3/1

Interval information

Ratio	3/1
Factorization	3
Monzo	[0 1⟩
Size in cents	1901.955¢
Names	3rd harmonic, tritave, triple, perfect twelfth
Color name	w12, wa 12th
FJS name	[math]\displaystyle{ \text{P12} }[/math]
Special properties	harmonic, prime harmonic
Tenney norm (log2 nd)	1.58496
Weil norm (log2 max(n, d))	3.16993
Wilson norm (sopfr(nd))	3
[sound info]
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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.

Significance 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

Main article: EDT

When used as an interval of equivalence, 3/1 can be called the tritave. This is very xenharmonic since it does not assume octave equivalence, and instead tritave equivalence is assumed, 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 tritave-based system 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. 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 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)
• No-twos 31-limit – non-octave 31-limit system containing neither 2 nor primes higher than 31
• Tritave complement – the analogue for octave complement