Tenney norm: Difference between revisions
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b1, b2 need not be denoted by the prime counting function; clarify what p is |
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<math>\log_2 (nd)</math> | <math>\log_2 (nd)</math> | ||
The Tenney height of a [[monzo]] b = {{monzo| ''b''<sub> | The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] b = {{monzo| ''b''<sub>1</sub> ''b''<sub>2</sub> … ''b''<sub>π (''p'')</sub> }} is given by | ||
<math>\lVert W^{-1}b \rVert_1 \\ | <math>\lVert W^{-1} \vec b \rVert_1 \\ | ||
= \vert | = \vert b_1 \vert + \log_2 (3) \vert b_2 \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert \\ | ||
= \log_2 (2^{| | = \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 = {{val| 2 3 5 … ''p'' }}, | where W is the Tenney weighter such that, for the prime basis Q = {{val| 2 3 5 … ''p'' }}, | ||
Revision as of 23:30, 31 January 2022
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 … bπ (p)⟩ is given by
[math]\displaystyle{ \lVert W^{-1} \vec b \rVert_1 \\ = \vert b_1 \vert + \log_2 (3) \vert b_2 \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert \\ = \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 |