Tenney Height

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If p/q is a positive rational number reduced to its lowest terms, then the Benedetti height is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 (log2), of the Benedetti height, leading to Tenney height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.

The Tenney height of a monzo is given by

|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)

Examples

Interval name Frequency ratio monzo log2(Benedetti height)
unison 1/1 |0> 0
octave 2/1 |1> 1
just perfect fifth 3/2 |-1 1> log2(6) = 2.585
just major third 5/4 |-2 0 1> log2(20) = 4.322
harmonic seventh 7/4 |-2 0 0 1> log2(28) = 4.807