By Ted Fitzgerald, Allison Gibson, By Ted Fitzgerald, Allison Gibson, Kaitlin Spiegel, and Danny Spindler Kaitlin Spiegel, and Danny Spindler

Mathematics in Music

History History 500 BC – Greek mathematician Pythagoras 500 BC – Greek mathematician Pythagoras

experiments with changing the lengths of experiments with changing the lengths of strings to produce different tonesstrings to produce different tones

Plucking the strings creates vibrations which Plucking the strings creates vibrations which is what musical tones areis what musical tones are

Pythagoras discovered that two strings, Pythagoras discovered that two strings, one half as long as the other, would one half as long as the other, would produce the same tone, except at produce the same tone, except at different pitches or frequenciesdifferent pitches or frequencies

Making one string half as long as the Making one string half as long as the other causes it to vibrate twice as fast other causes it to vibrate twice as fast as the other (the frequency is doubled)as the other (the frequency is doubled)

The faster the vibration, the higher the The faster the vibration, the higher the pitchpitch

The scale…The scale… The distance between a note and another The distance between a note and another

vibrating twice as fast is called an octavevibrating twice as fast is called an octave In-between these two notes are 11 others In-between these two notes are 11 others

creating the twelve-note chromatic scalecreating the twelve-note chromatic scale The notes in the scale are:The notes in the scale are:

A, A#, B, C, C#, D, D#, E, F, F#, G ,G#A, A#, B, C, C#, D, D#, E, F, F#, G ,G#

HarmoniesHarmonies Pythagoreans found that only harmonious musical Pythagoreans found that only harmonious musical

intervals were produced by dividing a string so that intervals were produced by dividing a string so that the two resultant lengths were in the simple ratios the two resultant lengths were in the simple ratios of 2:1, 3:2 (shown below), 4:3, or 5:4. of 2:1, 3:2 (shown below), 4:3, or 5:4.

They called these intervals the octave, the fifth, They called these intervals the octave, the fifth, the fourth and the third. Thus the numbers 1, 2, 3, the fourth and the third. Thus the numbers 1, 2, 3, 4 and 5 could produce all of the musical intervals 4 and 5 could produce all of the musical intervals they considered pleasing.they considered pleasing.

Why is it that these combinations Why is it that these combinations of notes sound good together, of notes sound good together, while others make you cringe?while others make you cringe?

The Answer: The Answer: It all has to do with frequencies.It all has to do with frequencies.

Frequencies that match up at regular Frequencies that match up at regular intervals will create a pleasing sound.intervals will create a pleasing sound.

For example: Here is a C and a G played together.

Now here is a C and a F# played together.

Frequencies of some notes in C Frequencies of some notes in C major:major:

C- 261.6 HzC- 261.6 Hz

E- 329.6 HzE- 329.6 Hz

G- 392.0 HzG- 392.0 Hz

Ratio of E to C is about Ratio of E to C is about 5/45/4

That means that every That means that every 55thth wave of E matches wave of E matches up with every 4up with every 4thth wave wave of C, producing a of C, producing a pleasant sound.pleasant sound.

Vibrations and SoundVibrations and Sound

Sound is vibrations in the airSound is vibrations in the air Air is a collection of atoms and molecules Air is a collection of atoms and molecules

that are vibratingthat are vibrating The molecules are farther apart in air than in The molecules are farther apart in air than in

solids and liquidssolids and liquids Sound travels through the air at 340 m/secSound travels through the air at 340 m/sec Longitudinal WavesLongitudinal Waves

4 parts that make a sound4 parts that make a sound

AMPLITUDE- size of duration/ loudnessAMPLITUDE- size of duration/ loudness PITCH- corresponding to frequency of PITCH- corresponding to frequency of

vibrationvibration TIMBRE- shape of the frequency spectrum TIMBRE- shape of the frequency spectrum

of soundof sound DURATION- length of time you can hear the DURATION- length of time you can hear the

notenote

The Human EarThe Human Ear

The human ear responds to frequencies between The human ear responds to frequencies between 20 Hz and 20,000 Hz (frequencies below 20 Hz 20 Hz and 20,000 Hz (frequencies below 20 Hz can be felt but not heard)can be felt but not heard)

Cochlea divides sound into different components Cochlea divides sound into different components before sending it into nerve pathways– allows our before sending it into nerve pathways– allows our brain to hear the musicbrain to hear the music

Measurement of SoundMeasurement of Sound

Sound intensity is measured in decibels (dB)Sound intensity is measured in decibels (dB) Weakest sound we can hear isWeakest sound we can hear is

0 dB = 100 dB = 10-12-12 watts/m watts/m22

Adding ten decibels multiplies the intensity by Adding ten decibels multiplies the intensity by a power of 10a power of 10

Sine WavesSine Waves

Sound is measured in Sound is measured in sine waves. Why is sine waves. Why is this?this?

Proved by differential Proved by differential equation for simple equation for simple harmonic motion:harmonic motion:

kydt

yd

2

2

ktcy sin

ktBktAy sincos

Fourier SeriesFourier Series

The Fourier Series is a series that explains the The Fourier Series is a series that explains the vibrations of piano strings. Because the vibrating vibrations of piano strings. Because the vibrating strings have endpoints, or points where the strings have endpoints, or points where the strings vibrate around, we get this equation to strings vibrate around, we get this equation to explain the shape of the vibrating strings. explain the shape of the vibrating strings.

Fourier Series equation:Fourier Series equation:

Bernoulli's ArgumentBernoulli's Argument

Bernoulli believed that all strings vibrate in Bernoulli believed that all strings vibrate in the same time besides its fundamental tone. the same time besides its fundamental tone. The only difference of the vibrating strings The only difference of the vibrating strings would be that some are higher than others. would be that some are higher than others.

Therefore, according to Bernoulli, all Therefore, according to Bernoulli, all vibrating strings vibrate at the same time, vibrating strings vibrate at the same time, but some vibrate in larger or smaller areas, but some vibrate in larger or smaller areas, giving the note a different tone or pitch. giving the note a different tone or pitch.

Bernoulli’s Argument EquationsBernoulli’s Argument Equations

Bernoulli said the equation for a vibrating Bernoulli said the equation for a vibrating string is:string is:

However, given t = 0, the cosines will cancel However, given t = 0, the cosines will cancel out, thus leaving the equation:out, thus leaving the equation:

Thus, we prove that vibrations resemble sine Thus, we prove that vibrations resemble sine waves waves

How a Guitar WorksHow a Guitar Works

Calculating the fret positionsCalculating the fret positions

The general formula for the distance from the nut to the kth fret is

f(k)=length(1-1/r)

where f(k) is the distance to the kth fret from the nut, length is the total scale length and r is the ratio you want from the fret.

Let's say, for example, you measured a 12" from the nut to the 12th fret and that you want frets placed so as to give a scale of

1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1.

This is an example of a just major scale. The nut is at 1/1, the first fret is calculated by

f(1)=24(1-1/(9/8))=24(1-8/9)=24(1/9)=2.67",

the second fret is calculated by

f(2)=24(1-1/(5/4))=24(1-4/5)=24(1/5)=4.8".

The remaining calculations are similar. The octave is 2/1, so the fret calculation is

f(8)=24(1-1/(2/1))=24(1-1/2)=24(1/2)=12",

just as it should be.

The formula for equal tempered fret spacing is the same, but now instead of ratios, we use powers of two (unless you are using a non-octave scale).

The formula

f(k)=length(1-2^(-k/n))

where the carat is the power sign and n is the number of equal tempered steps to the octave. As an example, let's calculate a few fret measurements for 19tet with a 35" bass guitar scale.

The 0th fret is the nut, and the distance to the first fret is given by

f(1)=35(1-2^(-1/19))=35(1-.9642)=35(.0358)=1.25".

The distance to the 8th fret is

f(8)=35(1-2^(-8/19))=35(1-.7469)=35(.2531)=8.86".

Keep your figures accurate to four places and your final measurement should be accurate to 1/100". Notice that .2531 is very close to 1/4.

This is as it should be, since the 8th fret in 19 approximates the perfect fourth (4/3) very well, but is a little sharp. Plug 4/3 into the equation for just frets and you will get

length(1-3/4)=length(1/4)=length(.25).

Calculus AnimationCalculus Animation

http://www.musemath.com/flash/calculus.swfhttp://www.musemath.com/flash/calculus.swf

A little PoemA little Poem "What could be simpler? Four"What could be simpler? Four

scale-steps descend from Do.scale-steps descend from Do.Four such measures carry overFour such measures carry overthe course of four phases, then home.the course of four phases, then home.

... the theme swells... the theme swellsto four seasons, four compass points, four winds,to four seasons, four compass points, four winds,forcing forth the four corners of the worldforcing forth the four corners of the worldperfect for getting lost in....perfect for getting lost in....

What could be simpler? Not even musicWhat could be simpler? Not even musicyet, but only counting: Do, ti, la, sol....yet, but only counting: Do, ti, la, sol....

Everything that ever summered forth startsEverything that ever summered forth startsin identical springs, or four-note variationsin identical springs, or four-note variationson that repeated theme: four seasons,on that repeated theme: four seasons,four winds, four corners, four-chambered heart...four winds, four corners, four-chambered heart...

Look, speak, add to the variants (what couldLook, speak, add to the variants (what couldbe simpler?) now beyond control. How can we helpbe simpler?) now beyond control. How can we helpbut hitch our all to these mere four notes?"but hitch our all to these mere four notes?"

Constructing a 17 note scaleConstructing a 17 note scale

For 11/6 to be the xth For 11/6 to be the xth tone of the scale, . We tone of the scale, . We substitute in the value substitute in the value of p and solve for x. of p and solve for x. satisfies or .satisfies or .

For 11/6 to be the xth For 11/6 to be the xth tone of the scale, . We tone of the scale, . We substitute in the value substitute in the value of p and solve for x. of p and solve for x.

For 11/6 to be the xth For 11/6 to be the xth tone of the scale, . We tone of the scale, . We substitute in the value substitute in the value of p and solve for x. of p and solve for x.

Thus, for a 17 tone Thus, for a 17 tone scale, the ratio of 11/6 scale, the ratio of 11/6 is closest to the 15th is closest to the 15th tone above 1/1.tone above 1/1.

Circle of FifthsCircle of Fifths

The Willow Flute The Willow Flute

One end is open and the other One end is open and the other contains a slot into which the player contains a slot into which the player blows, forcing air across a notch in the blows, forcing air across a notch in the body of the flute. The resulting body of the flute. The resulting vibration creates standing waves vibration creates standing waves inside the instrument whose frequency inside the instrument whose frequency determines the pitch.determines the pitch.

Wave Equation and solutionWave Equation and solution uu((xx, , tt) ) = = sinsin nnððxx L b L b sinsin ananððtt L L + + c c coscos ananððtt L aL a2 2 ..22uu ..xx2 2 == ..22uu ..tt22

Corollary 2. If qk and pk are the coordinates of ek, k > 0, thenµ -pkqk<1q2.

BibliographyBibliographyBenson, Dave. “Mathematics and Music.” Online. Available: Benson, Dave. “Mathematics and Music.” Online. Available:

http://www.math.uga.edu/~djb/html/music-hq.pdfhttp://www.math.uga.edu/~djb/html/music-hq.pdf, 15 October 2003, 15 October 2003

Boyd-Brent, John. “Pythagoras: Music and Space.” Online. Available: Boyd-Brent, John. “Pythagoras: Music and Space.” Online. Available: http://www.aboutscotland.com/harmony/prop.htmlhttp://www.aboutscotland.com/harmony/prop.html , 24 May 2004 , 24 May 2004

Heimiller, Joseph. “Where Math Meets Music.” Online. Available: Heimiller, Joseph. “Where Math Meets Music.” Online. Available: http://www.musicmasterworks.com/WhereMathMeetsMusic.htmlhttp://www.musicmasterworks.com/WhereMathMeetsMusic.html, 24 May 2004, 24 May 2004

Woebcke, Carl. “Pythagoras and the Music of the Spheres.” Online. Available: Woebcke, Carl. “Pythagoras and the Music of the Spheres.” Online. Available: http://www.myastrologybook.com/Pythagoras-music-of-the-spheres.htmhttp://www.myastrologybook.com/Pythagoras-music-of-the-spheres.htm, 24 May 2004, 24 May 2004