149edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|149}} | |||
== 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]]. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|149}} | |||
=== | === Subsets and supersets === | ||
149edo is the 35th [[prime edo]]. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 67: | Line 68: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 140: | Line 141: | ||
|} | |} | ||
[[Category:Heinz]] | [[Category:Heinz]] | ||
Revision as of 06:18, 29 August 2023
| ← 148edo | 149edo | 150edo → |
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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.28 | +0.26 | -2.38 | -3.67 | -2.94 | -0.26 | +0.47 | -0.09 | +1.30 | -1.41 |
| Relative (%) | +0.0 | -15.9 | +3.3 | -29.6 | -45.5 | -36.6 | -3.2 | +5.9 | -1.1 | +16.1 | -17.5 | |
| Steps (reduced) |
149 (0) |
236 (87) |
346 (48) |
418 (120) |
515 (68) |
551 (104) |
609 (13) |
633 (37) |
674 (78) |
724 (128) |
738 (142) | |
Subsets and supersets
149edo is the 35th prime edo.
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
| Periods per 8ve |
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 |