Tenney norm: Difference between revisions
Much needed readability improvement |
Respell ratio as n/d and explicitly give the formula for ratios |
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If | 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 | The Tenney height of a ratio ''n''/''d'' is given by | ||
<math>\lVert W^{-1}b \ | <math>\log_2 (nd)</math> | ||
The Tenney height of a [[monzo]] b = {{monzo| ''b''<sub>π (2)</sub> ''b''<sub>π (3)</sub> … ''b''<sub>π (''p'')</sub> }} is given by | |||
<math>\lVert W^{-1}b \rVert_1 \\ | |||
= \vert b_{\pi (2)} \vert + \log_2 (3) \vert b_{\pi (3)} \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert \\ | |||
= \log_2 (2^{|b_{\pi (2)}|} \cdot 3^{|b_{\pi (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'' }}, | ||
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== Examples == | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval name | ! Interval name | ||
! Ratio ( | ! Ratio (''n''/''d'') | ||
! Monzo | ! Monzo | ||
! Tenney height | ! Tenney height | ||
|- | |- | ||
| | | Unison | ||
| [[1/1]] | | [[1/1]] | ||
| {{Monzo| 0 }} | | {{Monzo| 0 }} | ||
| 0 | | 0 | ||
|- | |- | ||
| | | Octave | ||
| [[2/1]] | | [[2/1]] | ||
| {{Monzo| 1 }} | | {{Monzo| 1 }} | ||
| 1 | | 1 | ||
|- | |- | ||
| | | Just perfect fifth | ||
| [[3/2]] | | [[3/2]] | ||
| {{Monzo| -1 1 }} | | {{Monzo| -1 1 }} | ||
| 2.585 | | 2.585 | ||
|- | |- | ||
| | | Just major third | ||
| [[5/4]] | | [[5/4]] | ||
| {{Monzo| -2 0 1 }} | | {{Monzo| -2 0 1 }} | ||
| 4.322 | | 4.322 | ||
|- | |- | ||
| | | Harmonic seventh | ||
| [[7/4]] | | [[7/4]] | ||
| {{Monzo| -2 0 0 1 }} | | {{Monzo| -2 0 0 1 }} | ||
| 4.807 | | 4.807 | ||
|} | |} | ||
Revision as of 13:18, 27 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 monzo b = [bπ (2) bπ (3) … bπ (p)⟩ is given by
[math]\displaystyle{ \lVert W^{-1}b \rVert_1 \\ = \vert b_{\pi (2)} \vert + \log_2 (3) \vert b_{\pi (3)} \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert \\ = \log_2 (2^{|b_{\pi (2)}|} \cdot 3^{|b_{\pi (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 |