Other Interesting Math/Music connections
This section is dedicated to other connections between mathematics and music
1. How does listening to music help me with math?
There have been numerous studies on this topic. For one, quite often listening to music also means interpreting it. And by interpreting it you often use certain math skills, which may not immediately improve your math skills, but will heighten your sense of awareness of certain mathematical aspects.
One study which comes up time and time again is the "Mozart Effect". A number of students were isolated, one group listening to music by Mozart, another listening to complete silence, and a third listening to meditation tapes. A short period of time after listening to said music, each group was giving a series of skill and academic tests. In almost all these tests, the Mozart group came out on top.
As stated in The Boston Globe, and
quoted at http://www.math.niu.edu/~rusin/uses-math/music,
"After listening for 10 minutes to a tape of Mozart's sonata
for two pianos in D major, K. 488, college students in that
earlier experiment scored approximately 9 points higher in IQ
tests of abstract spatial reasoning than subjects exposed to 10
minutes of silence or a
meditation tape. Spatial reasoning tasks, which are generally
processed by the brain's right hemisphere, involve the
orientation of shapes in space. Such tasks are relevant to a wide
range of endeavors, from higher mathematics and geometry to
architecture, engineering, drawing and playing chess."
2. How does playing music help me with math?
Ask any person who plays music and they will tell you that math ability is one of the most essential parts of playing or singing music. More so in the case of a person who plays an instrument such as percussion, or a bass line instrument, but still throughout all players. The ability to count off the time (especially when it's divided into 64th's!), or to transpose notes is key to the playing or singing of music. Also, in addition to playing/singing music, one is listening, and this reverts back to the previous topic.
3. How does being good at math make me good at playing/interpreting music?
Any student that is quick in math will be much more able to think on their feet when counting rhythms, or transposing music, or figuring out note values/time signatures. A keenness in math also makes it easier for a musician to count out the beats of written music. For example, if in one bar there are a number of 64th notes, dotted notes, and triplets, one with strong math abilities should be able to count it out crisply and cleanly. Another way strong math skills help a musician when interpreting music, or doing music theory, are in such examples as displayed on the other two main pages of this site. For example, if you are scoring music for a number of percussion instruments, or other noisemakers which are not necessarily typical of most music, you can use frequencies easily to figure out how they will sound. Say you are writing music for an orchestra, but in addition to this orchestra, for whatever reason, you decide to include a car horn which honks at a frequency of 660 Hz, and a starter's pistol which makes a sound which measures at 500 Hz, you will be able to select easily the nearest measurable note, and harmonise accordingly. These are just a few examples of how mathematical ability helps improve musical ability.
This is one somewhat fun part to this site. It is actually taken from www.geocities.com/Vienna/9349/. The creator of this site has used algorithms to take well known mathematical functions, equations, models, and values to create very pleasant-sounding music. The following is taken from that site (Called The Sound of Mathematics), and describes briefly how the files were created:
"For some compositions, I have used the following methods to keep numbers within a specifiable range. Here n is a natural number and r is the range (the number of discrete elements):
Conversion Convert n to base r.
Congruence Arithmetic Replace n by n mod r, i.e. the
remainder when n is divided by r.
Digit Sums Replace n by its digit sum, repeatedly if
necessary, until it is less than r
Digit Products Replace n by its digit product, repeatedly if
necessary, until it is less than r.
Periodic Functions Replace n by r / 2 cos n - 1 or r / 2 sin
n - 1 rounded off to the nearest integer.
Mapping of Numbers to Tones: The following methods have been used in various ways to map numbers to tones:
Duration:
All digits/numbers maps to a specified note value
Digits or groups of digits maps to specified note values.
The number n of consecutive equal digits/numbers maps to a tied n
multiple of a specified note value.
The number of digits m of a number maps to the note value 1/m.
Pitch:
Digits or groups of digits maps to specified pitches,
including a pause and change of octave.
The number n maps to the pitch n scale steps relative to the
pitch of the previously determined tone
Structure:
Digits of numbers of different powers maps to separate parts
Parts are initiated later in a piece by numbers satisfying
specified conditions
To download the MIDI file of Pi, click here
5. Why is the harp shaped the way it is?
This is an extremely complex question. To put it simply, one cannot imagine the size it would have to be if the string length were to double with every octave drop (frequency doubles or halfs with an octave up or down). It is put best by Mathematics and Music:
"If the harpist were at the origin and the harp on the positive x-axis, (the harpist sits at the high-note end) then the string at position x would have to have a length varying like y = A * 2^x (with distances along the x-axis measured in octaves). To make room for the harpist's body, the bottom of the string is not on the floor; rather, it's attached to the soundboard -- a straight but tilted plane. Thus the bottom of the string which is x octaves down from the highest note is elevated to a height of the form y = a * (1 - x/k) where a is the height of the bottom of the shortest string (a is about 5 feet) and k is the number of octaves until you reach the floor (about 6). We know now where the bottom of each string is, and we know how long it is; where's the top? Must be at y = a * (1 - x/k) + A * 2^x. We can even determine A here if we assume that the top of the bass strings is about equal to the height of the top string: we need y(k) = y(0), which is roughly A*2^k = a. So the shape of the top of the harp is something like A*( 2^k*(1-x/k) + 2^x )"