The relationship between mathematics and music makes music one of the richer realms to solidify math concepts in children. From integer operations to fractions, there’s a musical analogue.

For example, transposition (which is moving a composition from one key to another) can be thought of in several ways, according to the child’s level of math exposure. The way that transposition works is as follows: suppose you’re transposing from C Major to F Major. First, you count the gap between C and F on a keyboard—much like Mathnasium’s method of subtraction between two numbers. There are 5 semitones between C and F: C sharp, D, E flat, E, and F. Then, you add 5 semitones to every single note in the composition you’re transposing. Hence, if your composition in C Major begins with C, D, E, B, C, your transposed composition would begin with F, G, A, E, F.

For a younger student, transposition is like adding an integer to a sequence of numbers. If the keyboard’s middle C were labelled 0, the origin, then the keys above would be positive and the keys below, negative. Hence, transposing from C Major to F Major would be equivalent to adding (+ 5); however, it would also be equivalent to adding (− 7), due to the fact that the keyboard repeats itself every 12 semitones.

For an older student, transposition is like a phase shift to the right or left, with the equationy=f(x-k), where k is the number of semitones you’re adding. Moreover, the periodicity of the keyboard, in which adding (+ 5) semitones is equivalent to adding (− 7) semitones, reminds the student of the periodicity of sinusoidal curves, in which a phase shift of π/2 to the right is equivalent to a phase shift of − 3π/2 to the left, when applied to a sine function with period equal to 2π .

Meanwhile, rhythm, another aspect of music, may help children struggling with fractions. For example, the time signature 34, in which a quarter note equals one beat and there are 3 quarter notes in one bar, reminds the elementary student of the fraction 34 in which the fraction’s denominator gives the nomination of the fraction, in this case, the quarter, and the numerator gives the quantity of the denomination, 3 quarters. Also, if the student knows a certain bar of 34 features only one quarter note, then he or she implicitly realizes that only one-third of the bar has a note, and the remaining two-thirds of the bar have a rest.

More complex rhythms avail themselves to more advanced students. Take Debussy’s Arabesque No. 1, in which left handplays eighth notes (12 of a quarter note) concurrently with the right hand's triplets (13 of a quarter note). When first learning this piece, the music student innately subdivides a quarter-note beat into sixths (the least common denominator of 2 and 3), so that eighth notes take up three-sixths and triplets, two-sixths. This fractional subdivision is necessary in order to render the hemiola pattern exact. Later, once the student had achieved exactness, he or she no longer needs the rigorous subdivision of the sixths and can play in Debussy’s spirit of languorous Romanticism.

Yet, these are surface-level connections between math and music; more advanced topics such as group theory, topology, and Fourier analysis are presently being applied to music1. Even so, the ability and willingness to perform research at this level require a thorough understanding of elementary math concepts, an understanding that music promotes.

Source:
1 http://www.ams.org/samplings/math-and-music