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. | ||
The logarithm base two of the Benedetti height is the [[ | 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 [http://www.webcitation.org/6076Lm8r4 Giovanni Battista Benedetti] first proposed it as a measure of inharmonicity. It may be the first number-theoretic [[ | The name is based on the fact that the scientist, mathematician and music theorist [http://www.webcitation.org/6076Lm8r4 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 == | |||
{| class="wikitable center-all" | |||
! Interval | |||
{| class="wikitable" | ! 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 | |||
|} | |} | ||
[[Category: | |||
[[Category: | == See also == | ||
[[Category: | |||
[[Category: | * [[Kees height]] | ||
[[Category: | * [https://en.wikipedia.org/wiki/Giambattista_Benedetti Giambattista Benedetti - Wikipedia] | ||
[[Category: | |||
[[Category: | [[Category:Benedetti]] | ||
[[Category:Definition]] | |||
[[Category:Height]] | |||
[[Category:Measure]] | |||
[[Category:Psychoacoustics]] | |||
[[Category:Tenney]] | |||
[[Category:Theory]] | |||