Pitch vs. Frequency:

Size: px

Start display at page:

Download "Pitch vs. Frequency:"

Eunice Agatha Hodge
5 years ago
Views:

1 Pitch vs. Frequency: Pitch = human ear s perceptin f frequency f a sund vibratin Lw pitch lw frequency f vibratin/scillatin High pitch high frequency f vibratin/scillatin Q: Is the relatin between {perceived} pitch vs. frequency linear? e.g. a straight line y = mx + b relatin? A: N See figure belw: Define frequency units assciated with subjective pitch = mels analgus t Hz. - - Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

2 The Audible Frequency Range f Human Hearing (when yung): ( ) 0 Hz < f < 0 KHz 3 rders f magnitude As we grw lder, the ange f frequencies that we can hear decreases (bth high and lw frequencies mstly n the high frequency end ) Frequency ranges f musical instruments typically ~ 00 Hz t ~ few KHz e.g. guitar Lw E = 8 Hz High E = 330 Hz Pian highest nte is ~ 400 Hz Very little abve ~ 0 KHz (squeals & scrapes) The human ear needs t be able t perceive a sund fr minimum length f time t. In rder t determine a pitch i.e. pure/single-frequency tne the minimum duratin time t f the pure tne depends n its frequency: ( t ) min ( ) ( t ) ( ) Fr f ~ 00 Hz ~ 0 msec : t ~ 40 msec ~ 4 cycles Fr f ~ 000 Hz msec : t ~ 3 msec ~ 3 cycles min The minimum duratin time t fr human perceptin f a pitch is certainly in part due ur ear & brain prcessing, but fr lw frequencies especially, minimum time duratin is als due t the uncertainty principle f t, which tells us that a pure tne/single-frequency sine wave signal f finite duratin t in fact has a finite frequency spread f! Only as the time duratin t des f Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

This can be seen by taking the Furier transfrm f a finite-length {time duratin t } pure-tne/single frequency { f f } time dmain sinusidal signal p t p sin t t the frequency dmain p f : If p t psint

3 This can be seen by taking the Furier transfrm f a finite-length {time duratin t } pure-tne/single frequency { f f } time dmain sinusidal signal p t p sin t t the frequency dmain p f : If p t psint psin ft fr t t, and: p t 0 fr t t, and defining the {rectangular} windw functin w t 0fr t t t t, respectively, then: t t t t t t tt p f p t sin t dt p w t sin t sin t dt p sin t sin t dt Using the trignmetric identity sin Asin B csabcsab : t t sin t sin t p f p cs t cs t t t dt p t t t The sinc functin sinc sin x x x {n.b. sinc0 }, hence we can write p f as: sinc sinc p f pt t t The pwer spectral density functins S f p f (a frequency dmain pp quantity) fr infinite length and finite length sine-wave signals are shwn belw: Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

4 The sinc functin sinc t sin t t fr sinewave signals f {shrt} time duratin t,,3,4 where f and the crrespnding # f cycles f scillatin Nc t,, 3, 4 are shwn in the figure belw. Nte that the width f f the main peak (at f f ) depends inversely n the time duratin f the signal, due t the uncertainty principle f t. t Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

5 Human perceptin f pitch als depends {~ weakly} n the ludness f the sund. * Effect arises due t nn-linearities in the f & I respnse f the human ear. * Pitch (perceived f) changes as ludness increases see graph belw... * Effect exists nly fr pure/simple tnes (!!!) * Cmplex tnes shw n perceived pitch changes with ludness! (why??) Tw ears f same persn may NOT perceive sund f a given frequency as having the same pitch!!! = DIPLACUSIS happens nly fr diseased, and/r injured ears. Fr nrmal musical purpses, frequency and pitch are synnymus (usually) n.b. applies nly t peridic sunds. Sund pulses are made up f a cntinuum f frequencies, sund pulses are thus anharmnic and hence have n characteristic frequency and/r pitch Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

6 The human ear can discriminate changes in sund intensity levels/sund pressure levels/ludnesses f JND L L L ~/ db; Our ability t d s als depends n frequency and sund pressure level/ludness: A JND ~/ db change in sund intensity level crrespnds t a fractinal change in sund intensity f I /I ~ %. Thus, due t the ~ lgarithmic respnse f the human ear, it is nt terribly sensitive t changes in the ludness f sunds Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

7 The typical human ear can discern changes in pitch/frequency at the f ~ 3 Hz level in the frequency range ~ 30 Hz f 000 Hz. Again, has frequency dependence: Nte that: At very lw frequencies: D f f 3 30 = 0% ( semitnes), Whereas at higher frequencies: D f f = 0.3% ( 0. semitnes) A gd musician can discern frequency changes significantly smaller than this e.g. abve f 500 Hz: 0.03 semitne (i.e. D f f 000 = 0.% )!!! The human ear/brain is capable f detecting small changes in frequency!!! Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

The human ear/brain is capable f perceiving a fundamental even when n fundamental is actually present!!! This is the s-called missing fundamental effect.

in} a cnsequence f the nn-linear respnse in/inside the human ear itself, and/r a nn-linear respnse(s) in the human brain s prcessing

8 The human ear/brain is capable f perceiving a fundamental even when n fundamental is actually present!!! This is the s-called missing fundamental effect. This effect is {again} a cnsequence f the nn-linear respnse in/inside the human ear itself, and/r a nn-linear respnse(s) in the human brain s prcessing f frequency infrmatin whenever e.g. a quadratic nn-linear respnse exists (in any system), if tw signals A and B with frequencies fa and fb are input t that system, then sum and difference frequencies (fa+ fb) and fa fb are prduced! Thus, e.g. a nd harmnic f and a 3 rd harmnic 3f can prduce a missing fundamental frm the difference frequency, 3f f = f!!! Fr further details n distrtin, read Physics 406 Lecture Ntes n Thery f Distrtin I & II. Fr sme musical instruments e.g. the trumpet, the be and/r the bassn the nd (r even 3 rd and higher) harmnics can actually have a larger amplitude than that f the fundamental, hwever we perceive/hear the nte that is played n the trumpet (and/r be, bassn) as that f the fundamental!!! The harmnic spectra aka pwer spectral density functins. pp S f vs f and assciated {time-averaged} relative phase harmnic phasr plts are shwn belw e.g. fr the steadily-played ntes A4 (440.0 Hz) played n the be, and F (87.3 Hz) played n the bassn: Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

9 Nte that the vertical axes f Spp f vs. f are displayed n a lgarithmic scale Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

10 Legal Disclaimer and Cpyright Ntice: Legal Disclaimer: The authr specifically disclaims legal respnsibility fr any lss f prfit, r any cnsequential, incidental, and/r ther damages resulting frm the mis-use f infrmatin cntained in this dcument. The authr has made every effrt pssible t ensure that the infrmatin cntained in this dcument is factually and technically accurate and crrect. Cpyright Ntice: The cntents f this dcument are prtected under bth United States f America and Internatinal Cpyright Laws. N prtin f this dcument may be reprduced in any manner fr cmmercial use withut prir written permissin frm the authr f this dcument. The authr grants permissin fr the use f infrmatin cntained in this dcument fr private, nncmmercial purpses nly Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic. Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage