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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 ]
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).
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]
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. [ 1 ] [ verification needed ] [ clarification needed ] Often, such games have simple rules and match procedures, such as tic-tac-toe and dots and boxes .
A diminished fourth is enharmonically equivalent to a major third; that is, it spans the same number of semitones, and they are physically the same pitch in twelve-tone equal temperament. For example, B–D ♯ is a major third; but if the same pitches are spelled B and E ♭ , as occurs in the C harmonic minor scale , the interval is instead a ...
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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 ...
Likewise, 11 of the 12 perfect fourths are also in tune, but the remaining fourth is an augmented third (rather than a true fourth). Wolf intervals are an artifact of keyboard design, and keyboard players using a key that is actually in-tune with a different pitch than intended. [11]