Tenney norm

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James Tenney

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 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 ratio n/d is given by

[math]\displaystyle{ \log_2 (nd) }[/math]

The Tenney height of a p-limit monzo b = [b₁ b₂ … b_{π (p)}⟩ (π being the prime-counting function) is given by

[math]\displaystyle{ \lVert W^{-1} \vec b \rVert_1 \\ = \vert b_1 \vert + \vert b_2 \vert \log_2 (3) + \ldots + \vert b_{\pi (p)} \vert \log_2 (p) \\ = \log_2 (2^{|b_1|} \cdot 3^{|b_2|} \cdot \ldots \cdot p^{|b_{\pi (p)}|}) }[/math]

where W is the Tenney weighter such that, for the prime basis Q = ⟨2 3 5 … p],

[math]\displaystyle{ W = \operatorname {diag} (1/\log_2 (Q)) }[/math]

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