Math 5: Music and Sound. Homework 1due Fri Apr 6 . . . but best if do relevant questions after each lectureFirst install audacity onto your laptop and make sure you can record a sound, play it back. See ourSoftware page for help. Please do this before Tues X-hour, since then you can bring your laptop and try itout in class.1. (a) What is the perio d, and frequency, of the signal frac(100t) ? (you may assume t is measured inseconds, and frac means fractional part)(b) What is the period, and frequency, of the signal sin(100t) ?(c) Compare the frequency ratio 3:2 to that given by 7 equal-tempered semitones. Express theirdifference both as a percentage e rror (e.g. 999 Hz is 0.1% flat relative to 1000 Hz), and in cents.2. Given the modern A4 of 4 40 Hz, compute the frequencies of the following notes using eq ual-temperedtuning: C4 (careful: this is the first C below A4), A1, D#3, F6 (the soprano top F in Mozart’s TheMagic Flute).3. Find which (equal-tempered, modern) note names the following frequencies are nearest, and expresshow out o f tune they are from these notes, in cents.(a) Handel’s ‘A4’ tuning fork from 18th centur y which still survives and is tuned to 422.5 Hz (no, itdidn’t drift).(b) 256 Hz (which was proposed by scientists in 1939 as a standard for C4 [why, do you think?], andrejected, thankfully)(c) The ‘interference’ hum you sometimes hear at 60 Hz due to our AC electrical system.4. Download Mystery Sound 1 from the HW page.(a) Find the (couple of main) component frequencies as accurately as you can using audacity. [Hint:make sp e c trum fill the screen, use Log a nd the longest transform length 16384. audacity suggestsnote names but you have no idea how far off they are].(b) Express them as notes in the equal-tempered system with tuning errors from these notes in cents.(c) Roughly what musical interval do these frequencies produce? The interval is close to one involvingsmall integers—which ones?(d) BONUS: Do so me detective work and explain what this familiar sound is and why it is not asfamiliar as usual. . .5. Five functions, which we can interpre t as pressure vs time, are given below either as a graph or aformula. State which (and there may be none, or more than one) of them are. . .(a) Decreasing in amplitude and in frequency(b) Getting louder but at constant pitch (ignore any subtle pitch-loudness p erception issues for now!)(c) Increas ing in pitch but not in amplitude(d) Decreasing amplitude but at constant frequency16. The pr e ssure sig nal sin(880πt +π/3) can be w ritten as A sin(880πt)+B cos(880πt). Find the constantsA and B. Explain with reasons whether you expect this signal to so und the same as sin(880πt) to theear.7. Two sinusoidal tones of frequencies 400 Hz and 402 Hz are played together both with amplitude 1 .(a) Write down a formula for the combined (added) signal.(b) Describe what you would hear. [you might check this by generating then mixing two tone trackswith audacity but this is not required].(c) Use a trig identity to re-express this signal in a form more useful for understanding what you hear.Explain how the two terms in this formula correspo nd to aspects of what you hear.8. Draw a time axis labelled 0, 0.01, 0.02 etc up to 0.08 (think of this as measured in seconds).(a) Sketch (without using a computer, since you’ll want to practise for doing this in an exam) thegraphs of sin(100πt) and sin(2003πt) on these axes. [Hint: first get the ze ro-crossings right](b) What are the individual frequencies, and periods, of these functions?(c) Add to your sketch (in a differe nt color) an estimate of the sum of the functions.(d) What is the period of the sum? What musical interval would you hear when the signals are playedtogether?(e) What would the period of the sum be c ome if the second frequency was changed very slightly to33 Hz?9. Upload to our Aural Postings page an interesting sound (maybe r e c orded with audacity and saved inOGG format) illustrating a concept from the first week of class (explain how it illustrates