Xenharmonic Wiki

3/2: Difference between revisions

← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots
4,330 edits
Sintel (talk | contribs)
autopatrolled
1,547 edits
→In regular temperament theory: I'll do you one better
 
(31 intermediate revisions by 10 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
| Color name = w5, wa 5th
| FJS name = P5
| Sound = jid_3_2_pluck_adu_dr220.mp3
| Sound = jid_3_2_pluck_adu_dr220.mp3
}}
}}
{{Wikipedia|Perfect fifth}}
{{Wikipedia|Perfect fifth}}


'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Not only that, but there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically [[5/4]]- as consonant. 3/2 is the simple [[just intonation]] interval best approximated by [[12edo]], after the [[octave]].
'''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.


Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical. In such systems, and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In [[12edo]], and hence in most discussions these days, this is simplified to seven semitones, which is fitting seeing as 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. On the other hand, in just intonation, the perfect fifth consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.
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]].


Then there's the possibility of [[schismatic]] temperaments, which flatten the perfect fifth such that an approximated 5/4 is generated by stacking eight fifths downwards; however, without a notation system that properly accounts for the syntonic comma (such as [[Syntonic-Rastmic Subchroma Notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]].
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]].


Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].  Of the aforementioned systems, the latter is particularly noteworthy in regards to [[telicity]] as while 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.
=== 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:


== Approximations by EDOs ==
[[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.
The following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
 
[[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"
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
|-
! [[EDO]]
! [[Edo]]
! class="unsortable" | deg\edo
! class="unsortable" | Deg\edo
! Absolute <br> error ([[Cent|¢]])
! Absolute <br>error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! Relative <br>error (%)
! &#8597;
! &#x2195;
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
! class="unsortable" | Equally accurate <br>multiples
|-
|  [[12edo|12]]  ||  7\12  || 1.9550 || 1.9550 || &darr; || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[17edo|17]]  || 10\17 || 3.9274 || 5.5637 || &uarr; ||  
|  [[12edo|12]]  ||   7\12 || 1.955 || 1.955 || &darr; || [[24edo|14\24]], [[36edo|21\36]]
|-
|-
|  [[29edo|29]]  ||  17\29 || 1.4933 || 3.6087 || &uarr; ||  
|  [[17edo|17]]  ||  10\17  || 3.927 || 5.564 || &uarr; ||  
|-
|-
|  [[41edo|41]]  ||  24\41 || 0.4840 || 1.6537 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|  [[29edo|29]]  ||  17\29 || 1.493 || 3.609 || &uarr; ||  
|-
|-
|  [[53edo|53]]  ||  31\53 || 0.0682 || 0.3013 || &darr; || [[106edo|62\106]], [[159edo|93\159]]
|  [[41edo|41]]  ||  24\41 || 0.484 || 1.654 || &uarr; || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
|-
|  [[65edo|65]]  ||  38\65 || 0.4165 || 2.2563 || &darr; || [[130edo|76\130]], [[195edo|114\195]]
|  [[53edo|53]]  ||  31\53 || 0.068 || 0.301 || &darr; || [[106edo|62\106]], [[159edo|93\159]]
|-
|-
|  [[70edo|70]]  ||  41\70 || 0.9021 || 5.2625 || &uarr; ||  
|  [[65edo|65]]  ||  38\65 || 0.416 || 2.256 || &darr; || [[130edo|76\130]], [[195edo|114\195]]
|-
|-
|  [[77edo|77]]  ||  45\77 || 0.6563 || 4.2113 || &darr; ||  
|  [[70edo|70]]  ||  41\70 || 0.902 || 5.262 || &uarr; ||  
|-
|-
|  [[89edo|89]]  ||  52\89 || 0.8314 || 6.1663 || &darr; ||  
|  [[77edo|77]]  ||  45\77 || 0.656 || 4.211 || &darr; ||  
|-
|-
|  [[94edo|94]]  ||  55\94 || 0.1727 || 1.3525 || &uarr; || [[188edo|110\188]]
|  [[89edo|89]]  ||  52\89 || 0.831 || 6.166 || &darr; ||  
|-
|-
| [[111edo|111]] ||  65\111 || 0.7477 || 6.9162 || &uarr; ||  
| [[94edo|94]] ||  55\94  || 0.173 || 1.352 || &uarr; || [[188edo|110\188]]
|-
|-
| [[118edo|118]] ||  69\118 || 0.2601 || 2.5575 || &darr; ||  
| [[111edo|111]] ||  65\111 || 0.748 || 6.916 || &uarr; ||  
|-
|-
| [[135edo|135]] ||  79\135 || 0.2672 || 3.0062 || &uarr; ||  
| [[118edo|118]] ||  69\118 || 0.260 || 2.557 || &darr; ||  
|-
|-
| [[142edo|142]] ||  83\142 || 0.5466 || 6.4675 || &darr; ||  
| [[135edo|135]] ||  79\135 || 0.267 || 3.006 ||&uarr; ||  
|-
|-
| [[147edo|147]] ||  86\147 || 0.0858 || 1.0512 || &uarr; ||  
| [[142edo|142]] ||  83\142 || 0.547 || 6.467 || &darr; ||  
|-
|-
| [[171edo|171]] || 100\171 || 0.2006 || 2.8588 || &darr; ||  
| [[147edo|147]] || 86\147 || 0.086 || 1.051 || &uarr; ||  
|-
|-
| [[176edo|176]] || 103\176 || 0.3177 || 4.6600 || &uarr; ||  
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || &darr; ||  
|-
|-
| [[183edo|183]] || 107\183 || 0.3157 || 4.8138 || &darr; ||  
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || &uarr; ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.0450 || 0.7500 || &uarr; ||  
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || &darr; ||  
|-
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || &uarr; ||
|}
|}


<references/>
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"
{| class="wikitable sortable"
|+Comparison of 3/2 approximations and "fifth classes", with 3/2 = 701.955 cents.
|+ style="font-size: 105%;" | Comparison of the fifths of edos 5 to 31
(from 5-EDO to 31-EDO, no subsets of 12-EDO.)
![[EDO]]
!degree
!cents
!fifth category
!error
|-
|-
|[[5edo]]
! Edo
|3/5
! Degree
|720
! Cents
|pentatonic EDO
! Edo Category
|  +18.045 ¢
! Error (¢)
|-
|-
|[[7edo]]
| [[5edo]]
|4/7
| 3\5
|685.714
| 720.000
|perfect EDO
| Pentatonic edo
|  -16.241 ¢
|  +18.045
|-
|-
|[[8edo]]
| [[7edo]]
|5/8
| 4\7
|750
| 685.714
|supersharp EDO
| perfect edo
| +48.045 ¢
| −16.241
|-
|-
|[[9edo]]
| [[8edo]]
|5/9
| 5\8
|666.667
| 750.000
|superflat EDO
| supersharp edo
|  -35.288 ¢
|  +48.045
|-
|-
|[[10edo]]
| [[9edo]]
|6/10
| 5\9
|720
| 666.667
|pentatonic EDO
| superflat edo
| +18.045 ¢
| −35.288
|-
|-
|[[11edo]]
| [[10edo]]
|6/11
| 6\10
|654.545
| 720.000
|superflat EDO
| pentatonic edo
|  -47.41 ¢
|  +18.045
|-
|-
|[[12edo]]
| [[11edo]]
|7/12
| 6\11
|700
| 654.545
|diatonic EDO
| superflat edo
| -1.955 ¢
| −47.41
|-
|-
|[[13edo]]
| [[12edo]]
|8/13
| 7\12
|738.462
| 700.000
|supersharp EDO
| diatonic edo
| +36.507 ¢
| −1.955
|-
|-
|[[14edo]]
| [[13edo]]
|8/14
| 8\13
|685.714
| 738.462
|perfect EDO
| supersharp edo
|  -16.241 ¢
|  +36.507
|-
|-
|[[15edo]]
| [[14edo]]
|9/15
| 8\14
|720
| 685.714
|pentatonic EDO
| perfect edo
| +18.045 ¢
| −16.241
|-
|-
|[[16edo]]
| [[15edo]]
|9/16
| 9\15
|675
| 720.000
|superflat EDO
| pentatonic edo
|  -26.955 ¢
|  +18.045
|-
|-
|[[17edo]]
| [[16edo]]
|10/17
| 9\16
|705.882
| 675.000
|diatonic EDO
| superflat edo
| +3.927 ¢
| −26.955
|-
|-
|[[18edo]]
| [[17edo]]
|11/18
| 10\17
|733.333
| 705.882
|supersharp EDO
| diatonic edo
|  +31.378 ¢
|  +3.927
|-
|-
|[[19edo]]
| [[18edo]]
|11/19
| 11\18
|694.737
| 733.333
|diatonic EDO
| supersharp edo
|  -7.218 ¢
|  +31.378
|-
|-
|[[20edo]]
| [[19edo]]
|12/20
| 11\19
|720
| 694.737
|pentatonic EDO
| diatonic edo
| +18.045 ¢
| −7.218
|-
|-
|[[21edo]]
| [[20edo]]
|12/21
| 12\20
|685.714
| 720.000
|perfect EDO
| pentatonic edo
|  -16.241 ¢
|  +18.045
|-
|-
|[[22edo]]
| [[21edo]]
|13/22
| 12\21
|709.091
| 685.714
|diatonic EDO
| perfect edo
| +7.136 ¢
| −16.241
|-
|-
|[[23edo]]
| [[22edo]]
|13/23
| 13\22
|678.261
| 709.091
|superflat EDO
| diatonic edo
|  -23.694 ¢
|  +7.136
|-
|-
|[[24edo]]
| [[23edo]]
|14/24
| 13\23
|700
| 678.261
|diatonic EDO
| superflat edo
| -1.955 ¢
| −23.694
|-
|-
|[[25edo]]
| [[24edo]]
|15/25
| 14\24
|720
| 700.000
|pentatonic EDO
| diatonic edo
| +18.045 ¢
| −1.955
|-
|-
|[[26edo]]
| [[25edo]]
|15/26
| 15\25
|692.308
| 720.000
|diatonic EDO
| pentatonic edo
|  -9.647 ¢
|  +18.045
|-
|-
|[[27edo]]
| [[26edo]]
|16/27
| 15\26
|711.111
| 692.308
|diatonic EDO
| diatonic edo
| +9.156 ¢
| −9.647
|-
|-
|[[28edo]]
| [[27edo]]
|16/28
| 16\27
|685.714
| 711.111
|perfect EDO
| diatonic edo
|  -16.241 ¢
|  +9.156
|-
|-
|[[29edo]]
| [[28edo]]
|17/29
| 16\28
|703.448
| 685.714
|diatonic EDO
| perfect edo
| +1.493 ¢
| −16.241
|-
|-
|[[30edo]]
| [[29edo]]
|17/30
| 17\29
|720
| 703.448
|pentatonic EDO
| diatonic edo
|  +18.045 ¢
|  +1.493
|-
|-
|[[31edo]]
| [[30edo]]
|18/31
| 18\30
|696.774
| 720.000
|diatonic EDO
| pentatonic edo
| -5.181 ¢
|  +18.045
|-
| [[31edo]]
| 18\31
| 696.774
| diatonic edo
| −5.181
|}
|}
* The many and various 3/2 approximations in different EDOs can be classified as (after [[Kite Giedraitis]]):
** '''superflat''' EDO - fifth is narrower than 686 cents.
** '''perfect''' EDO - fifth is 686 cents wide (and 4/7 steps).
** '''diatonic''' EDO - fifth is between 686.1 - 719.9 cents wide.
** '''pentatonic''' EDO - fifth is exactly 720 cents wide.
** '''supersharp''' EDO - fifth is wider than 720 cents.


== See also ==
== See also ==
* [[4/3]] – its [[octave complement]]
* [[4/3]] – its [[octave complement]]
* [[Fifth complement]]
* [[Fifth complement]]
* [[Edf]] – tunings which equally divide 3/2
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
* {{OEIS| A060528 }} – a list of EDOs with increasingly better approximations of 3:2 (and by extension 4:3)
* {{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| 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:Fifth]]
[[Category:Superparticular]]
[[Category:Taxicab-2 intervals]]
[[Category:Octave-reduced harmonics]]
Retrieved from "https://en.xen.wiki/w/3/2"