Tenney norm: Difference between revisions
Cmloegcmluin (talk | contribs) expand article to include historical terminology, and adjust other material accordingly |
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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. | 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. | ||
= Names = | == Names == | ||
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>. | |||
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> | |||
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>. | |||
== Computation == | |||
=== Ratio form === | |||
The Tenney height of a ratio ''n''/''d'' is given by | The Tenney height of a ratio ''n''/''d'' is given by | ||
<math>\log_2 (nd)</math> | <math>\log_2 (nd)</math> | ||
== Vector form == | === Vector form === | ||
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> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | 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> }} (π being the [[Wikipedia: prime-counting function|prime-counting function]]) is given by | ||
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<math>W = \operatorname {diag} (1/\log_2 (Q))</math> | <math>W = \operatorname {diag} (1/\log_2 (Q))</math> | ||
= Examples = | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval name | ! Interval name | ||