3/2
| Interval information |
reduced,
reduced harmonic
(Shannon, [math]\sqrt{nd}[/math])
[sound info]
3/2, the just perfect fifth, is a very consonant interval, due to the numerator and denominator of its ratio being very small numbers. Only the octave and the compound fifth have smaller numbers.
Properties
For harmonic 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 music of the world. 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.
A 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 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, ups and downs, FJS and color notation.
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.
In regular temperament theory
Because 3/2 has very low harmonic entropy, 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:
Meantone temperament flattens the fifth from just 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.
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.
Schismatic temperament flattens the fifth very slightly such that the diminished fourth generated by stacking eight fourths approximates 5/4. Thus a triad with 5/4 is written as C F♭ G (unless the notation has accidentals for 81/80, e.g. C vE G).
Approximations by edos
12edo approximates 3/2 to within only 2¢. 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¢ and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.
| Edo | deg\edo | Absolute Error (¢) |
Relative
Error (%) |
↕ | Equally accurate
multiples |
|---|---|---|---|---|---|
| 12 | 7\12 | 1.955 | 1.955 | ↓ | 14\24, 21\36 |
| 17 | 10\17 | 3.927 | 5.564 | ↑ | |
| 29 | 17\29 | 1.493 | 3.609 | ↑ | |
| 41 | 24\41 | 0.484 | 1.654 | ↑ | 48\82, 72\123, 96\164 |
| 53 | 31\53 | 0.068 | 0.301 | ↓ | 62\106, 93\159 |
| 65 | 38\65 | 0.416 | 2.256 | ↓ | 76\130, 114\195 |
| 70 | 41\70 | 0.902 | 5.262 | ↑ | |
| 77 | 45\77 | 0.656 | 4.211 | ↓ | |
| 89 | 52\89 | 0.831 | 6.166 | ↓ | |
| 94 | 55\94 | 0.173 | 1.352 | ↑ | 110\188 |
| 111 | 65\111 | 0.748 | 6.916 | ↑ | |
| 118 | 69\118 | 0.260 | 2.557 | ↓ | |
| 135 | 79\135 | 0.267 | 3.006 | ↑ | |
| 142 | 83\142 | 0.547 | 6.467 | ↓ | |
| 147 | 86\147 | 0.086 | 1.051 | ↑ | |
| 171 | 100\171 | 0.200 | 2.859 | ↓ | |
| 176 | 103\176 | 0.318 | 4.660 | ↑ | |
| 183 | 107\183 | 0.316 | 4.814 | ↓ | |
| 200 | 117\200 | 0.045 | 0.750 | ↑ |
Edos can be classified by their approximation of 3/2 as:
- Superflat edos have fifths narrower than 4\7 = ~686¢
- Perfect edos have fifths of exactly 4\7
- Diatonic edos have fifths between 4\7 and 3\5 = 720¢
- Pentatonic have fifths of exactly 3\5
- Supersharp edos have fifths wider than 3\5
| Edo | Degree | Cents | Edo 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 |
See also
- 4/3 – its octave complement
- Fifth complement
- Edf – tunings which equally divide 3/2
- 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)