Math 5: Music and Sound. Homework 2due Fri Apr 13 . . . but best if do relevant questions after each lectureYou will need to download the praat software (see our Software page) to do high-quality spectrograms.We will have an X-hour on this on Tuesday.For the listening questions I suggest you use headphones (at low volume) rather than the tiny laptopsp e akers. This will allow you to hear timbre much better.1. Let’s assume you hear beats only when two pure tone frequencies are 15 Hz apart or less. Compute thefrequency of the highest note that produces beats when played together with the note one semitoneabove it. What musical pitch is this neares t?12. In lecture 4 (4/4/ 07) I claimed that any function g(t) could be written as the sum of an even and oddsymmetric function. (This was needed for Fourier series). Prove this cla im by:(a) showing why e(t) :=12(g(t) + g(−t)) is always even for any g(t),(b) finding a similar formula for a function o(t) that is always odd,(c) showing why adding e(t) and o(t) gives g(t).(d) Tell me what e(t) and o(t) are for the function g(t) = (1 − t)2(don’t forget to expand a nd simplifyyour answer).3. Sketch using the mouse the following two periodic graphs into the online Falstad Fourier applet,reasonably accurately. For each, c omment on the harmonic content (strength o f high harmonics) a ndthe effect on the timbre you hear.What do you conclude about the harmonic content of discontinuous (jumpy) graphs vs smooth graphs?4. Here we study whether the phases of har monic content are important for timbre.(a) For review, compute amplitude C and phase φ such that C sin(ωt + φ) is equal to the sum3 sin(ωt) + 4 cos(ωt). [You don’t need to know ω to do this; it works for all frequencies].(b) Take a periodic signal such as sawtooth in the Falstad Fourier applet. Check Mag/Phase View toview ‘magnitudes’ (amplitude coefficients c1, c2, . . .) and phases (φ1, φ2, . . .). Adjust a bunch ofphases: does the waveform change? (not, a little, a lot?)1Maybe this explains why musically, semitone intervals are almost unheard-of at low pitches? They are more common(although not very) for high pitches1(c) Does the timbre change when phases are changed? (not, a little, a lot?) The best way to testthis is to switch off the sound while you change the phas e s, then switch it on and off to comparethe new timbre against the old. (If you don’t switch on and off, you’ll hear some overall volumechanges and may misinterpret this).5. Compute the ‘missing fundamental’ frequency that you would probably) hear if pure tones at thefollowing freq uencies were played together: 44 1 Hz, 588 Hz, 735 Hz, and 882 Hz. State the harmonicnumbers (i.e. 2nd, 3rd, etc) for each frequency. [Bonus: how does this relate to repetition frequencyof the combined signal?]6. Measure the first few partials pres e nt in the following two sounds (only consider strong partials, notrandom little bumps). The sounds are on the HW page; use audacity spectrum tool once you’vehighlighted a relevant part of the sound (read hints on the Software page). For each sound, answerthis: i) are the partials harmonically related? Zoom in o n that bit of signal so you can see theoscillations: ii) how close to p e riodic is it?(a) The first note the trumpet plays in ‘Mahler’s Fifth Symphony (trumpet opening)’,(b) Any part of ’Great Paul bell, St Paul’s Cathedral, London’.7. For each of these three spectrograms (made with praat), decide which fro m the list (i)-(v) it could be,and explain why. [If stuck: look at spectrograms of your own human-generated sounds for compariso n!]Time is horizontal, frequency vertical.a)b) c)Here a re the possibilities: (i) hiss, (ii) a single musical pitch which changes timbre, (iii) a rising musicalpitch, (iv) a falling musical pitch, (v) three different musical pitches played one after another.8. The FBI has given you an audio file (Telephone touch tones, on the HW page) recorded by a hiddenmicrophone. There is a lot of background noise. Using the information on touch tone codes from class,use a spectrogram (e.g. with praat) to identify the 10-digit telephone number. Then Google it to findthe lo c ation o f the crime!9. Basics of tuning systems.(a) If you started at C, climbed up by a perfect fifth (3:2 ) three times, what note of the diatonic scalewould you get to, and what musical interval does it form with the original C?(b) Co mpare the Pythagorean ratio for the above interval with the e qual-tempered equivalent, ex-pressing the error in cents.(c) Compute the ratio between the major third occurring in the Pythago rean scale and that occurringnaturally in the har monic ser ies (this ratio is the so-called Syntonic comma). Express it in cents.Where does the equal-tempered major third fall relative to these two other major