Xenharmonic Wiki

3/2: Difference between revisions

← Older edit
VisualWikitext
Spt3125 (talk | contribs)
autopatrolled, Sysops
523 edits
Sintel (talk | contribs)
autopatrolled
1,547 edits
→In regular temperament theory: I'll do you one better
 
(58 intermediate revisions by 15 users not shown)
Line 1: Line 1:
{{Infobox Interval
{{interwiki
| JI glyph = [[File:perfect_fifth.png|x48px]]
| de =
| Ratio = 3/2
| en =  
| Monzo = -1 1
| es =
| Cents = 701.95500
| ja =  
| ko =  
| ro = 3/2 (ro)
}}
{{Infobox interval
| Name = just perfect fifth
| Name = just perfect fifth
| Color name = w5, wa 5th
| Sound = jid_3_2_pluck_adu_dr220.mp3
| Sound = jid_3_2_pluck_adu_dr220.mp3
| Color name = w5, wa 5th
}}
}}
{{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—specifically [[5/4]]—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 (↑) or flat (↓) 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


'''3/2''' is the [[frequency ratio]] of the [[just perfect fifth]]. What tunes it well, is one of variants of [[12edo]] or [[17edo]] (such as [[24edo]], [[34edo]] and 36edo). Other edos tune it well too (5, 7, 29, 41, 53, 200). But not all edos are like this. 35edo is great for 2, 5, 7, 9, 11 and 17 but fails on 3.
{| 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 ==
== See also ==
* [[Gallery of Just Intervals]]
* [[4/3]] – its [[octave complement]]
* [[4/3]], the [[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:3-limit]]
[[Category:Fifth]]
[[Category:interval]]
[[Category:Taxicab-2 intervals]]
[[Category:pythagorean]]
[[Category:ratio]]
[[Category:superparticular]]
Retrieved from "https://en.xen.wiki/w/3/2"