78edo: 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:MasonGreen1|MasonGreen1]] and made on <tt>2016-06-18 10:43:29 UTC</tt>.<br>~~

~~: The original revision id was <tt>585778963</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">The 78 equal division divides the octave into 78 equal parts of size 15.385 [[cent]]s each. It tempers out 2048/2025 in the [[5-limit]]; 875/864 and 2401/2400 in the [[7-limit]]; and 100/99, 385/384 and 1375/1372 in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[Diaschismic family|keen temperament]].

~~Much like~~ [[~~100edo|100bddd~~]]~~, the 78ddd val can be used~~ to ~~construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of~~ the ~~tempered perfect fifth is 707.~~7 ~~cents. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3~~- ~~and 5-limits over the 7~~-limit, ~~while still ensuring that no basic 7-limit intervals other than~~ the ~~tritones are more than 16 cents off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be~~ [[~~https:~~/~~/en.wikipedia.org/wiki/Augmented-fourths_tuning~~|~~tuned in tritones~~]]~~. The appeal of this scale is that it is less xenharmonic than~~ [[~~22edo~~]] is, for ~~listeners accustomed to 12edo. In particular,~~ the ~~163.6¢ "flat minor whole tone"~~ of ~~22edo is now 169~~.~~2¢, making it more clearly a //whole// tone (albeit noticeably flat), rather than a neutral second.</pre></div>~~

== Theory ==

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

78edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]], inherited from [[39edo]], is quite large for the size of the system.

<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>78edo</title></head><body>The 78 equal division divides the octave into 78 equal parts of size 15.385 <a class="wiki_link" href="/cent">cent</a>s each. It tempers out 2048/2025 in the ~~<a class="wiki_link" href="/~~5-limit~~">5-limit</a>~~; 875/864 and 2401/2400 in the ~~<a class="wiki_link" href="/~~7-limit~~">7-limit</a>~~; and 100/99, 385/384 and 1375/1372 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 11-limit ~~<a class="wiki_link" href="/Diaschismic%20family">~~keen temperament~~</a>~~.~~<br />~~

~~<br />~~

This tuning [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

Much like ~~<a class="wiki_link" href="/~~100edo~~">~~100bddd~~</a>~~, the ~~78ddd~~ val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 ~~cents~~. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note ~~2MOS~~ generated in this way could be used to build straight-fretted guitars that would be ~~<a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/~~Augmented-~~fourths_tuning" rel="nofollow">~~tuned in tritones~~</a>~~. The appeal of this scale is that it is less xenharmonic than ~~<a class="wiki_link" href="/~~22edo~~">22edo</a>~~ is, for listeners accustomed to 12edo. In particular, the 163.~~6¢ &quot;~~flat minor whole tone~~&quot;~~ of 22edo is now 169.2¢, making it more clearly a ~~<em>~~whole~~</em>~~ tone (albeit noticeably flat), rather than a neutral second.~~&lt~~;/~~body><~~/~~html><~~/~~pre><~~/~~div>~~

Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 78 factors into {{factorization|78}}, 78edo has subset edos {{EDOs| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.

== Intervals ==

== Scales ==

* [[Maeve Gutierrez|Gutierrez Moonglade scale]]

== Instruments ==

[[Lumatone mapping for 78edo|Lumatone mappings for 78edo]] are available.

[[Category:Keen]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025)

@@ 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:MasonGreen1|MasonGreen1]] and made on <tt>2016-06-18 10:43:29 UTC</tt>.<br>
-: The original revision id was <tt>585778963</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">The 78 equal division divides the octave into 78 equal parts of size 15.385 [[cent]]s each. It tempers out 2048/2025 in the [[5-limit]]; 875/864 and 2401/2400 in the [[7-limit]]; and 100/99, 385/384 and 1375/1372 in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[Diaschismic family|keen temperament]].
-Much like [[100edo|100bddd]], the 78ddd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 cents. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|tuned in tritones]]. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6¢ "flat minor whole tone" of 22edo is now 169.2¢, making it more clearly a //whole// tone (albeit noticeably flat), rather than a neutral second.</pre></div>
+== Theory ==
-<h4>Original HTML content:</h4>
+edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]], inherited from [[39edo]], is quite large for the size of the system.
-<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;78edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 78 equal division divides the octave into 78 equal parts of size 15.385 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It tempers out 2048/2025 in the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;; 875/864 and 2401/2400 in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;; and 100/99, 385/384 and 1375/1372 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 11-limit &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;keen temperament&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
+This tuning [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.
-Much like &lt;a class="wiki_link" href="/100edo"&gt;100bddd&lt;/a&gt;, the 78ddd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 cents. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow"&gt;tuned in tritones&lt;/a&gt;. The appeal of this scale is that it is less xenharmonic than &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is, for listeners accustomed to 12edo. In particular, the 163.6¢ &amp;quot;flat minor whole tone&amp;quot; of 22edo is now 169.2¢, making it more clearly a &lt;em&gt;whole&lt;/em&gt; tone (albeit noticeably flat), rather than a neutral second.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Much like [[100edo|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w|Augmented-fourths tuning|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.
+=== Odd harmonics ===
+{{Harmonics in equal|78}}
+=== Subsets and supersets ===
+Since 78 factors into {{factorization|78}}, 78edo has subset edos {{EDOs| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.
+== Intervals ==
+{{Interval table}}
+== Scales ==
+* [[Maeve Gutierrez|Gutierrez Moonglade scale]]
+== Instruments ==
+[[Lumatone mapping for 78edo|Lumatone mappings for 78edo]] are available.
+[[Category:Keen]]
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025)