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70edo: Difference between revisions

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**Imported revision 238331361 - Original comment: **
 
→Notation: added interval mappings of 70p and 70cd
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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:genewardsmith|genewardsmith]] and made on <tt>2011-06-23 01:48:25 UTC</tt>.<br>
: The original revision id was <tt>238331361</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 //70 equal division// divides the octave into 70 equal parts of 14.143 cents each. It was singled out by William Stoney in his article "Theoretical Possibilities for Equally Tempered Systems" (in the book //The Cpmputer and Music//) as one of the six best systems of size 72 or smaller, along with 72, 58, 53, 65 and 41. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since.


The patent val for 70edo tempers out 2028/2025, making it a diaschismic system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out 32805/32768. In the 7-limit, the patent val tempers out 126/125, 5120/5103 and 2430/2401. The 70cd val tempers out 225/224 and 3125/3087 instead. The alternative mapping begans to make more sense in the 11-limit and higher, where the patent val tempers out 99/98 and 121/120 in the 11-limit, 169/168 and 352/351 in the 13-limit, and 221/220 in the 17-limit. 70cd on the other hand, with flat 5 and 7, tempers out 100/99 and 245/242 in the 11-limit, 105/104 and 196/195 in the 13-limit, and 154/153 and 170/169 in the 17-limit. 70 also makes sense as a no 5 or 7 system, tempering out 131769/131072 in the 11-limit, 352/351 and 2197/2187 in the 13-limit, and 289/288 and 1089/1088 in the 17-limit.
== Theory ==
This tuning was singled out by [[William Stoney]] in his article "Theoretical Possibilities for Equally Tempered Systems" (in the book [https://monoskop.org/images/c/c3/Lincoln_Harry_B_ed_The_Computer_and_Music_1970.pdf The Computer and Music]) as one of the six best systems of size 72 or smaller, along with [[72edo|72]], [[65edo|65]], [[58edo|58]], [[53edo|53]], and [[41edo|41]]. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since, despite its excellent perfect fifth, which is the 4th number in the convergent sequence to the [[Argent tuning|silver ratio]], following [[29edo]], [[12edo]], and [[5edo]] and preceding [[169edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth, and this perfect fifth, 41\70, is the true center of the diatonic tuning spectrum, as it is the [[geometric mean]] of [[5edo|3\5edo]] and [[7edo|4\7edo]].  


The 17-limit [[k*N subgroups|2*70]] subgroup, on which 70 is tuned like 140, is 2.3.25.35.11.13.17.</pre></div>
The [[patent val]] for 70edo [[tempering out|tempers out]] [[2048/2025]], making it a [[diaschismic]] system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out [[32805/32768]]. In the [[7-limit]], the patent val tempers out [[126/125]], [[2430/2401]] and [[5120/5103]], and provides the optimum patent val for the [[kumonga]] temperament. The 70c val tempers out [[50/49]], making it a tuning for [[doublewide]] even better than the optimal patent val. The 70cd val tempers out [[225/224]] and [[3125/3087]] instead. The alternative mapping begins to make more sense in the [[11-limit]] and higher, where the patent val tempers out [[99/98]] and [[121/120]] in the 11-limit, [[169/168]] and [[352/351]] in the [[13-limit]], and [[221/220]] in the [[17-limit]]. 70cd on the other hand, with flat 5 and 7, tempers out [[100/99]] and [[245/242]] in the 11-limit, [[105/104]] and [[196/195]] in the 13-limit, and [[154/153]] and [[170/169]] in the 17-limit. 70 also makes sense as a no-5 or -7 system, tempering out [[131769/131072]] in the 11-limit, [[352/351]] and [[2197/2187]] in the 13-limit, and [[289/288]] and [[1089/1088]] in the 17-limit.
<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;70edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;70 equal division&lt;/em&gt; divides the octave into 70 equal parts of 14.143 cents each. It was singled out by William Stoney in his article &amp;quot;Theoretical Possibilities for Equally Tempered Systems&amp;quot; (in the book &lt;em&gt;The Cpmputer and Music&lt;/em&gt;) as one of the six best systems of size 72 or smaller, along with 72, 58, 53, 65 and 41. These other systems have had notice paid to them, but the same does not seem to be true of 70, which seems to have been ignored ever since.&lt;br /&gt;
The 17-limit [[k*N subgroups|2*70]] subgroup, on which 70 is tuned like [[140edo]], is 2.3.25.35.11.13.17.
&lt;br /&gt;
 
The patent val for 70edo tempers out 2028/2025, making it a diaschismic system. An alternative mapping is 70c, with a flat rather than a sharp major third, tempering out 32805/32768. In the 7-limit, the patent val tempers out 126/125, 5120/5103 and 2430/2401. The 70cd val tempers out 225/224 and 3125/3087 instead. The alternative mapping begans to make more sense in the 11-limit and higher, where the patent val tempers out 99/98 and 121/120 in the 11-limit, 169/168 and 352/351 in the 13-limit, and 221/220 in the 17-limit. 70cd on the other hand, with flat 5 and 7, tempers out 100/99 and 245/242 in the 11-limit, 105/104 and 196/195 in the 13-limit, and 154/153 and 170/169 in the 17-limit. 70 also makes sense as a no 5 or 7 system, tempering out 131769/131072 in the 11-limit, 352/351 and 2197/2187 in the 13-limit, and 289/288 and 1089/1088 in the 17-limit.&lt;br /&gt;
=== Prime harmonics ===
&lt;br /&gt;
{{Harmonics in equal|70|columns=9|intervals=prime}}
The 17-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*70&lt;/a&gt; subgroup, on which 70 is tuned like 140, is 2.3.25.35.11.13.17.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|70|columns=9|intervals=prime|start=10|collapsed=true|title=Approximation of prime harmonics in 70edo (continued)}}
 
=== Subsets and supersets ===
Since 70 factors into {{factorization|70}}, 70edo has subset edos {{EDOs| 2, 5, 7, 10, 14, and 35 }}. 140edo, which doubles it, provides good correction for its approximation to harmonics 5 and 7.
 
== Intervals ==
{{Interval table}}
 
== Notation ==
=== Ups and downs notation ===
 
70edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
 
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:70-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 230 106 [[55/54]]
rect 230 80 350 106 [[33/32]]
default [[File:70-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:70-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 716 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 230 106 [[55/54]]
rect 230 80 350 106 [[33/32]]
default [[File:70-EDO_Revo_Sagittal.svg]]
</imagemap>
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|70}}
{{Q-odd-limit intervals|69.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 70cd val mapping}}
 
== Scales ==
=== Kukula's 2.3.13 70edo MOS ===
In July 2025, composer and theorist [[James Kukula]] created a 17-tone [[MOS scale]] for his piece in the [[Monthly Tunings]] project. The scale is generated by stacking the interval 33\70, 17 times, then [[octave reduction|octave reducing]] the result. It is designed to approximate the 2.3.13 [[subgroup]] very accurately. He discusses it in his blog post titled ''[https://interdependentscience.blogspot.com/2025/07/stepping-outside.html Stepping Outside]''.
 
; Subsets
* Kukula-Lambeth tridecimal neutral octatonic{{idiosyncratic}}: 8 13 8 12 8 9 8 4
* Kukula-Lambeth tridecimal neutral heptatonic{{idiosyncratic}}: 8 13 8 12 8 9 12
 
== Instruments ==
 
A [[Lumatone mapping for 70edo]] is available.
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/I42JQ98WPZI ''microtonal improvisation in 70edo''] (2025)
* [https://www.youtube.com/shorts/gp42aya9TL0 ''Drift Away - Steven Universe (microtonal cover in 70edo)''] (2025)
* [https://www.youtube.com/watch?v=Uc0c-AYq-zM ''Waltz in 70edo''] (2025)
* [https://www.youtube.com/watch?v=-NEq3jKHjzs ''Improv in 70edo''] (2025)
* [https://www.youtube.com/shorts/ZDU75YabROE ''70edo prelude''] (2025)
 
; [[James Kukula]]
* [https://app.box.com/s/i9vvgcplpmksrm8rubz2h3asee2cnr26 ''piece from Stepping Outside''] (2025)
 
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=dyEH-pP2dJU ''Improv in 70edo (Polymicrotonal 10edo+14edo Scale)''] (2025)
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/70edo"