Applied Psychoacoustics Lecture 4: Pitch & Timbre Perception Jonas Braasch Homework 2: Raw Data Homework 2: Mean Homework 2: Mean+STD Mean and Standard Deviation mean Standard deviation Homework 2: Mean vs. Median Mean, blue Median, red Homework 2: Median Statistics for non-Gaussian distributions Median: is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half.probability distribution Quartiles: first quartile (designated Q1) = lower quartile = cuts off lowest 25% of data = 25th percentilepercentile second quartile (designated Q2) = median = cuts data set in half = 50th percentilemedianpercentile third quartile (designated Q3) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile percentile Homework 2: Median Homework 2: Median+Quartiles Homework 2: Mean vs. Median Mean, blue Median, red Homework 2: Median+Quartiles HW 2: Median+Quartiles+Range HW2: left ear vs. right ear left ear, blue right ear, red Euclid ( BC) Some sounds are higher pitched, being composed of more frequent and more numerous motions Contents Pitch perception Pure Tones Place and Rate Theory Complex Tones Timbre IntervalSemitonesFreq. Ratio Prime01:1 Minor second116:15 Major second29:8 Minor third36:5 Major third45:4 Perfect Fourth54:3 Augmented fourth Diminished fifth 645:32 64:45 Perfect Fifth73:2 Minor sixth88:5 Major sixth95:3 Minor seventh1016:9 Major seventh1115:8 Perfect Octave122:1 IntervalEqual temp.Just intonation Prime00 Minor second Major second Minor third Major third Perfect Fourth Augmented fourth Diminished fifth Perfect Fifth Minor sixth Major sixth Minor seventh Major seventh Perfect Octave1200 cents Equal Temperament One semitone equals: 12 2=1.0595=5.9463% One cent: 1200 2=1.0006=0.059% Perfect fifth: 1.5=700 cent=50% Perfect Octave: 2=1200 cent=100% Psychometric Function Describes the relationship between a physical parameter and its psychological correlate Example: Phon-Sone conversion Weber-Fechner Law The earliest scientific approach to measuring a psychometric function Ernst H. Weber ( ) investigated just noticeable differences (JNDs) for lifting weights with the hand. The subjects were blindfolded and the weight was gradually increased until they were able to detect a difference. He noticed that the JNDs were proportional to the overall weight. (e.g., if the JND for a 100 g weigth was 10 g, the JND for a 1000 g weight was 100 g). If the mass is doubled, the threshold is also doubled. Gustav T. Fechner ( ) later developed the Weber-Fechner Law from Webers findings: S=klog(I/I 0 ) With I the physical parameter (Intensity), S its psychophysical correlate, and k a constant, and I 0 the detection threshold of I. The JND is then: dS=klog(dI/I) Weber-Fechner Law Fechners indirect scales 0 sensation units (0 JND of sensation) stimulus intensity at absolute detection threshold 1 sensation unit (1 JND of sensation) stimulus intensity that is 1 difference threshold above absolute threshold 2 sensation units (2 JND of sensation) stimulus intensity that is 1 difference threshold above the 1-unit stimulus Fechners Law Pitch Pitch is often thought to be perceived logarithmically: Frequencyoctave 440 Hz1 st 880 Hz2 nd 1760 Hz3rd But for other psychophysical correlates, this logarithmic relationship does not hold true Stevens Power Law Stevens was able to provide a general formula to relate sensation magnitudes to stimulus intensity: S = aI m Here, the exponent m denotes to what extent the sensation is an expansive or compressive function of stimulus intensity. The purpose of the coefficient a is to adjust for the size of the unit of measurement. log S = m log(I-I0) + log a Examples for Stevens Power Law Examples for Stevens Power Law Exponents and now in the log-log space Definition of Pitch Pitch is that attribute of auditory sensation in terms of which sounds may be ordered on a scale extending from low to high. Pitch depends mainly on the frequency content of the sound stimulus, but it also depends on the sound pressure and the waveform of the stimulus. ANSI standard 1994 The mel scale Stevens, Volkmann & Newmann, 1937 Five listeners were asked to judge a the frequency of a second sinusoidal tone generator to be perceived half the Magnitude of the first oscillator with constant frequency (method of adjustment) Sound was switched between both oscillators (2-s interval) 60 dB SPL Stevens, Volkmann & Newmann, 1937 Mel Scale - Raw Data Stevens, Volkmann & Newmann, 1937 Geometric means for five observers, and average error for 2 listeners Def.: 1000 mels= 1000 Hz at 40 dB Stevens, Volkmann & Newmann, 1937 Solid line: mel scale /2.83; Black squares: integrated difference limens; open circles: relative location of the resonant positions on the basilar membrane Size of Musical interval in terms of Mels Stevens, Volkmann & Newmann, 1937 Hz/mel conversion To convert f hertz into m mel use: m = log e (1 + f / 700). And the inverse: f = 700(e m / 1). Frequency JNDs Different symbols show different studies (Fig.:Terhardt 1998) Frequency Difference Limens Wier et al., 1977 Frequency Difference Limens At low sound pressure levels (