# Tenney Height

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 |