val

A map from a rational tone group to the integers, which respects multiplication.

If h is a val, then:

h(a*b) = h(a) + h(b);

h(1) = 0; and

h(1/a) = -h(a).

If we write the rational number "a" as a = 2^e2 * 3^e3 * ... * p^ep [that is, if we prime-factor it], we may denote it by a row vector [e2, e3, ...., ep]. In that case we denote a val by a column vector of integers of the same dimension.

In the language of abstract algebra, a val is a homomorphism from the tone group to the integers.

[from Gene Ward Smith, Yahoo tuning-math message 2569 (Thu Jan 10, 2002 11:11 pm)

---

(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