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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 ]
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n 4 = n × n × n × n. Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.
This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p−1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form.
Equal temperament by definition is such that A ♭ and G ♯ are at the same level. 1 ⁄ 4-comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 1 ⁄ 3-comma meantone produces the "just" minor third (6:5, 316 cents, a syntonic comma higher than the Pythagorean one of 294 ...
Quartal harmony normally works with a combination of perfect and augmented fourths. Diminished fourths are enharmonically equivalent to major thirds, so they are uncommon. [33] For example, the chord C–F–B is a series of fourths, containing a perfect fourth (C–F) and an augmented fourth/tritone (F–B).
These three intervals and their octave equivalents, such as the perfect eleventh and twelfth, are the only absolute consonances of the Pythagorean system. All other intervals have varying degrees of dissonance, ranging from smooth to rough. The difference between the perfect fourth and the perfect fifth is the tone or major second.
"The augmented-fourth interval is the only interval whose inverse is the same as itself. The augmented-fourths tuning is the only tuning (other than the 'trivial' tuning C–C–C–C–C–C) for which all chords-forms remain unchanged when the strings are reversed. Thus the augmented-fourths tuning is its own 'lefty' tuning." [23]
The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100. [3] In modern language, Nicomachus states without proof that every perfect number is of the form 2 n − 1 ( 2 n − 1 ) {\displaystyle 2^{n-1}(2^{n}-1)} where 2 n − 1 {\displaystyle 2^{n ...