Tenney height
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
[math]\log_2 (nd)[/math]
Vector form
The Tenney height of a p-limit monzo m = [m1 m2 … mπ (p)⟩ (π being the prime-counting function) is given by
[math]\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)}|})[/math]
where H is the transformation matrix such that, for the prime basis Q = ⟨2 3 5 … p],
[math]H = \operatorname {diag} (\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 |
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.[1][2][3] This terminology was also used in Paul Erlich's paper A Middle Path[4].
See also
Notes
- ↑ John Cage and the Theory of Harmony. James Tenney.
- ↑ On the Conception and Measure of Consonance. Alex Wand.
- ↑ A Signal-Based Model of Teleology in Tonal Music. Mark André Brand. p. 28. "Tenney's measure of harmonic distance (Hd) is thus singled out as perhaps his most 'crucial development', affording him the means towards 'compactness'. His is a Manhattan, rather than Euclidean metric, defined as Hd (a/b) = klog(ab), with a/b the maximally reduced ratio representing the frequency difference, and k = 1 indicating measure in octaves."
- ↑ Wherein Erlich writes: "This is why, in Tenney’s terminology, the taxicab distance an interval traverses in his lattice is the 'Harmonic Distance' of that interval."