Music Theory
In music theory, the just intonation of the diatonic scale involves regular numbers: the pitches in a single octave of this scale have frequencies proportional to the numbers in the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly consecutive regular numbers. Thus, for an instrument with this tuning, all pitches are regular-number harmonics of a single fundamental frequency. This scale is called a 5-limit tuning, meaning that the interval between any two pitches can be described as a product 2i3j5k of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.
5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music: Honing & Bod (2005) list 31 different 5-limit scales, drawn from a larger database of musical scales. Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers. Euler's tonnetz provides a convenient graphical representation of the pitches in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar grid. Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that pitch ratios based on primes larger than 5 cannot be consonant. However the equal temperament of modern pianos is not a 5-limit tuning, and some modern composers have experimented with tunings based on primes larger than five.
In connection with the application of regular numbers to music theory, it is of interest to find pairs of regular numbers that differ by one. There are exactly ten such pairs (x, x+1) and each such pair defines a superparticular ratio (x + 1)/x that is meaningful as a musical interval. These intervals are 2/1 (the octave), 3/2 (the perfect fifth), 4/3 (the perfect fourth), 5/4 (the just major third), 6/5 (the just minor third), 9/8 (the just major tone), 10/9 (the just minor tone), 16/15 (the just diatonic semitone), 25/24 (the just chromatic semitone), and 81/80 (the syntonic comma).
Read more about this topic: Regular Number
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