Interval
An interval is the difference in pitch between two notes.
Human pitch perception is logarithmic, therefore an interval can be described with a frequency ratio or a logarithmic measure of that ratio, such as cents.
A rational interval is an interval whose frequency ratio is a rational number. Its size in cents is then necessarily irrational[1], unless the interval happens to be an octave or some multiple of an octave. A tuning system based exclusively on rational intervals is said to be in just intonation. Conversely, an irrational interval is an interval whose frequency ratio is an irrational number. In that case, however, its cents value may or may not be rational. An interval with rational cents is always irrational, unless it's an octave or some multiple of an octave. But some intervals have both irrational ratios and irrational cents.
Another property is harmonic entropy, a measure of concordance, which is usually associated with consonance and dissonance.
See also
References
- ↑ See example on Wikipedia: Irrational number#Logarithms. A full proof would rely on the fundamental theorem of arithmetic to generalize the results to all pairs of coprime natural numbers.