3/1
The 3rd harmonic, tritave, triple, or perfect twelfth is the interval of frequency ratio 3/1. It is perhaps the most consonant interval after the octave, with frequency ratio 2/1. For this reason, it is used as an equave in some nonoctave systems, such as the Bohlen–Pierce scale.
| Interval information |
tritave,
triple,
perfect twelfth
prime harmonic
[sound info]
It is the second prime harmonic, after 2/1 and before 5/1.
Importance of prime 3
The octave-reduced 3rd harmonic is the perfect fifth 3/2, and the octave complement of 3/2 is the perfect fourth 4/3. The perfect fifth and fourth are considered essential in western music theory, and in 12edo, stacking them makes the circle of fifths/fourths. The perfect fifth is often used as the base for constructing chords, such as the classical major triad 1–5/4–3/2 (4:5:6). The perfect fourth can also be used as a base in chords, such as 1–7/6–4/3 (6:7:8), which deviates from traditional harmony.
In just intonation, 3/1 is the first prime harmonic that adds pitch classes besides the unison, octave, and multiples of the octave. Pythagorean tuning, also known as the 3-limit, is the subset of just intonation containing all intervals where the only prime factors are 2 and 3. Pythagorean tuning generates the pentic and diatonic scales, and is often used as a system for interval classification in just intonation.
As an interval of equivalence
When used as an interval of equivalence, 3/1 can be called the tritave. This is very xenharmonic since it assumes tritave equivalence instead of octave equivalence, so that 1/1, 3/1, and 9/1 are considered the same pitch class. Typically tritave-equivalent systems base harmony off of only odd harmonics, for example with the 3:5:7 triad as analogous to 4:5:6.
An example of a system that is typically treated as tritave-based is the Bohlen–Pierce scale. The equal-tempered version of the Bohlen–Pierce scale is 13edt, or 13 equal divisions of the tritave. Systems can be constructed analogously to octave-equivalent harmony, for example the 9-note lambda scale, which can be considered analogous to diatonic.
Etymology
The term tritave was coined by John Pierce[1]. It was derived from the word octave by replacing the perceived prefix octo- (eight, for the eighth degree of the diatonic scale) by tri- (three, for 3/1). However, the oct in octave is not a prefix, but part of the single-morpheme word derived from Latin octavus ("eighth"). In this sense, tritave is more of a contraction of tri- and octave than anything else. As such, the term usually refers to 3/1 as an interval of equivalence; in other contexts, it is more often called the perfect twelfth (after the 12th degree of the diatonic scale).
Triple is a proposed term which relates itself to the ancient Greek concept of multiples. It also fixes the problem of using part of the word octave.
Since the enneatonic 4L 5s⟨3/1⟩ ("Lambda") scale is the BP substitute for the diatonic scale, the term decade[2] or decim[citation needed] (tenth degree of the Lambda scale) has been proposed as an alternative to tritave, though decade almost always refers to ten times the frequency (10/1) in audio engineering.
See also
- EDT (equal divisions of the tritave/twelfth)
- 3/2 – its octave reduced form
- Twelfth complement – the analogue for octave complement