Math and music have always been considered
closely connected in many ways.
• It is widely believed that students who do well in music also excel in math.
• Let’s take a look at some of the basic components of music and see what math has to do with them.
Rhythm is to Music as Numbers are to Math
Rhythm measures time
• 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
• When you play a few different notes together or even repeat the
same note on an instrument, you create something called rhythm.
Pythagoras
• The Greek octave had a mere five notes.
• Pythagoras pointed out that each note was a fraction of a string.
• 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
Ratios
• Pythagoras was excited by the idea that these ratios were made up
of the numbers 1,2,3,4, and 5.
• Why ?
• Pythagoras imagined a "music of the spheres" that was
created by the universe.
• The 18th century music of J. S. Bach, has mathematical
undertones, so does the 20th century music of Philip Glass.
Golden Ratio and Fibonacci
• It is believed that some composers wrote their music using the
golden ratio and the Fibonacci numbers to assist them
• Golden Ratio: 1.6180339887
• Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21
From Then to Now
• So, how did we get the 12 notes scale out of these six notes ?
• Some unknown follower of Pythagoras tried applying these ratios
to the other notes on the scale.
• 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.
• 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 #.
• 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.
• 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.
• It is widely believed that students who do well in music also excel in math.
• Let’s take a look at some of the basic components of music and see what math has to do with them.
Rhythm is to Music as Numbers are to Math
Rhythm measures time
• 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
• When you play a few different notes together or even repeat the
same note on an instrument, you create something called rhythm.
Pythagoras
• The Greek octave had a mere five notes.
• Pythagoras pointed out that each note was a fraction of a string.
• 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
Ratios
• Pythagoras was excited by the idea that these ratios were made up
of the numbers 1,2,3,4, and 5.
• Why ?
• Pythagoras imagined a "music of the spheres" that was
created by the universe.
• The 18th century music of J. S. Bach, has mathematical
undertones, so does the 20th century music of Philip Glass.
Golden Ratio and Fibonacci
• It is believed that some composers wrote their music using the
golden ratio and the Fibonacci numbers to assist them
• Golden Ratio: 1.6180339887
• Fibonacci Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21
From Then to Now
• So, how did we get the 12 notes scale out of these six notes ?
• Some unknown follower of Pythagoras tried applying these ratios
to the other notes on the scale.
• 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.
• 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 #.
• 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.
• 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.
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