Math 5: Music and Sound. Midterm 12 hours, 7 questions, 55 points totalPlease show your working and pay attention to the indicated number of points available per question.The last page has useful informatio n.1. [8 points](a) What is the frequency of the pitch E♭6 in the equal-tempered system?(b) What musical pitch is the freq uency 350 Hz nearest, and what is the error from this pitch incents?(c) Construct the frequency ratio between E and C in the Pythagorea n C major (diatonic) scale.(Briefly show working)12. [7 points + bonus] An ice cream truck produces a constant pure tone at pitch A5 (880 Hz).(a) Assuming you and the truck are at res t, write a mathematical formula for the pressure signalheard as a function of time t in seconds.(b) If you stand still and the truck comes towards you at 41 m/s, what frequency do you hear?(c) Assume the truck is now stationary. What speed and direc tion (towards or away from the truck)would you have to be traveling at to lower the pitch to D5? (a perfect fifth; you may use 3:2).(d) BONUS: You and the truck are stationary but a very str ong wind o f 34 m/s is blowing in thedirection from the truck to you. What frequency do you hear?3. [7 points] Sketch spectrogr ams on the axes provided which could realistically match the followingsounds. Feel free to explain any features in words too:(a) A pure tone rising in pitch but getting quieter, then followed by a his sing sound (no apparentpitch)ft(b) A musical note with harsh timbre which is a periodic signal, followed by a bell-like note with nodefinite pitch.ft4. [5 points] A bell produces the following partials all at roughly equal amplitudes: 302, 781, 1168, 1560,2964. What ‘strike to ne’ frequency is per c e ived, and why?5. [11 points] A tuning fork is struck a nd produces a pure sinusoid at 300 Hz. A listener is a distance 2meters from the tuning fork.(a) Initially the tuning fork radiates 0 .005 W acoustic power in all directions. What intensity in dBdoes the listener hear?(b) The Q-fac tor of the tuning fork is 1000. What is the decay time?(c) How long since it was struck with the above initial strength does it take until the intensity at thelistener reaches the lower threshold of human hearing which is about 10−10W/m2at 300 Hz?(careful, not 10−12W/m2)(d) If mass is added to the prongs of the tuning fork so that their effective mass doubles (viewing thefork as a mass-spring oscillator), what frequency does the fork sound now?6. [7 points + bonus](a) Draw a space-time diagram s howing why a flutter echo is heard by a listener standing beside awall a distance L from another wall, when they produce a short sound such as a clap. Label thewalls and any sound pulses.(b) What period of signal is heard if the spacing between the walls is L = 10 meters?(c) BONUS: If the listener instead stands half way between the walls, what periodicity is heard?(draw a space-time diagram)7. [10 points + bonus] Random short-answer questions.(a) Explain briefly what is a Fourier series (you may use an equatio n). What kinds of functions doesit apply to?(b) State as clearly as you can what you will hear when pure tones of frequencies 500 and 507 Hz areplayed together, giving any re levant new fre quencies.(c) What frequency has a wavelength about equal to the size of the human ear (4 cm)?(d) According to the Helmholtz theory, state briefly why an octave is less dissonant than the majorseventh interval.(e) BONUS: When sounds are heard together their signals add. We learned that doubling the ampli-tude of a signal caused intensity to be multiplied by 4. However when two violins (for example)play the same music together the intensity is merely doubled relative to one violin. Resolve theparadox!Useful informationω = 2πfc = fλdB = 10 log10I10−12W/m2Q = πτTfobsf=11 − v/cor 1 + v/csin(a + b) = sin a cos b + cos a s in bsin a + sin b = 2 cos(a − b2) sin(a + b2)Intervals by number of semitones:1. minor second2. whole tone (major second)3. minor third4. major third5. perfect fourth6. tritone (augmented fourth)7. perfect fifth8. minor sixth9. major sixth10. minor seventh11. major seventh12. octaveThe standard musical pitch A4 is 440 HzYou can use the speed of sound as 340