Review Section IV

Chapter 13 The Loudness of Single and Combined Sounds

Useful Relationships  Energy and Amplitude, E  A 2  Intensity and Energy, I  E  Energy and Amplitude, E  A 2

Decibels Defined  When the energy (intensity) of the sound increases by a factor of 10, the loudness increases by 10 dB  = 10 log(I/I o )  = 10 log(E/E o )  = 20 log(A/A o )  Loudness always compared to the threshold of hearing

Decibels and Amplitude

Single and Multiple Sources  Doubling the amplitude of a single speaker gives an increased loudness of 6 dB (see arrows on last graph)  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

Threshold of Hearing  Depends on frequency Require louder source at low and high frequencies

Perceived Loudness  One sone when a source at 1000 Hz produces an SPL of 40 dB  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

Adding Loudness at Different Frequency As the pitch separation grows less, the combined loudness grows less. Critcal Bandwidth Note critical bandwidth plateau for small pitch separation, growing for lower frequencies. The sudden upswing in loudness at very small pitch separation caused by beats.

Upward Masking  Tendency for the loudness of the upper tone to be decreased when played with a lower tone. 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 Multiple Tones  Let S 1, S 2, S 3, … stand for the loudness of the individual tones. The loudness of the total noise partials is…

Closely Spaced Frequencies Produce Beats Audible Beats

Notes on Beats  Beat Frequency = Difference between the individual fre- quencies = f 2 - f 1  When the two are in phase the amplitude is momentarily doubled that of either component

Adding Sinusoids  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)

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.

Chapter 14 The Acoustical Phenomena Governing the Musical Relationships of Pitch

Other Ways Of Producing And Using Beats  Introduce a strong, single frequency (say, 400 Hz) source and a much weaker, adjustable frequency sound (the search tone) into a single ear. Vary the search tone from 400 Hz up. We hear beats at multiples of 400 Hz.

A Variation in the Experiment  Produce search tones of equal amplitude but 180° out of phase. Search tone now completely cancels single tone. Result is silence at that harmonic Each harmonic is silenced in the same way. How loud does each harmonic need to be to get silence of all harmonics?

Waves Out of Phase Superposition of these waves produces zero.

Loudness Required for Complete Cancellation  400 Hz 95 SPL Source Frequency  800 Hz 75 SPL  1200 Hz 75 SPL  1600 Hz 75 SPL  Harmonics are 20 dB or 100 times fainter than source (10% as loud)

Start with a Fainter Source  400 Hz89 SPLSource – ½ loudness  800 Hz63 SPL¼ as loud as above  1200 Hz57 SPL 1 / 8 as loud as above  1600 Hz51 SPL 1 / 16 as loud as above

…And Still Fainter Source  400 Hz 75 SPLSource  800 Hz 55 SPL  1200 Hz35 SPLToo faint  1600 Hz15 SPLToo faint  This example is appropriate to music.  Where do the extra tones come from? They are not real but are produced in the ear/brain

Heterodyne Components  Consider two tones (call them P and Q) From above we see that the ear/brain will produce harmonics at (2P), (3P), (4P), etc. Other components will also appears as combinations of P and Q Original Components Simplest Heterodyne Components Next-Appearing Heterodyne Components P(2P)(3P) (P + Q), (P – Q) (2P + Q), (2P – Q) (2Q + P), (2Q – P) Q(2Q)(3Q)

Heterodyne Beats  Beats can occur between closely space heterodyne components, or between a main frequency and a heterodyne component.  See the vibrating clamped bar example in text.

Driven System Response Natural Frequency, f o 2 nd Harmonic is f o 3 rd Harmonic is f o

Other Systems  More than one driving source We get higher amplitudes anytime heterodyne components approach the natural frequency.  Non-linear systems Load vs. Deflection curve is curved Heterodyne components always exist

Harmonic and Almost Harmonic Series HHarmonic Series composed of integer multiples of the fundamental PPartial frequencies are close to being integer multiples of the fundamental Always produce heterodyne components The components tend to clump around the harmonic partials. May sound like an harmonic series but “unclear”

Frequency - Pitch  Frequency is a physical quantity  Pitch is a perceived quantity  Pitch may be affected by whether… the tone is a single sinusoid or a group of partials heterodyne components are present, or noise is a contributor

The Equal-Tempered Scale  Each octave is divided into 12 equal parts (semitones)  Since each octave is a doubling of the frequency, each semitone increases frequency by  Each semitone is further divided into 100 equal parts called Cents The cent size varies across the keyboard (1200 cents/octave)

Calculating Cents  The fact that one octave is equal to 1200 cents leads one to the power of 2 relationship:  Or,

Frequency Value of Cent Through the Keyboard Frequency Hz/cent

The Unison and Pitch Matching  Consider two tones made up of the following partials Harmonic1234 Tone J Tone K Beat Frequency2468  Adjust the tone K until we are close to a match

Notes on Pitch Matching  As tone K is adjusted to tone J, the beat frequency between the fundamentals becomes so slow that it can not easily be heard.  We now pay attention to the beats of the higher harmonics. Notice that a beat frequency of ¼ Hz in the fundamental is a beat frequency of 1 Hz in the fourth harmonic.

Add the Heterodyne Components  In the vicinity of the original partials, clumps of beats are heard, which tends to muddy the sound. Eight frequencies near 250 Hz Seven near 500 Hz Six near 750 Hz Five near 1000 Hz.

Results  A collection of beats may be heard. Here are the eight components near 250 Hz sounded together.

The Octave Relationship Tone P Tone Q  As the second tone is tuned to match the first, we get harmonics of tone P, separated by 200 Hz.  Only tone P is heard

The Musical Fifth  A musical fifth has two tones whose fundamentals have the ratio 3:2. Tone M Tone N  Now every third harmonic of M is close to a harmonic of N

Results  We get clusters of frequencies separated by 100 Hz.  When the two are in tune, we will have the partials…  This is very close to a harmonic series of 100 Hz  The heterodyne components will fill in the missing frequencies.  The ear will invariably hear a single 100 Hz tone (called the implied tone).

Chapter 15 Successive Tones: Reverberations, Melodic Relationships, and Musical Scales

Audibility Time UUse a stopwatch to measure how long the sound is audible after the source is cut off AAgrees well with reverberation time Time for a sound to decay to 1/1000 th original level or 60 dB IIt is constant, independent of frequency, and unaffected by background noise

Advantages of Audibility Time  Only simple equipment required  Many sound level meters can only measure a decay of dB, not the 60 dB required by the definition  Sound level meters assume uniform decay of the sound, which may not be the case

Successive Tones  We can set intervals easily for successive tones (even in dead rooms) so long as the tones are sounded close in time.  Setting intervals for pure sinusoids (no partials) is difficult if the loudness is small enough to avoid exciting room modes.  At high loudness levels there are enough harmonics generated in the room and ear to permit good interval setting.  Intervals set at low loudness with large gaps between the tones tend to be too wide in frequency.

The Beat-Free Chromatic (or Just) Scale ChromaticScales Interval NameInterval RatioFrequency (beat-free) C 261 E3rd5/4327 F4th4/3349 G5th3/2392 AMajor 6th5/3436 Coctave2/1523

Harmonically Related Steps CG D CE FAB Notice the B and D are not harmonically related to C

Intervals with B and D 5th CG D CE FAB 4th 5th 3rd

Filling in the Scale 3rd 4th Minor 6 G C D CE FAB Notice that C#, Eb, and Bb come into the scheme, but Ab/G# is another problem.

Finding F# 3rd min3 C D C EF AB G

Equal Temperament  An octave represents a doubling of the frequency and we recognize 12 intervals in the octave. The octave is the only harmonic interval.  Make the interval  Using equal intervals makes the cents division more meaningful  The following table uses

Complete Scale Comparison Interval Ratio to Tonic Just Scale Ratio to Tonic Equal Temperament Unison Minor Second25/24 = Major Second9/8 = Minor Third6/5 = Major Third5/4 = Fourth4/3 = Diminished Fifth45/32 = Fifth3/2 = Minor Sixth8/5 = Major Sixth5/3 = Minor Seventh9/5 = Major Seventh15/8 = Octave2.0000

Chapter 16 Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano

Notes on the Just Scale Major Scale The D corresponds to the upper D in the pair found in Chapter 15. Also, the tones here (except D and B) were the same found in the beat-free Chromatic scale in Chapter 15. Here we use the lower D from chapter 15 and the upper Ab. Minor Scale

Notes on the Equal-Tempered Scale  The fifth interval is close to the just fifth = whereas the just fifth is 1.5  Only fifths and octaves are used for tuning Perfect fifth is…  Three times the frequency of the tonic reduced by an octave – f 5th = 1.5 f o  3*f o = 2*f 5th  Equal-tempered fifth is reduce 2 cents from the perfect fifth

Tuning by Fifths  Recall that the tonic contains the perfect fifth as one of the partials We tune by listening for beats Ex. The equal-tempered G 4 is 392 Hz  *C 4 Use perfect fifth rule  3(261.63) – 2(392.00) = 0.89 Hz This difference would be zero for a perfect fifth We tune listening for a beat frequency of slightly less than 1 Hz

Just and Equal-Tempered IntervalJust Equal- Tempered Cent Diff. Tonic Major 2 nd Major 3 rd Major 4 th Major 5 th Major 6 th Major 7 th Octave Minor 3 rd Minor 6 th

Use of the Previous Table  It is there to compare the two scales, not to memorize. Know how to generate the frequencies in the table  Just frequencies come by multiplying by whole number ratios  Equal-tempered frequencies come by multiplying by a power of

Notes  Certain intervals sound smoother (or rougher) than others. Notice particularly the Major 3 rd, 6 th, and 7 th and the minor intervals

The Circle of Fifths

Pythagorean Comma  Start from C and tune perfect 5ths all the way around to B#. C and B# are not in tune.  A perfect 5th is 702 cents = 8424 cents  An octave is 1200 cents = 8400 cents  = 24 cents = Pythagorean Comma

Well-Tempered Tuning  Many attempts distribute the Pythagorean Comma problem around the circle of Fifths in different ways to make the problem less obvious Werckmeister III – shrunken fifths  The interval is found by dividing the Pythagorean Comma into four equal parts (23.46/4 = 5.865). So instead of the perfect fifths being 702 cents, they are cents.

Werckmeister Circle of Fifths  Numbers in the intervals refer to differences from the perfect interval. The ¼ refers to ¼ of the Pythagorean Comma. Result is that transposing yields different moods

Physics of Vibrating Strings  Flexible Strings Frequencies of the harmonics depend directly on the tension and inversely on the length, density, and thickness of the string. f n = nf 1  Hinged Bars Frequencies of the harmonics depend directly on thickness of the bar and inversely on the length and density. f n = n 2 f 1

Real Strings  We need to combine the string and bar dependencies

Physics of Vibrating Strings The Termination Strings act more like clamped connections to the end points rather than hinged connections. The clamp has the effect of shortening the string length to L c. The effect of the termination is small.

Physics of Vibrating Strings The Bridge and Sounding Board We use a model where the string is firmly anchored at one end and can move freely on a vertical rod at the other end between springs F S is the string natural frequency F M is the natural frequency of the block and spring to which the string is connected. The string + mass acts as a simple string would that is elongated by a length C. The slightly longer length of the string gives a slightly lower frequency compared to what we would have gotten if the string were firmly anchored.

 Piano and harpsichord tuning is not marked by beat-free relationships, but rather minimum roughness relationships.  The intervals not longer are simple numerical values.  Larger sounding boards have overlapping resonances, which tend to dilute the irregularities. Thus grand pianos have a better harmonic sequence than studio pianos.