7edt: 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:genewardsmith|genewardsmith]] and made on <tt>2011-09-06 16:42:37 UTC</tt>.<br>
: The original revision id was <tt>251313288</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">[[toc|flat]]


=Properties=
== Theory ==
The 7 equal division of 3, the tritave, divides it into 7 equal parts  of 271.708 cents each, corresponding to 4.4165 edo. The step size is very close to the 271.509 cents of 7-limit [[Orwell|orwell temperament]] and also close to the 271.426 cents of 11-limit orwell. It is almost identical to 12\53, the [[53edo]] orwell generator which is 271.698 cents.
Since one step of 7edt approximates a [[7/6]] subminor third (4.84{{c}} sharp) quite nicely, three steps are almost exactly [[8/5]] (tempering out [[1728/1715]], the orwellisma), and four steps are very nearly [[15/8]] (tempering out [[2430/2401]], the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some [[7-limit]] harmony, along with some elements of the [[11-limit]], such as the [[11/8]] major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.


=Scale degrees of 7edt=
Due to the proximity of the step size with 7/6, 7edt supports [[orwell]] temperament. One step of 7edt is almost identical to 12\53, the [[53edo]] orwell generator, at about 271.698 cents. 7edt is also a good tuning for [[Electra]] temperament, with two steps of 7edt being a close approximation to [[15/11]].
|| Degrees || Cents || Approximate Ratio ||
|| 0 || 0 || [[1_1|1/1]] ||
|| 1 || 271.708 || [[7_6|7/6]] ||
|| 2 || 543.416 || [[15_11|15/11]], [[11_8|11/8]] ||
|| 3 || 815.124 || [[8_5|8/5]] ||
|| 4 || 1086.831 || [[15_8|15/8]] ||
|| 5 || 1358.539 || 11/5 ([[11_10|11/10]] plus an octave) ||
|| 6 || 1630.247 || 18/7 ([[9_7|9/7]] plus an octave) ||
|| 7 || 1901.955 || 3/1 ||


Since one step of 7edt is a sharp subminor (7/6) third, three steps are almost exactlty 8/5, four steps are very nearly 15/8 and six steps are a bit flat of 18/7, 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the blair comma.
=== Harmonics ===
{{Harmonics in equal|7|3|1|columns=15}}


=7n-edt Family:=
=== Subsets and supersets ===
[[14edt]]
7edt is the 4th [[prime equal division|prime edt]], after [[5edt]] and before [[11edt]].  
[[21edt]]
[[28edt]]
...</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;7edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Properties"&gt;Properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Scale degrees of 7edt"&gt;Scale degrees of 7edt&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#x7n-edt Family:"&gt;7n-edt Family:&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Properties&lt;/h1&gt;
The 7 equal division of 3, the tritave, divides it into 7 equal parts  of 271.708 cents each, corresponding to 4.4165 edo. The step size is very close to the 271.509 cents of 7-limit &lt;a class="wiki_link" href="/Orwell"&gt;orwell temperament&lt;/a&gt; and also close to the 271.426 cents of 11-limit orwell. It is almost identical to 12\53, the &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; orwell generator which is 271.698 cents.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Scale degrees of 7edt"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Scale degrees of 7edt&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{| class="wikitable center-1 right-2 right-3"
        &lt;td&gt;Degrees&lt;br /&gt;
|-
&lt;/td&gt;
! #
        &lt;td&gt;Cents&lt;br /&gt;
! Cents
&lt;/td&gt;
! [[Hekt]]s
        &lt;td&gt;Approximate Ratio&lt;br /&gt;
! Approximate ratios
&lt;/td&gt;
! [[Electra]] notation<br>({{nowrap|J {{=}} 1/1}})
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 0
        &lt;td&gt;0&lt;br /&gt;
| 0
&lt;/td&gt;
| 0
        &lt;td&gt;0&lt;br /&gt;
| [[1/1]]
&lt;/td&gt;
| J
        &lt;td&gt;&lt;a class="wiki_link" href="/1_1"&gt;1/1&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 1
    &lt;/tr&gt;
| 272
    &lt;tr&gt;
| 186
        &lt;td&gt;1&lt;br /&gt;
| [[7/6]]
&lt;/td&gt;
| K
        &lt;td&gt;271.708&lt;br /&gt;
|-
&lt;/td&gt;
| 2
        &lt;td&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;&lt;br /&gt;
| 543
&lt;/td&gt;
| 371
    &lt;/tr&gt;
| [[11/8]], [[15/11]]
    &lt;tr&gt;
| L
        &lt;td&gt;2&lt;br /&gt;
|-
&lt;/td&gt;
| 3
        &lt;td&gt;543.416&lt;br /&gt;
| 815
&lt;/td&gt;
| 557
        &lt;td&gt;&lt;a class="wiki_link" href="/15_11"&gt;15/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;&lt;br /&gt;
| [[8/5]]
&lt;/td&gt;
| M
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 4
        &lt;td&gt;3&lt;br /&gt;
| 1087
&lt;/td&gt;
| 743
        &lt;td&gt;815.124&lt;br /&gt;
| [[15/8]]
&lt;/td&gt;
| N
        &lt;td&gt;&lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 5
    &lt;/tr&gt;
| 1359
    &lt;tr&gt;
| 929
        &lt;td&gt;4&lt;br /&gt;
| [[11/5]]
&lt;/td&gt;
| O
        &lt;td&gt;1086.831&lt;br /&gt;
|-
&lt;/td&gt;
| 6
        &lt;td&gt;&lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;br /&gt;
| 1630
&lt;/td&gt;
| 1114
    &lt;/tr&gt;
| [[18/7]]
    &lt;tr&gt;
| P
        &lt;td&gt;5&lt;br /&gt;
|-
&lt;/td&gt;
| 7
        &lt;td&gt;1358.539&lt;br /&gt;
| 1902
&lt;/td&gt;
| 1300
        &lt;td&gt;11/5 (&lt;a class="wiki_link" href="/11_10"&gt;11/10&lt;/a&gt; plus an octave)&lt;br /&gt;
| [[3/1]]
&lt;/td&gt;
| J
    &lt;/tr&gt;
|}
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1630.247&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/7 (&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt; plus an octave)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.955&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:Orwell]]
Since one step of 7edt is a sharp subminor (7/6) third, three steps are almost exactlty 8/5, four steps are very nearly 15/8 and six steps are a bit flat of 18/7, 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the blair comma.&lt;br /&gt;
[[Category:Subminor third]]
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="x7n-edt Family:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;7n-edt Family:&lt;/h1&gt;
&lt;a class="wiki_link" href="/14edt"&gt;14edt&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/21edt"&gt;21edt&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/28edt"&gt;28edt&lt;/a&gt;&lt;br /&gt;
...&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 13:40, 23 February 2025

← 6edt 7edt 8edt →
Prime factorization 7 (prime)
Step size 271.708 ¢ 
Octave 4\7edt (1086.83 ¢)
Consistency limit 3
Distinct consistency limit 3

7 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 7edt or 7ed3), is a nonoctave tuning system that divides the interval of 3/1 into 7 equal parts of about 272 ¢ each. Each step represents a frequency ratio of 31/7, or the 7th root of 3.

Theory

Since one step of 7edt approximates a 7/6 subminor third (4.84 ¢ sharp) quite nicely, three steps are almost exactly 8/5 (tempering out 1728/1715, the orwellisma), and four steps are very nearly 15/8 (tempering out 2430/2401, the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony, along with some elements of the 11-limit, such as the 11/8 major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.

Due to the proximity of the step size with 7/6, 7edt supports orwell temperament. One step of 7edt is almost identical to 12\53, the 53edo orwell generator, at about 271.698 cents. 7edt is also a good tuning for Electra temperament, with two steps of 7edt being a close approximation to 15/11.

Harmonics

Approximation of harmonics in 7edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -113 +0 +45 -69 -113 -108 -68 +0 +89 -76 +45 -93 +50 -69 +91
Relative (%) -41.7 +0.0 +16.7 -25.5 -41.7 -39.9 -25.0 +0.0 +32.9 -27.9 +16.7 -34.3 +18.5 -25.5 +33.4
Steps
(reduced) 		4
(4) 		7
(0) 		9
(2) 		10
(3) 		11
(4) 		12
(5) 		13
(6) 		14
(0) 		15
(1) 		15
(1) 		16
(2) 		16
(2) 		17
(3) 		17
(3) 		18
(4)

Subsets and supersets

7edt is the 4th prime edt, after 5edt and before 11edt.

Intervals

# Cents Hekts Approximate ratios Electra notation
(J = 1/1)
0 0 0 1/1 J
1 272 186 7/6 K
2 543 371 11/8, 15/11 L
3 815 557 8/5 M
4 1087 743 15/8 N
5 1359 929 11/5 O
6 1630 1114 18/7 P
7 1902 1300 3/1 J
Retrieved from "https://en.xen.wiki/index.php?title=7edt&oldid=182807"