Tenney norm: Difference between revisions
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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. Alternative terms used include '''harmonic distance''' ('''HD'''). | 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. Alternative terms used include '''harmonic distance''' ('''HD'''). | ||
== Computation == | == Computation == | ||
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=== Vector form === | === Vector form === | ||
The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] | The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] m = {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | ||
<math>\lVert H \vec | <math>\lVert H \vec m \rVert_1 \\ | ||
= \vert | = \vert m_1 \vert + \vert m_2 \vert \log_2 (3) + \ldots + \vert m_{\pi (p)} \vert \log_2 (p) \\ | ||
= \log_2 (2^{| | = \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 = {{val| 2 3 5 … ''p'' }}, | where H is the transformation matrix such that, for the prime basis Q = {{val| 2 3 5 … ''p'' }}, | ||
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== Examples == | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval | ! Interval Name | ||
! Ratio (''n''/''d'') | ! Ratio (''n''/''d'') | ||
! Monzo | ! Monzo | ||
! Tenney | ! Tenney Height | ||
|- | |- | ||
| Unison | | Unison | ||
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| 4.807 | | 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''.<ref>[https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theory of Harmony'']. James Tenney. </ref><ref>[https://zh.booksc.eu/book/68954431/f87a1d ''On the Conception and Measure of Consonance'']. Alex Wand. </ref><ref>[https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y ''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'') = ''k''log(''ab''), with ''a''/''b'' the maximally reduced ratio representing the frequency difference, and ''k'' = 1 indicating measure in octaves."</ref> This terminology was also used in [[Paul Erlich]]'s paper [[A Middle Path]]<ref>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."</ref>. | |||
== See also == | == See also == | ||
* [[Generalized Tenney | * [[Generalized Tenney norms and Tp interval space]] | ||
== Notes == | == Notes == | ||