Tenney norm: Difference between revisions

Line 1:

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.

Line 10:

=== Vector form ===

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

Line 22:

Line 24:

== Examples ==

{| class="wikitable center-2"

|-

! Interval Name

! Ratio (''n''/''d'')

Line 54:

Line 57:

== 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 ==

Line 60:

Line 63:

== Notes ==

[[Category:Consonance and dissonance]]

@@ Line 1: / Line 1: @@
 {{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.
@@ Line 10: / Line 10: @@
 === 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>
@@ Line 22: / Line 24: @@
 == Examples ==
 {| class="wikitable center-2"
+|-
 ! Interval Name
 ! Ratio (''n''/''d'')
@@ Line 54: / Line 57: @@
 == 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 ==
@@ Line 60: / Line 63: @@
 == Notes ==
+<references />
 [[Category:Consonance and dissonance]]