1. The team verified the relationship between music and math by measuring the frequencies of notes played on an instrument. 2. Their results supported the hypotheses that the frequency of each octave is twice the previous octave, the logarithmic distance between notes is constant, and simpler frequency ratios produce more harmonious intervals. 3. Potential improvements included conducting the experiment in a quiet room and maintaining a fixed distance between the microphone and instrument.

1. M&M Music and Math
Dimitri Lo -z3372021
Johnathan Lee – z3421088
Sanjiv Kumar -z3401648
Lab day/time: Tuesday 11 am

2. 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 .

3. Hypothesis
1. 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).

4. 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

5. Results
Hypothesis 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 . 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.

6. 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).

7. 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

8. Plot 1: Log frequency distance from
previous note plot

9. Graph of Note vs Frequency
• Notes follow an exponential relationship
• Verifies the fact that the logarithmic distance between 2 adjacent notes
is constant

10. 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.

11. 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.

12. 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.