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

Piston Experiment  Clearly P  1/V V = (¼  D 2 )L Atmospheric Pressure D More than Atm. Pressure L

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

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

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  A 2  On the receiving end intensity is proportional to energy, or I  E I  A 2

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 1/10 th bel, or decibels  When the energy (intensity) of the sound increases by a factor of 10, the loudness increases by 10 dB

Decibel Scale  For intensities  = 10 log(I/I o )  For energies  = 10 log(E/E o )  For amplitudes  = 20 log(A/A o )

Threshold of Hearing  The I o or E o or A o refers to the intensity, energy, or amplitude of the sound wave for the threshold of hearing I o = W/m 2 Loudness levels always compared to threshold Relative measure  SPL (Sound Pressure Level) atm. = dynes/cm 2 One part in 3.53 billion

Common Loud Sounds 160 Jet engine - close up 150 Snare drums played hard at 6 inches away Trumpet peaks at 5 inches away 140 Rock singer screaming in microphone (lips on mic) 130 Pneumatic (jack) hammer Cymbal crash Planes on airport runway120 Threshold of pain - Piccolo strongly played Fender guitar amplifier, full volume at 10 inches away Power tools110 Subway (not the sandwich shop) 100 Flute in players right ear - Violin in players left ear

Common Quieter Sounds 90 Heavy truck traffic Chamber music80 Typical home stereo listening level Acoustic guitar, played with finger at 1 foot away Average factory 70 Busy street Small orchestra 60Conversational speech at 1 foot away Average office noise50 Quiet conversation40 Quiet office30 Quiet living room20 10Quiet recording studio 0Threshold of hearing for healthy youths

Loudness/Amplitude Ratios LoudnessAmplitude (Decibels)Factor LoudnessAmplitude (Decibels)Factor , ,000,000

Amplitude vs. Loudness

Quantifying the Sense of Scale Sound Level (at 1000 Hz) Amplitude Ratio Loudness Threshold of Hearing 10 dB Tuning Fork10040 dB Mezzo-Forte10,00080 dB Threshold of Pain1,000, dB

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

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 p a, p b, p c, etc. the net pressure amplitude is

Example  Let p a = 5, p b = 2, and p c = 1 Only slightly greater than the one source at 5.

Threshold of Hearing

Hearing Response  Horizontal axis in octaves  Low frequency response is poor  The range of reasonable sensitivity is 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

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

Response at constant SPL

Observations  Broad peak (almost a level plateau) from 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

Equalizer Settings

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

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

Building a Narrow Band Noise 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.

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…  f = 1 octaveL = 24 sones  f = ½ octaveL = 20 sones  f = 0L = 16 sones

Adding Loudness at Different Frequency Lower tone 300 Hz Lower tone 200 Hz Lower tone 100 Hz

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 FrequencyCritical Bandwidth > 280 Hz1/3 octave (major third) /3 octave (minor sixth) < 180Hz1 octave

Adding a Harmonic Series  Consider the set of frequencies – each at 13 sones 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

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

Examples FrequencyApparent Loudness sones sones 17 sones sones sones 19 sones sones sones 19.5 sones Notice that upward masking is greater at higher frequencies.

Upward Masking Arithmetic  Rough formula for calculating the loudness of up to 8 harmonically related tones  Let S 1, S 2, S 3, … stand for the loudness of the individual tones. The loudness of the total noise partials is…

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)  S tnp = = sones

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.

Notes on Beats  Beat Frequency = Difference between the individual frequencies = f 2 - f 1  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 as pitch separation becomes small

Beat Loudness

Increase Pitch Separation  When the frequency difference reached 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.

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

Sinusoidal Addition  Masking (one tone reducing the amplitude of another) is greatly reduced in a room S tsp = S 1 + S 2 + S 3 + …. Total sinusoidal partials (tsp versus tnp)

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) FrequencyPerceived Loudness sones sones sones S tsp = = 27 sones

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

Sound K (cont’d)  S tnp = 12 + (0.75*13.5) + (0.5*13) = 29 sones  Different formulas are needed for noise and sinusoidal waves Central FrequencyPerceived Loudness sones sones sones

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.

Saxophone Experiment  Note written G 3 has fundamental at Hz Sound Q produced with regular mouthpiece Sound R produced with a modified mouthpiece

Different Mouthpieces

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

Perceived Loudness HarmonicLoudness QR Total The new mouthpiece makes the sax 1.33 times as loud (72/54)

Sound Level Meter

Design Specs  Mimics what our ears receive Frequency (Hz) Reduction from Original - Type A Reduction from Original - Type B Reduction from Original - Type C

Three Types  Type A Weights are chosen to model the ear response to an SPL of 40 dB  Type B Weights are chosen to model the ear response to an SPL of 70 dB  Type C Weights are chosen to model the ear response to an SPL of 100 dB

Meter Shortcomings  Cannot account for upward masking  Cannot account for beats  It measures dB, not sones (not necessarily one-to-one)

dB Compared to Sones