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== Theory ==
== Theory ==
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. 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]], [[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.
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.  
 
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.
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.
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=== Subsets and supersets ===
=== Subsets and supersets ===
149edo is the 35th [[prime edo]].
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 ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -236 149 }}
| {{Monzo| -236 149 }}
| {{mapping| 149 236 }}
| {{Mapping| 149 236 }}
| +0.405
| +0.405
| 0.405
| 0.405
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| 2.3.5
| 2.3.5
| 78732/78125, {{monzo| -34 20 1 }}
| 78732/78125, {{monzo| -34 20 1 }}
| {{mapping| 149 236 346 }}
| {{Mapping| 149 236 346 }}
| +0.232
| +0.232
| 0.411
| 0.411
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| 2.3.5.7
| 2.3.5.7
| 1029/1024, 3136/3125, 19683/19600
| 1029/1024, 3136/3125, 19683/19600
| {{mapping| 149 236 346 418 }}
| {{Mapping| 149 236 346 418 }}
| +0.386
| +0.386
| 0.445
| 0.445
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| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 19683/19600
| 385/384, 441/440, 3136/3125, 19683/19600
| {{mapping| 149 236 346 418 515 }}
| {{Mapping| 149 236 346 418 515 }}
| +0.521
| +0.521
| 0.481
| 0.481
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 351/350, 385/384, 441/440, 676/675, 847/845
| 351/350, 385/384, 441/440, 676/675, 847/845
| {{mapping| 149 236 346 418 515 551 }}
| {{Mapping| 149 236 346 418 515 551 }}
| +0.567
| +0.567
| 0.451
| 0.451
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 273/272, 351/350, 385/384, 441/440, 676/675, 847/845
| 273/272, 351/350, 385/384, 441/440, 676/675, 847/845
| {{mapping| 149 236 346 418 515 551 609 }}
| {{Mapping| 149 236 346 418 515 551 609 }}
| +0.495
| +0.495
| 0.453
| 0.453
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Heinz]]
| [[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
<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:Heinz]]
[[Category:Heinz]]

Latest revision as of 07:15, 18 June 2026

← 148edo 149edo 150edo →
Prime factorization 149 (prime)
Step size 8.05369 ¢ 
Fifth 87\149 (700.671 ¢)
Semitones (A1:m2) 13:12 (104.7 ¢ : 96.64 ¢)
Consistency limit 17
Distinct consistency limit 17

149 equal divisions of the octave (abbreviated 149edo or 149ed2), also called 149-tone equal temperament (149tet) or 149 equal temperament (149et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 149 equal parts of about 8.05 ¢ each. Each step represents a frequency ratio of 21/149, or the 149th root of 2.

Theory

149edo is the smallest division which is uniquely consistent through the 17-odd-limit. It has a general flat tendency, with the fifth 1.28 ¢ 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, 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 ominous in the 13- and 17-limit.

It is also usable in the 23-limit, only missing 19/11, 21/11, and their octave complements 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

Approximation of prime harmonics in 149edo
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. 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

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 8ve 		Generator* Cents* 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 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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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