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 Ratio (p/q) Monzo Tenney height log2(p*q)
unison 1/1 [0 0 log2(1)
octave 2/1 [1 1 log2(1)
just perfect fifth 3/2 [-1 1 2.585 log2(6)
just major third 5/4 [-2 0 1 4.322 log2(20)
harmonic seventh 7/4 [-2 0 0 1 4.807 log2(28)

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