Just intonation

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

In Western musical practice, bowed instruments such as violins, violas, cellos, and double basses are tuned using pure fifths or fourths. In contrast, keyboard instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this impractical. Some instruments of fixed pitch, such as electric pianos, are commonly tuned using equal temperament, in which all intervals other than octaves consist of irrational-number frequency ratios. Acoustic pianos are usually tuned with the octaves slightly widened, and thus with no pure intervals at all.

The phrase "just intonation" is used both to refer to one specific version of a 5-limit diatonic intonation, that is, Ptolemy's intense diatonic, as well to a whole class of tunings which use whole number intervals derived from the harmonic series. In this sense, "just intonation" is differentiated from equal temperaments and the "tempered" tunings of the early renaissance and baroque, such as Well temperament, or Meantone temperament. Since 5-limit has been the most prevalent just intonation used in western music, western musicians have subsequently tended to consider this scale to be the only version of just intonation. In principle, there are an infinite number of possible "just intonations," since the harmonic series is infinite.

Just intonations are categorized by the notion of limits. The limit refers to the highest prime number fraction included in the intervals of a scale. All the intervals of any 3 limit just intonation will be multiples of 3. So ⁠ 6 / 5 ⁠ is included in 5 limit, because it has 5 in the denominator. If a scale uses an interval of 21:20, it is a 7 limit just intonation, since 21 is a multiple of 7. The interval ⁠ 9 / 8 ⁠ is a 3 limit interval because both numerator and denominator are multiples of 3 and 2. It is possible to have a scale that uses 5 limit intervals but not 2 limit intervals, i.e. no octaves, such as Wendy Carlos's alpha and beta scales. It is also possible to make diatonic scales that do not use fourths or fifths (3 limit), but use 5 and 7 limit intervals only. Thus, the notion of limit is a helpful distinction, but certainly does not tell us everything there is to know about a particular scale.

Pythagorean tuning, or 3 limit tuning, allows ratios including the numbers 2 and 3 and their powers, such as 3:2, a perfect fifth, and 9:4, a major ninth. Although the interval from C to G is called a perfect fifth for purposes of music analysis regardless of its tuning method, for purposes of discussing tuning systems musicologists may distinguish between a perfect fifth created using the 3:2 ratio and a tempered fifth using some other system, such as meantone or equal temperament.

5-limit tuning encompasses ratios additionally using the number 5 and its powers, such as 5:4, a major third, and 15:8, a major seventh. The specialized term perfect third is occasionally used to distinguish the 5:4 ratio from major thirds created using other tuning methods. 7 limit and higher systems use higher prime number partials in the overtone series (e.g. 11, 13, 17, etc.)

Commas are very small intervals that result from minute differences between pairs of just intervals. For example, the (5 limit) 5:4 ratio is different from the Pythagorean (3 limit) major third (81:64) by a difference of 81:80, called the syntonic comma. The septimal comma, the ratio of 64:63, is a 7 limit interval which is the distance between the Pythagorean semi-ditone, ⁠ 32 / 27 ⁠, and the septimal minor third, 7:6 , since ${\displaystyle \ \left({\tfrac {\ 32\ }{27}}\right)\div \left({\tfrac {\ 7\ }{6}}\right)={\tfrac {\ 64\ }{63}}~.}$

A cent is a measure of interval size. It is logarithmic in the musical frequency ratios. The octave is divided into 1200 steps, 100 cents for each semitone. Cents are often used to describe how much a just interval deviates from 12 TET. For example, the major third is 400 cents in 12 TET, but the 5th harmonic, 5:4 is 386.314 cents. Thus, the just major third deviates by −13.686 cents.

Pythagorean tuning has been attributed to both Pythagoras and Eratosthenes by later writers, but may have been analyzed by other early Greeks or other early cultures as well. The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts.^[1]

During the second century AD, Claudius Ptolemy described a 5-limit diatonic scale in his influential text on music theory Harmonics, which he called "intense diatonic".^[2] Given ratios of string lengths 120, ⁠112+1/2⁠, 100, 90, 80, 75, ⁠66+2/3⁠, and 60,^[2] Ptolemy quantified the tuning of what would later be called the Phrygian scale (equivalent to the major scale beginning and ending on the third note) – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9.

Ptolemy describes a variety of other just intonations derived from history (Pythagoras, Philolaus, Archytas, Aristoxenus, Eratosthenes, and Didymus) and several of his own discovery / invention, including many interval patterns in 3-limit, 5-limit, 7-limit, and even an 11-limit diatonic.

Non-Western music, particularly that built on pentatonic scales, is largely tuned using just intonation. In China, the guqin has a musical scale based on harmonic overtone positions. The dots on its soundboard indicate the harmonic positions: ⁠1/8⁠, ⁠1/6⁠, ⁠1/5⁠, ⁠1/4⁠, ⁠1/3⁠, ⁠2/5⁠, ⁠1/2⁠, ⁠3/5⁠, ⁠2/3⁠, ⁠3/4⁠, ⁠4/5⁠, ⁠5/6⁠, ⁠7/8⁠.^[3] Indian music has an extensive theoretical framework for tuning in just intonation.^{[citation needed]}

The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.

The 5-limit diatonic major scale is tuned in such a way that major triads on the tonic, subdominant, and dominant are tuned in the proportion 4:5:6, and minor triads on the mediant and submediant are tuned in the proportion 10:12:15. Because of the two sizes of wholetone – 9:8 (major wholetone) and 10:9 (minor wholetone) – the supertonic must be microtonally lowered by a syntonic comma to form a pure minor triad.

The 5-limit diatonic major scale (Ptolemy's intense diatonic scale) on C is shown in the table below:^{[4][5][6]: 78 [7]}

In this example the interval from D up to A would be a wolf fifth with the ratio 40⁄27, about 680 cents, noticeably smaller than the 702 cents of the pure 3⁄2 ratio. This is mentioned by Schenker in reference to the teaching of Bruckner.^[8]

For a justly tuned diatonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. It would include a tuning of 9:5 for the subtonic. For example, on A: