# 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. It is also known as *log product complexity*.

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) |