1. Musimatics: Mathematics of Classic Western Harmony - Pythagoras to Bach to Fourier to Today
Robert J. Marks II

2. a2 + b2 = c2 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

3. The music of the spheres.
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.

4. 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.”

5. 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:
Flat Earth
Heliocentrism

6. Pythagorean Music

7. Pythagorean Music

8. Pythagorean Music !
Tone pairs were most pleasing when the length of the strings were ratios of small numbers.

9. Pythagorean Music
Pentatonic Scale
1 9/8 5/4 3/2 5/3

10. Pythagorean Music C E G B 1 5/4 3/2 15/8 9/8 4/3 5/3 2 D F A C
Major Scale
C E G B
/ / /8
9/ / /
D F A C

11. 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
F A C E 
4/3 8/ /5
3/ /3 16/9 8/3
G B D F

12. Music of the Spheres

13. 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.

14. Vibrating String

15. Newton’s Second Law:
Dividing:

16. The Wave Equation

17. The Wave Equation
Boundary Conditions:
Solution is the
Fourier series.

18. Fourier Series Solution

19. Harmonics
: Same as for a vibrating air column
m = 3
m = 4
m = 5
m = 6

20. This is the initial condition.
Harmonics
This is the initial condition.

21. Harmonics

22. All Bugle Tunes Based on These Four Harmonics
Taps
m = 3
m = 4
m = 5
m = 6

23. Revelry
m = 3
m = 4
m = 5
m = 6
Taps

24. Harmonics
m = 3
m = 4
m = 5
m = 6

25. 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.

26. Building Harmonies subdominant tonic dominant A 5/3 F 4/3 C 1 F 2/3

27. Adjusting Octaves subdominant tonic dominant A 5/3 F 4/3 C 1 E 5/4 C 1

28. Order the Numbers: A Major Scale!
C E G B
/ / /8
9/ / /
D F A C
Ratios of small numbers!

29. Multiply by 2n anytime! It simply changes the octave – not the note.
Circle of Fifths
Divide
by 3
Multiply by 3
Multiply by 2n anytime! It simply changes the octave – not the note.
Divide by 5
Multiply by 5

30. Circle of Fifths
5/3
Multiply by 5 =A
Divide by 3

31. Circle of Fifths
53/27 1
125
27/53
= 1.02
=125/128
= 0.98 5 25

32. Major Scale by the Numbers...
num
den
ratio
log2(ratio) C 1
1.0000
0.0000 Db D 9 8
1.1250
0.1699 Eb E 5 4
1.2500
0.3219 F 3
1.3333
0.4150 F# G 2
1.5000
0.5850 Ab A
1.6667
0.7370 Bb B 15
1.8750
0.9069
2.0000

33. Major Scale by the Numbers...
cool!
log2(ratio)

34. Building Sub-Harmonies
subdominant
tonic
dominant
D 1/15
F 1/12
B 1/9
F 1/6
F 1/3
A 1/5
C 1/4
F 1/3
C 1/2
C 1
E 3/5
G 3/4
C 1
G 3/2
G 3

35. Adjust by Octaves & Order: Minor Scale (Fm)
F A C E 
4/3 8/ /5
3/ /9 16/15 8/3
G B D F
Ratios of small numbers!

36. Add the new notes... num den ratio log2(ratio) C 1 1.000 Db 16 15
num
den
ratio
log2(ratio) C 1
1.000 Db 16 15
1.067 D 9 8
1.125 Eb 6 5
1.200 E 4
1.250 F 3
1.333 F# G 2
1.500 Ab
1.600 A
1.667 Bb
1.778 B
1.875
2.000

37. Both Scales by the Numbers...
log2(ratio)

38. We have all ratios of small numbers.
num 2 3 5 9 15
1/1 C
3/2 G
5/4 E
9/8 D
15/8 B
den
4/3 F
5/3 A
8/5 Ab
6/5 Eb
9/5 ?
16/9 Bb
10/9 ?
16/15 Db
REDUNDANT
Some fractions between 1 and 2 you can make with 2,3 and 5 raised to small powers.

39. Two Values for B B B
9/5
16/9

40. We have all ratios of small numbers.
num 2 3 5 9 15
1/1 C
3/2 G
5/4 E
9/8 D
15/8 B
den
4/3 F
5/3 A
8/5 Ab
6/5 Eb
9/5 Bb
16/9 Bb
10/9 D
16/15 Db
1.125 
1.800
1.111 
1.778
REDUNDANT
Can’t modulate between keys
Inconsistent
bummer

41. Solution: Temper the Notes
log2(ratio)
Make the line fit exactly...

42. Solution: Temper the Notes
Divide the octave interval geometrically into 12 equally spaced intervals.
Solution:
Sanity check: n=12 gives an octave.

43. Tempered Frequency The ratio of frequency of two notes is 21/12.
The standard is A above middle C = 440 Hz.

44. Bach’s “Well-Tempered Clavier”
Written in all 12 major keys and all 12 minor keys.
( )
Bb minor

45. How many chromatic steps are there between frequency f1 and f2 ?
Half Steps Between 2 Frequencies
How many chromatic steps are there between frequency f1 and f2 ? ! ?
( )

46. Measuring Intervals in Cents! ?
100 cents = 1 chromatic step
( )

47. Circle of Fifths 1 312/219 3/2 311/217 32/23 310/215 312/219=1.013643
23 cents from 1
3/2
311/217
32/23
310/215
33/24
39/214
34/26
38/212
35/27
37/211
36/29
ratio
half steps
cents C
1.0000
0.0000 Db
1.0679
1.1369 14 D
1.1250
2.0391 4 Eb
1.2014
3.1760 18 E
1.2656
4.0782 8 F
1.3515
5.2151 22 F#
1.4238
6.1173 12 G
1.5000
7.0196 2 Ab
1.6018
8.1564 16 A
1.6875
9.0587 6 Bb
1.8020 20 B
1.8984 10
2.0273 23

48. Clockwise Circle of Fifths
ERROR

49. How Close is Bach to Pythagoras?
num
den
ratio
12*log2(ratio)
Tempered
cents C 1
1.000 Db 16 15
1.067 12 D 9 8
1.125 2 4 Eb 6 5
1.200 3 E
1.250
-14 F
1.333 -2 F# G
1.500 7 Ab
1.600 14 A
1.667
-16 Bb
1.778 10 -4 B
1.875 11
-12
2.000
How Close is Bach to Pythagoras?

50. Q: How Close is Bach to Pythagoras?
A: Pretty close.
One Chromatic Step
cents

51. Stringed Instrument Calibration
Recall:

52. Stringed Element Calibration

53. Fret Calibration

54. Harmony vs Melody Tradeoff
Indian Music: Sitar

55. The tempered scale, derived for harmony, is used for dissonant music.
Charles Ives
Irony
Dissonance
Schoenberg

56. Are there other “good’ scales?
How about 19 notes per octave?

57. 19 notes per octave... Does it Work?
19 Notes
Pythagoras
12 Notes
Ratio #
Cents
Num
Den C 1
1.0000 Db 2
-15 16 15
1.0667
1.0595 12 D 3 14 9 8
1.1250
1.1225 4 Eb 5 6
1.2000
1.1892 E 7
1.2500
1.2599
-14 F -7
1.3333
1.3348 -2 10 G 11
1.5000
1.4983 Ab 13
1.6000
1.5874 A
1.6667
1.6818
-16 Bb
1.7778
1.7818 -4 B 17
1.8750
1.8877
-12 18 19
2.0000

58. Brother Ray & Cherry Pop Tarts.
19 notes per octave vs
Error in cents with respect to the Pythagorean scale:
What sounds best?
Brother Ray & Cherry Pop Tarts.

59. Finish
“Truth on a Bus”