Just intonation

Any time an interval is sounded without acoustical beats it is in just intonation. The sound is also described as pure. The frequency of each note in a pure interval will correspond to the whole number ratios in the harmonic series. In the harmonic series on C, the 1st and 2nd notes form an octave in a 2:1 ratio. The fifth between the G and C is in a 3:2 ratio. The fourth is a 4:3 ratio. When its frequency is doubled, A 440 Hertz sounds an octave higher at 880 Hz. The pitch sounds an octave lower when the frequency is halved to 220 Hz. Just intonation also describes a tuning system that contains five or more pure intervals in an octave. There have been many attempts to construct scales composed completely of justly tuned intervals. ==History==

Musicians instinctively perform in just intonation when possible. Singers and string players gravitate towards pure intervals. Brass players default to just tuning when possible. In Ancient Greece, intervals like the octave, fourth, and fifth were recognized as consonances. Using a monochord, Pythagoras discovered that simple fractions of the string length correspond to these consonant intervals. Pythagoras' ratios reflected a naturally sounding collection of overtones known as the harmonic series. When two notes are sounded together, the resulting interval is perceived as more consonant when their overtones are in accordance. Pythagoras and Eratosthenes are credited with a solution that became known as Pythagorean tuning. However, the system is in evidence in much older Babylonian artifacts. Ptolemy and Didymus the Musician developed their own versions of the system. In China, the guqin draws on just intonation for its tuning system. Indian music has an extensive theoretical framework for tuning in just intonation. Just intonation fettered music to a limited range of harmony and keys. Emulating its pure sound was impractical. Johann Sebastian Bach was so adept at retuning his harpsichord, he could do it in fifteen minutes. With its division of the octave into twelve identical steps based on a ratio of the 12th root of 2 (≈1.0595), equal temperament uses irrational numbers to create a rational system. Just intonation generally relies on rational numbers to generate irrational systems. In the 20th century, many composers returned to just intonation. Some developed their own scales or instruments in order to use the tuning. Harry Partch, Lou Harrison, La Monte Young, Terry Riley, John Adams, and Glenn Branca are just a few of the contemporary composers that used just intonation. Computers greatly aided the continuing quest for just intonation. ==Scales==

Pythagorean tuning relies on the just intonation of fifths to create a scale. The intervals are tuned in the same way violinists tune their open strings. By creating a series of fifths, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments. When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is wide of its destination, which should be seven octaves higher than the initial note. Additionally, any Pythagorean scale with more than five notes has inherent tuning problems, particularly with thirds. A solution is to begin with a major triad that uses the 5:4 just major third as a reference for the remaining notes. This scale allows for the just major third in its natural 5:4 ratio. The ratios of just intonation can be governed by three prime numbers: 2, 3, 5. By this classification, the Pythagorean scale is a 3-limit scale. A scale with the 5:4 major third is in 5-limit tuning. Modern composers expanded the limit to 7, which creates far more complex tuning solutions. Partch experimented with prime number limits as high as 17. ==Notation==

's notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C. Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation. Moritz Hauptmann developed a system of notation to describe scales. Hermann von Helmholtz adapted it in On the Sensations of Tone as a Physiological Basis for the Theory of Music (1877). The system used a combination of + and - signs in addition to subscript numbers. Carl Eitz developed a similar system which was adapted by J. Murray Barbour. Superscript numbers indicate the number of syntonic commas to apply to the tuning. The basic just intonation scale appears as C0 – D0 – E−1 – F0 – G0 – A−1 – B−1 – C0. Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration. ==Audio examples==

• • An A-major scale, followed by three major triads, and then a progression of fifths in just intonation. • An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. The beating in this file may be more noticeable after listening to the above file. • A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound. • A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just intonation between the two chords. In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent. The square waveform makes the difference between equal temperament and just intonation more obvious. • • ==See also==