Benedetti height
The Benedetti height is a simple height function which measures the complexity of a JI interval. 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.
Computation
Ratio form
The Benedetti height of a ratio n/d is given by
[math]nd[/math]
Vector form
The Benedetti height of a p-limit monzo m = [m1 m2 … mπ (p)⟩ (π being the prime-counting function) is given by
[math]2^{\lVert H \vec m \rVert_1} \\ = 2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|}[/math]
where H is the transformation matrix such that, for the prime basis Q = ⟨2 3 5 … p],
[math]H = \operatorname {diag} (\log_2 (Q))[/math]
Examples
| Ratio | Monzo | Benedetti height |
|---|---|---|
| 1/1 | [0⟩ | 1 |
| 2/1 | [1⟩ | 2 |
| 3/2 | [-1 1⟩ | 6 |
| 6/5 | [1 1 -1⟩ | 30 |
| 9/7 | [0 2 0 -1⟩ | 63 |
| 13/11 | [0 0 0 0 -1 1⟩ | 143 |
History and terminology
Benedetti height was named by Gene Ward Smith sometime before 2011. 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.
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]