Mathematics and Society - OS X Lion Server

BACKGROUND:»LJJ:::J'~:.,:: ~.:.., fJ·iJ: ~.:;. § X':,·· V······ I'·:" []] 00·" ..~! {} !r;~ :.;...:; n~ :.\:'.:; i~ i~ i! :U! ({[...... ~:.... .' ...:. '::;:. :.... :~:.'. .: ....:. ' .. '.. :...Tunes can be composed using familiar number sequences for themes. Possibilitiesare Fibonacci numbers, Lucas numbers, square numbers, triangular numbers, primes, evens,multiples and factors. (The booklet Fibonacci and Lucas Numbers in the Houghton MifflinEnrichment Series gives interesting properties of Fibonacci and Lucas numbers.)The process involves writing down the sequence of numbers and converting it to noteson a scale. Let's use the Fibonacci numbers as an example:I, I, 2, 3, 5, 8, 13, 21, 34, 55, 89,Each number is found by adding the previous two numbers. (1 + 15, 3 + 5 = 8, • • .)2, 1 + 2 3, 2 + 3 =The first five numbers of the Fibonacci sequence could easily represent positionson a given scale, but what should be done about the numbers 8, 13, •••? Most notesused in songs belong to one of seven places on a major melodic scale. For example, ifwe wanted to write a song in the key of C, most of the notes would be included in theset of C D E F GAB. The symbols or letters could also be assigned numbers accordingto their position:CDEFGAB1234567Since most number patterns use more than the first seven numbers, we need to find asystem that will convert any given number to a number less than 8. Starting on C in thekey of C, the first and eighth notes are both C, the second and ninth are both D, and soOn because of the octave arrangement of scales. We can substitute 7 for any number thatis a multiple of 7. For any other number, substitute the remainder when the number isdivided by 7. The Fibonacci sequence is then converted to the following sequence ofnumbers ..1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,1, 1, 2, 3, 5, 1, 6, 7, 6, 6, 5,Notes are then written in the new sequence, with whatever rhythm seems appropriate.Here is one possible theme from Fibonacci:~At0Yo COULD BE WRITTEt-l AIJ OC,AVE00 HIGHER;:R LOWER. ./_~00I. 0I/ ~ till I-{( ""\ I._.- ~ I 0....... ./~ I'•• --I(fJ- -......,0, T\.n:''''E~NO,':':>~More measures can be written by extending the number sequence, by repeating a partof the sequence, by reversing the order of the sequence and so on. Motions and Musicgives more ideas that can be used to create additional related measures of music.Extension:Try composing music with 1the digits of the decimal representation of arepeating decimal like - or an irrational number like ~.~ ~ 3.141592653589793238462643383279502488409716939937510•••••••••305