109edo: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-31 02:10:42 UTC</tt>.<br>~~

~~: The original revision id was <tt>288887283</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">**109edo** is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; 245/243, 2401/2400 and 65625/65536 in the [[7-limit]]; 385/384, 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[Tetracot family|octacot temperament]], and 109ef provides an excellent tuning for 11- and 13-limit octacot.

~~109edo is the 29th [[prime numbers|prime]] edo.</pre></div>~~

== Theory ==

~~<h4>Original HTML content:</h4>~~

109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

~~<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"><html><head><title>~~109edo</title></head><body><strong>109edo</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into 109 parts of 11.009 <a class="wiki_link" href="/cent">cent</a>s each. It <a class="wiki_link" href="/tempering~~%20out">~~tempers out~~</a>~~ 20000/19683 in the ~~<a class="wiki_link" href="/~~5-limit~~">5-limit</a>~~; 245/243, 2401/2400 and 65625/65536 in the ~~<a class="wiki_link" href="/~~7-limit~~">7-limit</a>~~; 385/384, 1375/1372, and 4000/3993 in the ~~<a class="wiki_link" href="/~~11-limit~~">11-limit</a>~~. It provides the ~~<a class="wiki_link" href="/optimal%20patent%20val">~~optimal patent val~~</a>~~ for 7-limit ~~<a class="wiki_link" href="/Tetracot%20family">~~octacot temperament~~</a>~~, and 109ef provides an excellent tuning for 11- and 13-limit octacot.~~<br~~ /~~>~~

~~<br />~~

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.

109edo is the 29th ~~<~~a ~~class~~="~~wiki_link~~" ~~href~~="/~~prime%20numbers">prime<~~/~~a> edo~~.~~<~~/~~body><~~/~~html><~~/~~pre><~~/~~div>~~

=== Prime harmonics ===

=== Subsets and supersets ===

109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.

=== Nonoctave temperaments ===

Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==

== Music ==

; [[Francium]]

* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning

* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]

== See also ==

* [[109-7-comma]]

@@ Line 1: / Line 1: @@
-<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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-31 02:10:42 UTC</tt>.<br>
-: The original revision id was <tt>288887283</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">**109edo** is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; 245/243, 2401/2400 and 65625/65536 in the [[7-limit]]; 385/384, 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[Tetracot family|octacot temperament]], and 109ef provides an excellent tuning for 11- and 13-limit octacot.
-edo is the 29th [[prime numbers|prime]] edo.</pre></div>
+== Theory ==
-<h4>Original HTML content:</h4>
+edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
-<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;109edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;109edo&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into 109 parts of 11.009 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; 20000/19683 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;; 245/243, 2401/2400 and 65625/65536 in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;; 385/384, 1375/1372, and 4000/3993 in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 7-limit &lt;a class="wiki_link" href="/Tetracot%20family"&gt;octacot temperament&lt;/a&gt;, and 109ef provides an excellent tuning for 11- and 13-limit octacot.&lt;br /&gt;
-&lt;br /&gt;
+edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.
-edo is the 29th &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo.&lt;/body&gt;&lt;/html&gt;</pre></div>
+=== Prime harmonics ===
+{{Harmonics in equal|109|columns=16}}
+=== Subsets and supersets ===
+edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.
+=== Nonoctave temperaments ===
+Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].
+== Intervals ==
+{{Interval table}}
+== Music ==
+; [[Francium]]
+* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
+* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]
+== See also ==
+* [[109-7-comma]]