Problems

7.6 Problems

Section 7.1

Section 7.2

Metal bars.
Let's say that you have a metal bar that is 19.5 cm long and vibrates at A = 440 Hz.
a) How much would you have to cut off the bar to raise the pitch by a Just whole step (9/8)?
b) Before I cut off the full amount found in a), I first cut off just 0.75 mm (= 0.075 cm). What is the new frequency and could I hear the change in pitch?

Intervals.
If you keep going up by perfect fifths, you never reach a perfect octave. This is the basic problem with all methods of tuning. We will look at the minor third, in the same way.
a) The Pythagorean minor third corresponds to 32/27. How many Pythagorean minor thirds are in an octave? Can you tell the difference between the octave formed by this interval and a perfect octave?
b) Repeat a) for a Just minor third, 6/5.
c) Repeat a) for an Equal Temper minor third, 3 half-steps.

Smallest possible interval in a scale.
a) Using the Pythagorean system for generating notes on a scale, one could add notes endlessly. However, at some point you would no longer to hear the difference between the notes. How many notes could you just distinguish in a half-step? (Or, how many just-noticeable-differences make up a half step?)
b) What would be the total number of notes in one octave in this system?

Just-noticeable-difference for the major third.
a) Start with a pitch of A = 440 Hz. In the Pythagorean scale, a major third is two whole steps. What is the frequency of a Pythagorean major third above A = 440 Hz?
b) In an equal temper scale what is the frequency of a major third above A = 440 Hz?
c) In the Just scale what is the frequency of a major third above A = 440 Hz?
d) Can you hear the difference between these pitches?

Just-noticeable-difference for the perfect fifth.
Repeat the previous problem for an interval of a perfect fifth.

Doppler shift.
a) You are running down a football field at 5 meter/second towards a band playing a note at 500 Hz. What note would you hear?
b) At 500 Hz, what would be the smallest change in frequency that would be necessary for you to hear a difference? (I.e. the just-noticeable-difference). Could you hear the change in frequency found in part a)?

Doppler shift in speakers.
I read about an effect in speakers which seems peculiar: the claim was that if a speaker is sounding loudly at 100 Hz, it is moving back and forth with a high velocity. At the same time, if the speaker is also sounding a note at 1000 Hz, the higher sound can be Doppler shifted by the motion at the lower frequency, distorting the pitch. We want to estimate this effect.
a) Assume the speaker is vibrating at 100 Hz with an amplitude of 1 cm. This is a very large amplitude. What would be the maximum speed of the surface of the speaker?
b) Now consider a tone at 1000 Hz. What would be the Doppler shifted frequency of this note using the speed from a) and the moving source Doppler shift?
c) You can detect a frequency change of at most 1/200 of the original frequency. Could you hear the frequency shift from the Doppler effect?

Can you run fast enough to hear a Doppler shift?
a) A very good sprinter can run at 10 m/sec. If you run at this speed away from an alarm sounding a frequency of 1200 Hz, what pitch would you hear?
b) Can you detect this change in pitch? What is the Just-Noticeable-Difference?
c) Would the shift in frequency be greater than a half-step?

Doppler shift/sailboats.
You are in a sailboat sailing down wind. Because the boat is traveling with the wind you feel no breeze even though the wind is blowing at 20 miles/hour with respect to the ground. You are moving directly away from the shore where your friend is playing a note on a tuba at 120 Hz. (Note: this problem is about sound waves, not water waves!)
a) With respect to the medium, who is the source, who is the observer and who is moving?
b) What frequency of sound does the observer hear?
c) Could you hear the change in pitch due to the Doppler shift?

Critical bandwidth
The Critical Bandwidth of the ear is about 19% of the frequency of the tone heard.
a) If you start with a frequency of 256 Hz, how much must a second tone be so that the critical bandwidths of the two tones do not overlap?
b) What is the nearest interval to these two tones?

"Consonant" scale.
a) Suppose you wanted to construct a scale where no two notes ever had overlapping critical bands so that you could play any two notes without producing dissonant intervals. How many notes would be in the scale?
b) The scale in the previous question works for a xylophone but not for a regular instrument like a violin. Why would a any two notes of the "consonant" scale sound consonant with a xylophone but not a violin?

Hearing perception.
The critical bandwidth of the ear is 0.190 times the frequency, except that it is never less than 90 Hz.
a) Start at A = 440 Hz. What would your ear hear if you play this and the note one half-step higher?
b) What would your ear hear if you play this A and the note two half-steps higher?
c) What would your ear hear if you play this A and the note four half-steps higher?
d) Now start at A = 110 Hz. What would your ear hear if you play this and the note one half-step higher?
e) What would your ear hear if you play this lower A and the note two half-steps higher?
f) What would your ear hear if you play this lower A and the note four half-steps higher?

2-D Cavity as a musical instrument.
Heres what would happen if we made a musical instrument out of a 2-D air cavity instead of the normal 1-D cavity.
a) Fill in the following table for the overtones of a rectangular cavity. You are given the fundamental in each direction. Find the simple overtones and the combination tones.

	f₁ = 500 Hz	f₂ =
g₁ = 540 Hz	(f₁,g₁) =	(f₂,g₁) =
g₂ =	(f₁,g₂) =	(f₂,g₂) =

b) Which pairs of frequencies are within a critical bandwidth and what are their beat frequencies?
c) How would all of these frequencies sound if played together?

2-D Cavity as a musical instrument.
Heres what would happen if we made a musical instrument out of a 2-D air cavity instead of the normal 1-D cavity. The dimensions of the cavity are 30 cm by 40 cm.
a) Fill in the following table for the fundamental in each direction and all of the overtones (simple and combination) of the rectangular cavity.

	f₁ =	f₂ =
g₁ =	(f₁,g₁) =	(f₂,g₁) =
g₂ =	(f₁,g₂) =	(f₂,g₂) =

b) Which pairs of frequencies are within a critical bandwidth and what are their beat frequencies?
c) How would all of these frequencies sound if played together?

Siren.
A police car with a siren at 800 Hz is coming at you at 25 miles/hour. However, the sound from the siren also reflects off of a building behind the car and reaches you. You are not moving.

a) What frequency do you hear from the siren directly?
b) What frequency do you hear from the reflected sound?
c) What is the beat frequency between these two frequencies and how would they pair sound to you?

Hearing.
Assume that the ear responds in the following way:

a) Roughly, what is the center frequency and what is the critical bandwidth?
b) What is the just-noticeable-difference (jnd) for this center frequency?
c) Sketch a response curved shifted to higher frequency by about two half steps. Do the critical bandwidths of the two curves overlap?

Hearing.
Assume that the ear responds in the following way:

a) From the graph, what is the center frequency and, roughly, what is the critical bandwidth?
b) Is this critical bandwidth consistent with our 19% definition? Why or why not?
c) Sketch a response curved shifted to a frequency lower by two half steps. Do the critical bandwidths of the two curves overlap?

Section 7.3

Quarter tones.
a) Arabic music uses scales with many more notes per octave than Western music. Thus, the scale has quarter-steps. This interval corresponds to a ratio of 1.0293. Show that two quarter-steps equals a half step.
b) If you play a note at 500 Hz could you distinguish this pitch from a note one quarter-step higher?
c) What would you hear if you played these two notes together?

Hearing perception and the Arabic scale.
a) Arabic music uses scales with many more notes per octave than Western music. Thus, the scale has quarter-steps. This interval corresponds to a ratio of 1.0293. Show that two quarter-steps equals a half step.
b) If you play a pure tone at 300 Hz could you distinguish this pitch from a note one quarter-step higher? How would these two notes sound if you played them together?
c) If you play a pure tone at 800 Hz could you distinguish this pitch from a note one quarter-step higher? How would these two notes sound if you played them together?

Diminished fifth.
The interval of a diminished fifth is Ö 2 and there are exactly two diminished fifths in an octave.
a) Show why this is.
b) Determine the amount of dissonance between two notes a diminished fifth apart.

Beats and beat frequencies.
a) Middle A on the piano has a frequency of 440 Hz. The lowest note on the piano is an A four octaves lower. What is the frequency of this note?
b) What is the frequency of the note one half step higher than the lowest A? What is the beat frequency between these two notes?
c) How would the ear perceive this beat frequency?

Section 7.4

Critical bandwidths and JNDs within a complex note
Start with a note at 525 Hz. Assume that the overtone series of the note is a complete harmonic series.
a) At what harmonic do adjacent harmonics lie within a critical bandwidth?
b) At what harmonic do adjacent harmonics lie within a Just-Noticeable-Difference? Can you hear that high a pitch.

Tuning pianos/beats.
This problem comes from a discussion I had with my piano tuner as he tunes my piano and describes a method for precisely tuning octaves.
a) Start with A = 220 Hz. What is the frequency of the note (F) a major third lower, according to the Just scale?
b) What is the frequency of the first common overtone of these two notes?
c) What is the frequency of the note a major third (four half-steps) lower than A = 220 Hz, according to the Equal Temper scale?
d) What is the beat frequency of between the common overtones found in part b)? (By common overtone I mean those that are with 10 Hz of each other.)
e) Finally, what is the first common overtone between the note found in part c) and A = 440 Hz and what is the beat frequency?

By comparing the results from d) and e) you can determine if the A = 220 Hz and the A = 440 Hz form a perfect octave.

Using beats for precise tuning.
The differences between the Equal Temper and Pythagorean scale are just on the threshold of human perception. So, they would be hard to tune precisely. However, using beats, these intervals can be tuned precisely, as you will see in this problem.
a) Start at F = 171 Hz. Find the frequency for the note "A", which is a major third higher, in each tunings:
Pythagorean =
Equal Temper =
b) What is the Just Noticeable Difference at this frequency? Can you hear the difference between these?
c) The fifth harmonic of the F and the fourth harmonic of the A are very close. What are the beat frequencies between these harmonics in the two different scales? By listening for this beat frequency you can tune to one scale or the other.

Consonance/dissonance of a metal bar.
The overtones of a metal bar are at 2.76f and 5.40f.
a) What are the overtones of a bar with a fundamental frequency of 220 Hz?
b) What is the fundamental frequency and overtones of a bar tuned on octave higher?
c) Will these bars sound consonant or dissonant when struck at the same time?

Overtone series
A peculiar instrument has an overtone series given by:

f, where f is the fundamental and n = 0, 1, 2, 3 (Note: 2⁰ = 1).
a) If the fundamental frequency of a note is 525 Hz, what are the next 3 overtones?
b) Now find the pitch and overtones of a note a Perfect Fifth higher.
c) If I play these two notes together, how will they sound? Justify your answer: what are the beat frequencies and how would they sound to you.

Pan Flutes.
One class project was a pan flute made out of open-open tubes and it was claimed that the end correction is not important. Assume two tubes are cut so that they make an interval of an octave. The longer tube has a length of 38.98 cm and the shorter one has a length of 19.49 cm. The radius of the tubes is 1 cm.
a) Ignoring the end correction, what are the fundamental frequencies of the two tubes?
b) How would these notes sound (consonant, dissonant, beats, etc.) if they were played together? (Remember, the tubes will produce complex tones.)
c) What are the actual frequencies of the two tubes, if you take into account the end correction? (Remember, it is an open-open tube).
d) How would the actual notes found in part c) sound if they were played together? Is the end correction important?

Eight Note Scale.
One project based a xylophone on a scale made up of 8 equal "half steps". The ratio for this "half step" is then 1.0905.
a) Consider an interval made up of three of these "half-steps". What is the ratio for this interval? What would be the Just version of this interval? (I.e. what ratio of integers less than 10 comes closest to this value?)
b) Fill in the following table. On the second line, enter the harmonics of 220 Hz. On the third line, enter the fundamental of a note three "half-steps" higher in this scale, and its harmonics. Don't go past 1500 Hz.

Fundamental	2f	3f	4f	5f	6f
220 Hz

c) Does this interval sound consonant or dissonant? Indicate which pairs of overtones, if any, lie within a Critical Bandwidth and state how the ear would perceive these pairs.

Perfect fifth.
a) Consider a note with a fundamental of 440 Hz (middle A). What is the frequency of the note which is an interval of a perfect fifth higher?
b) What is the first common harmonic of these two notes?
c) An augmented fifth is one half step higher than a perfect fifth. What is the frequency of an augmented fifth above 440 Hz?
d) What is the beat frequency between the same harmonics found in part b) of the fundamental and the augmented fifth? How would the ear perceive this beat frequency?

Perfect fourth.
a) Take a note with a fundamental frequency of 420 Hz. What note is an interval of a fourth higher?
b) What are the frequencies of the overtones that these notes have in common? (Assume that the overtone series of each note is a complete harmonic series.)
c) If the second note is out of tune by being 3 Hz too high, what is the beat frequency between the first common overtones?

Major third.
a) Start with a pitch of 500 Hz. Find the frequency of the pitch a major third higher according to the Just scale. Now calculate all of the harmonics of each note up to 2600 Hz. Compare the frequencies of the two harmonic series and find the closest intervals. Are any less than a critical bandwidth (1.190)?
b) Repeat a) for a major third according to the Pythagorean scale. Which interval would you consider to be more consonant?

Major sixth.
a) The Just major sixth is a factor of 5/3. What is the frequency of a Just major sixth above "A" = 440 Hz?
b) The equal temper major sixth is 9 half-steps. What is the frequency of a equal temper major sixth above 440 Hz?
c) Can you distinguish these two frequencies (i.e. what is the Just-Noticeable-Difference)?
d) What is the first common harmonic between the "A" and the Just major sixth?
e) Considering just this harmonic, how would the equal temper major sixth sound when played with the "A"?

Augmented sixth.
The Pythagorean augmented sixth is formed by going up by Perfect Fifths 10 times and coming down by octaves. This ends up being a ratio of 59,049/32,768!
a) Obviously, some people did not like this definition. The Just augmented sixth is formed by a ratio of two integers, both less than 10, that comes the closest to the Pythagorean definition. What ratio defines the Just augmented sixth?
b) Fill in the following table. On the second line, enter the harmonics of 220 Hz. On the third line, enter the fundamental of a note an augmented sixth higher and enter its harmonics. Don't go past 2000 Hz.

Fundamental	2f	3f	4f	5f	6f	7f	8f	9f
220 Hz

c) Does the interval of the augmented sixth sound consonant or dissonant? Indicate which pairs of overtones, if any, lie within a Critical Bandwidth and state how the ear would perceive these pairs.

Diminished fifth.
a) If you go up by a fifth and down by a half-step you get a diminished fifth. What is a diminished fifth above 220 Hz?
b) Find the first three harmonics, including the fundamental, of 220 Hz and the first three harmonics of the diminished fifth.
c) What pair of overtones (one from each note) are the closest together and how would you perceive these two frequencies?

Pitch perception and the "Missing Fundamental".
The following table gives several examples of which harmonics are present for a variety of (imaginary!) instruments. In each case, determine what pitch you would hear if the fundamental frequency is f = 350 Hz:

	Pitch heard	1f	2f	3f	4f	5f	6f	7f	8f
a)		X	X		X	X		X
b)					X		X		X
c)			X	X	X		X		X
d)				X			X
e)				X		X		X

Missing fundamental
When you play the second harmonic on a violin the pitch seems to go up by an octave. The concept of the missing fundamental would seem to say that you should still hear the fundamental as the pitch of the note. Why does the pitch go up and why does not the missing fundamental apply here? What would be the difference for open-closed instruments?

Consonance and dissonance between different instruments.
It is possible that intervals sound more or less consonant depending on which instruments are playing the notes in the interval. Consider an equal temper major third.
a) Find all of the conflicting harmonics if violins play each of the two notes.
b) Now assume that a clarinet plays the lower note and a violin plays the upper note. How does this situation compare to part a)?
c) Now the clarinet plays the upper note and the violin plays the lower note. Is this more or less consonant than the interval in part b)?
d) Finally, assume clarinets play both notes. Are there any conflicting harmonics?

Complex tones: violins and clarinets.
A violin has a full harmonic series, while a clarinet has a partial harmonics series (1f, 3f, 5f, ).
a) For a violin, fill in the following table for the first 6 harmonics of E = 660 Hz and the harmonics of the note a Perfect Fifth higher. Which harmonics do these notes have in common?
b) If the second note were played 10 Hz too low, fill in the last line in the table. Would the interval sound consonant or dissonant?
c) If a clarinet played the note is part b) while a violin still played the first note, would it sound consonant or dissonant?

	f	2f	3f	4f	5f	6f
First note	600 Hz
Perfect Fifth
Out of tune

Section 7.5

Difference tones
If you play a note and a major third above it, what is the difference tone and what interval is it below the original note?

Difference tones in a major triad
A major triad consists of a primary note (called the tonic), a second note a Just Major third higer and a third note a Perfect Fifth above the primary. Show that all of the difference tones between the fundamentals of each note occur at one or more octaves below the tonic. In other words, all of the difference tones in a major triad reinforce the tonic note.