250/243: Difference between revisions

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**Imported revision 613052253 - Original comment: **
→Approximation: 24edo itself does not qualify as a chromium tuning
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 250/243
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2017-05-17 11:02:37 UTC</tt>.<br>
| en = 250/243
: The original revision id was <tt>613052253</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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<h4>Original Wikitext content:</h4>
{{Infobox Interval
<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">&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/250_243|Deutsch]]&lt;/span&gt;
| Name = porcupine comma, maximal diesis
**250/243**, known as the **porcupine [[comma]]** or maximal diesis, is an interval of size 49.166 [[Cent|cents]]. It is the amount by which two [[10_9|minor whole tones]] exceed a minor third, that is, (10/9)^2/(6/5). Tempering it out leads to [[5-limit]] [[Porcupine family|porcupine temperament]].</pre></div>
| Color name = y<sup>3</sup>1, triyo 1sn,<br>y<sup>3</sup>M, triyoma
<h4>Original HTML content:</h4>
| Comma = yes
<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;250_243&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/250_243"&gt;Deutsch&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
}}
&lt;strong&gt;250/243&lt;/strong&gt;, known as the &lt;strong&gt;porcupine &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;&lt;/strong&gt; or maximal diesis, is an interval of size 49.166 &lt;a class="wiki_link" href="/Cent"&gt;cents&lt;/a&gt;. It is the amount by which two &lt;a class="wiki_link" href="/10_9"&gt;minor whole tones&lt;/a&gt; exceed a minor third, that is, (10/9)^2/(6/5). Tempering it out leads to &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Porcupine%20family"&gt;porcupine temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
'''250/243''' is known as the '''porcupine comma''' or the '''maximal diesis'''. Measuring about 49{{cent}}, it is a [[medium comma]]. It is the amount by which two [[10/9|minor whole tones]] exceed a minor third, that is, (10/9)<sup>2</sup>/(6/5). It is also the difference between [[25/24]] and [[81/80]], the two smallest [[5-limit]] [[superparticular]] ratios, and between three syntonic commas and the [[2187/2048|Pythagorean apotome]], putting it on the [[Syntonic&ndash;chromatic equivalence continuum]].
 
== Temperaments ==
Tempering it out leads to the [[5-limit]] [[porcupine]] temperament. See [[porcupine family]] for the family of rank-2 temperaments where it is tempered out.  
 
== Approximation ==
250/243 is very close to one step of [[24edo]], which is the quarter tone that is exactly the half of [[12edo]] semitone. Therefore, if 250/243 is not tempered and instead is treated as an identity where it is equated with 1/24th of the octave, it serves as the period in the [[chromium]] temperament. (However, note that 24edo itself maps 250/243 inconsistently, and so chromium temperament starts at [[72edo]].) Thus in the framework of this temperament and the tuning systems associated with it, [[Eliora]] proposes the name ''chromium quartertone''.  
 
[[Category:Porcupine]]
[[Category:Commas named after compositions]]

Latest revision as of 02:17, 28 May 2026

Interval information
Ratio 250/243
Factorization 2 × 3-5 × 53
Monzo [1 -5 3⟩
Size in cents 49.16614¢
Names porcupine comma,
maximal diesis
Color name y31, triyo 1sn,
y3M, triyoma
FJS name [math]\displaystyle{ \text{A1}^{5,5,5} }[/math]
Special properties reduced
Tenney norm (log2 nd) 15.8906
Weil norm (log2 max(n, d)) 15.9316
Wilson norm (sopfr(nd)) 32
Comma size medium
S-expression S102⋅S11
Open this interval in xen-calc

250/243 is known as the porcupine comma or the maximal diesis. Measuring about 49 ¢, it is a medium comma. It is the amount by which two minor whole tones exceed a minor third, that is, (10/9)2/(6/5). It is also the difference between 25/24 and 81/80, the two smallest 5-limit superparticular ratios, and between three syntonic commas and the Pythagorean apotome, putting it on the Syntonic–chromatic equivalence continuum.

Temperaments

Tempering it out leads to the 5-limit porcupine temperament. See porcupine family for the family of rank-2 temperaments where it is tempered out.

Approximation

250/243 is very close to one step of 24edo, which is the quarter tone that is exactly the half of 12edo semitone. Therefore, if 250/243 is not tempered and instead is treated as an identity where it is equated with 1/24th of the octave, it serves as the period in the chromium temperament. (However, note that 24edo itself maps 250/243 inconsistently, and so chromium temperament starts at 72edo.) Thus in the framework of this temperament and the tuning systems associated with it, Eliora proposes the name chromium quartertone.

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