User:Akselai/FM scale: Difference between revisions
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An '''FM scale of the second kind''' is a function from the ''real numbers'' to musical intervals. Since these are usually continuous functions, it is meaningless to talk about scale steps of an FM scale. | An '''FM scale of the second kind''' is a function from the ''real numbers'' to musical intervals. Since these are usually continuous functions, it is meaningless to talk about scale steps of an FM scale. | ||
It is defined as the integral of the FM function: | It is defined as the integral of the FM function: | ||
<math>\text{FM}( | <math>\text{FM}(t) = \displaystyle \int_0^t f(x) dx</math> | ||
The x-th "scale step" in such a scale is called the x-th '''spec''' (pl. '''specs'''), which comes from the phrase "tone spectrum". | The x-th "scale step" in such a scale is called the x-th '''spec''' (pl. '''specs'''), which comes from the phrase "tone spectrum". | ||
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== Properties == | == Properties == | ||
An FM scale is aperiodic if and only if some of the <math>a_i</math> | An FM scale is aperiodic if and only if some of the <math>a_i</math> are irrational multiples of ''π''. | ||
Since | Since | ||