mel-scale

The mel-scale was created and is maintained for the sake of signal-processing — specifically speech recognition, audio analysis, and music information retrieval. For whose sake? Signal-processing. The scale is a tool designed to serve this domain's need for perceptually-uniform frequency representation. SERVES direction (servant→master) is correct.

The mel scale is a perceptual frequency scale: m = 2595 * log10(1 + f/700). It maps physical frequency (Hz) to perceived pitch. Remove the concept of frequency and the mel scale has nothing to map from — it stops operating entirely. This is a present-tense constitutive dependency (Law 8), not just historical.

mel-scale IS a specific kind of frequency-scale: it maps physical frequencies to a perceptual scale where equal distances correspond to equal musical intervals. A competent speaker calls mel-scale 'a frequency scale (perceptual).' Valid specific→general per Law 9.

MFCC derives from the mel-scale: the mel-scale (Stevens et al. 1937) predates MFCC by decades. MFCC uses triangular mel-scale filterbanks as its core frequency warping mechanism. Which came first? Mel-scale — unambiguously historical.

Mel-filterbank (1980s, newer) derives from mel-scale (Stevens et al. 1937, older). The filterbank is built by applying triangular windows spaced according to the mel scale. Which-came-first: mel-scale predates mel-filterbank by decades.

The mel-scale (perceptual frequency scale, 1930s Stevens/Volkmann/Newman, refined 1970s) predates the mel-frequency-spectrogram (1980s). The mel-frequency-spectrogram is built by warping the spectrogram onto the mel scale — which-came-first: mel-scale is older and fed into mel-frequency-spectrogram.

MFC maps the spectrum onto the mel-scale before computing the cepstrum. The mel-scale warping is essential — remove it and MFC collapses to ordinary cepstrum. The removal test passes (Law 8).

MFCC computation requires a mel-scale filterbank as its core mechanism. The mel-scale determines how frequency bins are warped before the DCT. Remove the mel-scale concept and MFCC loses its defining characteristic — there is no mel-scale filterbank to warp the spectrum. This is a genuine operational dependency.

The mel-frequency spectrum maps spectral content onto the mel scale — a perceptual frequency scale. Removal test: remove mel-scale as a concept and the mel-frequency spectrum has no frequency axis to map onto; its entire purpose (perceptual frequency warping) disappears. Constitutive dependency.

The mel filterbank is parameterized by the mel-frequency scale; removing the scale removes the filterbank's defining spacing and center frequencies. The filterbank depends on the scale for its operation.