Perfect fifths in the context of "Tritone"

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio ${\displaystyle 3:2}$. This is chosen because it is the next harmonic of a vibrating string, after the octave (which is the ratio ${\displaystyle 2:1}$), and hence is the next most consonant "pure" interval, and the easiest to tune by ear. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth), which is ≈ 702 cents wide.

The system dates back to Ancient Mesopotamia;. (See Music of Mesopotamia § Music theory.) It is named, and has been widely misattributed, to Ancient Greeks, notably Pythagoras (sixth century BC) by modern authors of music theory. Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium" and "tonus" in Latin (256:243 × 9:8 × 9:8), to Eratosthenes. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords [may be considered] a dissonance."

A mandolin (Italian: mandolino, pronounced [mandoˈliːno]; literally "small mandola") is a stringed musical instrument in the lute family and is generally plucked with a pick. It most commonly has four courses of doubled strings tuned in unison, thus giving a total of eight strings. A variety of string types are used, with steel strings being the most common and usually the least expensive. The courses are typically tuned in an interval of perfect fifths, with the same tuning as a violin (G3, D4, A4, E5). Also, like the violin, it is the soprano member of a family that includes the mandola, octave mandolin, mandocello and mandobass.

There are many styles of mandolin, but the three most common types are the Neapolitan or round-backed mandolin, the archtop mandolin and the flat-backed mandolin. The round-backed version has a deep bottom, constructed of strips of wood, glued together into a bowl. The archtop, also known as the carved-top mandolin, has an arched top and a shallower, arched back both carved out of wood. The flat-backed mandolin uses thin sheets of wood for the body, braced on the inside for strength in a similar manner to a guitar. Each style of instrument has its own sound quality and is associated with particular styles of music. Neapolitan mandolins feature prominently in European classical music and in traditional music like the Andean music of Peru. Archtop instruments are common in American folk music and bluegrass music. Flat-backed instruments are commonly used in Irish, British, and Brazilian folk music, and Mexican estudiantinas.