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

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

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.

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: Flat Earth 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 Pentatonic Scale 1 9/8 5/4 3/2 5/3

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 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 F A C E  4/3 8/5 1 6/5 3/2 4/3 16/9 8/3 G B D F

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

Newton’s Second Law: Dividing:

The Wave Equation

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

Fourier Series Solution

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

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

Harmonics

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

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

Harmonics m = 3 m = 4 m = 5 m = 6

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 tonic dominant A 5/3 F 4/3 C 1 F 2/3

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

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!

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

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

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

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

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

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

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 1.000 Db 16 15   num den ratio log2(ratio) C 1 1.000 Db 16 15 1.067 0.093109404 D 9 8 1.125 0.169925001 Eb 6 5 1.200 0.263034406 E 4 1.250 0.321928095 F 3 1.333 0.415037499 F# G 2 1.500 0.584962501 Ab 1.600 0.678071905 A 1.667 0.736965594 Bb 1.778 0.830074999 B 1.875 0.906890596 2.000

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

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.

Two Values for B B B 9/5 16/9

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

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

Solution: Temper the Notes Divide the octave interval geometrically into 12 equally spaced intervals. Solution: 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.

Bach’s “Well-Tempered Clavier” Written in all 12 major keys and all 12 minor keys. (1685-1750) Bb minor

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 ? ! ? (1685-1750)

Measuring Intervals in Cents ! ? 100 cents = 1 chromatic step (1685-1750)

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 10.1955 20 B 1.8984 11.0978 10 2.0273 12.2346 23

Clockwise Circle of Fifths ERROR

How Close is Bach to Pythagoras?   num den ratio 12*log2(ratio) Tempered cents C 1 1.000 0.000000 Db 16 15 1.067 1.117313 12 D 9 8 1.125 2.039100 2 4 Eb 6 5 1.200 3.156413 3 E 1.250 3.863137 -14 F 1.333 4.980450 -2 F# G 1.500 7.019550 7 Ab 1.600 8.136863 14 A 1.667 8.843587 -16 Bb 1.778 9.960900 10 -4 B 1.875 10.882687 11 -12 2.000 12.000000 How Close is Bach to Pythagoras?

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

Stringed Instrument Calibration Recall:

Stringed Element Calibration

Fret Calibration

Harmony vs Melody Tradeoff Indian Music: Sitar

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

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

19 notes per octave... Does it Work?   19 Notes Pythagoras 12 Notes Ratio # Cents Num Den C 1 1.0000 1.037155044 Db 1.075690586 2 -15 16 15 1.0667 1.0595 12 D 1.115657918 3 14 9 8 1.1250 1.1225 4 1.157110237 Eb 1.20010272 5 6 1.2000 1.1892 E 1.244692589 7 1.2500 1.2599 -14 1.290939198 F 1.338904101 -7 1.3333 1.3348 -2 1.388651143 1.440246538 10 G 1.493758962 11 1.5000 1.4983 1.549259642 Ab 1.606822453 13 1.6000 1.5874 A 1.666524013 1.6667 1.6818 -16 1.728443787 Bb 1.792664192 1.7778 1.7818 -4 B 1.85927071 17 1.8750 1.8877 -12 1.928351996 18 19 2.0000

Brother Ray & Cherry Pop Tarts. 19 notes per octave vs. 12... Error in cents with respect to the Pythagorean scale: What sounds best? Brother Ray & Cherry Pop Tarts.

Finish “Truth on a Bus” http://eceserv0.ece.wisc.edu/~sethares/mp3s/truthonabus.html