The tempered Fourier transform

3 2 = 2 7.01955/12 ≅ 2 7/12 . The set of harmonics (l = 1, 2, 3), 1/Kl = (29, 14, 13) ≅ (25/12, 27/12, 2), when scaled by 2am/12, m = 1, 2, 3, 4, constitutes a logarithmically uniform set, flm, of 12 frequency classes per octave (8 va). The tempered Fourier transform samples a record x(t) into four independent parallel sequences xm(i) at sampling rates 2m/4. In each 8 va, n, of each sequence, m, three Fourier coefficients, Xlmn(j) = Σ xmn(j − i)W(i/NKl)exp − j(2πi/Kl) are computed, where W(i/NKl) is a window N cycles long, and xmn(j) = Σ hkxmn+1(2j − k) is the data sequence in the nth 8 va after it has been low-pass filtered and decimated from the (n + 1) 8 va above by a half-band filter with impulse response h. With a suitable value of N, the array Xlmn is statistically and dynamically equivalent to a 112–8 va spectral analysis of x. Examples and variants of the method will be discussed.