Tenney norm

[math]\displaystyle{ \def\hs{\hspace{-3px}} \def\lvsp{{}\mkern-5.5mu}{} \def\rvsp{{}\mkern-2.5mu}{} \def\llangle{\left\langle\lvsp\left\langle} \def\lllangle{\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llllangle{\left\langle\lvsp\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\rvsp\right\rangle} \def\rrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\abs#1{\left|{#1}\right|} \def\norm#1{\left\|{#1}\right\|} \def\floor#1{\left\lfloor{#1}\right\rfloor} \def\ceil#1{\left\lceil{#1}\right\rceil} \def\round#1{\left\lceil{#1}\right\rfloor} \def\rround#1{\left\lfloor{#1}\right\rceil} }[/math] The Tenney norm, otherwise known as harmonic distance (HD) or Tenney height, is commonly used as a measure of complexity for just 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 norm.

Computation

Ratio form

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

$$\log_2 (nd) $$

Vector form

The Tenney norm of a p-limit monzo m = [m₁ m₂ … m_{π (p)}⟩ (π being the prime-counting function) is given by

$$ \begin{align} \norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\ &= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) \end{align} $$

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 norm
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.

See also

• Generalized Tenney norms and T_p interval space