8edt: 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}}
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-11-20 10:03:35 UTC</tt>.<br>
: The original revision id was <tt>277362058</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">8EDT


As the double of [[4edt]], harmonically, it is analogous to 10edo in that the harmonic chain is doubled. However, doing so does not make it a schismatic temperament like 10edo, because the full 3:5:7 triad is already present in 4edt. In any case, as a multiple of 4edt, I will say it is the widest variety of "Black-Extraterrestrial-Tree" temperament.
== Theory ==
As the double of [[4edt]], it is the analog of [[10edo]] being the double of [[5edo]]. However, the full 3:5:7 triad is already present in 4edt which is unlike the situation in 10edo where 4:5:6 gains a new better approximation than the sus4 triad in [[5edo]]. More precisely, 8edt is [[enfactoring|enfactored]] in the 3.5.7 subgroup.


What it does introduce are flat pseudooctaves and sharp 3:2's, making it related to 5edo melodically.
What it does introduce are flat [[2/1]] pseudo-octaves and sharp [[3/2]] perfect fifths, making it related to 5edo melodically. It is equivalent to 5edo with the [[3/1]] made just, by compressing the octave by 11.3 cents, which has the side effect of bringing the step size slightly closer to [[8/7]].


0: 1/1 0.000 unison, perfect prime
=== Harmonics ===
1: 237.744 cents 237.744
{{Harmonics in equal|8|3|1|}}
2: 475.489 cents 4/3
{{Harmonics in equal|8|3|1|intervals=prime}}
3: 713.233 cents 713.233
 
4: 950.978 cents 5/3
== Interval table ==
5: 1188.722 cents 1188.722
{{Interval table}}
6: 1426.466 cents 1426.466
 
7: 1664.211 cents 1664.211
[[Category:Macrotonal]]
8: 3/1 1901.955 perfect 12th</pre></div>
<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;8edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;8EDT&lt;br /&gt;
&lt;br /&gt;
As the double of &lt;a class="wiki_link" href="/4edt"&gt;4edt&lt;/a&gt;, harmonically, it is analogous to 10edo in that the harmonic chain is doubled. However, doing so does not make it a schismatic temperament like 10edo, because the full 3:5:7 triad is already present in 4edt. In any case, as a multiple of 4edt, I will say it is the widest variety of &amp;quot;Black-Extraterrestrial-Tree&amp;quot; temperament.&lt;br /&gt;
&lt;br /&gt;
What it does introduce are flat pseudooctaves and sharp 3:2's, making it related to 5edo melodically.&lt;br /&gt;
&lt;br /&gt;
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
1: 237.744 cents 237.744&lt;br /&gt;
2: 475.489 cents 4/3&lt;br /&gt;
3: 713.233 cents 713.233&lt;br /&gt;
4: 950.978 cents 5/3&lt;br /&gt;
5: 1188.722 cents 1188.722&lt;br /&gt;
6: 1426.466 cents 1426.466&lt;br /&gt;
7: 1664.211 cents 1664.211&lt;br /&gt;
8: 3/1 1901.955 perfect 12th&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 18:42, 8 March 2025

← 7edt 8edt 9edt →
Prime factorization 23
Step size 237.744 ¢ 
Octave 5\8edt (1188.72 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 4

8 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 8edt or 8ed3), is a nonoctave tuning system that divides the interval of 3/1 into 8 equal parts of about 238 ¢ each. Each step represents a frequency ratio of 31/8, or the 8th root of 3.

Theory

As the double of 4edt, it is the analog of 10edo being the double of 5edo. However, the full 3:5:7 triad is already present in 4edt which is unlike the situation in 10edo where 4:5:6 gains a new better approximation than the sus4 triad in 5edo. More precisely, 8edt is enfactored in the 3.5.7 subgroup.

What it does introduce are flat 2/1 pseudo-octaves and sharp 3/2 perfect fifths, making it related to 5edo melodically. It is equivalent to 5edo with the 3/1 made just, by compressing the octave by 11.3 cents, which has the side effect of bringing the step size slightly closer to 8/7.

Harmonics

Approximation of harmonics in 8edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -11 +0 -23 +67 -11 -40 -34 +0 +55 -110 -23
Relative (%) -4.7 +0.0 -9.5 +28.0 -4.7 -17.0 -14.2 +0.0 +23.3 -46.1 -9.5
Steps
(reduced) 		5
(5) 		8
(0) 		10
(2) 		12
(4) 		13
(5) 		14
(6) 		15
(7) 		16
(0) 		17
(1) 		17
(1) 		18
(2)
Approximation of prime harmonics in 8edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -11 +0 +67 -40 -110 +77 +88 -105 +40 +114 -1
Relative (%) -4.7 +0.0 +28.0 -17.0 -46.1 +32.2 +36.9 -44.1 +16.8 +48.0 -0.6
Steps
(reduced) 		5
(5) 		8
(0) 		12
(4) 		14
(6) 		17
(1) 		19
(3) 		21
(5) 		21
(5) 		23
(7) 		25
(1) 		25
(1)

Interval table

Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 237.7 162.5 7/6, 8/7, 9/8, 15/13, 17/15, 20/17, 22/19
2 475.5 325 4/3, 9/7, 13/10, 17/13, 21/16
3 713.2 487.5 3/2, 20/13
4 951 650 7/4, 12/7, 17/10, 19/11
5 1188.7 812.5 2/1
6 1426.5 975 7/3, 9/4, 16/7, 20/9
7 1664.2 1137.5 8/3, 13/5, 18/7, 21/8
8 1902 1300 3/1
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