# 3/2

(Redirected from Perfect fifth)
 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 $\text{P5}$ 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 Harmonic entropy(Shannon, $\sqrt{nd}$) ~3.42385 bits https://en.xen.wiki/w/File:Jid_3_2_pluck_adu_dr220.mp3[sound info] open this interval in xen-calc
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3/2, the just perfect fifth, is the second 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. 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 simplest just intonation interval to be very well approximated by 12edo, after the octave.

Producing a chain of just perfect fifths yields Pythagorean tuning. Since log2(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. Nevertheless, even in xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the moment of symmetry scales created by this tuning—specifically in the TAMNAMS system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators. An example of such a scale is the familiar Pythagorean diatonic scale.

Meanwhile, meantone temperaments 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 also 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 5-limit just intonation, the just 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.

There are also superpyth (or "superpythagorean") temperaments, which sharpen the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6. This also means that intervals such as A–G or C–B♭ approximate 7/4 instead of 9/5.

Then there is 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 ups and downs notation or Syntonic-Rastmic Subchroma notation), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with 8192/6561, and this in turn results in common chords such as conventional Major and Minor triads being awkward to notate.

Some tunings which have better (in terms of closeness to just intonation) approximations of the perfect fifth than in 12edo are 29edo, 41edo, and 53edo. Of the aforementioned systems, 53edo 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.

## 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 ()
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 edo approximations of 3/2 and "fifth classes" (from 5edo to 31edo)
Edo Degree Cents Fifth 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 17\30 720.000 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 edos have fifths narrower than 686 cents.
• Perfect or heptatonic edos have fifths 68547 cents wide (and 4/7 steps).
• Diatonic edos have fifths between 68547 and 720 cents wide.
• Pentatonic have fifths exactly 720 cents wide.
• Supersharp edos have fifths wider than 720 cents.