Math and Music

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  • There are two constant values in music. The first is that the A note that is 9 white keys below middle C has a frequency of 440 hz. The second constant value in music is the 12th root of 2 (1.0594630943593...) which is the ratio of the frequencies between half tones. So, the frequency of A# is 440 × 1.059... = 466.16376... The frequency of B is 466.1637 × 1.0594 = 493.8833. After you do this 12 times you end up with A an octave higher which equals 880hz. Doubling the frequency creates a note an octave higher. Reversely, dividing the frequency in half creates a note an octave lower.
  • There are two constant values in music. The first is that the A note that is 9 white keys below middle C has a frequency of 440 hz. The second constant value in music is the 12th root of 2 (1.0594630943593...) which is the ratio of the frequencies between half tones. So, the frequency of A# is 440 × 1.059... = 466.16376... The frequency of B is 466.1637 × 1.0594 = 493.8833. After you do this 12 times you end up with A an octave higher which equals 880hz. Doubling the frequency creates a note an octave higher. Reversely, dividing the frequency in half creates a note an octave lower.
  • The first person to make the connection between math and music was Pythagoras of Samos, a famous philosopher and cult leader who lived most of the time in southern Italy in 5th century BC.For Pythagoras, ratios were everything. He believed every value could be expressed as a fraction (he was wrong, but that is a whole different story). He also is the first to believe in the idea that mathematics is everywhere.
  • He also thought that there were five planets that moved along similar ratios and that all this meant something (ultimately the universe turns out to be irrational, which may explain a lot). A wonderful idea that inspired many composers.
  • Math and Music

    1. 1. Math and Music<br />How are they related?<br />A Think Quest<br />By Ms. Cogley<br />
    2. 2. Math & Music<br />Math and music have always been considered closely connected in many ways.<br />It is widely believed that students who do well in music also excel in math. <br />Let’s take a look at some of the basic components of music and see what math has to do with them. <br />
    3. 3. Rhythm is to Music as Numbers are to Math<br />Rhythm measures time<br />Measure is the space between two bar lines on the staff that represents the division of time by which air and movement of music are regulated<br />When you play a few different notes together or even repeat the same note on an instrument, you create something called rhythm.<br />
    4. 4. "Give me an A" = 440hz<br />Music is made up of sound. <br />Sound is made from repeating sound waves. <br />The musical pitch of each note has a corresponding frequency measured physically in hz (hertz) or cycles per second. <br />There are some important mathematical relationships between the notes played in music and the frequency of those notes.<br />
    5. 5. A table of Frequencies<br />
    6. 6. Pythagoras<br />The Greek octave had a mere five notes. <br />Pythagoras pointed out that each note was a fraction of a string. <br />Example: Lets say you had a string that played an A. The next note is 4/5 the length (or 5/4 the frequency) which is approximately a C. The rest of the octave has the fractions 3/4 (approximately D), 2/3 (approximately E), and 3/5 (approximately F), before you run into 1/2 which is the octave A<br />
    7. 7. Ratios<br />Pythagoras was excited by the idea that these ratios were made up of the numbers 1,2,3,4, and 5.<br />Why?<br />Pythagoras imagined a "music of the spheres" that was created by the universe. <br />The 18th century music of J. S. Bach, has mathematical undertones, so does the 20th century music of Philip Glass.<br />
    8. 8. Golden Ratio and Fibonacci<br />It is believed that some composers wrote their music using the golden ratio and the Fibonacci numbers to assist them<br />Golden Ratio: 1.6180339887<br />Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21<br />
    9. 9. From Then to Now<br />So, how did we get the 12 notes scale out of these six notes? <br />Some unknown follower of Pythagoras tried applying these ratios to the other notes on the scale.<br />For example, B is the result of the 2/3 ratio note (E) applied to itself. 2/3 * 2/3 = 4/9 which lies between octave A (1/2) and octave C (4/10). To put B in the same octave we multiply 4/9 by two to arrive at 8/9. G is produced backward from A. As B is a full tone above A at a string ratio of 8/9, we can create a missing tone below A by lengthening the string to a ratio of 9/8. To add G to the same octave we apply 9/8 to 1/2 (octave A) and by multiplication we get 9/16 as the ratio to G. <br />BUT! There was a problem, however, if you performed this transformation a third time. The 12 tone octave created by starting with an A was different than the 12 tone octave created when you started with an A#. <br />Which means that two harps (or pianos, or any other instrument) tuned to different keys would sound out of tune with one another. Also, music written in one scale could not be transposed easily into another because it would sound quite different.<br />The solution was created around the time of Bach. A "well tempered" scale was created by using the 2 to the 1/12th power ratio mentioned above. Using an irrational number to fix music based on ratios, Pythagoras probably rolled over in his grave.<br />
    10. 10. Your Turn<br />I would like for you to research mathematics and it’s connection to music<br />Conduct a survey of students who played an instrument, for how long, and their grades in mathematics<br />I want to know how you feel about the two and their connection.<br />
    11. 11. References<br />http://en.wikipedia.org/wiki/Fibonacci_number<br />http://library.thinkquest.org/4116/Music/music.htm<br />http://www.musicmasterworks.com/ConsonanceComplication/TheComplicationWithConsonance.htm<br />http://www.mathhiker.com/archives/496<br />http://www.musicmasterworks.com/WhereMathMeetsMusic.html<br />

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