M&M Music and Math

Dimitri Lo -z3372021

Johnathan Lee – z3421088

Sanjiv Kumar -z3401648

Lab day/time: Tuesday 11 am

Project Overview

AIM: 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 . f0

o , f0- = 2-n . f0

o)

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.

microphone

USB cable

Logger Pro

Results

Hypothesis 1:n=given note

superoctave = 2n × frequency abovesuboctave = 2-n × frequency below( f0

+= 2n . f0o , f0

- = 2-n . f0o)

In note A, the frequency of the superoctave was about 2n times the frequency of the given note and the frequency of the suboctave was about 2-n times the frequency of the given note.

• Suboctave: 2-n . f0o

• Superocatve: 2n . f0o

Hypothesis 2

Each 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 3

The 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 from previous 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 4

Simpler ratios between frequencies of notes result in a more

concordant 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 notes

result 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 . f0

o

• Frequency of a suboctave is: f0- = 2-n . f0

o)

• 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.