93edo: 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:genewardsmith|genewardsmith]] and made on <tt>2012-05-29 01:48:29 UTC</tt>.<br>~~

~~: The original revision id was <tt>340355226</tt>.<br>~~

== Theory ==

~~: The revision comment was: <tt></tt><br>~~

Since {{nowrap|93 {{=}} 3 × 31}}, 93edo is a [[contorted]] [[31edo]] through the [[7-limit]]. In the 11-limit the [[patent val]] [[tempering out|tempers out]] [[4000/3993]] and in the 13-limit [[144/143]], [[1188/1183]], and [[364/363]]. It provides the [[optimal patent val]] for the 11-limit [[31st-octave_temperaments#Prajapati|prajapati]] and 13-limit [[31st-octave_temperaments#Kumhar|kumhar]] temperaments and the 11- and 13-limit [[Meantone family#Trimean|trimean]] ({{nowrap|43 & 50}}) temperament, and is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{nowrap|{{frac|2|7}}-[[64/63|septimal comma]]}} superpyth very well.

~~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>~~

Since 93edo has good approximations of [[13/1|13th]], [[17/1|17th]] and [[19/1|19th]] [[harmonic]]s unlike 31edo (as 838.710{{c}}, 103.226{{c}}, and 296.774{{c}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.

~~<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">The 93 equal division divides the octave into 93 equal parts of 12.903 cents each.~~ 93 = 3 * 31, ~~and 93~~ is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&50 temperament.</~~pre><~~/~~div>~~

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

=== Odd harmonics ===

~~<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>93edo</title></head><body>The 93~~ equal ~~division divides the octave into 93 equal parts of 12.903 cents each.~~ 93 = 3 * 31, ~~and 93 is a contorted 31 through~~ the 7 limit. In the 11-~~limit the patent val tempers out 4000/3993 and in the 13~~-limit ~~144/143, 1188/1183 and 364/363~~. ~~It provides the optimal patent val for the 11-limit prajapati and~~ 13~~-limit kumhar temperaments, and the 11 and~~ 13 ~~limit 43&amp;50 temperament~~.~~&lt~~;/~~body><~~/~~html><~~/~~pre><~~/~~div>~~

=== No-3 approach ===

If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.

== Intervals ==

== Scales ==

* Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)

* Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)

* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)

* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)

== Instruments ==

A [[Lumatone mapping for 93edo]] is available.

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)

== See also ==

* [[93edo and stretched hemififths]]

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-05-29 01:48:29 UTC</tt>.<br>
-: The original revision id was <tt>340355226</tt>.<br>
+== Theory ==
-: The revision comment was: <tt></tt><br>
+Since {{nowrap|93 {{=}} 3 × 31}}, 93edo is a [[contorted]] [[31edo]] through the [[7-limit]]. In the 11-limit the [[patent val]] [[tempering out|tempers out]] [[4000/3993]] and in the 13-limit [[144/143]], [[1188/1183]], and [[364/363]]. It provides the [[optimal patent val]] for the 11-limit [[31st-octave_temperaments#Prajapati|prajapati]] and 13-limit [[31st-octave_temperaments#Kumhar|kumhar]] temperaments and the 11- and 13-limit [[Meantone family#Trimean|trimean]] ({{nowrap|43 &amp; 50}}) temperament, and is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{nowrap|{{frac|2|7}}-[[64/63|septimal comma]]}} superpyth very well.
-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>
+Since 93edo has good approximations of [[13/1|13th]], [[17/1|17th]] and [[19/1|19th]] [[harmonic]]s unlike 31edo (as 838.710{{c}}, 103.226{{c}}, and 296.774{{c}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
-<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">The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 93 = 3 * 31, and 93 is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&amp;50 temperament.</pre></div>
-<h4>Original HTML content:</h4>
+=== Odd harmonics ===
-<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;93edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 93 = 3 * 31, and 93 is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&amp;amp;50 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
+{{Harmonics in equal|93}}
+=== No-3 approach ===
+If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.
+== Intervals ==
+{{Interval table}}
+== Scales ==
+* Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
+* Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
+* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
+* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)
+== Instruments ==
+A [[Lumatone mapping for 93edo]] is available.
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)
+== See also ==
+* [[93edo and stretched hemififths]]