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Musimatics: Mathematics of Classic
Western Harmony - Pythagoras to
Bach to Fourier to Today
Robert J. Marks II
Pythagoras (~570 BC)
Pythagoras’ Theorem
a
b
(a+b)2 = c2 + 4 (½ a b )
a2 + b2 + 2ab = c2 + 2ab
Thus:
a2 + b2 = c2
c
Pythagorean Cult
The music of the
spheres.
A cult formed around Pythagoras. They taught…
(1) that at its deepest level, reality is
mathematical in nature,
(2) that philosophy can be used for
spiritual purification,
(3) that the soul can rise to union
with the divine,
(4) that certain symbols have a
mystical significance, and
(5) that all brothers of the order
should observe strict loyalty and
secrecy.
Aristotle on Pythagoras
Aristotle wrote: “The
Pythagorean ...
having been
brought up in the
study of
mathematics,
thought that things
are numbers ... and
that the whole
cosmos is a scale
and a number.”
The Pythagorean Cult
An Example of Math/Science being extrapolated
outside of its proper domain. Examples…
(1) Determinism from Physics.
(2) Social applications of Darwinism.
(3) Relativism from Relativity.
There are also cases of religion being
extrapolated outside of its proper
domain. Examples:
1. Flat Earth
2. Heliocentrism
Pythagorean Music
Pythagorean Music
Pythagorean Music
Tone pairs were most
pleasing when the length
of the strings were ratios
of small numbers.
!
Pythagorean Music
1
9/8 5/4 3/2 5/3
Pentatonic
Scale
Pythagorean Music
Major Scale
C
E
G
B
1
5/4
3/2
15/8
9/8
4/3
5/3
2
D
F
A
C
Pythagorean Music
F
A
C
E
4/3
8/5
1
6/5
3/2
4/3 16/9 8/3
G
B
D
F
Minor Scale
Music of the Spheres
Jean Baptiste Joseph Fourier
(1768 - 1830)
 Contemporary
of Napoleon
 A founder of Egyptology
 First Suggested the
Greenhouse effect
 Fourier series.
–
Laplace & Lagrange were on
his examining committee.
Vibrating String
tan 1  T1
T  T1 cos 1  T2 cos 2
Newton’s Second Law:
2 y
T1 sin 1  T2 sin  2  x 2
t
Dividing:
T1 sin 1 T2 sin 2 x  2 y


T1 cos  T1 cos 
T t 2
x  2 y
tan 1  tan  2 
T t 2
y
x
y
tan 1  T1
x
1  y

x  x
x x
x x
y    2 y


2

x
T

t
x x
x
2 y  2 y

2
x
T t 2
The Wave Equation
 y 1  y
 2 2
2
x
c t
2
2
c
T

The Wave Equation
 y 1  y
 2 2
2
x
c t
2
2
c
Boundary Conditions:
y (0, t )  y ( L, t )  0
Solution is the
Fourier series.
T

Fourier Series Solution
 y 1  y
 2 2
2
x
c t
2
2
c
T

  mct    mx 
y ( x, t )   Am cos 
 sin 
 ;0  x  L
 L   L 
m 1

  mx 
y ( x,0)   Am sin 
 ;0  x  L
 L 
m 1

Harmonics : Same as for a vibrating air column
  mx 
y ( x,0)   Am sin 
 ;0  x  L
 L 
m 1

m=3
m=4
m=5
m=6
Harmonics
  mx 
y ( x,0)   Am sin 
 ;0  x  L
 L 
m 1

This is the initial
condition.
y
x0
x
xL
Harmonics
  mx 
y ( x,0)   Am sin 
 ;0  x  L
 L 
m 1

All Bugle Tunes Based on These Four Harmonics
Taps
m=3
m=4
m=5
m=6
m=6
m=5
m=4
Taps
m=3
Revelry
Harmonics
1

3
f  constant
f  3 fo
m=3
f  3 fo
m=4
m=5
m=6
f0
o
o
Harmoneous Assumptions
The smaller the
number of the
harmonic, the more
harmony.
 Multiply or dividing by
powers of 2 gives you
the same note in a
different octave.

Building Harmonies
subdominant
A 5/3
F 4/3
C 1
F 2/3
F 1/3
tonic
dominant
E
C
G
C
C
B 15
G 12
D 9
G 6
G 3
5
4
3
2
1
Adjusting Octaves
subdominant
A 5/3
F 4/3
C 1
F 4/3
F 4/3
tonic
E
C
G
C
C
5/4
1
3/2
1
1
dominant
B 15/16
G 3/2
D 9/8
G 3/2
G 3/2
Order the Numbers:
A Major Scale!
C
E
G
B
1
5/4
3/2
15/8
9/8
4/3
5/3
2
D
F
A
C
Ratios of small numbers!
Circle of
Divide
by 3
Divide
by 5
Multiply
by 3
Multiply
by 5
Fifths
Multiply by
2n anytime!
It simply
changes
the octave
– not the
note.
Circle of
5/3
=A
Fifths
Multiply
by 5
Divide
by 3
Circle of
27/53
Fifths
125
53/27
1
=125/128
= 1.02
5
25
= 0.98
Major Scale by the Numbers...
num
den
ratio
log2(ratio)
1
1
1.0000
0.0000
9
8
1.1250
0.1699
E
5
4
1.2500
0.3219
F
4
3
1.3333
0.4150
3
2
1.5000
0.5850
5
3
1.6667
0.7370
B
15
8
1.8750
0.9069
C
2
1
2.0000
1.0000
C
Db
D
Eb
F#
G
Ab
A
Bb
Major Scale by the Numbers...
cool!
1
log2(ratio)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
C
Db D
Eb E
F
F#
G
Ab A
Bb
B
C
Building Sub-Harmonies
subdominant
D 1/15
F 1/12
B 1/9
F 1/6
F 1/3
tonic
A 1/5
C 1/4
F 1/3
C 1/2
C 1
dominant
E 3/5
G 3/4
C 1
G 3/2
G 3
Adjust by Octaves & Order:
Minor Scale (Fm)
F
A
C
E
4/3
8/5
1
6/5
3/2
16/9 16/15 8/3
G
B
D
F
Ratios of small numbers!
Add the new notes...
num
den
ratio
log2(ratio)
C
1
1
1.000
0
Db
16
15
1.067
0.093109404
D
9
8
1.125
0.169925001
Eb
6
5
1.200
0.263034406
E
5
4
1.250
0.321928095
F
4
3
1.333
0.415037499
G
3
2
1.500
0.584962501
Ab
8
5
1.600
0.678071905
A
5
3
1.667
0.736965594
Bb
16
9
1.778
0.830074999
B
15
8
1.875
0.906890596
C
2
1
2.000
1
F#
Both Scales by the Numbers...
1
0.9
log2(ratio)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
C
Db D
Eb E
F
F#
G
Ab A
Bb
B
C
We have all ratios of small numbers.
num
2
den
2
1/1 C
3
4/3 F
5
8/5 Ab
9
16/9 Bb
15
16/15 Db
3
3/2 G
5
5/4 E
9
9/8 D
15
15/8 B
5/3 A
6/5 Eb
9/5 ?
10/9 ?
REDUNDANT
Some fractions between 1 and 2
you can make with 2,3 and 5
raised to small powers.
Two Values for B
B
B
We have all ratios of small numbers.
num
2
den
2
1/1 C
3
4/3 F
5
8/5 Ab
9
16/9 Bb
15
16/15 Db
Inconsistent
bummer
3
3/2 G
5
5/4 E
9
9/8 D
15
15/8 B
5/3 A
6/5 Eb
9/5 Bb
10/9 D
REDUNDANT
1.111
 1.125
1.778
 1.800
Can’t
modulate
between
keys
Solution: Temper the Notes
1
0.9
log2(ratio)
0.8
0.7
0.6
0.5
Make the
line fit
exactly...
0.4
0.3
0.2
0.1
0
C
Db D
Eb E
F
F#
G
Ab A
Bb
B
C
Solution: Temper the Notes
Divide the octave interval
geometrically into 12 equally
spaced intervals.
Solution:
fn  f0 2
n
12
Sanity check: n=12 gives
an octave.
Tempered Frequency
•The ratio of
frequency of
two notes is
21/12.
•The standard
is A above
middle C = 440
Hz.
12
2 f n  f n 1
12
2  1.05946...
Bach’s “Well-Tempered Clavier”
Written in all 12
major keys and
all 12 minor
keys.
(1685-1750)
Bb minor
Half Steps Between 2
Frequencies
?
How many chromatic
steps are there between
frequency f1 and f2 ?
 f2 
n  log12 2  
 f1 
(1685-1750)
Measuring Intervals in Cents
?
100 cents = 1 chromatic step
f c1 
1200
(1685-1750)
1200
2 fc ;
2  1.0005777895...
 f2 
cents  log1200 2  
 f1 
1
Circle of Fifths
312/219
3/2
311/217
32/23
310/215
312/219=1.013643
23 cents from 1
39/214
33/24
34/26
38/212
35/27
37/211
36/29
Clockwise Circle of Fifths
ERROR
ratioto12*log2(ratio)
Hownum
Close den
is Bach
C Pythagoras?
1
1
1.000
0.000000
Tempered cents
0
0
Db
16
15
1.067
1.117313
1
12
D
9
8
1.125
2.039100
2
4
Eb
6
5
1.200
3.156413
3
16
E
5
4
1.250
3.863137
4
-14
F
4
3
1.333
4.980450
5
-2
G
3
2
1.500
7.019550
7
2
Ab
8
5
1.600
8.136863
8
14
A
5
3
1.667
8.843587
9
-16
Bb
16
9
1.778
9.960900
10
-4
B
15
8
1.875
10.882687
11
-12
C
2
1
2.000
12.000000
12
0
F#
Q: How Close is Bach to
Pythagoras?
cents
One Chromatic Step
A: Pretty close.
Stringed Instrument Calibration
Recall: f  constant
fn  f0 2
f0
n
0

 fn
0
2
n
12
n
n

12
Stringed Element Calibration
f0
0
 fn
n 1

n
 constant
n
2
1

12
1

0
Fret Calibration
2

1
2

1
12

0
2

2
12
19

0
2

19
12
2
1

12
Harmony vs Melody Tradeoff
Indian Music:
Sitar
Irony
The tempered scale,
derived for harmony, is
used for dissonant music.
Dissonance
Schoenberg
Are there other “good’ scales?
How about 19 notes per
octave?
fn  f0 2
n
19
19 notes per octave... Does it Work?
19 Notes
Ratio
#
Cents
1
0
0
1.037155044
1
Db
1.075690586
2
-15
16
D
1.115657918
3
14
1.157110237
4
1.20010272
5
1.244692589
6
1.290939198
7
1.338904101
8
1.388651143
9
1.440246538
10
1.493758962
11
1.549259642
12
Ab
1.606822453
A
C
Eb
E
F
G
Bb
B
C
12
Notes
Pythagoras
Num
Den
Ratio
Ratio
Cents
1.0000
C
1.0000
0
15
1.0667
Db
1.0595
12
9
8
1.1250
D
1.1225
4
0
6
5
1.2000
Eb
1.1892
16
7
5
4
1.2500
E
1.2599
-14
-7
4
3
1.3333
F
1.3348
-2
7
3
2
1.5000
G
1.4983
2
13
-7
8
5
1.6000
Ab
1.5874
14
1.666524013
14
0
5
3
1.6667
A
1.6818
-16
1.728443787
15
1.792664192
16
-14
16
9
1.7778
Bb
1.7818
-4
1.85927071
17
15
15
8
1.8750
B
1.8877
-12
1.928351996
18
2
19
0
2
1
2.0000
C
2.0000
0
19 notes per octave vs. 12...
Error in cents with respect to the Pythagorean scale:
19 Cents
12 Cents
20.0
15.0
10.0
5.0
0.0
-5.0
C
Db D
Eb E
F
G
-10.0
-15.0
-20.0
What sounds best?
Brother Ray & Cherry Pop Tarts.
Ab A
Bb B
Finish
“Truth
on a Bus”
http://eceserv0.ece.wisc.edu/~sethares/mp3s/truthonabus.html