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]]

Revision as of 19:55, 24 September 2021

149edo is the equal division of the octave into 149 equal parts of 8.054 cents 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 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.

Prime harmonics

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Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-236 149⟩ [⟨149 236]] +0.405 0.405 5.03
2.3.5 78732/78125, [-34 20 1⟩ [⟨149 236 346]] +0.232 0.411 5.11
2.3.5.7 1029/1024, 3136/3125, 19683/19600 [⟨149 236 346 418]] +0.386 0.445 5.53
2.3.5.7.11 385/384, 441/440, 3136/3125, 19683/19600 [⟨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 [⟨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 [⟨149 236 346 418 515 551 609]] +0.495 0.453 5.62

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
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
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