Benedetti height: Difference between revisions
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The '''Benedetti height''' of a positive rational number | The '''Benedetti height''' of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', 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 [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm. | ||
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== Examples == | == Examples == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! Interval | ! Interval | ||
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== History == | == History == | ||
Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956#20956</ref>. | Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956#20956</ref>. | ||
== See also == | == See also == | ||
* [[Kees height]] | * [[Kees height]] | ||
* [ | * [[Wikipedia: Giambattista Benedetti]] | ||
== References == | == References == | ||
<references/> | <references/> | ||
Revision as of 08:43, 6 December 2022
The Benedetti height of a positive rational number n/d reduced to lowest terms (no common factor between n and d) is equal to nd, 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 |
History
Benedetti height was named by Gene Ward Smith sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same height function. Due to pushback from Paul Erlich (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well[1].