225edo: Difference between revisions

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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:hstraub|hstraub]] and made on <tt>2011-06-22 07:23:30 UTC</tt>.<br>
 
: The original revision id was <tt>238144507</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
225edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with rather large errors in [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. It has three mappings possible for the 7-limit:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* {{val| 225 357 522 632 }} ([[patent val]]),
<h4>Original Wikitext content:</h4>
* {{val| 225 '''356''' 522 '''631''' }} (225bd),
<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">=Scales=
* {{val| 225 357 '''523''' 632 }} (225c).
[[bossier8]]
 
[[bossier11]]
Using the patent val, it tempers out 20000/19683 and 2109375/2097152 in the 5-limit; 3125/3087, 10976/10935, and 589824/588245 in the 7-limit.
[[bossier14]]</pre></div>
 
<h4>Original HTML content:</h4>
Using the 225bd val, it tempers out 78732/78125 (sensipent comma) and {{monzo| -52 27 4 }} in the 5-limit; 225/224, 177147/175000, and 40353607/40000000 in the 7-limit.
<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;225edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
 
&lt;a class="wiki_link" href="/bossier8"&gt;bossier8&lt;/a&gt;&lt;br /&gt;
Using the 225c val, it tempers out 131072000/129140163 (rodan comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 2401/2400, 4375/4374, and 2097152/2066715 in the 7-limit.
&lt;a class="wiki_link" href="/bossier11"&gt;bossier11&lt;/a&gt;&lt;br /&gt;
 
&lt;a class="wiki_link" href="/bossier14"&gt;bossier14&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Odd harmonics ===
{{Harmonics in equal|225}}
 
== Scales ==
* [[Bossier scales]]
 
[[Category:Bossier]]

Latest revision as of 13:46, 24 March 2025

← 224edo 225edo 226edo →
Prime factorization 32 × 52
Step size 5.33333 ¢ 
Fifth 132\225 (704 ¢) (→ 44\75)
Semitones (A1:m2) 24:15 (128 ¢ : 80 ¢)
Dual sharp fifth 132\225 (704 ¢) (→ 44\75)
Dual flat fifth 131\225 (698.667 ¢)
Dual major 2nd 38\225 (202.667 ¢)
Consistency limit 3
Distinct consistency limit 3

225 equal divisions of the octave (abbreviated 225edo or 225ed2), also called 225-tone equal temperament (225tet) or 225 equal temperament (225et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 225 equal parts of about 5.33 ¢ each. Each step represents a frequency ratio of 21/225, or the 225th root of 2.

Theory

225edo is inconsistent to the 5-odd-limit and higher limits, with rather large errors in harmonics 3, 5, 7, 11, and 13. It has three mappings possible for the 7-limit:

  • ⟨225 357 522 632] (patent val),
  • ⟨225 356 522 631] (225bd),
  • ⟨225 357 523 632] (225c).

Using the patent val, it tempers out 20000/19683 and 2109375/2097152 in the 5-limit; 3125/3087, 10976/10935, and 589824/588245 in the 7-limit.

Using the 225bd val, it tempers out 78732/78125 (sensipent comma) and [-52 27 4⟩ in the 5-limit; 225/224, 177147/175000, and 40353607/40000000 in the 7-limit.

Using the 225c val, it tempers out 131072000/129140163 (rodan comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 2401/2400, 4375/4374, and 2097152/2066715 in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 225edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.04 -2.31 +1.84 -1.24 -1.98 +2.14 -0.27 +1.71 +1.15 -1.45 +1.06
Relative (%) +38.3 -43.4 +34.5 -23.3 -37.2 +40.1 -5.0 +32.1 +21.6 -27.1 +19.9
Steps
(reduced) 		357
(132) 		522
(72) 		632
(182) 		713
(38) 		778
(103) 		833
(158) 		879
(204) 		920
(20) 		956
(56) 		988
(88) 		1018
(118)

Scales

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