5/4: Difference between revisions

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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 = Naturterz

~~: This revision was by author [[User:Andrew_Heathwaite~~|~~Andrew_Heathwaite]] and made on <tt>2011-09-14 19:55:54 UTC<~~/~~tt>.<br>~~

| en = 5/4

~~: The original revision id was <tt>254149974</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>~~

}}

~~<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¢, 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

}}

~~See:~~ [[~~Gallery of Just Intervals|Galley of Just Intervals~~]]~~</pre></div>~~

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".

~~<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"><html><head><title>5_4</title></head><body>In <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a>, 5/4 is the frequency ratio between the 5th and 4th ~~harmonics~~. Measuring about 386.3¢, it is about 13.7¢ away from ~~<a class="wiki_link" href="/~~12edo~~">12edo</a>~~'s major third of ~~400¢~~. It has a distinctive ~~&quot;~~sweet~~&quot;~~ sound, and has been described as more ~~&quot;~~laid back~~&quot;~~ 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 ~~<a class="wiki_link" href="/81_64">~~81/64~~</a>~~ by the syntonic comma of ~~<a class="wiki_link" href="/81_80">~~81/80~~<~~/~~a>~~, which measures about 21.5¢. 81/64 and 5/4 are both just intonation ~~&quot;~~major thirds,~~&quot;~~ 81/64 having a more active and discordant quality, 5/4 sounding more &~~amp~~;~~quot~~;~~restful~~&~~amp~~;~~quot~~;.<~~br /~~>

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).

<~~br /~~>

~~See:~~ <~~a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"~~>~~Galley of Just Intervals~~</a><~~/body~~><~~/html~~></~~pre>~~</~~div~~>

== 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 (↑) or flat (↓).

{| 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¢]])

! ↕

! 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 || ↓ ||

|-

| [[28edo|28]] || 9\28 || 0.5994 || 1.3987 || ↓ || [[56edo|18\56]], [[84edo|27\84]], [[112edo|36\112]], [[140edo|45\140]]

|-

| [[31edo|31]] || 10\31 || 0.7831 || 2.0229 || ↑ || [[62edo|20\62]], [[93edo|30\93]]

|-

| [[34edo|34]] || 11\34 || 1.9216 || 5.4445 || ↑ ||

|-

| [[53edo|53]] || 17\53 || 1.4081 || 6.2189 || ↓ ||

|-

| [[59edo|59]] || 19\59 || 0.1270 || 0.6242 || ↑ || [[118edo|38\118]], [[177edo|57\177]]

|-

| [[87edo|87]] || 28\87 || 0.1068 || 0.7744 || ↓ || [[174edo|56\174]]

|-

| [[90edo|90]] || 29\90 || 0.3530 || 2.6471 || ↑ || [[180edo|58\180]]

|-

| [[115edo|115]] || 37\115 || 0.2268 || 2.1731 || ↓ ||

|-

| [[121edo|121]] || 39\121 || 0.4631 || 4.6701 || ↑ ||

|-

| [[143edo|143]] || 46\143 || 0.2997 || 3.5718 || ↓ ||

|-

| [[146edo|146]] || 47\146 || 0.0123 || 0.1502 || ↓ ||

|-

| [[149edo|149]] || 48\149 || 0.2635 || 3.2714 || ↑ ||

|-

| [[152edo|152]] || 49\152 || 0.5284 || 6.6930 || ↑ ||

|-

| [[171edo|171]] || 55\171 || 0.3488 || 4.9704 || ↓ ||

|-

| [[199edo|199]] || 64\199 || 0.3841 || 6.3691 || ↓ ||

|-

| [[643edo|643]] || 207\643 || 0.0004 || 0.0235 || ↑ ||

|}

== 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]]

== Notes ==

[[Category:Third]]

[[Category:Major third]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{interwiki
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| de = Naturterz
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-14 19:55:54 UTC</tt>.<br>
+| en = 5/4
-: The original revision id was <tt>254149974</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>
+}}
-<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¢, 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}}
-See: [[Gallery of Just Intervals|Galley of Just Intervals]]</pre></div>
+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".
-<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;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;, 5/4 is the frequency ratio between the 5th and 4th harmonics. Measuring about 386.3¢, 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 5-limit harmony. It is distinguished from the Pythagorean 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;
+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).
-&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>
+== 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;).
+{| 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; ||
+|}
+== 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]]
+== Notes ==
+<references/>
+[[Category:Third]]
+[[Category:Major third]]