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

Speed

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

things:

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!

BOO HISS NO ONE LIKES 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:

EQUAL TEMPERAMENT

But first lets discuss the ratio-based

Pythagorean Tuning. Greek Scale Pythagoras was also a budding music

theorist

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?

Well,

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:

x*2^(1/12)

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

Major: WWHWWWH

Asymetric

Uses every note name

Key

Most used

CIRCLE OF FIFTHS Intervals

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?

Chords

A group of pitches played together

Consonant or dissonant

Triads:

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

CMaj

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

Activity

-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

-Percussion:

-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:

(2/((L)^5))^1/2

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#

Repeats....

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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# Music (Theory) and Math

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