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. | 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. | ||
= Names = | == 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>. | |||
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> | |||
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 height of a ratio ''n''/''d'' is given by | ||
<math>\log_2 (nd)</math> | <math>\log_2 (nd)</math> | ||
== Vector form == | === 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 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 | ||
| Line 30: | Line 23: | ||
<math>W = \operatorname {diag} (1/\log_2 (Q))</math> | <math>W = \operatorname {diag} (1/\log_2 (Q))</math> | ||
= Examples = | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval name | ! Interval name | ||
Revision as of 00:40, 6 March 2022
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.
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.[1][2][3] This terminology was also used in Paul Erlich's paper A Middle Path[4].
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 b = [b1 b2 … bπ (p)⟩ (π being the prime-counting function) is given by
[math]\displaystyle{ \lVert W^{-1} \vec b \rVert_1 \\ = \vert b_1 \vert + \vert b_2 \vert \log_2 (3) + \ldots + \vert b_{\pi (p)} \vert \log_2 (p) \\ = \log_2 (2^{|b_1|} \cdot 3^{|b_2|} \cdot \ldots \cdot p^{|b_{\pi (p)}|}) }[/math]
where W is the Tenney weighter such that, for the prime basis Q = ⟨2 3 5 … p],
[math]\displaystyle{ W = \operatorname {diag} (1/\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 |
- ↑ 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."