User:Akselai/FM scale: Difference between revisions
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Suppose for each value ''i'' we provide an '''FM mapping''', that is, we perturb the scale step indices from integers to nearby real numbers. This is done using a cumulative sum over the FM function. | Suppose for each value ''i'' we provide an '''FM mapping''', that is, we perturb the scale step indices from integers to nearby real numbers. This is done using a cumulative sum over the FM function. | ||
The ''i''-th scale step will be mapped to the <math>A_i</math>-th spec, where <math>A_i = \displaystyle \sum_{1 \leq | The ''i''-th scale step will be mapped to the <math>A_i</math>-th spec, where <math>A_i = \displaystyle \sum_{1 \leq j \leq i} 1+f(j)</math>. By smoothing out the scale steps using the sigmoid function, the FM scale becomes <math>\text{FM}(x) = \displaystyle\sum_{1 \leq i \leq x}\sigma(x - A_i).</math> | ||
Setting ''k'' = ∞ gives a discrete scale, with unequal spec ranges corresponding to equal steps. | Setting ''k'' = ∞ gives a discrete scale, with unequal spec ranges corresponding to equal steps. | ||
== Examples == | == Examples == | ||