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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 corresponds to 10.2571 [[edo]].  
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-07-08 15:32:51 UTC</tt>.<br>
: The original revision id was <tt>80630037</tt>.<br>
: The revision comment was: <tt>added intervals in cents</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">=Equal division of the perfect fifth (3/2) into 6 equal parts=
(a.k.a. "6th root of 3/2")


==Compositions==  
== Theory ==
[[http://www.seraph.it/XenoTunes3.html|Metashakti]] by Carlo Serafini
6edf is related to the [[miracle]] temperament, which [[tempering out|tempers out]] [[225/224]] and [[1029/1024]] in the 7-limit.


==Intervals==  
=== Harmonics ===
|| degrees || cents ~ cents octave-reduced ||
{{Harmonics in equal|6|3|2}}
|| 0 || 0 (perfect unison, 1:1) ||
|| 1 || 117 ||
|| 2 || 234 ||
|| 3 || 351 ||
|| 4 || 468 ||
|| 5 || 585 ||
|| 6 || 702 (just perfect fifth, 3:2) ||
|| 7 || 819 ||
|| 8 || 936 ||
|| 9 || 1053 ||
|| 10 || 1170 ||
|| 11 || 1287 ~ 87 ||
|| 12 || 1404 ~ 204 (just major whole tone/ninth, 9:4) ||
|| 13 || 1521 ~ 321 ||
|| 14 || 1638 ~ 438 ||
|| 16 || 1872 ~ 672 ||
|| 17 || 1988 ~ 788 ||
|| 18 || 2106 ~ 906 (Pythagorean major sixth, 27:16) ||
|| 19 || 2223 ~ 1023 ||
|| 20 || 2340 ~ 1140 ||
|| 21 || 2457 ~ 57 ||
|| 22 || 2574 ~ 174 ||
|| 23 || 2691 ~ 291 ||
|| 24 || 2808 ~ 408 (Pythagorean major third, 81:64) ||</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;6edf&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Equal division of the perfect fifth (3/2) into 6 equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Equal division of the perfect fifth (3/2) into 6 equal parts&lt;/h1&gt;
(a.k.a. &amp;quot;6th root of 3/2&amp;quot;)&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Equal division of the perfect fifth (3/2) into 6 equal parts-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Compositions&lt;/h2&gt;
&lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&gt;Metashakti&lt;/a&gt; by Carlo Serafini&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Equal division of the perfect fifth (3/2) into 6 equal parts-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
        &lt;td&gt;degrees&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;cents ~ cents octave-reduced&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0 (perfect unison, 1:1)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;117&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;234&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;468&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;585&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;702 (just perfect fifth, 3:2)&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;819&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;936&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1053&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1170&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1287 ~ 87&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1404 ~ 204 (just major whole tone/ninth, 9:4)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1521 ~ 321&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1638 ~ 438&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1872 ~ 672&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1988 ~ 788&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2106 ~ 906 (Pythagorean major sixth, 27:16)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2223 ~ 1023&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2340 ~ 1140&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2457 ~ 57&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2574 ~ 174&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2691 ~ 291&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2808 ~ 408 (Pythagorean major third, 81:64)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximate Ratios
! [[1L 3s (fifth-equivalent)|Neptunian]]<br>Notation
|-
| 0
| 0
| [[1/1]]
| C
|-
| 1
| 117
| [[16/15]], [[15/14]]
| C#
|-
| 2
| 234
| [[8/7]]
| Db
|-
| 3
| 351
| [[11/9]], [[27/22]]
| D
|-
| 4
| 468
| [[21/16]]
| E
|-
| 5
| 585
| [[7/5]], [[45/32]]
| F
|-
| 6
| 702
| [[3/2]]
| C
|-
| 7
| 819
| [[8/5]], [[21/13]]
| C#
|-
| 8
| 936
| [[12/7]], [[55/32]]
| Db
|-
| 9
| 1053
| [[11/6]]
| D
|-
| 10
| 1170
| [[49/25]], [[160/81]], [[2/1]]
| E
|-
| 11
| 1287
|
| F
|-
| 12
| 1404
| 9/4
| C
|-
| 13
| 1521
|
| C#
|-
| 14
| 1638
|
| Db
|-
| 15
| 1755
|
| D
|-
| 16
| 1872
|
| E
|-
| 17
| 1988
|
| F
|-
| 18
| 2106
| 27/8
| C
|-
| 19
| 2223
|
| C#
|-
| 20
| 2340
|
| Db
|-
| 21
| 2457
|
| D
|-
| 22
| 2574
|
| E
|-
| 23
| 2691
|
| F
|-
| 24
| 2808
| 81/16
| C
|}
 
== Music ==
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/metashakti.mp3 ''Metashakti'']
 
; [[XэнкøрX]]
* [https://youtu.be/OSQljL4ANf8 "The Blame Game"] from ''State of the World (XLP)'' (2023)
 
[[Category:Listen]]
 
{{todo|expand}}

Latest revision as of 08:55, 19 December 2024

← 5edf 6edf 7edf →
Prime factorization 2 × 3
Step size 116.993 ¢ 
Octave 10\6edf (1169.93 ¢) (→ 5\3edf)
Twelfth 16\6edf (1871.88 ¢) (→ 8\3edf)
Consistency limit 3
Distinct consistency limit 3
Special properties

6 equal divisions of the perfect fifth (abbreviated 6edf or 6ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 6 equal parts of about 117 ¢ each. Each step represents a frequency ratio of (3/2)1/6, or the 6th root of 3/2. It corresponds to 10.2571 edo.

Theory

6edf is related to the miracle temperament, which tempers out 225/224 and 1029/1024 in the 7-limit.

Harmonics

Approximation of harmonics in 6edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -30.1 -30.1 +56.8 +21.5 +56.8 +24.0 +26.8 +56.8 -8.6 -56.6 +26.8
Relative (%) -25.7 -25.7 +48.6 +18.4 +48.6 +20.5 +22.9 +48.6 -7.3 -48.4 +22.9
Steps
(reduced) 		10
(4) 		16
(4) 		21
(3) 		24
(0) 		27
(3) 		29
(5) 		31
(1) 		33
(3) 		34
(4) 		35
(5) 		37
(1)

Intervals

# Cents Approximate Ratios Neptunian
Notation
0 0 1/1 C
1 117 16/15, 15/14 C#
2 234 8/7 Db
3 351 11/9, 27/22 D
4 468 21/16 E
5 585 7/5, 45/32 F
6 702 3/2 C
7 819 8/5, 21/13 C#
8 936 12/7, 55/32 Db
9 1053 11/6 D
10 1170 49/25, 160/81, 2/1 E
11 1287 F
12 1404 9/4 C
13 1521 C#
14 1638 Db
15 1755 D
16 1872 E
17 1988 F
18 2106 27/8 C
19 2223 C#
20 2340 Db
21 2457 D
22 2574 E
23 2691 F
24 2808 81/16 C

Music

Carlo Serafini
XэнкøрX
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