Tenney norm: Difference between revisions

If p/q is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the [[Benedetti height]], leading to Tenney [[height]]. In either form it is widely used as a [[measure of inharmonicity]] and/or complexity for intervals. It is also known as ''log product complexity''.

The '''Tenney height''' of a [[monzo]] b = {{monzo| ''b''_{π (2)} ''b''_{π (3)} … ''b''_{π (''p'')} }} is given by

<math>\lVert W^{-1}b \rVert = \vert b_{\pi (2)} \vert + \log_2 (3) \vert b_{\pi (3)} \vert + \ldots + \log_2 (p) \vert b_{\pi (p)} \vert = \log_2 (2^{|b_{\pi (2)}|} \cdot 3^{|b_{\pi (2)}|} \cdot \ldots \cdot p^{|b_{\pi (p)}|})</math>

! Ratio (p/q)

| unison

| octave

| just perfect fifth

| just major third

| harmonic seventh