Benedetti height
The Benedetti height of a positive rational number N/D reduced to lowest terms (no common factor between N and D) is equal to N*D, the product of the numerator and denominator. In general mathematics it is known as product complexity.
The logarithm base two of the Benedetti height is the Tenney height, or Tenney norm.
The name is based on the fact that the scientist, mathematician and music theorist Giovanni Battista Benedetti first proposed it as a measure of inharmonicity. It may be the first number-theoretic height function ever defined for any purpose.
Examples
| Interval | Benedetti height | Tenney height |
|---|---|---|
| 1/1 | 1 | 0 |
| 2/1 | 2 | 1 |
| 3/2 | 6 | 2.585 |
| 6/5 | 30 | 4.907 |
| 9/7 | 63 | 5.977 |
| 13/11 | 143 | 7.160 |