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. Alternative terms used include '''harmonic distance''' ('''HD'''). | 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'''). | ||
== Computation == | == Computation == | ||
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=== Vector form === | === Vector form === | ||
The Tenney height of a [[Harmonic limit|''p''-limit]] [[monzo]] | 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 | ||
<math>\lVert H \vec | <math>\lVert H \vec m \rVert_1 \\ | ||
= \vert | = \vert m_1 \vert + \vert m_2 \vert \log_2 (3) + \ldots + \vert m_{\pi (p)} \vert \log_2 (p) \\ | ||
= \log_2 (2^{| | = \log_2 (2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|})</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 Q = {{val| 2 3 5 … ''p'' }}, | ||
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== Examples == | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval | ! Interval Name | ||
! Ratio (''n''/''d'') | ! Ratio (''n''/''d'') | ||
! Monzo | ! Monzo | ||
! Tenney | ! Tenney Height | ||
|- | |- | ||
| Unison | | Unison | ||
| Line 55: | Line 52: | ||
| 4.807 | | 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 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>. | |||
== See also == | == See also == | ||
* [[Generalized Tenney | * [[Generalized Tenney norms and Tp interval space]] | ||
== Notes == | == Notes == | ||
Revision as of 13:31, 28 August 2023
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).
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{ \lVert H \vec m \rVert_1 \\ = \vert m_1 \vert + \vert m_2 \vert \log_2 (3) + \ldots + \vert m_{\pi (p)} \vert \log_2 (p) \\ = \log_2 (2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|}) }[/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
- ↑ John Cage and the Theory of Harmony. James Tenney.
- ↑ On the Conception and Measure of Consonance. Alex Wand.
- ↑ 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) = klog(ab), with a/b the maximally reduced ratio representing the frequency difference, and k = 1 indicating measure in octaves."
- ↑ 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."