What most fundamentally differentiates a pure tone, for example a note played on a tuning fork, from a tone played on an instrument, are the harmonics belonging to the tone played on an instrument. When played by an ordinary instrument, each note is characterized built of a fundamental frequency plus, in lesser intensity, the frequencies of 1/2, 1/3, 1/4, …1/n of the fundamental frequency. The series 1, 1/2,…1/n… is called the harmonic series. So tones are built from the harmonic series. For a given tone, x0 (with a corresponding fundamental frequency), the only tones on a discrete Western instrument like the piano, whose fundamental frequency is part of the harmonic series of x0 are the tones an integer number of octaves from x0. Intervals of an octave are factors of 2 apart in the harmonic series of frequencies. For example, if 1/4*x corresponds to the frequency associated with A, then 1/8*x is A shifted up by an octave. However, the other tones of the harmonic series of a frequency whose tones corresponds to x0 can be approximated on a piano.
David Cope (1997) suggests the concept of interval strength, in which an interval’s strength, consonance, or stability is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps-Meyer law.