Just intonation

In music, just intonation (sometimes abbreviated as JI) or pure intonation is the tuning of musical intervals as (small) whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals and chords are aggregates of harmonic series partials and may be seen as sharing a (lower) implied fundamental. For example, a tone with a frequency of 300 Hz and another with a frequency of 200 Hz are both multiples of 100 Hz (100 × 3 and 100 × 2 respectively). Their interval is, therefore, an aggregate of the second and third partials of the harmonic series of an implied fundamental frequency 100 Hz. In music, just intonation (sometimes abbreviated as JI) or pure intonation is the tuning of musical intervals as (small) whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals and chords are aggregates of harmonic series partials and may be seen as sharing a (lower) implied fundamental. For example, a tone with a frequency of 300 Hz and another with a frequency of 200 Hz are both multiples of 100 Hz (100 × 3 and 100 × 2 respectively). Their interval is, therefore, an aggregate of the second and third partials of the harmonic series of an implied fundamental frequency 100 Hz. Without context, 'just intonation' typically refers to 5-limit just intonation, where ratios only contain powers of the prime numbers 2, 3, and 5. American composer Ben Johnston proposed the term extended just intonation for composition involving ratios that contain prime numbers beyond 5 (7, 11, 13, etc.). Just intonation may be contrasted and compared with standard 12-tone equal temperament, which dominates Western instruments of fixed pitch (e.g., piano or organ) and default MIDI tuning on electronic keyboards. In equal temperament, all intervals are defined as an integer power of the basic step – the equal-tempered semitone, whose ratio is 2 12 : 1 {displaystyle {sqrt{2}}:1} (100 cents) – so two notes separated by the same number of steps share the same frequency ratio. Except for the doubling of frequencies (one or more octaves), all intervals are, in fact, irrational and may not be expressed as a ratio of whole numbers. Just intonation, on the other hand, suggests many microtonally differentiated sizes of intervals, which stem from different regions of the harmonic series. For example, the major third has three standard tunings in 7-limit just intonation – 9:7 (435.08 cents), 81:64 (407.82 cents), and 5:4 (386.31 cents). Pythagorean tuning, the first tuning system to be theoretically elaborated, is a system in which all tones are generated using ratios of prime numbers 2 and 3 as well as their powers. The most basic of these is the ratio 3:2 itself, called the perfect fifth. Pythagorean tuning is, in this sense, a spiral of cycling fifths. The justly tuned perfect fifth with the ratio 3:2 (701.96 cents wide), however, is not equivalent to the modern equal-tempered perfect fifth on the piano with ratio 2 7 / 12 {displaystyle 2^{7/12}} (700.00 cents wide). Rather, it is larger than the equal-tempered fifth by the small interval of a twelfth of the Pythagorean comma 531441 524288 12 {displaystyle {sqrt{frac {531441}{524288}}}} (1.96 cents). A stack of 12 justly tuned perfect fifths, therefore, does not arrive at the same pitch class it began with. This new pitch class is one full Pythagorean comma 'higher' than the starting pitch class, demonstrating how a continuous spiral of Pythagorean perfect fifths will generate an infinite collection of unique pitch classes within a frequency range. In Pythagorean tuning, the most consonant intervals are the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) are complex and comparably much more dissonant than the smoother sounding intervals with simpler ratios obtained from a tuning system than introduces powers of the prime number 5. The 5-limit major and minor thirds have ratios 5:4 and 6:5 respectively. The difference between the Pythagorean major third and the 5-limit major third – sometimes referred to as the Ptolemaic major third – is known as the syntonic comma and has the ratio of 81:80 (21.51 cents). During the second century AD, Claudius Ptolemy described a 5-limit diatonic scale in his influential text on music theory Harmonics, which he called 'tense diatonic'. Given ratios of string lengths 120, 112 ​1⁄2, 100, 90, 80, 75, 66 ​2⁄3, and 60, Ptolemy quantified the consonant tuning of what would today be called the major scale beginning and ending on the mediant – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9. 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.