5/4: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| de = Naturterz
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2013-05-19 18:35:51 UTC</tt>.<br>
| en = 5/4
: The original revision id was <tt>432738890</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>
| ro = 5/4 (ro)
<h4>Original Wikitext content:</h4>
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<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">In [[Just Intonation]], **5/4** is the frequency ratio between the 5th and 4th harmonics. Measuring about 386.3[[Cent|¢]], it is about 13.7¢ away from [[12edo]]'s major third of 400¢. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81_64|81/64]] by the syntonic comma of [[81_80|81/80]], which measures about 21.5¢. 81/64 and 5/4 are both just intonation "major thirds," 81/64 having a more active and discordant quality, 5/4 sounding more "restful".
{{Infobox Interval
| Name = just major third, classic(al) major third, ptolemaic major third
| Color name = y3, yo 3rd
| Sound = jid_5_4_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major third}}


In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated here melodically in singing into a resonant udderbot (from the fundamental up to 5 and then noodling between 5 and 4). [[file:5-4.mp3]] Hear it?
In [[5-limit]] [[just intonation]], '''5/4''' is the [[frequency ratio]] between the 5th and 4th [[harmonic]]s. It has been called the '''just major third''', '''classic(al) major third''', or '''ptolemaic major third'''<ref>For reference, see [[5-limit]].</ref> to distinguish it from other intervals in that neighborhood. Measuring about 386.3 [[cent|¢]], it is about 13.7{{c}} away from [[12edo]]'s major third of 400{{c}}. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for [[5-limit]] harmony. It is distinguished from the [[Pythagorean]] major third of [[81/64]] by the syntonic comma of [[81/80]], which measures about 21.5{{c}}, and from the Pythagorean diminished fourth of [[8192/6561]] by the [[schisma]], which measures about 1.95{{c}}. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".  


5/4 converted to cents (¢): 1200 * log (5/4) / log (2) = 386.314...
In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in [[:File: 5-4.mp3]] melodically in singing into a resonant [[udderbot]] (from the fundamental up to 5 and then noodling between 5 and 4).


When two notes sound a 5/4 together,
== Approximations by edos ==
Following [[edo]]s (up to 200, and also 643) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).


5/4 the interval, like all //intervals//, refers to a //relation// between two pitches. We speak of this relation (one pitch beating 5/4 times as fast as the other) after we are able to distinguish it from other relations.
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
|-
|  [[25edo|25]]  ||  8\25  || 2.3137 || 4.8202 || &darr; ||
|-
|  [[28edo|28]]  ||  9\28  || 0.5994 || 1.3987 || &darr; || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]
|-
|  [[31edo|31]]  || 10\31  || 0.7831 || 2.0229 || &uarr; || [[62edo|20\62]], [[93edo|30\93]]
|-
|  [[34edo|34]]  || 11\34  || 1.9216 || 5.4445 || &uarr; ||
|-
|  [[53edo|53]]  || 17\53  || 1.4081 || 6.2189 || &darr; ||
|-
|  [[59edo|59]]  || 19\59  || 0.1270 || 0.6242 || &uarr; || [[118edo|38\118]], [[177edo|57\177]]
|-
|  [[87edo|87]]  || 28\87  || 0.1068 || 0.7744 || &darr; || [[174edo|56\174]]
|-
|  [[90edo|90]]  || 29\90  || 0.3530 || 2.6471 || &uarr; || [[180edo|58\180]]
|-
| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || &darr; ||
|-
| [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || &uarr; ||
|-
| [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || &darr; ||
|-
| [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || &darr; ||
|-
| [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || &uarr; ||
|-
| [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || &uarr; ||
|-
| [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || &darr; ||
|-
| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || &darr; ||
|-
| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || &uarr; ||
|}


In relation to 12 tone equal, 5/4 is about 13.7¢ flatter than the 4th degree (400¢). 5/4 the interval has been called the //perfect major third// to distinguish it from the other intervals in that neighborhood.
== See also ==
* [[8/5]] – its [[octave complement]]
* [[6/5]] – its [[fifth complement]]
* [[16/15]] – its [[fourth complement]]
* [[5/2]] – the interval up one [[octave]] which sounds even more [[consonant]]
* [[Ed5/4]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]


==5/4 quotes==  
== Notes ==
got any?
<references/>


See: [[Gallery of Just Intervals|Galley of Just Intervals]]</pre></div>
[[Category:Third]]
<h4>Original HTML content:</h4>
[[Category:Major third]]
<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;5_4&lt;/title&gt;&lt;/head&gt;&lt;body&gt;In &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt;, &lt;strong&gt;5/4&lt;/strong&gt; is the frequency ratio between the 5th and 4th harmonics. Measuring about 386.3&lt;a class="wiki_link" href="/Cent"&gt;¢&lt;/a&gt;, it is about 13.7¢ away from &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;'s major third of 400¢. It has a distinctive &amp;quot;sweet&amp;quot; sound, and has been described as more &amp;quot;laid back&amp;quot; than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; harmony. It is distinguished from the &lt;a class="wiki_link" href="/Pythagorean"&gt;Pythagorean&lt;/a&gt; major third of &lt;a class="wiki_link" href="/81_64"&gt;81/64&lt;/a&gt; by the syntonic comma of &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, which measures about 21.5¢. 81/64 and 5/4 are both just intonation &amp;quot;major thirds,&amp;quot; 81/64 having a more active and discordant quality, 5/4 sounding more &amp;quot;restful&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated here melodically in singing into a resonant udderbot (from the fundamental up to 5 and then noodling between 5 and 4). &lt;!-- ws:start:WikiTextFileRule:2:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/5-4.mp3?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@5-4.mp3&amp;quot; title=&amp;quot;File: 5-4.mp3&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/5-4.mp3/30382423/5-4.mp3" onclick="ws.common.trackFileLink('/file/view/5-4.mp3/30382423/5-4.mp3');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/audio/mpeg.png" height="32" width="32" alt="5-4.mp3" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/5-4.mp3/30382423/5-4.mp3" onclick="ws.common.trackFileLink('/file/view/5-4.mp3/30382423/5-4.mp3');" class="filename" title="5-4.mp3"&gt;5-4.mp3&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/5-4.mp3"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/5-4.mp3/30382423/5-4.mp3"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;402 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:2 --&gt; Hear it?&lt;br /&gt;
&lt;br /&gt;
5/4 converted to cents (¢): 1200 * log (5/4) / log (2) = 386.314...&lt;br /&gt;
&lt;br /&gt;
When two notes sound a 5/4 together,&lt;br /&gt;
&lt;br /&gt;
5/4 the interval, like all &lt;em&gt;intervals&lt;/em&gt;, refers to a &lt;em&gt;relation&lt;/em&gt; between two pitches. We speak of this relation (one pitch beating 5/4 times as fast as the other) after we are able to distinguish it from other relations.&lt;br /&gt;
&lt;br /&gt;
In relation to 12 tone equal, 5/4 is about 13.7¢ flatter than the 4th degree (400¢). 5/4 the interval has been called the &lt;em&gt;perfect major third&lt;/em&gt; to distinguish it from the other intervals in that neighborhood.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-5/4 quotes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;5/4 quotes&lt;/h2&gt;
got any?&lt;br /&gt;
&lt;br /&gt;
See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Galley of Just Intervals&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 13:37, 16 April 2025

Interval information
Ratio 5/4
Factorization 2-2 × 5
Monzo [-2 0 1⟩
Size in cents 386.3137¢
Names just major third,
classic(al) major third,
ptolemaic major third
Color name y3, yo 3rd
FJS name [math]\displaystyle{ \text{M3}^{5} }[/math]
Special properties superparticular,
reduced,
reduced harmonic
Tenney height (log2 nd) 4.32193
Weil height (log2 max(n, d)) 4.64386
Wilson height (sopfr(nd)) 9

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

In 5-limit just intonation, 5/4 is the frequency ratio between the 5th and 4th harmonics. It has been called the just major third, classic(al) major third, or ptolemaic major third[1] to distinguish it from other intervals in that neighborhood. Measuring about 386.3 ¢, it is about 13.7 ¢ away from 12edo's major third of 400 ¢. It has a distinctive "sweet" sound, and has been described as more "laid back" than its 12edo counterpart. Providing a novel consonance after 3, it is the basis for 5-limit harmony. It is distinguished from the Pythagorean major third of 81/64 by the syntonic comma of 81/80, which measures about 21.5 ¢, and from the Pythagorean diminished fourth of 8192/6561 by the schisma, which measures about 1.95 ¢. 81/64 and 5/4 are both just intonation "major thirds", 81/64 having a more active and discordant quality, 5/4 sounding more "restful".

In the context of the harmonic series, 5/4 can be heard between the 4th and 5th member of the series, demonstrated in File: 5-4.mp3 melodically in singing into a resonant udderbot (from the fundamental up to 5 and then noodling between 5 and 4).

Approximations by edos

Following edos (up to 200, and also 643) contain good approximations[2] of the interval 5/4. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓).

Edo deg\edo Absolute
error (¢) 		Relative
error (r¢) 		↕ Equally acceptable multiples [3]
25 8\25 2.3137 4.8202 ↓
28 9\28 0.5994 1.3987 ↓ 18\56, 27\84, 36\112, 45\140
31 10\31 0.7831 2.0229 ↑ 20\62, 30\93
34 11\34 1.9216 5.4445 ↑
53 17\53 1.4081 6.2189 ↓
59 19\59 0.1270 0.6242 ↑ 38\118, 57\177
87 28\87 0.1068 0.7744 ↓ 56\174
90 29\90 0.3530 2.6471 ↑ 58\180
115 37\115 0.2268 2.1731 ↓
121 39\121 0.4631 4.6701 ↑
143 46\143 0.2997 3.5718 ↓
146 47\146 0.0123 0.1502 ↓
149 48\149 0.2635 3.2714 ↑
152 49\152 0.5284 6.6930 ↑
171 55\171 0.3488 4.9704 ↓
199 64\199 0.3841 6.3691 ↓
643 207\643 0.0004 0.0235 ↑

See also

Notes

  1. ↑ For reference, see 5-limit.
  2. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  3. ↑ Super-edos up to 200 within the same error tolerance
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