299edo: Difference between revisions

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**Imported revision 240476789 - Original comment: **
 
 
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<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>2011-07-08 02:09:16 UTC</tt>.<br>
 
: The original revision id was <tt>240476789</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
In the 5-limit, 299et [[tempering out|tempers out]] the [[kleisma]], 15625/15552, in the [[7-limit]] [[10976/10935]], in the [[11-limit]] [[385/384]]; and in the [[13-limit]] [[325/324]], [[625/624]] and [[676/675]]. It provides the [[optimal patent val]] for the 13-limit rank-3 [[enlil]] temperament, and the rank-4 temperament tempering out 325/324 and 385/384. 
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>
=== Prime harmonics ===
<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 //299 equal division// divides the octave into 299 parts of 4.013 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, in the 7-limit 10976/10935, in the 11-limit 385/384; and in the 13-limit 325/324, 625/624 and 676/675.</pre></div>
{{Harmonics in equal|299}}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;299edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;299 equal division&lt;/em&gt; divides the octave into 299 parts of 4.013 cents each. In the 5-limit it tempers out the kleisma, 15625/15552, in the 7-limit 10976/10935, in the 11-limit 385/384; and in the 13-limit 325/324, 625/624 and 676/675.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 299 factors into 13 × 23, 299edo contains [[13edo]] and [[23edo]] as subsets.
 
== 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| 474 -299 }}
| {{Mapping| 299 474 }}
| −0.1218
| 0.1218
| 3.04
|-
| 2.3.5
| 15625/15552, {{monzo| 80 -49 -1 }}
| {{Mapping| 299 474 694 }}
| +0.0665
| 0.2844
| 7.09
|-
| 2.3.5.7
| 10976/10935, 15625/15552, 823543/819200
| {{Mapping| 299 474 694 839 }}
| +0.1925
| 0.3291
| 8.20
|-
| 2.3.5.7.11
| 385/384, 6250/6237, 10976/10935, 12005/11979
| {{Mapping| 299 474 694 839 1034 }}
| +0.2399
| 0.3092
| 7.70
|-
| 2.3.5.7.11.13
| 325/324, 385/384, 625/624, 10648/10647, 10976/10935
| {{Mapping| 299 474 694 839 1034 1106 }}
| +0.2779
| 0.2948
| 7.34
|-
| 2.3.5.7.11.13.17
| 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619
| {{Mapping| 299 474 694 839 1034 1106 1222 }}
| +0.2595
| 0.2767
| 6.89
|-
| 2.3.5.7.11.13.17.19
| 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444
| {{Mapping| 299 474 694 839 1034 1106 1222 1270 }}
| +0.2424
| 0.2627
| 6.54
|}
 
=== 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
| 25\299
| 100.33
| 1323/1250
| [[Heptacot]] (7-limit)
|-
| 1
| 79\299
| 317.06
| 6/5
| [[Hanson]]
|-
| 1
| 124\299
| 497.66
| 4/3
| [[Cotoneum]] (7-limit)
|-
| 1
| 124\299
| 505.69
| 75/56
| [[Marfifths]]
|}
<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:Enlil]]
[[Category:Keenanismic]]

Latest revision as of 12:15, 20 May 2026

← 298edo 299edo 300edo →
Prime factorization 13 × 23
Step size 4.01338 ¢ 
Fifth 175\299 (702.341 ¢)
Semitones (A1:m2) 29:22 (116.4 ¢ : 88.29 ¢)
Consistency limit 7
Distinct consistency limit 7

299 equal divisions of the octave (abbreviated 299edo or 299ed2), also called 299-tone equal temperament (299tet) or 299 equal temperament (299et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 299 equal parts of about 4.01 ¢ each. Each step represents a frequency ratio of 21/299, or the 299th root of 2.

Theory

In the 5-limit, 299et tempers out the kleisma, 15625/15552, in the 7-limit 10976/10935, in the 11-limit 385/384; and in the 13-limit 325/324, 625/624 and 676/675. It provides the optimal patent val for the 13-limit rank-3 enlil temperament, and the rank-4 temperament tempering out 325/324 and 385/384.

Prime harmonics

Approximation of prime harmonics in 299edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.39 -1.03 -1.60 -1.49 -1.73 -0.61 -0.52 +1.83 +1.86 -1.22
Relative (%) +0.0 +9.6 -25.7 -39.9 -37.0 -43.1 -15.1 -13.0 +45.5 +46.4 -30.5
Steps
(reduced) 		299
(0) 		474
(175) 		694
(96) 		839
(241) 		1034
(137) 		1106
(209) 		1222
(26) 		1270
(74) 		1353
(157) 		1453
(257) 		1481
(285)

Subsets and supersets

Since 299 factors into 13 × 23, 299edo contains 13edo and 23edo as subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [474 -299⟩ [⟨299 474]] −0.1218 0.1218 3.04
2.3.5 15625/15552, [80 -49 -1⟩ [⟨299 474 694]] +0.0665 0.2844 7.09
2.3.5.7 10976/10935, 15625/15552, 823543/819200 [⟨299 474 694 839]] +0.1925 0.3291 8.20
2.3.5.7.11 385/384, 6250/6237, 10976/10935, 12005/11979 [⟨299 474 694 839 1034]] +0.2399 0.3092 7.70
2.3.5.7.11.13 325/324, 385/384, 625/624, 10648/10647, 10976/10935 [⟨299 474 694 839 1034 1106]] +0.2779 0.2948 7.34
2.3.5.7.11.13.17 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619 [⟨299 474 694 839 1034 1106 1222]] +0.2595 0.2767 6.89
2.3.5.7.11.13.17.19 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444 [⟨299 474 694 839 1034 1106 1222 1270]] +0.2424 0.2627 6.54

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 25\299 100.33 1323/1250 Heptacot (7-limit)
1 79\299 317.06 6/5 Hanson
1 124\299 497.66 4/3 Cotoneum (7-limit)
1 124\299 505.69 75/56 Marfifths

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

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