Meantone temperament in the context of "Pythagorean tuning"

A regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth (700 cents), which is the basis behind the circle of fifths.

When only two generators are needed, with one of them the octave, this is sometimes called a "linear temperament". The best-known example of linear temperaments are meantone temperaments, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave. Other linear temperaments include the schismatic temperament of Hermann von Helmholtz and miracle temperament.

In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament."

Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.

Quarter-comma meantone, or ⁠1/4⁠-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning (frequency ratio 3:2); the result is ${\displaystyle {\tfrac {3}{2}}\times \left({\tfrac {80}{81}}\right)^{1/4}={\sqrt[{4}]{5}}\approx 1.49535,}$ or a fifth of 696.578 cents. (The 12th power of that value is 125, whereas 7 octaves is 128, and so falls 41.059 cents short.) This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds (with a frequency ratio equal to 5:4). It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible". Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

In classical music from Western culture, a diminished third (Play) is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A♯ to C, and from A to C♭ are diminished thirds, two semitones wide. Being diminished, it is considered a dissonant interval.

In 12-tone equal temperament a diminished third is enharmonic with the major second, both having a value of 200 cents. However, in meantone tunings with fifths flatter than 700 cents, the diminished third is wider than the major second. In 19 equal temperament it is in fact enharmonically equivalent to an augmented second, both having a value of 252.6 cents. In 31 equal temperament it has a more typical value of 232.3 cents. In a twelve-note keyboard tuned in a meantone tuning from E♭ to G♯, the dimininished third appears between C♯ and E♭, and again between G♯ and B♭.