Tenney norm: Difference between revisions

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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 norm.

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''').~~

~~== 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>.

== Computation ==

=== Ratio form ===

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

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

~~<math>~~\log_2 (nd)~~</math>~~

$$\log_2 (nd) $$

=== 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

The Tenney norm 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 ~~b \rVert_1 \\~~

$$

= \~~vert b_1 \vert~~ + \~~vert b_2 \vert~~ \log_2 (3) + \ldots + \~~vert b_~~{\pi (p)} ~~\vert~~ \log_2 (p) \\

\begin{align}

= \log_2 (2^{~~|b_1|~~} \cdot 3^{~~|b_2|~~} \cdot \ldots \cdot p^{~~|b_~~{\pi (p)}|})~~</math>~~

\norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \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)

\end{align}

$$

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>~~

$$ H = \operatorname {diag} (\log_2 (Q)) $$

== Examples ==

{| class="wikitable"

{| class="wikitable center-2"

|-

! Interval name

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

! Monzo

! Tenney ~~height~~

! Tenney norm

|-

| Unison

Line 55:

| 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 {{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 ==

* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T''p'' interval space]]

* [[Generalized Tenney Norms and Tp Interval Space]]

== References ==

== ~~Notes~~ ==

[[Category:~~Regular temperament theory]]~~

[[Category:Consonance and dissonance]]

~~[[Category:Psychoacoustics~~]]

[[Category:Harmonic entropy]]

[[Category:~~Tenney~~]]

[[Category:Interval complexity measures]]

[[Category:~~Benedetti]]~~

[[Category:Tenney-weighted measures]]

~~[[Category:Consonance]]~~

~~[[Category:Dissonance]]~~

~~[[Category:Height]]~~

~~[[Category:Measure~~]]

@@ 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 norm.
-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''').
-== 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>.
 == Computation ==
 === Ratio form ===
-The Tenney height of a ratio ''n''/''d'' is given by
+The Tenney norm of a ratio ''n''/''d'' is given by
-<math>\log_2 (nd)</math>
+$$\log_2 (nd) $$
 === 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 norm 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 b \rVert_1 \\
+$$
-= \vert b_1 \vert + \vert b_2 \vert \log_2 (3) + \ldots + \vert b_{\pi (p)} \vert \log_2 (p) \\
+\begin{align}
-= \log_2 (2^{|b_1|} \cdot 3^{|b_2|} \cdot \ldots \cdot p^{|b_{\pi (p)}|})</math>
+\norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\
+&= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)
+\end{align}
+$$
-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>
+$$ H = \operatorname {diag} (\log_2 (Q)) $$
 == Examples ==
-{| class="wikitable"
+{| class="wikitable center-2"
+|-
 ! Interval name
 ! Ratio (''n''/''d'')
 ! Monzo
-! Tenney height
+! Tenney norm
 |-
 | Unison
@@ Line 55: / Line 55: @@
 | 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 {{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 ==
+* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]]
-* [[Generalized Tenney Norms and Tp Interval Space]]
+== References ==
+<references />
-== Notes ==
-[[Category:Regular temperament theory]]
+[[Category:Consonance and dissonance]]
-[[Category:Psychoacoustics]]
 [[Category:Harmonic entropy]]
-[[Category:Tenney]]
+[[Category:Interval complexity measures]]
-[[Category:Benedetti]]
+[[Category:Tenney-weighted measures]]
-[[Category:Consonance]]
-[[Category:Dissonance]]
-[[Category:Height]]
-[[Category:Measure]]