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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:vaisvil|vaisvil]] and made on <tt>2011-11-28 23:15:17 UTC</tt>.<br>
: The original revision id was <tt>279979454</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">[[media type="custom" key="10763046"]]


11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.
== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.


From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.
From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].


Tuning in scala format is as follows:
The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].


! E:\cakewalk\scales\11_of_tritave.scl
=== Harmonics ===
!
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
11 in tritave
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}
!
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1


=== Subsets and supersets ===
11edt is the fifth [[prime equal division|prime edt]], following [[7edt]] and before [[13edt]], so it does not contain any nontrivial subset edts.


== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios*
! [[Arcturus]]<br>enneatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[10/9]], [[11/10]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[15/11]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[18/11]], [[33/20]]
| N
|-
| 6
| 1037.4
| 708.6
| [[9/5]], [[11/6]], [[20/11]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
| [[9/4]], [[11/5]]
| P
|-
| 9
| 1556.1
| 1062.9
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| Q
|-
| 10
| 1729.0
| 1181.0
| [[8/3]], [[11/4]]
| R
|-
| 11
| 1902.0
| 1300.0
| [[3/1]]
| J
|}
<nowiki/>* As a 2.3.5.11-subgroup temperament


== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves


Mozart's [[http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3|sonata #11 in A Major K331 in 11 EDT]] (using a 11 =&gt; 12 key mapping so octaves become tritaves)
=== 21st century ===
[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]]
; [[Chris Vaisvil]]
[[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3|Molly's Playground]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&gt; [[http://chrisvaisvil.com/?p=1716| information about the piece]]</pre></div>
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play]
<h4>Original HTML content:</h4>
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]
<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;11edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10763046?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;10763046&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&gt;
 
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
== See also ==
&lt;br /&gt;
* [[7edo]] – relative edo
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.&lt;br /&gt;
* [[18ed6]] – relative ed6
&lt;br /&gt;
From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.&lt;br /&gt;
&lt;br /&gt;
Tuning in scala format is as follows:&lt;br /&gt;
&lt;br /&gt;
! E:\cakewalk\scales\11_of_tritave.scl&lt;br /&gt;
!&lt;br /&gt;
11 in tritave&lt;br /&gt;
!&lt;br /&gt;
172.90500&lt;br /&gt;
345.81000&lt;br /&gt;
518.71500&lt;br /&gt;
691.62000&lt;br /&gt;
864.52500&lt;br /&gt;
1037.43000&lt;br /&gt;
1210.33500&lt;br /&gt;
1383.24000&lt;br /&gt;
1556.14500&lt;br /&gt;
1729.05000&lt;br /&gt;
3/1&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Mozart's &lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3" rel="nofollow"&gt;sonata #11 in A Major K331 in 11 EDT&lt;/a&gt; (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&gt;Frozen Time Occupies Wall Street&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt;&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&gt; information about the piece&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3" rel="nofollow"&gt;Molly's Playground&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1716" rel="nofollow"&gt; information about the piece&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 12:48, 26 May 2025

← 10edt 11edt 12edt →
Prime factorization 11 (prime)
Step size 172.905 ¢ 
Octave 7\11edt (1210.34 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 5

11 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 11edt or 11ed3), is a nonoctave tuning system that divides the interval of 3/1 into 11 equal parts of about 173 ¢ each. Each step represents a frequency ratio of 31/11, or the 11th root of 3.

Theory

11edt can be seen as a very stretched version of 7edo, with octaves sharpened by 10.3 cents. The octave stretching makes the 3/2 perfect fifth in better tune, while preserving a just 3/1 tritave.

From a no-2 point of view, 11edt has a 5/3 major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic 7/1, which is off by almost half a step (or about a semitone), which causes it to temper out 49/45 in the 7-limit. 11edt is at the extreme end of arcturus temperament, defined by tempering out 15625/15309 in the 3.5.7 subgroup. It gives an equalized interpretation for the sub-arcturus mos scale.

The 11th harmonic, 11/1, only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out 35/33 and 77/75 in the 11-limit. In the 3.5.11 subgroup, it tempers out 125/121.

Harmonics

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(0) 		14
(3) 		16
(5) 		18
(7) 		19
(8) 		21
(10) 		22
(0) 		23
(1) 		24
(2) 		25
(3)
Approximation of harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced) 		26
(4) 		26
(4) 		27
(5) 		28
(6) 		28
(6) 		29
(7) 		29
(7) 		30
(8) 		30
(8) 		31
(9) 		31
(9) 		32
(10)

Subsets and supersets

11edt is the fifth prime edt, following 7edt and before 13edt, so it does not contain any nontrivial subset edts.

Intervals

# Cents Hekts Approximate ratios* Arcturus
enneatonic notation (J = 1/1)
0 0.0 0.0 1/1 J
1 172.9 118.1 10/9, 11/10 J#, Kb
2 345.8 236.2 5/4, 6/5, 11/9, 27/22 K
3 518.7 354.3 4/3, 15/11 L
4 691.6 472.4 3/2 M
5 864.5 590.5 5/3, 18/11, 33/20 N
6 1037.4 708.6 9/5, 11/6, 20/11 N#, Ob
7 1210.3 826.7 2/1 O
8 1383.2 944.8 9/4, 11/5 P
9 1556.1 1062.9 5/2, 12/5, 22/9, 27/11 Q
10 1729.0 1181.0 8/3, 11/4 R
11 1902.0 1300.0 3/1 J

* As a 2.3.5.11-subgroup temperament

Music

Modern renderings

Wolfgang Amadeus Mozart

21st century

Chris Vaisvil
  • Frozen Time Occupies Wall Street (2011) – blog | play
  • Molly's Playground (2011) – blog | play

See also

Retrieved from "https://en.xen.wiki/index.php?title=11edt&oldid=197763"