3/2
| Interval information |
reduced,
reduced harmonic
(Shannon, [math]\sqrt{nd}[/math])
[sound info]
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 (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 edos have fifths narrower than 686 cents.
- Perfect or heptatonic edos have fifths 6854⁄7 cents wide (and 4/7 steps).
- Diatonic edos have fifths between 6854⁄7 and 720 cents wide.
- Pentatonic have fifths exactly 720 cents wide.
- Supersharp edos have fifths wider than 720 cents.
See also
- 4/3 – its octave complement
- Fifth complement
- Edf
- 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 log2(3)
- OEIS: A206788 – denominators of the semiconvergents to log2(3)