1. Applied Psychoacoustics Lecture 4: Loudness Jonas Braasch

2. Definition Loudness Loudness is the quality of a sound that is the primary psychological correlate of physical intensity. Loudness is also affected by parameters other than intensity, including: frequency bandwidth and duration.

3. How can we measure loudness We can present a sinusoidal tone to our subject and ask them to report on the loudness.

4. Scales of Measurement nominalordinalintervalratio Non-numeric scaleScale with greater than, equal and less than attributes, but indeterminate intervals between adjacent scale values equal intervals between adjacent scale values, but no rationale zero point Scale has a rationale zero point e.g., color, gendere.g., rank order of horse race finalist e.g., the difference between 1 and 2 is equal to the difference of 101 and 102 e.g., Temperature in Fahrenheit e.g., the ratio of 4 to 8 is equal to the ratio of 8 to 16. e.g., Temperature in Kelvin.

5. Measuring loudness in phon At 1 kHz, the loudness in phon equals the sound pressure level in dB SPL. At all other frequencies the loudness the corresponding dB SPL value is determined by adjusting the level until the loudness is equally high to the reference value at 1kHz (so-called equal loudness curves or Fletcher-Munson curves). Loudness measured in Phon is based on an ordinal scale

6. Typical goal of our measurement Determine the relationship between a physical scale (e.g., sound intensity) and the psychophysical correlate (e.g., loudness)

7. Measurement Methods method of adjustment or constant response –The subject is asked to adjust the stimulus to a fulfill certain task (e.g., adjust the stimulus to be twice as loud). method of constant stimuli –Report on given stimulus (e.g., to what extent it is louder or less loud then the previous stimulus).

8. Equal loudness curve Detection threshold

10. Loudness in Sone Stevens proposed in 1936 to measure loudness on a ratio scale in a unit he called Sone. He defined 1 Sone to be 40 phons. The rest can be derived from measurements. Do you have an idea how he could have done it?

11. 1. Measure the Sone scale at 1 kHz e.g., by asking the subject to adjust the level of the sound stimulus (1-kHz sinusoidal tone) such that it is twice, 4 times … as loud as the Reference stimulus at 1 Sone (40 phons).

12. Relationship between Sones and Phons At 1-kHz this is also the psychometric function between Sone and dB SPL

13. 2. Measure the Sone scale at all other frequencies The easiest way is to follow the equal loudness contours. If they are labeled with the correct Sone value at 1 kHz, they are still valid.

14. 2 Sone 1 Sone 4 Sone 8 Sone 16 Sone 32 Sone 64 Sone Equal loudness curve

15. Psychometric Function

16. 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.

17. Examples for Steven’s Power Law

18. Examples for Steven’s Power Law Exponents

19. … and now in the log-log space

20. An introduction to Signal Detection Theory Let us come back to our initial example of determining the Absolute Threshold of Hearing, but this time we choose a constant stimulus approach. How can we do it?

21. Measuring the ATH one more time For example, we can present the stimulus at different levels ask the subject each time whether he or she perceived an auditory event or not. We might naively assume that our psychometric function will look like this:

22. Measuring the ATH one more time Sound pressure level Number of correct responses [%] Hearing threshold 100% 0% … would be nice, but in reality

23. … it looks like this Log-normalized stimulus intensity (e.g, sound pressure level) Probability for correct response 50 % threshold 75 % threshold In signal detection theory, we explain this variation with internal noise in the central nervous system

24. Definition of the correct response hitmiss False alarmCorrect reject Positive response Negative response Stimulus present Stimulus not present Sometimes it is better to rather accept a false alarm (e.g., fire detector) while other times it is better to accept a miss (e.g., non-emergency surgery cases)

25. Just noticeable differences (JNDs) Is the smallest value of a stimulus variation (e.g., sound intensity) that we are able to detect. A classic example is the work of Fechner measuring the JNDs for lifting different weights

26. Weber-Fechner’s Law In one of his classic experiments, Weber gradually increased the weight that a blindfolded man was holding and asked him to respond when he first felt the increase. Weber found that the response was proportional to a relative increase in the weight. That is to say, if the weight is 1 kg, an increase of a few grams will not be noticed. Rather, when the mass is increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold is also doubled. This kind of relationship can be described by a differential equation as, – where dp is the differential change in perception, dS is the differential increase in the stimulus and S is the stimulus at the instant. A constant factor k is to be determined experimentally. Integrating the above equation gives – where C is the constant of integration, ln is the natural logarithm.constant of integrationnatural logarithm To determine C, put p = 0, i.e. no perception; then – where S0 is that threshold of stimulus below which it is not perceived at all. Therefore, our equation becomes – The relationship between stimulus and perception is logarithmiclogarithmic

27. Fechner’s 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

28. Fechner’s Law

29. Determination whether difference is perceivable Internal response to stimulus #1, e.g., perceived loudness Internal response to stimulus #2

32. Receiver Operating Characteristic (ROC)

33. Discriminability index: d’=m s -m m /(   ), if  s =  m

36. Excitation pattern for 1-kHz sinusoid from Delgutte An excitation pattern for a 1-kHz Sinusoid with a level of 70-dB SPL. The data was calculated from the response of several single neurons. For each neuron, the SPL was recorded that was needed to receive the same discharge rate compared to the discharge rate for the reference condition (1-kHz, 70 dB SPL). The thick line plots the resulting level of the CF tone for many neurons. Of course, for neurons with a CF of 1 kHz, the level has to be 70 dB, because here the CF is identical to the reference frequency. The further the CF is apart from the reference frequency, the lower the SPL that is needed to excite the neuron with similar discharge rate, because the neuron becomes less sensitive to excitation at 1 kHz. The thin curve describes the threshold SPL for many neurons at their CF (Bump at 3 kHz not clear).

40. Zwicker loudness model N=  N’ m m=1 24 Overall loudness

41. dB (A), roughly 35 phons dB (B) dB (C) Equal Loudness Contours

42. Frequency weighting for dBA and dBC from: Salter, Acoustics, 1998

44. from: W. J. Cavanaugh, Acoustics-General Principles, 1988 Typical Exterior Sound Sources

45. from: W. J. Cavanaugh, Acoustics-General Principles, 1988 Different Weighting Schemes

46. Relative Subjective Changes Cavanaugh & Wilkens, Architectural Acoustics, 1998