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Music (Theory) and Math

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Shawn Hogan

on 23 March 2011

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Transcript of Music (Theory) and Math

Music and Math

Louis Markowitz and Shawn Hogan Measure
Measure - unit of music, smallest division of a meter
Represented in a fraction
Numerator is beats per measure
Denominator is note value that recieves the beat
To find how many of a certain value fit in a bar of 4/4: increase exponent of 2.
2^0 =1, one whole note Meter
Repeating pattern of beats
Builds rhythms
Strong and weak accented beats
Douple Meter - 4/4, 2/4
Triple Meter - 3/4, 6/8
Compound Meter - 5/8, 7/8, 12/8
The Basics
A note is a a pitch that can be measured and timed
Organized notes = music
Note Values
Rest - silence
Whole, half, quarter, eighth, sixteenth Tempo
Measured in BPM
Metronome mark
hey guys if you can read this i'm about to punch you Temperament- Scale Building: We've been talking about the 12 tone western scale,
Which is an equally tempered system What does this mean? To understand that we must first understand a few
Cents Are a measure of an interval,
this is a logorithmic measurement,
so that every interval always has the same
measure in cents (as opposed to frequency which
changes). The equation for converting a ratio
(which can either be a number, or a fraction
or based on the 2 fundamental frequencies)
to cents is: F2/F1 =2^(¢/1200)

Where F2 is the second higher frequency, and F1 is the lower/fundamental frequency. To go the other way the equation is: ¢= 1200*(ln(F2/F1)/ln2) Next we must understand the comma!

A constant of music is that the frequency of
the octave above a note should be twice that note The answer my friends:

But first lets discuss the ratio-based
Pythagorean Tuning. Greek Scale Pythagoras was also a budding music
The Greek Scale was the foundation for his work
It was a five-tone (pentatonic) scale:

If we call the first note A (440 Hz)
and use modern note notation the scale looks like this (ALL NOTE NAMES ARE APPROX): 5/4 4/3 3/2 5/3 A C D E F A 2/1 (F2/F1) Wait a minute.... There are only 5 notes? Actually, Some follower of Pythagoras
decided to apply this ratios to each other to find
other ratios and therefore named other parts of the scale. To find B, The ratio of A to E was applied to itself.

(3/2*3/2)= 9/4 (This is a Major Ninth)

This lies between the Ratio of an ocatve higher than A
8/4 and Octave higher than C 10/4 (5/4*2).
If you divide it by 2 (9/4), you get the ratio (9/8) which is in the origional scale.
In modern music, B is between A and C. (semitones like A# and Bb complicate this idea. Let's look at B as an example.... Sounds pretty cool man,
Why don't we use this?
If you start on a different pitch
say A#. It didn't sound the same as
starting in A, and the scales didn't convert
well because the ratios affected the intervals. This is an example of a just tempered system,
which is a system in which the intervals maintain
exact integer based ratios. Also the ratios caused semitones
to be somewhat unweildy: Here is how the whole sytem breaks down
with all the semitones factored in (notice this one is in the key of C,
the ratios are the same but the frequencies are much different): The difference between a
chromatic and diatonic semtione is large:
about 24 cents. This is well above JND. JND= Just Noticeable Difference
It is defined as five cents and it is the smallest interval
the human ear can differentiate between. So.... wait a new york minute!
What's Equal Temperament?
Equal Temperament:
A temperament system (used today) which seeks to
equally spread the comma throught the scale,
which leads to an intune scale, and all slightly out
of tune intervals (in that the ratios aren't perfect). Allows
any instrument to play in any key, and allows all keys to be
transposable. E.T. (phone home)
has 3 constants:

A4= 440 Hz
An octave is 2/1 or 1200 cents.
A half stem/semitone is 100 cents or the twelth root of 2. So......
In equal temperament the intervals are all equal

In pythagorean temperament the ratios are all perfect. This entire system is actually based off of a
12:9:8:6 ratio stringed instrument.
Credited to Jacobus of Liege. It is called a quadrichord.
You get very perfect ratios,
an Octave (12:6)
2 fifths (12:8, 9:6)
2 fourths (12:9, 8:6)
Major Second (9:8) x^12=2,
2^(1/12)=x This means that to find a half step higher than a note
multiply it by the twelfth root of two:
To go lower divide:
x/(2^(1/2)) Harder Topic:
Lucy Tuning
note: both ratio and cents
work to measure intervals Pitch
Pitches are sound waves
Hertz = 6.875x2^((3+MIDIPITCH)/12)
MIDI Pitch - middle C is 60, C# is 61 etc..
Middle amount of frequencies = more musical the pitch
Glass breaking vs. Alarm beep
Highness and lowness of frequency is inversely proportional to wavelength
To find octave, multiply hertz by 2 What is the hertz of F above middle C? Consonance and Dissonance
Consonance - pleasing mix of notes
Repeating pattern of sound matchups
C and G: 2nd C matches 3rd D
Dissonance - not so pleasing
Matchups are not so regular
C and F# - no noticable regular pattern
13 or up in ratio -> dissonance Scales
Group of pitches that can be used to make music.
Chromatic Scale: all 12 pitches
Diatonic Scale: set of 7 + octave Chromatic Scale
Every interval is a half-step
Every note on the piano
Symmetric Diatonic Scale
Uses every note name
Most used
Distance between any two notes
Half Step (semitone) -smallest
Whole Step (whole tone)
11 intervals from each note
In diatonic scale - 7 intervals (kinda)
Half Steps
Ratio is
Ratio increases exponentially
Sounds even Diatonic Intervals
If your root is C (the C scale has no sharps or flats)...
A second up from C is D
A third up from C is?
A group of pitches played together
Consonant or dissonant
Major: 1, 3, 5; most consonant, 6:5:4
Minor: 1, b3, 5; less consonant, 15: 12: 10
Augmented: 1, 3, #5; dissonant
Diminished: 1, b3, b5; dissonant
How would you figure out ratios for diminished and augmented? How do chords fit into a song
Every note in a diatonic scale has a triad to go along with it
I ii iii IV V vi viio I
In the key of C what triad is the fifth?
Ever heard a I IV V? YES.

Major Triad Example
C-E about 4/5
G-E about 5/4
G-C about 3/2 Geometry in Composition
Restatement in Geometric patterns = recognize
Modulation = Transformation
Intervals remain constant
-Piano and Bell groups:
-Compose 4 bars of music
-Use only white keys
-Write out the rhythms, write down the note names
-Use a geometric pattern

-Write a 4 bar rhythm, use at least one of each of the note values we talked about

-You have 5-8 minutes, GO! Inversions
If C to F is a fourth, what is F to C?
Subtract from 9 to find inversion however,
if you apply
a rational ratio to a fundamental
frequency, the octaves eventually
go out of tune. Lucy tuning is a system of
temperament which seeks to address
the fundamental puzzle of music.

How to elegantly temper a scale, without
getting certain keys out of tune, and without
completely ruining the intervals' integrity. Instead of defining itself
by a rational ratio like pythagorus,
it uses two special ratios which are
constant like in E.T, however these ratios are
based on..... yep, pi.

Various mathemeticians/musicians
develeped this (it is named for Charles Lucy, who
advocated it from the writings of John Harrison who was a famous english genius who discovered many mathimatical concepts such as latitude and longitude). Harrison described two "notes" which are
actually ratios.

The first called the Larger note (L)
is the 2 times pi root of 2. (2^(1/(2π)) which equals a ratio of 1.116633 or 190.9858 cents
(compare to a whole tone in E.T. which is 200 cents). The second note called the Lesser/Smaller note (s)

is the ratio of:

or in words: Half the difference between 5 Larger notes and an
octave. Using your Calculator,
see if you can find this ratio
(hint: to write the larger ratio
enter 2^(1/2π)) What did you get? The answer is:
approx. 1.073344 ,
which when converted to cents
is about 122.5354¢ In LucyTuning,
The fifth is defined as
3L+s (much like a normal fifth
is 3 whole tones and one semitone).
This is an interval of 695.4¢.
And the E.T. fifth is 700¢...

JND? Circle of fifths/fourths in Lucytuning: This is
Equal Temperament.
This is LucyTuning A,E,B,F#,C#,G#,D#
A#,E#,B#,F##,C##, G##,D## Fourths spiral the same
way but with flats. Intervals closer together on this spiral,
ex. F## and C##, tend to sound more consonant. Now we're going to
discuss the last topic of
our presentation:
Harmonics! Harmonics are notes,
that develop from a fundamental
pitch, because of the physics of
how a string or chamber vibrates. N=harmonic # The velocity of the string is constant in
a harmonic series.

And the pitch and string proportions work in a
very just (remember this one???) way.

So let us assume, that we have an E5 string.
This string vibrates at 659.26, but for ease we'll
round this to 660Hz.

When n=1 (or the fundamental)
The whole string is vibrating (1/1) and the frequency is in a ratio of 1/1.
Obvious right?
Ok then, n=2
The string is vibrating at 1/2 it's length
and it has 2/1, the frequency (it's an octave!!!!)
so now we have an E6 vibrating at 1320Hz.

And away we go n=3, we have 1/3 the length,
and 3x the frequency which gives us an octave plus a fifth,
or a B.
Here's something for you to do!
Do N=4,
find the string ratio,
freqency ratio to the fundamental
Frequency of the new note,
cents to the fundamental.
Can you figure out the interval? So n=4?
So the string is vibrating at a 1/4 of
its length.

It has 4x the frequency of the fundamental
which is a frequency of 2640Hz
which is 2 octaves or 2400 cents. Funnily enough,
harmonics tend to follow a more just
system than the equally tempered
western scale. So as the harmonics
get higher, they go slightly out of tune to
the equally tempered system. Thanks For Listening,
We hope we've shown you the deep
mathematical concepts embedded in music theory! Also, This is nature's scale! The way sound works actually defines this scale, and notice the consonance in all the notes! It's all rational and Pythagoras would love it! I l
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