149edo: Difference between revisions

← 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

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VisualWikitext

Latest revision as of 07:15, 18 June 2026

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 2^1/149, 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
Error	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
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-26 14:16:07 UTC</tt>.<br>
-: The original revision id was <tt>214221250</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //149 equal division// divides the octave into 149 equal parts of 8.054 cents each. It provides the [[optimal patent val]] for 11- 13- and 17-limit [[Sensipent family|heinz temperament]] and the 13-limit rank three temperament [[Gamelismic family|ominous]]. It has a generally 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.
-is a prime number.</pre></div>
+== Theory ==
-<h4>Original HTML content:</h4>
+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{{c}} flat, but the major third is a quarter of a cent sharp.
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;149edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;149 equal division&lt;/em&gt; divides the octave into 149 equal parts of 8.054 cents each. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 11- 13- and 17-limit &lt;a class="wiki_link" href="/Sensipent%20family"&gt;heinz temperament&lt;/a&gt; and the 13-limit rank three temperament &lt;a class="wiki_link" href="/Gamelismic%20family"&gt;ominous&lt;/a&gt;. It has a generally 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.&lt;br /&gt;
-&lt;br /&gt;
+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.
-is a prime number.&lt;/body&gt;&lt;/html&gt;</pre></div>
+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.
+=== Prime harmonics ===
+{{Harmonics in equal|149}}
+=== Subsets and supersets ===
+edo 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 ==
+{| 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 }}
+| {{Mapping| 149 236 }}
+| +0.405
+| 0.405
+| 5.03
+|-
+| 2.3.5
+| 78732/78125, {{monzo| -34 20 1 }}
+| {{Mapping| 149 236 346 }}
+| +0.232
+| 0.411
+| 5.11
+|-
+| 2.3.5.7
+| 1029/1024, 3136/3125, 19683/19600
+| {{Mapping| 149 236 346 418 }}
+| +0.386
+| 0.445
+| 5.53
+|-
+| 2.3.5.7.11
+| 385/384, 441/440, 3136/3125, 19683/19600
+| {{Mapping| 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
+| {{Mapping| 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
+| {{Mapping| 149 236 346 418 515 551 609 }}
+| +0.495
+| 0.453
+| 5.62
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! 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
+| [[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]]
+|}
+<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]]

149edo: Difference between revisions

Latest revision as of 07:15, 18 June 2026

Contents

Theory

Prime harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

Navigation menu

149edo: Difference between revisions

Latest revision as of 07:15, 18 June 2026

Theory

Prime harmonics

Subsets and supersets

Regular temperament properties

Rank-2 temperaments

Navigation menu

Search