Upload
sanjiv-kumar
View
143
Download
2
Tags:
Embed Size (px)
Citation preview
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.