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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 =
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-09-14 16:05:36 UTC</tt>.<br>
| en =
: The original revision id was <tt>162625261</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>
| ko =
<h4>Original Wikitext content:</h4>
| ro = 3/2 (ro)
<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">See [[Just Perfect Fifth]].</pre></div>
}}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;3_2&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Just%20Perfect%20Fifth"&gt;Just Perfect Fifth&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| Name = just perfect fifth
| Color name = w5, wa 5th
| Sound = jid_3_2_pluck_adu_dr220.mp3
}}
{{Wikipedia|Perfect fifth}}
 
'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers. Only the [[2/1|octave]] and the [[3/1|tritave]] have smaller numbers.
 
== Properties ==
For harmonic [[timbre|timbres]], the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it's easy to hear if it's mistuned.
 
== Usage ==
Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[Approaches to musical tuning|music of the world]]. [[Historical temperaments|Historically]], European music treated the perfect fifth as consonant long before it treated the major third&mdash;specifically [[5/4]]&mdash;as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.
 
A [[Chain of fifths|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]]. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are pythogorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[Ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].
 
Music using unusual intervals can be very disorienting. The presence of perfect fifths can provide a "ground" that make it less so. Some composers deliberately use tunings that lack fifths, to make their music sound more [[xenharmonic]].
 
=== In regular temperament theory ===
Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:
 
[[Meantone]] temperament flattens the fifth from just (to around 695 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.
 
[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor 7th 16/9 approximates 7/4 instead of 9/5.
 
* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710c tuning when prioritizing the thirds, or a ~715c tuning when prioritizing 7/4.
 
[[Schismatic|Schismic]] temperament adjusts the fifth such that the ''diminished'' fourth generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most simply flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).
 
* Garibaldi temperament is an extension of schismatic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.
 
== Approximations by edos ==
12edo approximates 3/2 to within only 2{{c}}. [[29edo]], [[41edo]], and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 [[telic]] system, 53edo is notably a 3-strong 3-2 telic system.
 
The following edos (up to 200) approximate 3/2 to within both 7{{c}} and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (&uarr;) or flat (&darr;) fifth.
 
{| 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 (%)
! &#x2195;
! class="unsortable" | Equally accurate <br>multiples
|-
|  [[12edo|12]]  ||  7\12  || 1.955 || 1.955 || &darr; || [[24edo|14\24]], [[36edo|21\36]]
|-
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || &uarr; ||
|-
|  [[29edo|29]]  ||  17\29  || 1.493 || 3.609 || &uarr; ||
|-
|  [[41edo|41]]  ||  24\41  || 0.484 || 1.654 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
|  [[53edo|53]]  ||  31\53  || 0.068 || 0.301 || &darr; || [[106edo|62\106]], [[159edo|93\159]]
|-
|  [[65edo|65]]  ||  38\65  || 0.416 || 2.256 || &darr; || [[130edo|76\130]], [[195edo|114\195]]
|-
|  [[70edo|70]]  ||  41\70  || 0.902 || 5.262 || &uarr; ||
|-
|  [[77edo|77]]  ||  45\77  || 0.656 || 4.211 || &darr; ||
|-
|  [[89edo|89]]  ||  52\89  || 0.831 || 6.166 || &darr; ||
|-
|  [[94edo|94]]  ||  55\94  || 0.173 || 1.352 || &uarr; || [[188edo|110\188]]
|-
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || &uarr; ||
|-
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || &darr; ||
|-
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||&uarr; ||
|-
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || &darr; ||
|-
| [[147edo|147]] ||  86\147 || 0.086 || 1.051 || &uarr; ||
|-
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || &darr; ||
|-
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || &uarr; ||
|-
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || &darr; ||
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || &uarr; ||
|}
 
Edos can be classified by their approximation of 3/2 as:
* '''Superflat''' edos have fifths narrower than {{nowrap|4\7 {{=}} ~686{{c}}}}
* '''Perfect''' edos have fifths of exactly 4\7
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap|3\5 {{=}} 720{{c}}}}
* '''Pentatonic''' have fifths of exactly 3\5
* '''Supersharp''' edos have fifths wider than 3\5
 
{| class="wikitable sortable"
|+ style="font-size: 105%;" | Comparison of the fifths of edos 5 to 31
|-
! Edo
! Degree
! Cents
! Edo Category
! Error (¢)
|-
| [[5edo]]
| 3\5
| 720.000
| Pentatonic edo
|  +18.045
|-
| [[7edo]]
| 4\7
| 685.714
| perfect edo
| −16.241
|-
| [[8edo]]
| 5\8
| 750.000
| supersharp edo
|  +48.045
|-
| [[9edo]]
| 5\9
| 666.667
| superflat edo
| −35.288
|-
| [[10edo]]
| 6\10
| 720.000
| pentatonic edo
|  +18.045
|-
| [[11edo]]
| 6\11
| 654.545
| superflat edo
| −47.41
|-
| [[12edo]]
| 7\12
| 700.000
| diatonic edo
| −1.955
|-
| [[13edo]]
| 8\13
| 738.462
| supersharp edo
|  +36.507
|-
| [[14edo]]
| 8\14
| 685.714
| perfect edo
| −16.241
|-
| [[15edo]]
| 9\15
| 720.000
| pentatonic edo
|  +18.045
|-
| [[16edo]]
| 9\16
| 675.000
| superflat edo
| −26.955
|-
| [[17edo]]
| 10\17
| 705.882
| diatonic edo
|  +3.927
|-
| [[18edo]]
| 11\18
| 733.333
| supersharp edo
|  +31.378
|-
| [[19edo]]
| 11\19
| 694.737
| diatonic edo
| −7.218
|-
| [[20edo]]
| 12\20
| 720.000
| pentatonic edo
|  +18.045
|-
| [[21edo]]
| 12\21
| 685.714
| perfect edo
| −16.241
|-
| [[22edo]]
| 13\22
| 709.091
| diatonic edo
|  +7.136
|-
| [[23edo]]
| 13\23
| 678.261
| superflat edo
| −23.694
|-
| [[24edo]]
| 14\24
| 700.000
| diatonic edo
| −1.955
|-
| [[25edo]]
| 15\25
| 720.000
| pentatonic edo
|  +18.045
|-
| [[26edo]]
| 15\26
| 692.308
| diatonic edo
| −9.647
|-
| [[27edo]]
| 16\27
| 711.111
| diatonic edo
|  +9.156
|-
| [[28edo]]
| 16\28
| 685.714
| perfect edo
| −16.241
|-
| [[29edo]]
| 17\29
| 703.448
| diatonic edo
|  +1.493
|-
| [[30edo]]
| 18\30
| 720.000
| pentatonic edo
|  +18.045
|-
| [[31edo]]
| 18\31
| 696.774
| diatonic edo
| −5.181
|}
 
== See also ==
* [[4/3]] – its [[octave complement]]
* [[Fifth complement]]
* [[Edf]] – tunings which equally divide 3/2
* [[Gallery of just intervals]]
* {{OEIS| A060528 }} – sequence of edos with increasingly better approximations of 3/2 (and by extension 4/3)
* {{OEIS| A005664 }} – denominators of the convergents to log<sub>2</sub>(3)
* {{OEIS| A206788 }} – denominators of the semiconvergents to log<sub>2</sub>(3)
 
[[Category:Fifth]]
[[Category:Taxicab-2 intervals]]
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