Tenney norm: Difference between revisions

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{{Wikipedia| James Tenney }}
{{Wikipedia| James Tenney }}
 
{{texops}}
The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. 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 the Tenney height.  
The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. 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 the Tenney height.  


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=== Vector form ===
=== Vector form ===
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
The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] {{nowrap|'''m''' {{=}} {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w|prime-counting function}}) is given by


<math>\lVert H \vec m \rVert_1 \\
<math>
= \vert m_1 \vert + \vert m_2 \vert \log_2 (3) + \ldots + \vert m_{\pi (p)} \vert \log_2 (p) \\
\norm{H \vec m}_1 \\
= \log_2 (2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|})</math>
= \abs{m_1} + \abs{m_2} \abs{\log_2 (3)} + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\
= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)
</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 {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }}}},  


<math>H = \operatorname {diag} (\log_2 (Q))</math>
<math>H = \operatorname {diag} (\log_2 (Q))</math>
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== Examples ==
== Examples ==
{| class="wikitable center-2"
{| class="wikitable center-2"
|-
! Interval Name
! Interval Name
! Ratio (''n''/''d'')
! Ratio (''n''/''d'')
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== History and terminology ==
== 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>.   
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 {{nowrap|Hd(''a''/''b'') {{=}} ''k'' log(''ab'')}}, with ''a''/''b'' the maximally reduced ratio representing the frequency difference, and {{nowrap|''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 ==
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== Notes ==
== Notes ==
<references />


[[Category:Consonance and dissonance]]
[[Category:Consonance and dissonance]]

Revision as of 18:29, 8 May 2025

English Wikipedia has an article on:

[math]\displaystyle{ \def\hs{\hspace{-3px}} \def\lvsp{{}\mkern-5.5mu}{} \def\rvsp{{}\mkern-2.5mu}{} \def\llangle{\left\langle\lvsp\left\langle} \def\lllangle{\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llllangle{\left\langle\lvsp\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\rvsp\right\rangle} \def\rrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\abs#1{\left|{#1}\right|} \def\norm#1{\left\|{#1}\right\|} \def\floor#1{\left\lfloor{#1}\right\rfloor} \def\ceil#1{\left\lceil{#1}\right\rceil} \def\round#1{\left\lceil{#1}\right\rfloor} \def\rround#1{\left\lfloor{#1}\right\rceil} }[/math] The Tenney norm, otherwise known as harmonic distance (HD) or Tenney height, is commonly used as a measure of complexity for just 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 the Tenney height.

Computation

Ratio form

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

[math]\displaystyle{ \log_2 (nd) }[/math]

Vector form

The Tenney height of a p-limit monzo m = [m1 m2 … mπ (p)⟩ (π being the prime-counting function) is given by

[math]\displaystyle{ \norm{H \vec m}_1 \\ = \abs{m_1} + \abs{m_2} \abs{\log_2 (3)} + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\ = \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right) }[/math]

where H is the transformation matrix such that, for the prime basis Q = ⟨2 3 5 … p],

[math]\displaystyle{ H = \operatorname {diag} (\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

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.[1][2][3] This terminology was also used in Paul Erlich's paper A Middle Path[4].

See also

Notes

  1. ↑ John Cage and the Theory of Harmony. James Tenney.
  2. ↑ On the Conception and Measure of Consonance. Alex Wand.
  3. ↑ 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."
  4. ↑ 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."
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