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

Mathematics in Music. By Ted Fitzgerald, Allison Gibson, Kaitlin Spiegel, and Danny Spindler. History. 500 BC – Greek mathematician Pythagoras experiments with changing the lengths of strings to produce different tones Plucking the strings creates vibrations which is what musical tones are.

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

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  1. Mathematics in Music By Ted Fitzgerald, Allison Gibson, Kaitlin Spiegel, and Danny Spindler

  2. History • 500 BC – Greek mathematician Pythagoras experiments with changing the lengths of strings to produce different tones • Plucking the strings creates vibrations which is what musical tones are

  3. Pythagoras discovered that two strings, one half as long as the other, would produce the same tone, except at different pitches or frequencies • Making one string half as long as the other causes it to vibrate twice as fast as the other (the frequency is doubled) • The faster the vibration, the higher the pitch

  4. The scale… The distance between a note and another vibrating twice as fast is called an octave In-between these two notes are 11 others creating the twelve-note chromatic scale The notes in the scale are: A, A#, B, C, C#, D, D#, E, F, F#, G ,G#

  5. Harmonies • Pythagoreans found that only harmonious musical intervals were produced by dividing a string so that the two resultant lengths were in the simple ratios of 2:1, 3:2 (shown below), 4:3, or 5:4. • They called these intervals the octave, the fifth, the fourth and the third. Thus the numbers 1, 2, 3, 4 and 5 could produce all of the musical intervals they considered pleasing.

  6. Why is it that these combinations of notes sound good together, while others make you cringe?The Answer: It all has to do with frequencies.

  7. Frequencies that match up at regular 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.

  8. C- 261.6 Hz E- 329.6 Hz G- 392.0 Hz Ratio of E to C is about 5/4 That means that every 5th wave of E matches up with every 4th wave of C, producing a pleasant sound. Frequencies of some notes in C major:

  9. Vibrations and Sound • Sound is vibrations in the air • Air is a collection of atoms and molecules that are vibrating • The molecules are farther apart in air than in solids and liquids • Sound travels through the air at 340 m/sec • Longitudinal Waves

  10. 4 parts that make a sound • AMPLITUDE- size of duration/ loudness • PITCH- corresponding to frequency of vibration • TIMBRE- shape of the frequency spectrum of sound • DURATION- length of time you can hear the note

  11. The Human Ear • The human ear responds to frequencies between 20 Hz and 20,000 Hz (frequencies below 20 Hz can be felt but not heard) • Cochlea divides sound into different components before sending it into nerve pathways– allows our brain to hear the music

  12. Measurement of Sound • Sound intensity is measured in decibels (dB) • Weakest sound we can hear is 0 dB = 10-12 watts/m2 Adding ten decibels multiplies the intensity by a power of 10

  13. Sine Waves • Sound is measured in sine waves. Why is this? • Proved by differential equation for simple harmonic motion:

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

  15. Bernoulli's Argument • Bernoulli believed that all strings vibrate in the same time besides its fundamental tone. The only difference of the vibrating strings would be that some are higher than others. • Therefore, according to Bernoulli, all vibrating strings vibrate at the same time, but some vibrate in larger or smaller areas, giving the note a different tone or pitch.

  16. Bernoulli’s Argument Equations • Bernoulli said the equation for a vibrating string is: • However, given t = 0, the cosines will cancel out, thus leaving the equation: • Thus, we prove that vibrations resemble sine waves

  17. How a Guitar Works

  18. Calculating 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.

  19. 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).

  20. Calculus Animation • http://www.musemath.com/flash/calculus.swf

  21. A little Poem • "What could be simpler? Fourscale-steps descend from Do.Four such measures carry overthe course of four phases, then home. • ... the theme swellsto four seasons, four compass points, four winds,forcing forth the four corners of the worldperfect for getting lost in.... • What could be simpler? Not even musicyet, but only counting: Do, ti, la, sol.... • Everything that ever summered forth startsin identical springs, or four-note variationson that repeated theme: four seasons,four winds, four corners, four-chambered heart... • Look, speak, add to the variants (what couldbe simpler?) now beyond control. How can we helpbut hitch our all to these mere four notes?"

  22. For 11/6 to be the xth tone of the scale, . We substitute in the value of p and solve for x. satisfies or . For 11/6 to be the xth tone of the scale, . We substitute in the value of p and solve for x. For 11/6 to be the xth tone of the scale, . We substitute in the value of p and solve for x. Thus, for a 17 tone scale, the ratio of 11/6 is closest to the 15th tone above 1/1. Constructing a 17 note scale

  23. Circle of Fifths

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

  25. Wave Equation and solution • u(x, t) = sin • nðx • L b sin • anðt • L + c cos • anðt • L a2 .2u • .x2 = • .2u • .t2

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

  27. Bibliography Benson, Dave. “Mathematics and Music.” Online. Available: http://www.math.uga.edu/~djb/html/music-hq.pdf, 15 October 2003 Boyd-Brent, John. “Pythagoras: Music and Space.” Online. Available: http://www.aboutscotland.com/harmony/prop.html , 24 May 2004 Heimiller, Joseph. “Where Math Meets Music.” Online. Available: http://www.musicmasterworks.com/WhereMathMeetsMusic.html, 24 May 2004 Woebcke, Carl. “Pythagoras and the Music of the Spheres.” Online. Available: http://www.myastrologybook.com/Pythagoras-music-of-the-spheres.htm, 24 May 2004

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