Tenney norm: Difference between revisions
Jump to navigation
Jump to search
m Xenwolf moved page Tenney Height to Tenney height: Rule: only first letter capitalized. Reason: less typing when referenced |
reworked, still something todo with the introduction |
||
| Line 1: | Line 1: | ||
If p/q is a positive rational number reduced to its lowest terms, then the [[ | If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer pq. 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. | ||
The ''Tenney height'' of a [[ | The '''Tenney height''' of a [[monzo]] is given by | ||
<pre>|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre> | <pre>|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)</pre> | ||
==Examples== | == Examples == | ||
{| class="wikitable" | {| class="wikitable" | ||
! Interval name | |||
! Ratio (p/q) | |||
! Monzo | |||
! Tenney height | |||
! log2(p*q) | |||
|- | |- | ||
| unison | |||
| [[1/1]] | |||
| {{Monzo| 0 }} | |||
| 0 | |||
| log2(1) | |||
|- | |- | ||
| | | octave | ||
| | | [[2/1]] | ||
| | | | {{Monzo| 1 }} | ||
| | | 1 | ||
| log2(1) | |||
|- | |- | ||
| | | just perfect fifth | ||
| | | [[3/2]] | ||
| | | | {{Monzo| -1 1 }} | ||
| | | | 2.585 | ||
| log2(6) | |||
|- | |- | ||
| just major third | |||
| | | [[5/4]] | ||
| | | {{Monzo| -2 0 1 }} | ||
| | log2( | | 4.322 | ||
| log2(20) | |||
|- | |- | ||
| harmonic seventh | |||
| [[7/4]] | |||
| {{Monzo| -2 0 0 1 }} | |||
| 4.807 | |||
| log2(28) | |||
| | |||
| | |||
| | log2(28) | |||
|} | |} | ||
[[Category: | |||
[[Category: | == External links == | ||
[[Category: | * [https://en.wikipedia.org/wiki/James_Tenney James Tenney - Wikipedia] | ||
[[Category: | |||
[[Category: | [[Category:Benedetti]] | ||
[[Category: | [[Category:Consonance]] | ||
[[Category: | [[Category:Dissonance]] | ||
[[Category: | [[Category:Harmonic entropy]] | ||
[[Category:Height]] | |||
[[Category:Measure]] | |||
[[Category:Psychoacoustics]] | |||
[[Category:Theory]] | |||
[[Category:Todo:improve synopsis]] | |||
[[Category:Todo:reduce mathslang]] | |||
Revision as of 14:32, 12 June 2020
If p/q is a positive rational number reduced to its lowest terms, then the Benedetti height is the integer pq. 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.
The Tenney height of a monzo is given by
|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)
Examples
| Interval name | Ratio (p/q) | Monzo | Tenney height | log2(p*q) |
|---|---|---|---|---|
| unison | 1/1 | [0⟩ | 0 | log2(1) |
| octave | 2/1 | [1⟩ | 1 | log2(1) |
| just perfect fifth | 3/2 | [-1 1⟩ | 2.585 | log2(6) |
| just major third | 5/4 | [-2 0 1⟩ | 4.322 | log2(20) |
| harmonic seventh | 7/4 | [-2 0 0 1⟩ | 4.807 | log2(28) |