Benedetti height: Difference between revisions
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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. | 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. It is also known as ''product complexity''. | ||
The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm. | The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm. | ||
Revision as of 22:03, 16 August 2020
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. It is also 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 |