harmonic distance

For a ratio a/b, the logarithm of a*b. It is a measure of harmonic complexity developed by James Tenney.

This distance function is a special use of the Minkowski metric in a tonal space where the units along each of the axes are the logarithms of prime numbers.

[For example], the harmonic distance of the interval 7:9 is 2*log(3)+log(7).

[from John Chalmers, Divisions of the Tetrachord]

see also complexity

. . . . . . . . . . . . . . . . . . . . . . . . .

Paul Erlich commented (Tuning List, 2000.4.19),:

I wonder why, in this definition, Chalmers refers to the city-block metric as the "Minkowski metric". The only definitions of the Minkowski metric I've ever seen are the usual special relativistic space-time distance -- see

http://aci.mta.ca/Courses/Physics/4701/EText/MinkowskiMetric.html

or http://mathworld.wolfram.com/MinkowskiMetric.html

or http://www.treasure-troves.com/physics/MinkowskiMetric.html;

and a generalized Euclidean metric where the exponents 2 and 1/2 are replaced with p and 1/p respectively (p can be infinite, in which case one takes a limit):

http://www.cs.mcgill.ca/~scottyb/DistTrans/Project.html

or http://www.ctr.columbia.edu/~jrsmith/html/pubs/acmmm96/node8.html

or http://zx81.ncsa.uiuc.edu/camilleg/eviolin.html (a MIDI violin controller!)

So what Chalmers really means is, using the second definition, the Minkowski first-order metric. Subject to Chalmers' approval, I suggest we change that definition.

It also might help to give an example using a ratio with one even number, to emphasize that one doesn't ignore factors of 2 in this computation.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

John Chalmers replied (Tuning List, 2000.4.20),:

Go ahead and clarify the definition of city-block, aka Taxicab, aka Manhattan, aka Minkowski metric.

---

(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to the Microtonal Dictionary index return to my home page return to the Sonic Arts home page