Tenney norm: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! style="text-align:center;" | Interval name | |||
! style="text-align:center;" | Frequency ratio | |||
! style="text-align:center;" | monzo | |||
! style="text-align:center;" | log2(Benedetti height) | |||
|- | |- | ||
| | unison | | | unison | ||
| | 1/1 | | | 1/1 | ||
| | |0> | | |<nowiki> |0</nowiki>> | ||
| | 0 | | | 0 | ||
|- | |- | ||
| | octave | | | octave | ||
| | 2/1 | | | 2/1 | ||
| | |1> | | |<nowiki> |1</nowiki>> | ||
| | 1 | | | 1 | ||
|- | |- | ||
| | just perfect fifth | | | just perfect fifth | ||
| | 3/2 | | | 3/2 | ||
| | |-1 1> | | |<nowiki> |-1 1</nowiki>> | ||
| | log2(6) = 2.585 | | | log2(6) = 2.585 | ||
|- | |- | ||
| | just major third | | | just major third | ||
| | 5/4 | | | 5/4 | ||
| | |-2 0 1> | | |<nowiki> |-2 0 1</nowiki>> | ||
| | log2(20) = 4.322 | | | log2(20) = 4.322 | ||
|- | |- | ||
| | harmonic seventh | | | harmonic seventh | ||
| | 7/4 | | | 7/4 | ||
| | |-2 0 0 1> | | |<nowiki> |-2 0 0 1</nowiki>> | ||
| | log2(28) = 4.807 | | | log2(28) = 4.807 | ||
|} | |} | ||
Revision as of 15:47, 12 January 2019
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 | Frequency ratio | monzo | log2(Benedetti height) |
|---|---|---|---|
| unison | 1/1 | |0> | 0 |
| octave | 2/1 | |1> | 1 |
| just perfect fifth | 3/2 | |-1 1> | log2(6) = 2.585 |
| just major third | 5/4 | |-2 0 1> | log2(20) = 4.322 |
| harmonic seventh | 7/4 | |-2 0 0 1> | log2(28) = 4.807 |