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. It is ~~also~~ known as ''log product complexity''.

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 =

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>Original paper by Tenney: https://www.plainsound.org/pdfs/JC&ToH.pdf</ref><ref>https://zh.booksc.eu/book/68954431/f87a1d</ref><ref>https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y "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>.

= Computation =

== Ratio form ==

The Tenney height of a ratio ''n''/''d'' is given by

== Vector form ==

The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] b = {{monzo| ''b''1 ''b''2 … ''b''π (''p'') }} (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>

== Examples ==

= Examples =

{| class="wikitable"

! Interval name

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 {{Wikipedia| James Tenney }}
-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''.
+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 =
+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>Original paper by Tenney: https://www.plainsound.org/pdfs/JC&ToH.pdf</ref><ref>https://zh.booksc.eu/book/68954431/f87a1d</ref><ref>https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y "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>.
+= 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 [[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
@@ Line 17: / Line 30: @@
 <math>W = \operatorname {diag} (1/\log_2 (Q))</math>
-== Examples ==
+= Examples =
 {| class="wikitable"
 ! Interval name