3/2
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]\text{P5}[/math] |
Special properties | superparticular, reduced, reduced harmonic |
Tenney height (log_{2} nd) | 2.58496 |
Weil height (log_{2} max(n, d)) | 3.16993 |
Wilson height (sopfr (nd)) | 5 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.08242 bits |
[sound info] | |
open this interval in xen-calc |
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. 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 simplest just intonation interval to be very well 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. 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 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 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 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.
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 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 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.
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 | ↑ |
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 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
- 4/3 – its octave complement
- Fifth complement
- 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_{2}(3)
- OEIS: A206788 – denominators of the semiconvergents to log_{2}(3)