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149edo: Difference between revisions

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== Theory ==
== 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 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 35th [[prime EDO]].  
149edo is the 35th [[prime EDO]].  
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=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|149}}
{{Primes in edo|149}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -236 149 }}
| [{{val| 149 236 }}]
| +0.405
| 0.405
| 5.03
|-
| 2.3.5
| 78732/78125, {{monzo| -34 20 1 }}
| [{{val| 149 236 346 }}]
| +0.232
| 0.411
| 5.11
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 19683/19600
| [{{val| 149 236 346 418 }}]
| +0.386
| 0.445
| 5.53
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 19683/19600
| [{{val| 149 236 346 418 515 }}]
| +0.521
| 0.481
| 5.97
|-
| 2.3.5.7.11.13
| 351/350, 385/384, 441/440, 676/675, 847/845
| [{{val| 149 236 346 418 515 551 }}]
| +0.567
| 0.451
| 5.60
|-
| 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 }}]
| +0.495
| 0.453
| 5.62
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 3\149
| 24.16
| 686/675
| [[Sengagen]]
|-
| 1
| 16\149
| 128.86
| 14/13
| [[Tertiathirds]]
|-
| 1
| 18\149
| 144.97
| 49/45
| [[Swetneus]]
|-
| 1
| 24\149
| 193.29
| 28/25
| [[Luna]] / [[hemithirds]]
|-
| 1
| 29\149
| 233.56
| 8/7
| [[Slendric]]
|-
| 1
| 47\149
| 378.52
| 56/45
| [[Subpental]]
|-
| 1
| 55\149
| 442.95
| 162/125
| [[Sensipent]]
|-
| 1
| 57\149
| 459.06
| 125/96
| [[Majvam]]
|-
| 1
| 60\149
| 483.22
| 45/34
| [[Hemiseven]]
|-
| 1
| 61\149
| 491.28
| 3645/2744
| [[Fifthplus]]
|-
| 1
| 68\149
| 547.65
| 11/8
| [[Heinz]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Theory]]
[[Category:Heinz]]
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