# Tenney height

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The Tenney height, Tenney norm, or otherwise known as harmonic distance (HD), is widely used as a measure of inharmonicity and/or complexity for intervals. If n/d is a positive rational number reduced to its lowest terms, then the Benedetti height is the integer nd. Often it is more convenient instead to take the logarithm, usually base 2 (log2), of the Benedetti height, leading to the Tenney height.

## Computation

### Ratio form

The Tenney height of a ratio n/d is given by

$\log_2 (nd)$

### Vector form

The Tenney height of a p-limit monzo m = [m1 m2mπ (p) (π being the prime-counting function) is given by

$\lVert H \vec m \rVert_1 \\ = \vert m_1 \vert + \vert m_2 \vert \log_2 (3) + \ldots + \vert m_{\pi (p)} \vert \log_2 (p) \\ = \log_2 (2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|})$

where H is the transformation matrix such that, for the prime basis Q = 2 3 5 … p],

$H = \operatorname {diag} (\log_2 (Q))$

## Examples

Interval Name Ratio (n/d) Monzo Tenney Height
Unison 1/1 [0 0
Octave 2/1 [1 1
Just perfect fifth 3/2 [-1 1 2.585
Just major third 5/4 [-2 0 1 4.322
Harmonic seventh 7/4 [-2 0 0 1 4.807

## History and terminology

In general mathematics, this measurement is known as log-product complexity. With respect to microtonal tuning, this measurement was first described by James Tenney, who himself called it harmonic distance. This terminology was also used in Paul Erlich's paper A Middle Path.