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M n m presentation
M&M Music and Math Dimitri Lo -z3372021 Johnathan Lee – z3421088 Sanjiv Kumar -z3401648 Lab day/time: Tuesday 11 am
Project OverviewAIM: Verify existing relationship between music and math.INTRODUCTION:• Historical Context: Origins of western musical scale can be traced back to Ancient Greeks. Pythagoras was credited with finding relationship between concordant music intervals and simpler integer ratios.• Theories and principles being tested against the hypothesis: 1. f=1/T. 2. Superposition- sound waves combine their energies to form a single wave .
Hypothesis1. n=given note superoctave = 2n × frequency above suboctave = 2-n × frequency below ( f0+= 2n . f0o , f0- = 2-n . f0o)2. Each successive octave spans twice the frequency of the previous octave.3. The log2 frequency distance between adjacent nodes is 1/12. log2(fn)-log2(fn-1)= 1/12 (0.08333).4. Simpler ratios between frequencies of notes result in a more concordant and regular interval (combination of 2 notes).
Procedure• The microphone was connected to the logger pro.• The instrument was tuned and microphone placed near it.• The note was played and “collect” button was pressed on logger pro software to obtain the data.• The adjacent peaks of the sound pressure wave was observed and the time taken to travel between them (T) was noted.• Formula f=1/T was used to find the frequency. Logger Pro microphone USB cable
ResultsHypothesis 1:n=given note superoctave = 2n × frequency above suboctave = 2-n × frequency below ( f0+= 2n . f0o , f0- = 2-n . f0o) • Suboctave: 2-n . f0o • Superocatve: 2n . f0oIn note A, the frequency of the superoctave was about 2n times the frequencyof the given note and the frequency of the suboctave was about 2-n times thefrequency of the given note.
Hypothesis 2Each successive octave spans twice the frequency of the previous octave. • A3–A4 spans from 218 Hz to 440 Hz (span ≈ 220 Hz). • A4–A5 spans from 497 Hz to 974Hz (span ≈ 440 Hz).
Hypothesis 3The log2 frequency distance between adjacent nodes is 1/12. log2(fn)-log2(fn-1)= 1/12 (0.08333). Log Frequencies of Average Frequencies
Plot 1: Log frequency distance fromprevious note plot
Graph of Note vs Frequency• Notes follow an exponential relationship• Verifies the fact that the logarithmic distance between 2 adjacent notes is constant
Hypothesis 4Simpler ratios between frequencies of notes result in a moreconcordant and regular interval. Frequency Ratios • Concordant intervals (C and G) has a ratio close to 3:2 (which is a simple ratio). • Discordant intervals (C and C#) has a ratio close to 16:15 (a more complex ratio).The results confirm the fact that simpler ratios between frequencies of notesresult in a more concordant and regular interval.
IMPROVEMENTS• Conducting the experiment in a room without any additional sources of sound.• Fixing the microphone and ukulele so that their distance between them are constant which would prevent errors arising from varying distances.EXTENSIONS• Using other instruments with larger note spans to further support relationships verified.
CONCLUSION• Frequency of a superoctave is: f0+= 2n . f0o• Frequency of a suboctave is: f0- = 2-n . f0o)• Each successive octave spans twice the frequency range of the previous octave.• The log2 frequency distance between adjacent nodes is 1/12. log2(fn)-log2(fn-1)= 1/12 (0.08333).• Simpler ratios between frequencies of notes result in a more concordant and regular interval.