Math and music are related in many different forms. This can be seen through note lengths, scale intervals, the Harmonic Mean, and key signatures. Most of math used in music is in the form of an equation. These equations can be from do simple addition and subtraction to doing complex trigonometry applications. These can deal with anything from simple note values to the equation that relates frequency to pitch. .
In music there are simple equations used to determine the length of notes, like using a " " to add half of the original note value to the note. For instance if you have a quarter note, which is usually only one beat, and add a " " to it, it becomes one and a half beats. This is the equation used to determine the value: 1 + (1/2 * 1) = X. This applies to all notes and rests. This is probably one of the simplest math formulas used in music, but it is one of the most important ones.
The scale area of music can be very complicated because there are several different math ideas worked into it. Going up or down in a scale is expressed as a half step or a whole step. The half steps involve flats and sharps. Going up a step will make the note sharp, and going down a step will make a note flat. Going up or down a note using a whole step is like jumping from an "A" to a "B". In a major scale the pattern goes like this, assuming H is a half step and W is a whole step: W W H W W W H. In a minor scale the pattern follows: .
W H W W W H W. .
The Harmonic Mean was founded by a guy named Palladio. It involves intervals between notes and there differences. It states that the mean of three numbers will exceed one extreme and be exceeded by another extreme by the same fraction of the two extremes. It states that the mean of three numbers will exceed one extreme and be exceeded by another extreme by the same fraction of the two extremes, in other words: (b-a)/a = (c-b)/c. For example, if you aare given the numbers six, eight, and twelve the mean would be eight: (8-6)/6 = (12-8)/12.