3.5.7
 | 1029/1024, 3136/3125, 19683/19600
-| [{{val| 149 236 346 418 }}]
+| {{Mapping| 149 236 346 418 }}
 | +0.386
 | 0.445
@@ Line 43: / Line 52: @@
 | 2.3.5.7.11
 | 385/384, 441/440, 3136/3125, 19683/19600
-| [{{val| 149 236 346 418 515 }}]
+| {{Mapping| 149 236 346 418 515 }}
 | +0.521
 | 0.481
@@ Line 50: / Line 59: @@
 | 2.3.5.7.11.13
 | 351/350, 385/384, 441/440, 676/675, 847/845
-| [{{val| 149 236 346 418 515 551 }}]
+| {{Mapping| 149 236 346 418 515 551 }}
 | +0.567
 | 0.451
@@ Line 57: / Line 66: @@
 | 2.3.5.7.11.13.17
 | 273/272, 351/350, 385/384, 441/440, 676/675, 847/845
-| [{{val| 149 236 346 418 515 551 609 }}]
+| {{Mapping| 149 236 346 418 515 551 609 }}
 | +0.495
 | 0.453
@@ Line 65: / Line 74: @@
 === 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br>per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 94: / Line 104: @@
 | 193.29
 | 28/25
-| [[Luna]] / [[hemithirds]]
+| [[Hemithirds]]
 |-
 | 1
@@ Line 138: / Line 148: @@
 | [[Heinz]]
 |}
+<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
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]
 [[Category:Heinz]]