Just intonation is the tuning of a musical interval without beats. The result is an acoustically pure sound that resonates within the harmonic series. The simplest relationship between pitches in this series can be expressed as small whole number ratios.
Just intonation also describes any musical tuning system containing five or more pure intervals within an octave.[1]
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.
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.
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.[3] 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. Clashing overtones will result in acoustic beats.[4] 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. 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.[5]
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.[6] [7] Ptolemy and Didymus the Musician developed their own versions of the system.[8]
In China, the guqin draws on just intonation for its tuning system.[9] Indian music has an extensive theoretical framework for tuning in just intonation.[10]
20th century composers often returned to just intonation for inspiration. Many developed their own scales or instruments in order to use the tuning.[11] Computers greatly aided the continuing quest for just intonation.[12]
See also: Five-limit tuning, Ptolemy's intense diatonic scale and Pythagorean tuning. 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.[13] 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.[14] When extended to scales with more notes, several tuning problems arise particularly with thirds. An alternate solution is to begin with a tuned major triad as a reference for the remaining notes.
In his second century AD book Harmonics, Ptolemy calculated a "tense diatonic" scale with ratios of string lengths 120,, 100, 90, 80, 75,, and 60.[15] [16] Harry Partch described this scale as "one of the world's fundamentally beautiful tonal sequences".[17] Justly tuned scales often yield multiple versions of the same interval, which can be managed through notation.[18]
All of the ratios of just intonation are governed by four prime numbers: 1, 2, 3, and 5. Tuning solutions that rely on just this set of primes is sometimes called five-limit tuning.[19] Modern composers expanded the limit to 7, which creates far more complex tuning solutions.[20]
In Indian music, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch (dha), and further modifications may be made to all pitches excepting sa and pa.[21]
| Note | sa | re | ga | ma | pa | dha | ni | sa | |
|---|---|---|---|---|---|---|---|---|---|
| Ratio | 1:1 | 9:8 | 5:4 | 4:3 | 3:2 | 5:3 | 15:8 | 2:1 | |
| Cents | 0 | 204 | 386 | 498 | 702 | 884 | 1088 | 1200 |
Some accounts of Indian intonation system cite a given 12 swaras being divided into 22 shrutis.[22] [23] According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, shadja (sa), and the pure fifth, pancham (pa), are inviolate (known as achala[24] in Indian music theory):
| Note | C | D | D | D | D | E | E | E | E | F | F | F | F | G | A | A | A | A | B | B | B | B | C | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio | 1:1 | 256:243 | 16:15 | 10:9 | 9:8 | 32:27 | 6:5 | 5:4 | 81:64 | 4:3 | 27:20 | 45:32 | 729:512 | 3:2 | 128:81 | 8:5 | 5:3 | 27:16 | 16:9 | 9:5 | 15:8 | 243:128 | 2:1 | |
| Cents | 0 | 90 | 112 | 182 | 204 | 294 | 316 | 386 | 408 | 498 | 520 | 590 | 612 | 702 | 792 | 814 | 884 | 906 | 996 | 1018 | 1088 | 1110 | 1200 |
Moritz Hauptmann developed a system of notation to describe scales.[26] 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.[27]
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.[28] [29]
In the 1960s, Ben Johnston developed "Extended Just Intonation". He also used + and − signs in his notation.[30] [18]
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.[31] [32]
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.[33] [34] Sagittal notation uses arrows as accidentals. The size of the symbol indicates the size of the alteration.[35]
