3/1: Difference between revisions

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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]]. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen–Pierce]] scale.

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|circle of fifths/fourths]]. The perfect fifth is often used as the base for constructing chords, such as the classical major triad [[4:5:6|1–5/4–3/2]] (4:5:6). The perfect fourth can also be used as a base in chords, such as [[6:7:8|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 class]]es 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 thus often used as a system for interval classification.

== 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 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 harmonic]]s, 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 temperament|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 ==

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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]]. For this reason, it is used as an [[equave]] in some [[nonoctave]] systems, such as the [[Bohlen–Pierce]] scale.
+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|circle of fifths/fourths]]. The perfect fifth is often used as the base for constructing chords, such as the classical major triad [[4:5:6|1–5/4–3/2]] (4:5:6). The perfect fourth can also be used as a base in chords, such as [[6:7:8|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 class]]es 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 thus often used as a system for interval classification.
+== As an interval of equivalence ==
+{{Main|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 harmonic]]s, 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 temperament|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 ==