Math 5: Music and Sound. Homework 7due Fri May 18 . . . but best if do relevant questions after each lectureFor fo rmant analyses you will want to use praat with spectrogram settings 0-4000 Hz range, windowlength 0.01 s (this is short to blur the harmonics of the spoken pitch together but leave the overall vocaltract response shape), dynamic range 60 dB. You should also experiment with Show formants.1. Download the file noisebottle.wav which is recorded from a microphone being lowered into a bottle,while a stereo system emitted a constant random background noise at all frequencies.(a) Looking at the spectrogram, describe the change in the frequency spectrum. Why does thishapp e n? What is responsible for the strong est peak that appears at low frequency?(b) We defined the width ∆f of a re sonance peak as the size of the frequency interval where theresponse a mplitude is at least half its pea k. What dB change does dropping to half the amplitudecorrespond to? (Review)(c) Use this to estimate the Q factor, and therefore the decay time τ, fo r this low frequency resonantmode. (Bring up a spectrum graph for the entire in-bottle part. Ignore the other nearby noisypeaks and use the width o f the central one)2. Consider a closed-open pipe of length 0.17 m, a model for the vocal tract. Assume initially the pipehas constant width and there’s no end corr ection.(a) Compute the for mant frequencies F1 and F2.(b) If the mouth is opened, what happens to F1? F2?(c) If the pharynx is constricted about 6cm up from the vocal cords, what happens to F1 (note it’snearer an antinode)? F2?3. Download the file eeaa both.wav, which I made with my own vocal instrument.(a) Measure the pitch (fundamental frequency) a nd frequencies of formants F1 and F2 fo r the twosp oken vowels.(b) For the same vowels whispered, is there a pitch? Are there formants? If so do they have similarfrequencies to the sung case? (give only a qualitative discussion)4. Record your own voice speaking three vowels ‘ee’, ‘aa’, and ‘oo’1(a) Print out a spectrogram of your recording, label F1 and F2 for each vowel, and list their frequen-cies.(b) Now draw a new pair of axes: horizontally F1 (on a range 0-1000 Hz), vertically F2 (a range0-3000 Hz), and plot the points giving the first two formants of each vowel (each vowel gives onepoint).(c) Apply Convert - Change gender..., choosing Formant shift ratio to be 1.3 if you’re male or0.7 if you’re female . . . or subvert the concept of gender entirely and become a chipmunk. Examinethe new spectrogram: Did the pitch change? What happened to the formants?1Hold each for at least a second, and if you want, rather than changing suddenly, slide between them to watch formantschange gradually. You will want to put the mic close to your mouth but not get crackle or noise from air blowing on it;experiment until you get a clean signal.1If you produce something entertaining, upload it.5. Read the links on overtone singing and Tuvan throat singing on the website.(a) How far up the harmonic series (approximately what n) do you need to go so that adjacentharmonics are separated by a whole tone? By a semitone? Since melodies usually involve scalesin which such interva ls appear, this tells you typical Tuvan harmonic numbers.(b) A Tuvan singer wants to tune a formant to resonate at the 12th harmonic, but to give less than1/4 that intensity for both neighboring 11th and 13th harmonics. What Q factor do they need toproduce for this formant? (Assume their vocal cords pro duces equal intensities for all harmonics).(c) Draw a spe c trum that could b e observed if a Tuvan singer sings a fundamental of 100 Hz andtunes a formant as in part b).6. A room has dimensions 4 × 6 × 10 m3. Compute the reverberation time T60in the following situations:(a) The room is pe rfectly reflective (impossible in reality) but has windows open totalling area 5 m2.(b) The room is e verywhere lined with wood (use the absorbtion coefficients on p.222 at 1000 Hz)and the windows are closed (treat them the same as wo od).(c) The room is lined with wood everywhere except 5 m2of windows are opened.7. Measure the reverberation time T60of the Kemeny stairwell from the sound of a clap: Kemenystairwell clap.wavI sug gest you use the Show intensity featur e of praat and compute it from the time to drop by 30dB. Quote yo ur answer with an estimate on accuracy, e.g. 5 ± 0.7 sec.8. A shower cabinet ha s ac oustically-reflective walls and dimensions 2×1×1 m3. Compute the frequenciesof the seven modes la belled 000 through 111. Sketch the res ulting spectrum if they all were excited.These are the pitches that you hear resonate when singing in the shower. What is the musical intervalbetween the frequencies f011and