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 interval of equivalence in some non-octave systems.
| 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
Western music generally revolves around the principle of octave equivalence: notes an octave apart are often perceived as being the same chroma but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the tritave.
It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the tonotopic representation of the mammalian auditory system[dead link] is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence, especially when using timbres whose overtones consist of primarily or only odd harmonics such as clarinets, square waves, or triangle waves.
An example of a system that is typically treated as tritave-based is the Bohlen–Pierce (BP) scale, most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well). This system seems to have been the first such arrangement to be seriously studied and made into music. It was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.
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