Calculation of a constant Q spectral transform Judith

Calculation of a constant Q spectral transform Judith C. Brown Journal of the Acoustical Society of America, 1991 Jain-De, Lee

Outline � INTRODUCTION � CALCULATION � RESULTS � SUMMARY

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

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

INTRODUCTION � 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

INTRODUCTION � 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

INTRODUCTION � 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 32000 samples/s and the resolution is 31. 3 Hz(32000/1024=31. 25) The violin low end of the range is G 3(196 Hz) and the adjacent note is G#3(207. 65 Hz), the resolution is much greater than the frequency separation for two adjacent notes tuned

INTRODUCTION � The frequencies sampled by the discrete Fourier transform should be exponentially spaced � If we require quartertone spacing ◦ The variable resolution of at most ( 21/24 -1)= 0. 03 times the frequency ◦ A constant ratio of frequency to resolution f / δf = Q ◦ Here Q =f /0. 029 f= 34

CALCULATION � Quarter-tone spacing of the equal tempered scale , the frequency of the k th spectral component is fk = (21/24)k fmin Where f an upper frequency chosen to be below the Nyquist frequency fmin can be chosen to be the lowest frequency about which Information is desired � The resolution f / δf for the DFT, then the window size must varied

CALCULATION � For quarter-tone resolution Q = f / δf = f / 0. 029 f = 34 Where the quality factor Q is defined as f / δf bandwidth δf = f / Q Sampling rate S = 1/T � Calculate the length of the window in frequency fk N[k]= S / δfk = (S / fk)Q

CALCULATION � 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

CALCULATION

CALCULATION

RESULTS

RESULTS

RESULTS Constant Q transform of piano playing diatonic violin flute playing glissando scale playing from diatonic C 4 from (262 scale Hz) pizzicato to(880 Hz) C 5(523 from D 5 diatonic D 5(587 scale (587 Hz) to. C 4 with A 5 (262 vibrato Hz)from to. Hz) C 5 The G 3 (196 attack onto. D 5(587 G 5(784 Hz) is also visible (523 Hz)Hz) with increasing amplitude

SUMMARY � 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