21ed4: Difference between revisions

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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}} It represents every second step of [[21edo]].
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2015-01-13 06:04:42 UTC</tt>.<br>
 
: The original revision id was <tt>537096556</tt>.<br>
== Intervals ==
: The revision comment was: <tt></tt><br>
{{Interval table}}
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>
== 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">Music:
{{Harmonics in equal
[[http://chrisvaisvil.com/two-piano-preludes-for-two-sisters/|Sister equal]] by Chris Vaisvil</pre></div>
| steps = 21
<h4>Original HTML content:</h4>
| num = 4
<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;21ed4&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Music:&lt;br /&gt;
| denom = 1
&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/two-piano-preludes-for-two-sisters/" rel="nofollow"&gt;Sister equal&lt;/a&gt; by Chris Vaisvil&lt;/body&gt;&lt;/html&gt;</pre></div>
}}
{{Harmonics in equal
| steps = 21
| num = 4
| denom = 1
| start = 12
| collapsed = 1
}}
 
== Music ==
* [http://chrisvaisvil.com/two-piano-preludes-for-two-sisters/ Sister equal] by [[Chris Vaisvil]] — 3rd Song on that page is in 21ED4
 
[[Category:Macrotonal]]
{{todo|expand}}

Latest revision as of 21:18, 31 July 2025

← 19ed4 21ed4 23ed4 →
Prime factorization 3 × 7
Step size 114.286 ¢ 
Octave 11\21ed4 (1257.14 ¢)
Twelfth 17\21ed4 (1942.86 ¢)
Consistency limit 1
Distinct consistency limit 1

21 equal divisions of the 4th harmonic (abbreviated 21ed4) is a nonoctave tuning system that divides the interval of 4/1 into 21 equal parts of about 114 ¢ each. Each step represents a frequency ratio of 41/21, or the 21st root of 4. It represents every second step of 21edo.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 114.3 14/13, 15/14
2 228.6 17/15
3 342.9 17/14
4 457.1 13/10, 17/13, 22/17
5 571.4 7/5
6 685.7
7 800
8 914.3 17/10, 22/13
9 1028.6
10 1142.9
11 1257.1
12 1371.4 11/5, 20/9
13 1485.7
14 1600
15 1714.3
16 1828.6
17 1942.9
18 2057.1 23/7
19 2171.4
20 2285.7
21 2400

Harmonics

Approximation of harmonics in 21ed4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +57.1 +40.9 +0.0 -43.5 -16.2 -54.5 +57.1 -32.5 +13.7 -37.0 +40.9
Relative (%) +50.0 +35.8 +0.0 -38.0 -14.2 -47.7 +50.0 -28.4 +12.0 -32.4 +35.8
Steps
(reduced) 		11
(11) 		17
(17) 		21
(0) 		24
(3) 		27
(6) 		29
(8) 		32
(11) 		33
(12) 		35
(14) 		36
(15) 		38
(17)
Approximation of harmonics in 21ed4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +16.6 +2.6 -2.6 +0.0 +9.3 +24.7 +45.3 -43.5 -13.6 +20.1 -56.8
Relative (%) +14.5 +2.3 -2.2 +0.0 +8.2 +21.6 +39.7 -38.0 -11.9 +17.6 -49.7
Steps
(reduced) 		39
(18) 		40
(19) 		41
(20) 		42
(0) 		43
(1) 		44
(2) 		45
(3) 		45
(3) 		46
(4) 		47
(5) 		47
(5)

Music

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