321edo: Difference between revisions

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**Imported revision 268601160 - Original comment: **
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→Scales: Added a corresponding temperament for the Blastoff scale.
 
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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>2011-10-25 22:01:08 UTC</tt>.<br>
 
: The original revision id was <tt>268601160</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
321edo is in[[consistent]] in the [[5-odd-limit]]. The [[patent val]] [[Tempering out|tempers out]] [[2401/2400]], [[5120/5103]] and [[10976/10935]] in the 7-limit, supporting [[hemififths]]. In the 11-limit it tempers out [[385/384]] and 1375/1372, and in the 13-limit [[325/324]], [[352/351]], [[847/845]], [[2080/2079]] and [[4096/4095]], providing the [[optimal patent val]] for 11- and 13-limit [[akea]] temperament.
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>
=== Prime 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;white-space: pre-wrap ! important" class="old-revision-html">The 321 equal division divides the octave into 321 equal parts of 3.738 cents each. The patent val tempers out 2401/2400, 5120/5103 and 10976/10935 in the 7-limit, supporting hemififths temperament. In the 11-limit it tempers out 385/384 and 1375/1372, and in the 13-limit 325/324, 352/351, 847/845 and 2080/2079, providing the [[optimal patent val]] for 11- and 13-limit [[Hemifamity family#Akea|akea temperament]].</pre></div>
{{Harmonics in equal|321}}
<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;321edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 321 equal division divides the octave into 321 equal parts of 3.738 cents each. The patent val tempers out 2401/2400, 5120/5103 and 10976/10935 in the 7-limit, supporting hemififths temperament. In the 11-limit it tempers out 385/384 and 1375/1372, and in the 13-limit 325/324, 352/351, 847/845 and 2080/2079,  providing the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 11- and 13-limit &lt;a class="wiki_link" href="/Hemifamity%20family#Akea"&gt;akea temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
== Interval table ==
{{main|Table of 321edo intervals}}
 
== Scales ==
[[JUMBLE]]'s Blastoff scale (9L&nbsp;8s)
* 25\321
* 37\321
* 62\321
* 74\321
* 99\321
* 111\321
* 136\321
* 148\321
* 173\321
* 185\321
* 210\321
* 222\321
* 247\321
* 259\321
* 284\321
* 296\321
* 321\321
This scale may be thought of as a tuning of the [[Greenwoodmic temperaments#Secundly|secundly]] (or lower limit secund) temperament.
 
== Music ==
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=4HT5onKQaz0 ''Sun Through The Window''] (2023) – ambient
* [https://www.youtube.com/watch?v=cQoWVBbKnuA ''BLASTOFF!''] (2023) – synthwave, [[9L 8s]] scale
* [https://www.youtube.com/watch?v=e8hhlQSYqVY ''Infinity''] (2023)
* [https://www.youtube.com/watch?v=ZW7JyextZoM ''Greige City''] (2023) – synthwave, [[9L 8s]] scale
* [https://www.youtube.com/watch?v=yIfooPURtYE ''So Long, San Diego''] (2024)
* [https://www.youtube.com/watch?v=xI0HXnrlSlA ''Ten Day Forecast''] (2024) – ambient
 
; [[No Clue Music]]
* [https://www.youtube.com/watch?v=orjW0qIveqU ''Frozen Star''] (2024)
 
[[Category:Akea]]
[[Category:Listen]]

Latest revision as of 10:46, 23 May 2025

← 320edo 321edo 322edo →
Prime factorization 3 × 107
Step size 3.73832 ¢ 
Fifth 188\321 (702.804 ¢)
Semitones (A1:m2) 32:23 (119.6 ¢ : 85.98 ¢)
Consistency limit 3
Distinct consistency limit 3

321 equal divisions of the octave (abbreviated 321edo or 321ed2), also called 321-tone equal temperament (321tet) or 321 equal temperament (321et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 321 equal parts of about 3.74 ¢ each. Each step represents a frequency ratio of 21/321, or the 321st root of 2.

Theory

321edo is inconsistent in the 5-odd-limit. The patent val tempers out 2401/2400, 5120/5103 and 10976/10935 in the 7-limit, supporting hemififths. In the 11-limit it tempers out 385/384 and 1375/1372, and in the 13-limit 325/324, 352/351, 847/845, 2080/2079 and 4096/4095, providing the optimal patent val for 11- and 13-limit akea temperament.

Prime harmonics

Approximation of prime harmonics in 321edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.85 -1.27 -0.60 -1.79 +0.59 -0.28 +1.55 -0.24 -1.54 -1.11
Relative (%) +0.0 +22.7 -33.9 -16.1 -47.8 +15.9 -7.6 +41.5 -6.3 -41.2 -29.7
Steps
(reduced) 		321
(0) 		509
(188) 		745
(103) 		901
(259) 		1110
(147) 		1188
(225) 		1312
(28) 		1364
(80) 		1452
(168) 		1559
(275) 		1590
(306)

Interval table

Main article: Table of 321edo intervals

Scales

JUMBLE's Blastoff scale (9L 8s)

  • 25\321
  • 37\321
  • 62\321
  • 74\321
  • 99\321
  • 111\321
  • 136\321
  • 148\321
  • 173\321
  • 185\321
  • 210\321
  • 222\321
  • 247\321
  • 259\321
  • 284\321
  • 296\321
  • 321\321

This scale may be thought of as a tuning of the secundly (or lower limit secund) temperament.

Music

JUMBLE
No Clue Music
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