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Chapter 13

Chapter 13. The Loudness of Single and Combined Sounds. Four Important Musical Properties. Pitch (Chapter 5) Tone Color (Chapter 7 and others) Duration (Chapter 10 and 11) Loudness. D. Atmospheric Pressure. L. More than Atm. Pressure. Piston Experiment. Clearly P  1/V

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Chapter 13

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  1. Chapter 13 The Loudness of Single and Combined Sounds

  2. Four Important Musical Properties • Pitch (Chapter 5) • Tone Color (Chapter 7 and others) • Duration (Chapter 10 and 11) • Loudness

  3. D Atmospheric Pressure L More than Atm. Pressure Piston Experiment • Clearly P 1/V • V = (¼pD2)L

  4. Same Experiment with Sound • At the threshold of hearing for 1000 Hz • D = 0.006 cm (human hair) • L = 0.01 cm • Change in volume of our “piston” of one part in 3.5 billion.

  5. 100X Threshold of Hearing Developing a Sense of Scale Tuning Fork at 9 in 10,000 X Threshold Messo-Forte 1,000,000X Threshold Threshold of Pain

  6. Energy and Intensity • Energy is the unifying principle • heat, chemical, kinetic, potential, mechanical (muscles), and acoustical, etc. • For vibrational processes, energy is proportional to amplitude squared, or E  A2 • On the receiving end intensity is proportional to energy, or I  E • I  A2

  7. Loudness • When the energy (intensity) of the sound increases by a factor of 10, the loudness increases by 1 bel • Named for A. G. Bell • One bel is a large unit and we use 1/10th bel, or decibels • When the energy (intensity) of the sound increases by a factor of 10, the loudness increases by 10 dB

  8. Decibel Scale • For intensities • b = 10 log(I/Io) • For energies • b = 10 log(E/Eo) • For amplitudes • b = 20 log(A/Ao)

  9. Threshold of Hearing • The Io or Eo or Ao refers to the intensity, energy, or amplitude of the sound wave for the threshold of hearing • Io = 10-12 W/m2 • Loudness levels always compared to threshold • Relative measure • SPL (Sound Pressure Level) • 2.833 10-10 atm. = 0.000283 dynes/cm2 • One part in 3.53 billion

  10. 160 Jet engine - close up 150 Snare drums played hard at 6 inches awayTrumpet peaks at 5 inches away 140 Rock singer screaming in microphone (lips on mic) 130 Pneumatic (jack) hammer Cymbal crash Planes on airport runway 120 Threshold of pain - Piccolo strongly played Fender guitar amplifier, full volume at 10 inches away Power tools 110 Subway (not the sandwich shop) 100 Flute in players right ear - Violin in players left ear Common Loud Sounds

  11. 90 Heavy truck traffic Chamber music 80 Typical home stereo listening levelAcoustic guitar, played with finger at 1 foot away Average factory 70 Busy street Small orchestra 60 Conversational speech at 1 foot away Average office noise 50 Quiet conversation 40 Quiet office 30 Quiet living room 20 10 Quiet recording studio 0 Threshold of hearing for healthy youths Common Quieter Sounds

  12. Loudness/Amplitude Ratios Loudness Amplitude (Decibels) Factor 13 4.467 14 5.012 15 5.623 16 6.310 17 7.079 18 7.943 19 8.913 20 10.000 40 100.000 80 10,000 120 1,000,000 Loudness Amplitude (Decibels) Factor 0 1.000 1 1.122 2 1.259 3 1.413 4 1.585 5 1.778 6 1.995 7 2.239 8 2.512 9 2.818 10 3.162 11 3.548 12 3.981

  13. Amplitude vs. Loudness

  14. Sound Level (at 1000 Hz) Amplitude Ratio Loudness Threshold of Hearing 1 0 dB Tuning Fork 100 40 dB Mezzo-Forte 10,000 80 dB Threshold of Pain 1,000,000 120 dB Quantifying the Sense of Scale

  15. Loudness Arithmetic • To get the loudness at, say 97 dB • Split into 80 + 17 • From table 80 dB is an amplitude ratio of 10,000 • 17 dB is an amplitude ratio of 7.079 • 97 dB corresponds to 7.079*10,000 = 70,790 amplitude ratio

  16. Adding loudspeakers • Doubling the amplitude of a single speaker gives an increased loudness of 6 dB (table) • Two speakers of the same loudness give an increase of 3 dB over a single speaker • For sources with pressure amplitudes of pa, pb, pc, etc. the net pressure amplitude is

  17. Example • Let pa = 5, pb = 2, and pc = 1 Only slightly greater than the one source at 5.

  18. Threshold of Hearing

  19. Hearing Response • Horizontal axis in octaves • Low frequency response is poor • The range of reasonable sensitivity is 250 - 6000 Hz • Young people tend to have the same shaped curve, but the overall levels may be raised (less sensitive) • The high frequency response is worse as we age • Curve for threshold of pain looks the same, 120 dB the threshold of hearing

  20. Perceived Loudness • One sone when a source at 1000 Hz produces an SPL of 40 dB • Sones are usually additive

  21. Response at constant SPL

  22. Observations • Broad peak (almost a level plateau) from 250 - 500 Hz • Dips a bit at 1000 Hz before rising dramatically at 3000 Hz • Drops quickly at high frequency • The perceived loudness of a tone at any frequency about doubles when the SPL is raised 10 dB

  23. Equalizer Settings

  24. Single and Multiple Sources Relative Amplitude for Curve A Number of Sources for Curve B

  25. Notes • Need to almost triple the amplitude of a single source before the perceived loudness reaches two sones • The four-sone level occurs for an amplitude increase of 10X • Curve B adds multiple one sone sources • Add by square root rule • Need 10 to double the loudness • One player who can vary loudness is more effective than fixed loudness players

  26. Building a Narrow BandNoise Source • Make a number a sinusoidal tones closely spaced in frequency. • The loudness is equal to that of a single sinusoidal source of the same SPL at the central frequency.

  27. Adding Two Narrow Band Noise Sources • We have two noise sources – one at 300 Hz the other at 1200 Hz or more, each at 13 sones • Since the frequencies are far apart, they add to give 26 sones • As frequencies move closer together… Df = 1 octave L = 24 sones Df = ½ octave L = 20 sones Df = 0 L = 16 sones

  28. Lower tone 300 Hz Lower tone 200 Hz Lower tone 100 Hz Adding Loudness atDifferent Frequency

  29. Notes • The plateau at small pitch separation is interesting • We process closely spaced pitches as though they are indistinguishable in perceived loudness • Called Critical Bandwidth – notice that it grows at low frequency Frequency Critical Bandwidth > 280 Hz 1/3 octave (major third) 180 - 280 2/3 octave (minor sixth) < 180Hz 1 octave

  30. 300 600 Fifth (half octave) - these combine to 19.5 sones 900 Perfect fourth (five semitones) - these combine to 19 sones 1200 Major third (four semitones) - these combine to 17 sones 1500 Adding a Harmonic Series • Consider the set of frequencies – each at 13 sones

  31. Upward Masking • The upper tone's loudness tends to be masked by the presence of the lower tone.

  32. Examples Frequency Apparent Loudness 1200 13 sones 1500 4 sones 17 sones 900 13 sones 1200 6 sones 19 sones 600 13 sones 900 6.5 sones 19.5 sones Notice that upward masking is greater at higher frequencies.

  33. Upward Masking Arithmetic • Rough formula for calculating the loudness of up to 8 harmonically related tones • Let S1, S2, S3, … stand for the loudness of the individual tones. The loudness of the total noise partials is…

  34. Example • For the five harmonically related noise partials – each with loudness 13 sones • 300 Hz (13 sones) • 600 Hz (0.75*13 sones = 9.75 sones) • 900 Hz (0.5* 13 sones = 6.5 sones) • 1200 Hz (0.5* 13 sones = 6.5 sones) • 1500 Hz (0.3*13 sones = 3.9 sones) • Stnp = 13 + 9.75 + 6.5 + 6.5 + 3.9 = 39.65 sones

  35. Closely Spaced Frequencies Produce Beats Open two instances of the Tone Generator on the Study Tools page. Set one at 440 Hz and the other at 442 Hz and start each.

  36. Notes on Beats • Beat Frequency = Difference between the individual frequencies = f2 - f1 • When the two are in phase the amplitude is momentarily doubled that of either component • gives an increase in loudness of 50% • Notice increase in loudness on Fig. 13.6 as pitch separation becomes small

  37. Beat Loudness

  38. Increase Pitch Separation • When the frequency difference reached 5 - 15 Hz, the beat frequency is too great to hear the individual beats, but we hear a rolling sound with loudness between 16 and 19.7 sones.

  39. Beats – Two Sources • One or the other component may dominate in certain parts of the room • Beats are more prominent than in the single earphone experiment • Some will be able to hear both tones and the beat frequency in the middle • Only the beat frequency is heard with earphone experiments

  40. Sinusoidal Addition • Masking (one tone reducing the amplitude of another) is greatly reduced in a room Stsp = S1 + S2 + S3 + …. • Total sinusoidal partials (tsp versus tnp)

  41. Frequency Perceived Loudness 200 8.5 sones 400 10 sones 630 8.5 sones Experimental Verification • Two signals (call them J and K) are adjusted to equal perceived loudness • Sound J is composed of three sinusoids at 200, 400, and 630 Hz, each having an SPL of 70 dB (see Fig 13.4) Stsp = 8.5 + 10 + 8.5 = 27 sones

  42. Sound K • Sound K is composed of three equal-strength noise partials, each having sinusoidal components spread over 1/3-octave • Central frequencies of 200, 400, and 630 Hz • Adjust K to be as loud as J • Measured loudness 75 dB • Again using Fig 13.4

  43. Central Frequency Perceived Loudness 200 12 sones 400 13.5 sones 630 13 sones Sound K (cont’d) • Stnp = 12 + (0.75*13.5) + (0.5*13) = 29 sones • Different formulas are needed for noise and sinusoidal waves

  44. Notes • Noise is more effective at upward masking in room listening conditions • Upward masking plays little role when sinusoidal components are played in a room • The presence of beats adds to the perceived loudness • Beats are also possible for components that vary in frequency by over 100 Hz.

  45. Saxophone Experiment • Note written G3 has fundamental at 174.6 Hz • Sound Q produced with regular mouthpiece • Sound R produced with a modified mouthpiece

  46. Different Mouthpieces

  47. Results • Original instrument showed strong harmonics out to about 4 and then falling rapidly • Modified mouthpiece shows a weakened first harmonic, very strong second, and then strong harmonics 5, and 6

  48. Harmonic Loudness Q R 1 17 12 2 19 22 3 9 11 4 3 6 5 2 7 6 2 5 7 2.0 3.5 8 0.3 3.0 9 0.0 2.5 Total 54.3 72.0 Perceived Loudness The new mouthpiece makes the sax 1.33 times as loud (72/54)

  49. Sound Level Meter

  50. Frequency (Hz) Reduction from Original - Type A Reduction from Original - Type B Reduction from Original - Type C 100 0.1 0.56 1.0 200 0.28 0.28 1.0 500 - 2000 1.0 1.0 1.0 5000 0.7 0.7 0.6 Design Specs • Mimics what our ears receive

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