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The perfect fourth is a perfect interval like the unison, octave, and perfect fifth, and it is a sensory consonance. In common practice harmony, however, it is considered a stylistic dissonance in certain contexts, namely in two-voice textures and whenever it occurs "above the bass in chords with three or more notes". [ 2 ]
Archytas provided a rigorous proof that the basic musical intervals cannot be divided in half, or in other words, that there is no mean proportional between numbers in super-particular ratio (octave 2:1, fourth 4:3, fifth 3:2, 9:8). [12] [14] Archytas was also the first ancient Greek theorist to provide ratios for all 3 genera. [1]
Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem). Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem).
In musical set theory or atonal theory, complement is used in both the sense above (in which the perfect fourth is the complement of the perfect fifth, 5+7=12), and in the additive inverse sense of the same melodic interval in the opposite direction – e.g. a falling 5th is the complement of a rising 5th. [citation needed]
The interval of a perfect fourth is a foundational element of many genres of music, represented in music theory as the tonic and subdominant relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the interval of four in more active harmonic contexts. The typical number of movements in a ...
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A common voicing is the 1–5 perfect fifth (A), to which the octave can be added, 1-5-1 (B). A perfect fourth 5-1 (C) is also a power chord, as it implies the "missing" lower 1 pitch. Either or both of the pitches may be doubled an octave above or below (D is 5-1-5-1), which leads to another common variation, 5-1-5 (not shown).
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