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.^[1]

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.^[2] 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.^[3]

Just intonation also describes a tuning system that contains five or more pure intervals in an octave.^[1] There have been many attempts to construct scales composed completely of justly tuned intervals.^[4]: 18

Musicians instinctively perform in just intonation when possible. Singers and string players gravitate towards pure intervals.^[5] Barbershop quartets naturally sing in just intonation.^[6][7]

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.^[8] 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.^[9] Clashing overtones will result in acoustic beats.^[10] When an interval is performed without audible beats, it was historically described as pure or just.^[2]

Constructing a scale out of just intervals requires compromise.^[9]: 2 Because of the difficulty of justly tuning fixed pitch instruments, the manifold attempts to do so have been likened to a quest for the Holy Grail in its simultaneous futility and worthiness.^[4]: 18

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.^[11][12] Ptolemy and Didymus the Musician developed their own versions of the system.^[13]: 2

In China, the guqin draws on just intonation for its tuning system.^[14] Indian music has an extensive theoretical framework for tuning in just intonation.^[15]

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.^[13]: 191 Several musical temperaments were developed that standardized intervals, stabilizing musicmaking and enabling wider tonal adventures for composers. The system that became standard was equal temperament.^[16] 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.^[17]: 4

In the 20th century, many composers returned to just intonation. Some developed their own scales or instruments in order to use the tuning.^[18] 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.^{[19][20][21][22]} Computers greatly aided the continuing quest for just intonation.^[9]

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.^[24] By creating a series of fifths in the ratio 3:2, a justly tuned pentatonic scale can easily be formed. Pythagorean tuning was used on early Renaissance keyboard instruments.^[25]

When justly tuned fifths are stacked to generate all twelve chromatic tones, the final note in the series is short of its destination. This gap is the Pythagorean comma.^[26] Any Pythagorean scale with more than five notes has inherent tuning problems, particularly with thirds. An alternate solution is to begin with a tuned major triad as a reference for the remaining notes.^[23]

In his second century AD book Harmonics, Ptolemy calculated a "tense diatonic" scale with ratios of string lengths 120, ⁠112+1/2⁠, 100, 90, 80, 75, ⁠66+2/3⁠, and 60.^[27][28] Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".^[29]: 167 Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.^[30]: 77

All of the ratios of just intonation are governed by three prime numbers: 2, 3, and 5. Tuning solutions that rely on just this set of primes is sometimes called five-limit tuning.^[31] Modern composers expanded the limit to 7, which creates far more complex tuning solutions.^[32]

Moritz Hauptmann developed a system of notation to describe scales.^[35] 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.^[3]: 276

Carl Eitz developed a similar system which was adapted by J. Murray Barbour (1951). 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.^[36][37]

In the 1960s, Ben Johnston developed an extended just intonation. He also used + and − signs in his notation.^{[38][30]: 77–88}

Beginning in 2000, Marc Sabat and Wolfgang von Schweinitz developed an accidental system suggested by Helmholtz's work that incorporated Alexander John Ellis' invention of cents. Their system, Extended Helmholtz-Ellis JI Pitch Notation, adapts traditional accidentals combined with arrows to indicate alterations in tuning.^[33][39]

Composers like James Tenney employed just intonation by marking cents deviations from equal tempered pitches in his scores. Musicians often employ tuning devices during performances.^[40][41] Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.^[42]

• Just intonation^ⓘ An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.
• Equal temperament^ⓘ 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.
• Equal temperament and just intonation compared^ⓘ 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.
• Equal temperament and just intonation compared with square waveform^ⓘ 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.
• Just major third on C^ⓘ

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