In Search of the Imperfect Fifth (a non-technical explanation of modern tuning)

Temperament is the compromising of pure intervals of just intonation in the tuning of a musical instrument in order to satisfy broader requirements in the composition and performance of music. This was done to avoid problems caused by the Pythagorean comma, the difference in pitch between seven octaves (ratio 2:1) and twelve justly tuned perfect fifths (Ratio 3:2)

Modern Western music tends to make heavy use of triads, that is, a note, its third, and its fifth, which is called a major chord. On an instrument tuned in the Pythagorean way, this does not sound good, prompting Renaissance musical theorists to look at ways to mitigate the problems and expand the realm of possibilities in musical composition.

The main weakness of the Pythagorean system is that fifths will never map to a note that is a pure octave above the starting note, making it impossible to have an octave that is a closed loop. To understand why this is a serious problem, imagine a series of perfect fifths beginning on C. This would give: C, G, D, A, E, B, F sharp, C sharp, G sharp, D sharp, A sharp, E sharp, and B sharp. Using pure fifths, this B sharp is actually higher in pitch than C, seven octaves above the starting note. This necessitated having eleven pure-fifths and one small (wolf) fifth that sounded disturbingly dissonant.

In 1581, Vincenzo Galilei, the father of Galileo, advocated the 12-note equal temperament model, which is now commonplace in modern Western music. An octave is divided into twelve semitones of equal temperament, which is to say that the ratio of every note to its adjacent notes is the same, the twelfth root of two. The note A at 440Hz is commonly used as the reference point to tune to concert pitch because A has an integer number of vibrations per second. To a musician not particularly interested in theory, this would all boil down to the fact that each change in the note is a diatonic half-step and would sound evenly spaced in pitch intervals.

Equal temperament does result in a fifth that is slightly smaller than a pure fifth and a major third that is slightly larger than a pure major third, but the error is small enough not to be disturbing. In an equal temperament system where pure fifths are not used in the circle of fifths, the last B sharp would be an enharmonic equivalent, that is, equal in pitch, to C seven octaves above the initial starting note, giving a closed loop of fifths and octaves. To accomplish this, a detuning of 2 cents (a fiftieth of a semitone) from each fifth is required.

Is the two-cent detuning a problem?

The ratio difference between an equally tempered fifth and a pure fifth is tiny. 1.4983 instead of 1.5 (3/2), a discrepancy of 0.0017. The largest discrepancy from the pure note is the major seventh, 1.8897 instead of 1.8333 (11:6), a discrepancy of 0.0564. These tiny errors are almost certain to be overshadowed by imperfections in the intonation of the musical instrument being tuned. Even in the most perfectly precise guitar, there is going to be some stretching of the string when the musician pushes down on the fret.

A singer who can pitch with uncommon precision will consistently pitch to within 5 cents, which is far higher than the 0.02 semitone correction factor on the perfect fifth. Most people do not find the small errors inherent in human vocal performances disturbing; in fact, most people don’t notice them at all. As a point of interest, the human voice is not affected by the Pythagorean comma because a human vocal performer can instinctively avoid dissonance while pitching a note.

There was opposition to the equal temperament system of tuning, notably from J.S. Bach, who felt that compromising note intervals in this way was sullying the purity of the music being produced under the new system. Bach advocated a system called well temperament.

The various well-temperament tunings tried to strike a balance between having or staying close to pure intervals and not having wolf intervals. A well-tempered instrument might have several justly tuned perfect fifths in it and some smaller fifths, but not small enough to be wolf intervals. In such systems, tuning would be noticeably different in each key, but every key would still be usable. Because of this tuning variance between keys, different keys in well-temperament were considered to correlate with different colors and emotions.

While some musicians still advocate the use of well-temperament, at least for compositions created with it in mind, in the modern day, the vast majority of musicians and theorists unequivocally accept equal temperament as the standard system for tuning musical instruments. Equal temperament is by far the more straightforward system, not requiring retuning for music that changes key. It is also far better suited to music that is chromatic or harmonically complex for the same reason.

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