Math 5: Music and Sound FALL 2008: Final3 hours, 9 questions, 80 points totalTry to show working. Heed the p oints available for each question. Try the bonuses once the rest is ok.The last page has useful information. Good luck, have fun, and it was great to have you in the course!1. [9 points](a) What is the frequency of the pitch C8 (the highest note on the piano) in the equal-temperedsystem?(b) The second column of the touchtone keypad is encoded by frequency 1336 Hz. What musicalpitch is this nearest, and what is the error from this pitch in cents?(c) Construct the frequency ratio between C and the D above it in the Pythagorean tuned C major(diatonic) scale. (Briefly show working)(d) If you continued using this Pythagorean construction to compute all notes of the chromatic scale,what error in cents would occur when you eventually returned to (and compared against) yourstarting note?12. [10 points] An empty 3 liter soda bottle has a neck with radius 1 cm and effective length 3.06 cm.(a) Compute the frequency that sounds when someone (e.g. Mike Wu) blows across the bottle.(b) When the bottle opening is tapped, a pressure signal of the form Ae−t/τsin 2πft is produced. Ittakes 5 seconds for the intensity to drop by 60 dB. Compute the value of the decay time.(c) Someone now excites the bottle by sounding a pure tone. What range of frequencies would causeat least half the maximum response amplitude inside the bottle? (If you didn’t get parts a and/orb, give your answer in symb ols.) Sketch a response curve to illustrate this, labeling your axes.(d) How much water should be poured into the bottle to change the pitch by one octave? (does it goup or down?)23. [10 points] A speaker cabinet S and listener R are both 4 m from a wall, and 6 m apart, as shown inplan view below.(a) Sketch the direct and reflected sound paths from S to R and compute their travel times.6 m4 mSR(b) What is the lowest pure tone frequency emitted that would cause destructive interference of thesetwo paths, for this listener?(c) A second parallel wall is added 10 m to the left as shown below. Find two new travel times whichoccur. Show any geometric constructions used.6 m4 m6 mSR(d) [BONUS] If the speaker emits a loud but short click, what would you expect the tail (decayingpart) of the echo the listener hears to be like, and why? (diagrams help)34. [10 points] A child’s vocal tract behaves like a closed-open pipe 12 cm long.(a) Compute the lowest two formant frequencies assuming the tract is of uniform width.(b) Sketch graphs of a spectrum you might hear with this tract shape if the child were. . .singing with pitch 200 Hz:whispering:(be sure to label your axes and give some values on the horizontal axis)(c) The child now breathes harmless but dense sulphur hexaflouride which halves the speed of soundin their tract. Redo the new spectrum for the first case above. . .singing with pitch 200 Hz:Explain what has changed and what has not:(d) Returning to normal air, the child presses their tongue upwards to constrict the tract 1/3 of theway down from the mouth. What happens to their first two formants? (relative to part a)4(e) Relative to a uniform vocal tract, how could the child easily shape their tract to lower both F1and F2?5. [7 points](a) A musical note with pitch 200 Hz is accomanied by another, either a p er fect fifth above it, or aminor sixth above it. Compare the resulting dissonance for these two situations using Helmholtz’stheory (consider only below 1400 Hz)(b) What is the probable perceived pitch of a sound with partials at 432, 601, 900, 1199, 1435 Hz,and why?(c) [BONUS] How it can be that the piano sounds better when its octaves are tuned with ratio slightlygreater than 2:1 ?56. [9 points] Here you model the sound of the lowest string of the electric bass, which has length 0.8 m.(a) The string pro duces a note of E1 (41.25 Hz). What is the speed of transverse waves on this string?(b) The string is plucked 1/3 of the way up. Compute the excitation amplitudes of the first fourmodes of the string (are any zero?)(c) The amplified signal is from an electric pickup placed 1/4 of the way up. Use this and part b) tocompute (relative) amplitudes of the first four modes heard, and sketch the resulting spectrum.(d) A finger is now used to damp (lightly touch) the string 2/5 of the way up. What is the pitch ofthe sound produced? (note this is true regardless of where it is now plucked)67. [7 points] The following show actual spectrograms, with frequency (0 to 4000 Hz) vertical, and time(0 to 3 sec) horizontal. Describe pitch (has one?), decay, timbre, and use to state an instrument andmethod which could have produced it: (if stuck think about the spectra)a)b)c)Which of the above, if any, will be periodic signals (at least while they last), and why?78. [9 points] Short unrelated calculation problems.(a) What is the period of the signal sin(100πt + π/4) ? (as usual, t is time in seconds)(b) You are a stationary listener. How fast does a source of sound need to travel towards you so thatits pitch appears to change by an octave?(c) Compute the intensity in dB of an orchestra radiating 5 W acoustic power in all directions, at adistance of 100 m.(d) A small animal hears a pure tone of 680 Hz. The phase difference between the signal at its leftand right ears is then π/5 radians. From this find the smallest possible distance between its ears.(Ignore delays due to curvature of the head, i.e. assume straight-line travel of sound)89. [9 points] Explanation questions: points for correct and precise use of concepts. Diagrams can help.(a) Explain the difference between frequency and pitch.(b) Explain what a Fourier series is and what kind of signals it can and cannot represent.(c) Explain how a Tuvan throat singer produces a high-pitched ’whistle-like’ melody by singing.(d) Explain why the musical interval between a closed-open and open-op en pipe of the same lengthis not exactly an octave (is it bigger or smaller?)(e) [BONUS] In class we learned about two ways in which digital (as opposed to analog) soundrecording may change (distort) a signal in order to convert it to data. Describe one of these, andthe type of distortion produced.9Useful informationω = 2πfc = fλdB = 10 log10I10−12W/m2Q = πτTfobsf=11 − v/cor 1 + v/csin(a + b) = sin a cos b + cos a sin bsin a + sin b = 2 cos(a − b2) sin(a + b2)cstring=sTµfHelm=c2πraV lIntervals by number of semitones:1. minor second2. whole tone (major