Judith C. Brown Journal of the Acoustical Society of America,1991 Jain-De,Lee

 INTRODUCTION  CALCULATION  RESULTS  SUMMARY

 The work is based on the property that, for sounds made up of harmonic frequency components

 The positions of these frequency components relative to each other are the same independent of fundamental frequency

 The conventional linear frequency representation ◦ Rise to a constant separation ◦ Harmonic components vary with fundamental frequency  The result is that it is more difficult to pick out differences in other features ◦ Timbre ◦ Attack ◦ Decay

 The log frequency representation ◦ Constant pattern for the spectral components ◦ Recognizing a previously determined pattern becomes a straightforward problem  The idea has theoretical appeal for its similarity to modern theories ◦ The perception of the pitch–Missing fundamental

 To demonstrate the constant pattern for musical sound ◦ The mapping of these data from the linear to the logarithmic domain  Too little information at low frequencies and too much information at high frequencies  For example ◦ Window of 1024 samples and sampling rate of samples/s and the resolution is 31.3 Hz(32000/1024=31.25) The violin low end of the range is G3(196Hz) and the adjacent note is G # 3( Hz),the resolution is much greater than the frequency separation for two adjacent notes tuned

 The frequencies sampled by the discrete Fourier transform should be exponentially spaced  If we require quartertone spacing ◦ The variable resolution of at most ( 2 1/24 -1)= 0.03 times the frequency ◦ A constant ratio of frequency to resolution f / δf = Q ◦ Here Q =f /0.029f= 34

 Quarter-tone spacing of the equal tempered scale,the frequency of the k th spectral component is  The resolution f / δf for the DFT, then the window size must varied f k = (2 1/24 ) k f min Where f an upper frequency chosen to be below the Nyquist frequency f min can be chosen to be the lowest frequency about which Information is desired

 For quarter-tone resolution  Calculate the length of the window in frequency f k Q = f / δf = f / 0.029f = 34 Where the quality factor Q is defined as f / δf bandwidth δf = f / Q Sampling rate S = 1/T N[k]= S / δf k = (S / f k )Q

 We obtain an expression for the k th spectral component for the constant Q transform  Hamming window that has the form W[k,n]=α + (1- α)cos(2πn/N[k]) Where α = 25/46 and 0 ≤ n ≤ N[k]-1

Constant Q transform of violin playing diatonic scale pizzicato from G3 (196 Hz) to G5(784 Hz) Constant Q transform of violin playing D5(587 Hz) with vibrato Constant Q transform of violin glissando from D5 (587 Hz) to A5 (880Hz) Constant Q transform of flute playing diatonic scale from C4 (262 Hz) to C5 (523 Hz) with increasing amplitude Constant Q transform of piano playing diatonic scale from C4 (262 Hz) to C5(523 Hz) The attack on D5(587 Hz) is also visible

 Straightforward method of calculating a constant Q transform designed for musical representations  Waterfall plots of these data make it possible to visualize information present in digitized musical waveform