149edo: Difference between revisions

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== 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 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]]. 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.

=== Prime harmonics ===

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| 2.3

| {{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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| 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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| 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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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| 193.29

| 28/25

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

| [[Hemithirds]]

|-

| 1

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| [[Heinz]]

|}

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

[[Category:Heinz]]

@@ Line 3: / Line 3: @@
 == 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 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]]. 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.
 === Prime harmonics ===
@@ Line 24: / Line 24: @@
 | 2.3
 | {{monzo| -236 149 }}
-| [{{val| 149 236 }}]
+| {{mapping| 149 236 }}
 | +0.405
 | 0.405
@@ Line 31: / Line 31: @@
 | 2.3.5
 | 78732/78125, {{monzo| -34 20 1 }}
-| [{{val| 149 236 346 }}]
+| {{mapping| 149 236 346 }}
 | +0.232
 | 0.411
@@ Line 38: / Line 38: @@
 | 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 45: / Line 45: @@
 | 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 52: / Line 52: @@
 | 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 59: / Line 59: @@
 | 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 69: / Line 69: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 96: / Line 96: @@
 | 193.29
 | 28/25
-| [[Luna]] / [[hemithirds]]
+| [[Hemithirds]]
 |-
 | 1
@@ Line 140: / Line 140: @@
 | [[Heinz]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Heinz]]