Tenney norm: Difference between revisions

(31 intermediate revisions by 8 users not shown)

Line 1:

If p/q is a positive rational number reduced to its lowest terms, then the [[~~Benedetti_height|~~Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[~~log2|~~log2]]), of the ~~[[Benedetti_height|~~Benedetti height]], leading to Tenney ~~[[Height|height]]. In either form it is widely used as a [[measure_of_inharmonicity|measure of inharmonicity]] and/or complexity for intervals~~.

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.

The ''~~Tenney height~~'' ~~of a [[monzo|monzo]]~~ is given by

== Computation ==

=== Ratio form ===

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

~~<pre>|| |e2 e3 ... ep> || = |e2| + log2~~(3)~~|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>~~

$$\log_2 (nd) $$

==~~Examples~~==

=== Vector form ===

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

{| class="wikitable"

$$

\begin{align}

\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 {{nowrap| ''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }},

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

== Examples ==

{| class="wikitable center-2"

|-

! ~~style="text-align:center;" |~~ Interval name

! Interval name

! ~~style="text-align:center;" | Frequency ratio~~

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

! ~~style="text-align:center;" | monzo~~

! Monzo

! ~~style="text-align:center;" | log2(Benedetti height)~~

! Tenney norm

|-

| ~~| unison~~

| Unison

| | 1/1

| [[1/1]]

| ~~|<nowiki>~~ |0~~</nowiki>>~~

| {{Monzo| 0 }}

| | 0

| 0

|-

| ~~| octave~~

| Octave

| | 2/1

| [[2/1]]

| ~~|<nowiki>~~ |1~~</nowiki>>~~

| {{Monzo| 1 }}

| | 1

| 1

|-

| ~~| just~~ perfect fifth

| Just perfect fifth

| | 3/2

| [[3/2]]

| ~~|<nowiki>~~ |-1 1~~</nowiki>>~~

| {{Monzo| -1 1 }}

| ~~| log2(6) =~~ 2.585

| 2.585

|-

| ~~| just~~ major third

| Just major third

| | 5/4

| [[5/4]]

| ~~|<nowiki>~~ |-2 0 1~~</nowiki>>~~

| {{Monzo| -2 0 1 }}

| ~~| log2(20) =~~ 4.322

| 4.322

|-

| ~~| harmonic~~ seventh

| Harmonic seventh

| | 7/4

| [[7/4]]

| ~~|<nowiki>~~ |-2 0 0 1~~</nowiki>>~~

| {{Monzo| -2 0 0 1 }}

| ~~| log2(28) =~~ 4.807

| 4.807

|}

[[~~Category~~:~~benedetti~~]]

[[~~Category:consonance~~]]

== History and terminology ==

[[~~Category:dissonance~~]]

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>.

[[~~Category:harmonic_entropy~~]]

[[Category:~~height~~]]

== See also ==

[[Category:~~measure~~]]

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

[[Category:~~psychoacoustics~~]]

[[Category:~~theory~~]]

== References ==

[[Category:Consonance and dissonance]]

[[Category:Harmonic entropy]]

[[Category:Interval complexity measures]]

[[Category:Tenney-weighted measures]]

@@ Line 1: / Line 1: @@
-If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti_height|Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2|log2]]), of the [[Benedetti_height|Benedetti height]], leading to Tenney [[Height|height]]. In either form it is widely used as a [[measure_of_inharmonicity|measure of inharmonicity]] and/or complexity for intervals.
+{{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.
-The ''Tenney height'' of a [[monzo|monzo]] is given by
+== Computation ==
+=== Ratio form ===
+The Tenney norm of a ratio ''n''/''d'' is given by
-<pre>|| |e2 e3 ... ep&gt; || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre>
+$$\log_2 (nd) $$
-==Examples==
+=== Vector form ===
+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
-{| class="wikitable"
+$$
+\begin{align}
+\norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\
+&= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)
+\end{align}
+$$
+where ''H'' is the transformation matrix such that, for the prime basis {{nowrap| ''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }},
+$$ H = \operatorname {diag} (\log_2 (Q)) $$
+== Examples ==
+{| class="wikitable center-2"
 |-
-! style="text-align:center;" | Interval name
+! Interval name
-! style="text-align:center;" | Frequency ratio
+! Ratio (''n''/''d'')
-! style="text-align:center;" | monzo
+! Monzo
-! style="text-align:center;" | log2(Benedetti height)
+! Tenney norm
 |-
-| | unison
+| Unison
-| | 1/1
+| [[1/1]]
-| |<nowiki> |0</nowiki>&gt;
+| {{Monzo| 0 }}
-| | 0
+| 0
 |-
-| | octave
+| Octave
-| | 2/1
+| [[2/1]]
-| |<nowiki> |1</nowiki>&gt;
+| {{Monzo| 1 }}
-| | 1
+| 1
 |-
-| | just perfect fifth
+| Just perfect fifth
-| | 3/2
+| [[3/2]]
-| |<nowiki> |-1 1</nowiki>&gt;
+| {{Monzo| -1 1 }}
-| | log2(6) = 2.585
+| 2.585
 |-
-| | just major third
+| Just major third
-| | 5/4
+| [[5/4]]
-| |<nowiki> |-2 0 1</nowiki>&gt;
+| {{Monzo| -2 0 1 }}
-| | log2(20) = 4.322
+| 4.322
 |-
-| | harmonic seventh
+| Harmonic seventh
-| | 7/4
+| [[7/4]]
-| |<nowiki> |-2 0 0 1</nowiki>&gt;
+| {{Monzo| -2 0 0 1 }}
-| | log2(28) = 4.807
+| 4.807
 |}
-[[Category:benedetti]]
-[[Category:consonance]]
+== History and terminology ==
-[[Category:dissonance]]
+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>.
-[[Category:harmonic_entropy]]
-[[Category:height]]
+== See also ==
-[[Category:measure]]
+* [[Generalized Tenney norms and Tp interval space|Generalized Tenney norms and T<sub>''p''</sub> interval space]]
-[[Category:psychoacoustics]]
-[[Category:theory]]
+== References ==
+<references />
+[[Category:Consonance and dissonance]]
+[[Category:Harmonic entropy]]
+[[Category:Interval complexity measures]]
+[[Category:Tenney-weighted measures]]