Every pure musical tone has a frequency, the number of oscillations per second in the sound wave. Doubling the frequency corresponds to moving up one octave. A musical note consists of a base frequency or pitch, called the fundamental together with a series of harmonics, or oscillations whose frequencies are whole-number multiples of the fundamental frequency.
Each musical instrument has a different pattern of harmonics that give it an individual quality or character. Thus, the note A sounds quite distinct when played on an oboe or a clarinet: the fundamentals are the same, but the overtones or harmonics are different.
In western music, the octave, ranging from a tonic note to one of twice the frequency, has twelve distinct notes, the interval between successive notes being a semitone. There are many ways to determine the precise frequencies of the notes, no one scheme being ideal, the best scheme depending on the type of music.
Pythagoras discovered that a perfect fifth, with a frequency ratio of 3:2, is especially consonant. The entire musical scale can be constructed using only the ratios 2:1 (octaves) and 3:2 (fifths). Our perception of musical intervals is logarithmic, so adding intervals corresponds to multiplying frequency ratios.
In the tonic sol-fa scale the eight notes of the major scale are Do, Re, Mi, Fa, So, La, Ti, Do. Starting with “Do”, the ratio 3:2 brings us to “So”. Moving up another fifth, we have the ratio 9:4. Reducing this by 2 to remain within the octave, we get 9:8, the note “Re”. Moving up another fifth gives a ratio 27:16 and brings us to “La”. Continuing thus, we get all the “white notes” in the major scale. This is called Pythagorean tuning:
The Pythagoreans noticed that 2^19 ≈ 3^12, so going up twelve fifths, with ratio (3/2)^12 and down seven octaves (1/2)^7 gets you back (almost) to your starting point. The number 3^12/2^19 ≈ 1.01364 is called the Pythagorean comma. The frequencies of enharmonics such as F sharp and G flat, differ by this ratio.
Circle of Fifths
The Circle of Fifths is a diagram representing the relationship between musical pitches and key signatures. It is a geometric diagram showing the twelve tones of the chromatic scale. The Circle is useful in harmonising melodies and in building chords.
We start at the top with middle C, the base note of the scale of C major which has no sharps or flats. Moving clockwise to one o’clock we have G (1 sharp), then D (2 sharps) and so on to F sharp (which has six sharps) at 6 o’clock. Proceeding counter-clockwise from the top, we have F (1 flat), B flat (2 flats) and onward to G flat (six flats) at 6 o’clock. The notes F sharp and G flat are called enharmonics.
Triads and just intonation
The triad, three notes separated by 4 and 3 semitones, such as C–E–G, is of central importance in western music. In the tuning scheme of Pythagoras, the third (C–E) has a frequency ratio of 81:64. Generally, ratios with smaller numbers result in more pleasant sensations of sound. A more consonant sound is given by replacing 81:64 by 80:64 = 5:4. The three notes of the triad C–E–G are then in the ratio 4:5:6. Likewise, changing the sixth note (A or “La”) from 27:16 to 25:15 = 5:3 makes F–A–C a perfect triad with the frequency ratios 4:5:6. Finally, if the monstrous 243:128 is replaced by 240:128 = 15:8, we get a scheme of tuning called just intonation:
It is impossible to tune an instrument like a piano so that all fifths have perfect frequency ratios of 3:2. There are tuning schemes that make small compromises so that all ratios are close to the perfect values, and modulation or change between one key and another is possible. In such tempered scales, the ratios are all slightly imperfect, but close enough to be acceptable to the ear.
In western music it is essential to be able to change smoothly between keys without dissonant consequences. The scheme called equal temperament achieves this. The idea is to make all semitone intervals equal. There are twelve semitones in an octave, and adding intervals corresponds to multiplying frequency ratios. Since an octave has ratio 2:1, we need a number that yields 2 when multiplied by itself 12 times. This number is just the twelfth root of 2 or 2^(1/12) ≈ 1.059 and it is the key (!) to modern tuning. In this system, no harmonic relationships are perfect, but all are acceptable to the ear.
In equal temperament, the twelve steps all have identical frequency ratios. There are no enharmonics: F sharp and G flat are identical. In the perfect harmonic system, they are distinct notes. Equal temperament is a compromise, and there are those who argue that it is “like black & white compared to the colour of just intonation”. The best way to decide is to listen.
 Benson, D J, 2007: Music: A Mathematical Offering. Camb. Univ. Press. ISBN: 9-780-521-61999-8.
 Harkleroad, Leon, 2006: The Math behind the Music. Camb. Univ. Press. ISBN: 9-780-521-00935-5.
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