Earlier, the interplay between math and music was subtle and most likely more felt than thought. By the time of the ancient Greeks, music was defined and even restricted by the mathematics that dictated its theory. The relationship between the two disciplines has gone through many changes since then. Now, although mathematics still provides a basis for music theory, music is not thought of as a strictly mathematical discipline. The historical development of tuning theory, which involves the mathematical relationships that dictate the tuning of an instrument, sheds much light on the nature of this transition. For practical purposes, our discussion of this theory will center on the piano.

The piano keyboard consists of 88 white and black keys and a pattern that repeats every 12 keys. The portion the repeats contains 7 white keys and 5 black keys. The white keys are giving the letter names A through G, and the black keys are represented by a sharp symbol or a flat symbol, depending on the situation. For instance, the black key between C and D has two names: C-sharp and D-flat.

The distance between two adjacent keys on the piano is called a half step (for example, between C and C-sharp); two half steps make a whole step (for example, between C and D). A sharp raises a note a half step, while a flat lowers it a half step.

A scale is a series of musical tones with specified distances, or intervals, between them. If you start with any C on a piano keyboard and play all the white notes up to the next C, you will hear the familiar major scale on which many Western songs are based. The major scale is made up of two whole steps, a half step, three more whole steps, and another half step. The syllables do, re, mi, fa, sol, la, and ti are often used to name the seven tones of the major scale.

The frequencies, in Hz (vibrations per second), of the notes in this scale produced by a piano are the following (rounded to the nearest tenth):

Notice that the frequency of A₄ (440 Hz) is exactly twice that of A₃ (220 Hz). This is no mere coincidence. For any two notes that are an octave apart, the ratio of the frequency of the high note to the low note is always 2:1.

In addition, a special relationship exists between any two adjacent notes in the chromatic scale (a scale of 13 adjacent half step notes). Since all half steps in the chromatic scale sound the same distance apart, the frequencies of all the pitches should be somehow related. Close inspection reveals that the frequency of any note is the product of the frequency of the note before it and a constant factor. Simple arithmetic shows that factor to be 1.059463094.... That is, the frequencies of any two adjacent notes are in the ratio 1.059...:1.

Where does the number 1.059... come from? There are 12 half steps in an octave. A and A-sharp, for example, form an interval of a half step. Call the ratio of their frequencies h. Then, A-sharp/A = h and A-sharp = Ah. The following pattern emerges:

Within any octave, the frequency relationships are as follows:

The frequencies of the rest of the notes on the piano can be found by successively multiplying and dividing by h.

When an instrument is tuned to these frequencies, with the octave divided into 12 equal parts and the frequency ratio between half steps equal to 1.059..., it is said to be even-tempered, or tuned in equal temperament.

Equal temperament provides only one of many possible ways of assigning intervals to a scale. What is the advantage of dividing the octave into only 12 tones? After all the audible range of frequencies is approximately from 20 Hz to 20,000 Hz, and there are an infinite number of frequencies, and therefore tones, within this range, and also an infinite number of frequencies within an octave. Practically, a trained music ear can only distinguish about 100 different tones within an octave - not an infinite number. The actual number of tones used in musical composition and performance is much smaller; a simple melody consists of only a few notes, a piano keyboard has only 88 keys, and even a full symphony orchestra can produce only a finite number of audible pitches.

The number of notes in a scale varies from culture to culture. 2400-year-old sets of musical bells based on a 12-note scale have been discovered in China. And in Asian music, the octave is usually divided into 24 tones - within each interval of a half step there are two quarter tones. However, most cultures around the world divide the huge spectrum of tones in a n octave into scales consisting of 5 to 7 notes.

The technical definition of a tuning for an instrument is "a method for creating intervals that can be expressed in integer ratio," whereas temperament is "a modification of tuning that uses irrational numbers to express the ratios of some or all of its intervals." Before the advent of equal temperament, the methods for tuning instruments were truly tunings in the strict sense - the intervals were defined by ratios of integers.

Pythagoras' experiments with the monochord (a simple instrument consisting of a string stretched over a moveable bridge) led to a method for tuning instruments with intervals in integer ratio, known as Pythagorean tuning. The scale produced by this tuning, called the Pythagorean diatonic scale, was used for many years in the Western world. It can be derived from the monochord, and, consistent with Pythagorean doctrine, all of its intervals can be expressed as ratios of integers.

Let's build a diatonic scale based on a monochord whose string, when plucked, produces C₁. Three fourths of the string, when plucked, yields the fourth (fourth note) F₁; two thirds of the string yields the fifth (fifth note) G₁; and one half of the string yields the octave (eighth note) called C₂. Remember that the frequency ratio is the reciprocal of the string-length ratio.

Throughout history, mathematics has both thwarted and aided musical progress. The changing conception of what a musician is reflects the changing relationship between mathematics and music. Centuries ago, a musician was one who studied the mathematical relationships underlying the tonal relationships and judged music using reason rather than the ear. Today, a musician is anyone who writes or makes music, whether or not hi has a theoretical understanding of the mathematics behind it. Yet even though music is not longer a strictly mathematical discipline, mathematics will forever be inherent in music, and will continue to influence the evolution of music theory.