The Mathematics of Music
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Transcript The Mathematics of Music
The Mathematics of Music
Is music just numbers?
Are you listening to logic?
Is Music Mathematical?
Musicians? Mathematicians?
Bach? Mozart? Beethoven?
Einstein? Feynman?
What is music?
Ordered patterns of sound and
silence in a framework of time
(A. Renino)
Beethoven’s Ninth (1824)
Performed by Seattle Symphony
Listen!
The silences - the holes in the
sound - are part of the music.
John Cage’s notorious 4’33”!
4 minutes & 33 seconds
of silence, the pianist sitting at
the keyboard ready to play!
What is sound?
Pitch & Harmony
Rhythm & Tempo
Dynamics & Timbre
and more …
Pitch and Harmony
Pitch – the tones of the scale
The Sound of Music (1965)
Do-Re-Mi-Fa-So-La-Ti-Do
What is this pattern of tones?
It’s not evenly spaced.
Where did it come from?
Did someone invent it?
Pythagoras!
• The triangle guy!
• c2 = a2 + b2
c
a
b
Who was Pythagoras?
Greece, circa 569-475 BC.
Traveled widely.
India? Met the Buddha?
The Middle East
Pythagoreans
School? Religion? Cult?
Secret & Egalitarian
“At its deepest level, reality is
mathematical!”
All is Number
They attempted to explain
everything with mathematics,
including music.
Vibrating Strings
String lengths with small integer
ratios produce pleasing harmonies
2:1
3:2
4:3
Do-Do
Do-So
Do-Fa
How about Fa-So?
If low Do is length L,
Fa is 3L/4 and So is 2L/3
and high Do is L/2.
Fa-So, Fa-Do, So-Do
Fa-So = (3L/4)/(2L/3) = 9:8
Fa-Do = (3L/4)/(L/2) = 3:2
So-Do = (2L/3)/(L/2) = 4:3
(Omit L, invert ratios, divide)
Omit L, invert ratios, divide
Do-Fa
——— =
Do-So
3/4
——
2/3
9
= —
8
Wherefore the scale?
Start with Do-Fa-So-Do
Add other tones with 9:8 ratio
Do-Re = 9:8
Re-Mi = 9:8
Fa is next! What is Mi-Fa ratio?
Do-Fa = 4:3
Do-Mi = 9:8 * 9:8 = 81:64
Mi-Fa = (64/81)/(3/4) = 256/243
!?!?!
9:8 and 256:243
Basis of the Pythagorean scale:
9:8 - a second
256:243 - a hemitone
The hemitone “resyncs” Do-Fa.
Pythagorean (Major) Scale
Do Re Mi
Fa So La Ti
Do
9:8 9:8 256:243 9:8 9:8 9:8 256:243
The hemitone also “resyncs” Do-Do.
Do-Do
(9*9*256*9*9*9*256)/(8*8*243*8*8*8*243)
= (95*2562)/(2432*85)
= (310*216)/(310*215) = 2/1
Do-Do = 2:1 = octave.
Pythagorean Intervals
256:243 = Hemitone = Mi-Fa = Ti-Do
9:8 = Second = Do-Re = Re-Mi = Fa-So = So-La = La-Ti
4:3 = Fourth = Do-Fa = Re-So = Mi-La = So-Do
3:2 = Fifth = Do-So = Re-La = Mi-Ti = Fa-Do
Every tone is paired with another as a fourth or fifth.
No other 8-tone scale pattern accomplishes this.
Cumulative Ratios of Tones
Do Re Mi Fa So La Ti Do
1 9 81 4 3 27 243 2
1 8 64 3 2 16 128 1
S S H S S S
H
Pythagorean (Major) Scale
It’s what we want to hear when
playing melodies!
It’s the natural harmonies
of string lengths.
Is It the Only Scale?
The minor scale
The pentatonic scale
The whole note scale
and others ...
The Minor Scale
Pythagorean scale - start at La:
La Ti Do Re Mi Fa So La
S H S S H S S
Starting Point is Crucial
In any melody
one wants to return to a
starting point.
The Pentatonic Scale
Omit hemitones from Pythagorean:
Do Re Fa So La Do
9:8 32:27 9:8 9:8 32:27
The Whole-Note Scale
Omit hemitones from Pythagorean:
Do Re Mi So La Do
9:8 9:8 32:27 9:8 32:27
Piano Keyboard
Other Cultures - Other Scales
Indian Sitar Music
Ravi Shankar
Raga Miniature (1972)
Other Cultures - Other Scales
Indian sitar music uses a much
richer scale, many more tones,
and we’ll reconnect with
Ravi Shankar, later, sort of ...
Pitch and Harmony
Pythagorean harmony was based
upon two pitches with
Do-Fa, Do-So or Do-Do intervals,
Fouths, Fifths or Octaves.
Harmony - Chords (Triads)
Do-Mi-So - the tonic ”major”
Fa-La-Do - the 4th “major”
So-Ti-Re - the 5th “major”
The Major Chord
In the Pythagorean Scale,
each of these has the
same ratio of the string lengths:
1:1 - 81:64 - 3:2
Cumulative Ratios of Tones
1 9 81 4 3 27 243 2 9
1 8 64 3 2 16 128 1 4
S S H S S S
H S
Three Chord Harmony
Pythagorean (Ionian) Harmony
(major scale)
Predominant harmony of rock & pop
Louie, Louie (1956)
The Kingsmen (1963)
Written by Richard Berry
Originally performed by him
and the Pharaohs
My Girl (1964)
The Temptations
Written by Smokey Robinson
and Ronald White
Semi-Charmed Life (1997)
Third Eye Blind
Written by Stephan Jenkins
(Rolling Stone: Best song of the 1990s)
Sonata in C Major (1788)
Wolfgang Amadeus Mozart
Performed by Maria-Joao Pires
Probably “pop” music in its time.
Confession!
By talking of guitars and pianos,
I’ve been lying to you,
but only a teeny, tiny bit.
Problems with Pythagoras?
No scale or chord for Re.
The third tone is not there.
Let’s see why ...
Pythagorean Scale
Do Re Mi Fa So La Ti Do
1 9 81 4 3 27 243 2
1 8 64 3 2 16 128 1
A scale based on Re would be
Re Mi ??? So La Ti ??? Re
9 81 729 3 27 243 2187 9
8 64 512 2 16 128 1024 4
A scale based on Mi would be
Mi ??? ???? ??? Ti ???? ????? Mi
81 729 6561 324 243 2187 19683 162
64 512 4096 192 128 1024 8192 64
Pythagorean Scale
1 9 81 4 3 27 243 2 9
1 8 64 3 2 16 128 1 4
S S H S S S
H S
How to add the extra tones?
The missing tones are
between the seconds.
Maybe add a hemitone
between each second?
Pythagorean Comma
Up or down a hemitone (H)?
Up an H = 256/243
Down an H = 2187/2048
1.053498 vs 1.067871
Piano Keyboard
Black keys are
the extra tones
Well-Tempered Scales
Some kept Pythagorean as base
and added extra tones.
Some abandoned Pythagorus.
Well-Tempered Scales
The Just-Tempered Scale
The Mean-Tempered Scale
and others
Just-Tempered Scale
Do Re Mi Fa So La Ti Do
1 9 5 4 3 5 15 2
1 8 4 3 2 3 8 1
Just-Tempered Scale
1
1
1
1
9 5 4
8 4 3
9 81 4
8 64 3
3 5 15
2 3 8
3 27 243
2 16 128
2
1
2
1
Compare two scales in decimal
Pythagorean:
1.0 1.125 1.2656 1.3333
1.5
1.6875 1.8984 2.0
Just-Tempered:
1.0 1.125 1.25
1.5
1.6666 1.875
1.3333
2.0
Do, Re, Fa, So, Do are the same.
Other tones are significantly different.
Melody versus Harmony.
Do-Mi-So, Fa-La-Do, So-Ti-Re chords sound “justly” better.
But La and Ti sound “justly” worse in melodies.
Just-Tempered Scale
1 25 9 6 5 4 45 3 8 5 9 15 2
1 24 8 5 4 3 32 2 3 3 6 8 1
(How to fill in the gaps.)
Changing Do’s
Well-tempered scales locked
keyboards into a pre-tuned Do.
To change Do’s required re-tuning.
String instruments like violins did not.
The solution?
A mathematical compromise
invented over decades
by J.S.Bach and others ...
Equi-tempered Scale
Divide the octave into twelve
equally spaced tones.
What’s the ratio of each ...
The ratio of string lengths?
Call it R. Multiply by R 12 times must get 2 to “resync” the octave.
So … R12 = 2 … R = 122
The Chromatic Scale
Each fret on the guitar shortens the
string by a factor of 122.
This is called a half-step.
The Chromatic (12 tone) Scale
Adjacent keys on a piano
are half-steps.
Two half-steps are a whole-step.
Let’s Compare
Hemitone = 256/243 1.0535
Half-step = 122 1.0595
Second = 9/8 = 1.125
Whole-step = 62 1.1225
Redefine Major Scale
In whole-steps and half-steps:
W W H W W W H
Similar to Pythagorean:
S S H S S S H
Piano Keyboard
One may start a scale anywhere
Compare three scales in decimal
Pythagorean:
1.0 1.125 1.2656 1.3333
1.5
1.6875 1.8984 2.0
Equi-Tempered:
1.0 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2.0
Just-Tempered:
1.0 1.125 1.25
1.3333
1.5
1.6666 1.875
Do’s are right on. Re, Fa & So are close.
Mi & La are less close.
Ti is off by - the biggest difference.
Equi-tempered is between the two.
2.0
It’s a good compromise!
One may start a scale anywhere!
But Ti is almost a little too flat.
The trained ear can hear it!
It’s why musicians “pull-up” Ti.
What Would Pythagoras Think?
He’d be horrified!
He’d think it an obscenity!
The 122 is irrational!
Irrational Numbers
Not a ratio of whole numbers!
Discovered by Pythagoreans.
And it horrified them!
It contradicted “All is (whole) number!”
Irrational Numbers
… like 122 and 2
are not expressible as
whole number fractions!
Square Root of Two
(3/2)2 = 9/4 = 2.25 - Close
(17/12)2 = 289/144 2.007 - Closer
(41/29)2 = 1681/841 1.999 - Closer
But no fraction is exactly 2
Rationals vs Irrationals
17/12 = 1.41666666666666…
but
2 = 1.4142135623730950488…
(no pattern - never repeats)
Square Root of Two is Real
Pythagoras
2
c2 = a2 + b2
(2)2 = 12 + 12
1
2=1+1
1
Pythagorean Scandal
The discovery of irrational
numbers was a scandal.
The man who revealed it
to the outside was (?) killed.
A silly idea?
Whole numbers explain music.
Should whole numbers suffice
to explain everything?
Maybe not so silly ...
Look up on the wall behind me.
The Periodic Table.
What does it mean?
The Material World
All stuff is made of atoms!
Atoms are explained by whole
numbers - number of protons!
Pythagoras Confirmed!
The material world is explainable
by whole number quantities!
Not so silly after all.
Let’s address some myths ...
Fibonacci Sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
See the pattern?
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
F1 = 1, F2 = 1, Fn+1 = Fn + Fn-1
The next is the previous plus the prior.
The Keyboard & Fibonacci ?
Fibonacci and the Scale?
Coincidence and Nonsense.
Patterns without significance.
The scale was based on 2:1, 3:2, 4:3, 9:8,
256/243 and later fudged with 122 .
Golden Mean & the Scale?
How about phi? Limit(Fn+1/Fn)
= (1 + 5)/2 1.618
It’s NOT the basis of the scale ratios.
and Pleasing Proportions
BEWARE!!!!
There’s a lot of nonsense
out there
on the Internet!
Rhythm and Tempo
All about the beat!
Ordered patterns of sound and
silence in a framework of time.
Good Loving (1966)
The (Young) Rascals
(One of my Favorites)
Let’s listen …
Four Beat Rhythm
Most (nearly all) pop and rock
songs are four beat and
three chord music.
Rhythm and Tempo
Rhythm - pattern of the beat.
Tempo - pace of the beat.
Tempo is a “rate of change”.
Tempo may be fast or slow.
Norah Jones (2002)
She is amazingly inventive!
Her songs show an unusual
diversity of rhythms and tempos.
Don’t Know Why
Norah Jones
Written by Jesse Harris
Slow four beat music.
Cold Cold Heart
Norah Jones
Written by Hank WiIliams
Medium four beat music
Syncopated
Lonestar
Norah Jones
Written by Lee Alexander
Funky two beat music
I’ve Got to See You Again
Norah Jones
Written by Jesse Harris
Eight beat music?
Come Away with Me
Norah Jones
Written by Norah Jones
Three beat music
Turn Me On
Norah Jones
Written by J.D. Loudermilk
Two? Three? Six beat music!
The Nearness of You
Norah Jones
Written by Hoagy Carmichael
& New Washington
No beat ???
Take Five (1965)
Dave Brubeck Quartet
Written by Paul Desmond
Five Beat Rhythm
Sound is ...
Pitch & Harmony
Rhythm & Tempo
Dynamics & Timbre
Dynamics
Loudness and softness.
Forte and Piano.
We must get more physical.
(and we’ll re-understand pitch!)
Sound is ...
Vibrations in an elastic medium.
Like air - a fluid.
Vibrations caused by a plucked
string - vibrating at the same rate.
Sound is ...
Longitudinal “wave” vibrating
in the direction of motion …
plot the air pressure ...
Pitch
… corresponds to
how fast the wave vibrates,
the frequency of vibration
(a rate of change)
Pure Tone ( y = cos(x) )
Pure Tone Octave Higher (2:1)
Pure Tone Octave Lower (1:2)
Pitches
We hear pitches corresponding
to vibrational frequencies between
20/second and 20,000/second
(20 and 20,000 Hertz).
Loudness and Softness ...
… to the amount of
change in air pressure
Louder Pure Tone
Softer Pure Tone
A Burst of Tone
Harmony ...
… frequencies of vibration
are simple integer ratios
Tones in Do-So Harmony (3:2)
Tones in Do-Fa Harmony (4:3)
Timbre ...
… shape of the tone,
how exactly does the air pressure
change and repeat
A Pure Tone
A Violin Might Be
A Flute Might Be
Is music only ...
Pitch & Harmony
Rhythm & Tempo
Dynamics & Timbre
All describable mathematically
Higher and Higher (1967)
Jackie Wilson
Happiest Song ever Written!
But why? What’s special …
4 beats and 3 chords ...
Acknowledgements
Nicole Longmore - for help with PowerPoint and more
Cathy Hess - helpful conversations
Alfred Renino - Director, White Plains High School Band
Sir James Jeans - "Science and Music”, 1968
Encyclopedia Britannica
http://universe-review.ca/R12-03-wave.htm
http://goldennumber.net/music.htm
http://www.americanscientist.org/