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'''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. Systems excluding perfect fifths can thus sound more "xenharmonic".  
'''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 [[Wikipedia: Perfect fifth|corresponding Wikipedia article]] attests there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically [[5/4]]- as consonant. 3/2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].
Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically [[5/4]]- as consonant. 3/2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].


Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. 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. 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]].
Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. 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. 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]].

Revision as of 20:51, 16 November 2021

Interval information
Ratio 3/2
Factorization 2-1 × 3
Monzo [-1 1⟩
Size in cents 701.955¢
Name just perfect fifth
Color name w5, wa 5th
FJS name [math]\displaystyle{ \text{P5} }[/math]
Special properties superparticular,
reduced,
reduced harmonic
Tenney height (log2 nd) 2.58496
Weil height (log2 max(n, d)) 3.16993
Wilson height (sopfr(nd)) 5

[sound info]
Open this interval in xen-calc

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 corresponding Wikipedia article attests there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". Variations of the fifth (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically 5/4- as consonant. 3/2 is the simple JI interval best approximated by 12edo, after the octave.

Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. 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. 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.

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 the 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.

Approximations by EDOs

The following EDOs (up to 200) contain good approximations[1] of the interval 3/2. 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 [2]
12 7\12 1.9550 1.9550 ↓ 14\24, 21\36
17 10\17 3.9274 5.5637 ↑
29 17\29 1.4933 3.6087 ↑
41 24\41 0.4840 1.6537 ↑ 48\82, 72\123, 96\164
53 31\53 0.0682 0.3013 ↓ 62\106, 93\159
65 38\65 0.4165 2.2563 ↓ 76\130, 114\195
70 41\70 0.9021 5.2625 ↑
77 45\77 0.6563 4.2113 ↓
89 52\89 0.8314 6.1663 ↓
94 55\94 0.1727 1.3525 ↑ 110\188
111 65\111 0.7477 6.9162 ↑
118 69\118 0.2601 2.5575 ↓
135 79\135 0.2672 3.0062 ↑
142 83\142 0.5466 6.4675 ↓
147 86\147 0.0858 1.0512 ↑
171 100\171 0.2006 2.8588 ↓
176 103\176 0.3177 4.6600 ↑
183 107\183 0.3157 4.8138 ↓
200 117\200 0.0450 0.7500 ↑
  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance
Comparison of 3/2 approximations and "fifth classes", with 3/2 = 701.955 cents. (from 5-EDO to 31-EDO, no subsets of 12-EDO.)
EDO degree cents fifth category error
5edo 3/5 720 pentatonic EDO +18.045 ¢
7edo 4/7 685.714 perfect EDO -16.241 ¢
8edo 5/8 750 supersharp EDO +48.045 ¢
9edo 5/9 666.667 superflat EDO -35.288 ¢
10edo 6/10 720 pentatonic EDO +18.045 ¢
11edo 6/11 654.545 superflat EDO -47.41 ¢
12edo 7/12 700 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 pentatonic EDO +18.045 ¢
16edo 9/16 675 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 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 diatonic EDO -1.955 ¢
25edo 15/25 720 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 17/30 720 pentatonic EDO +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

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