3/1: Difference between revisions

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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.

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 often used as a system for interval classification in just intonation.

== As an interval of equivalence ==

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 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.
+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 often used as a system for interval classification in just intonation.
 == As an interval of equivalence ==