User:Hkm/Just intonation

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Just intonation (JI) or Rational intonation (RI) is an approach to musical tuning which uses the most concordant (or melding) intervals, which are found at rational ratios of frequencies.

Just intervals are precisely those intervals which achieve concordance through alignment of partials if the interval has harmonic timbre. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords where every comprising interval is a concordance. Just intervals are also precisely those intervals which achieve concordance by melding with each other by forming different harmonics of the same unplayed root. (JI multi-note chords formed from harmonics of the same root can be the most concordant chords.)

In the context of Western music theory prior to the 20th century, the term just intonation used alone usually refers to 5-limit tuning. Extended just intonation, a term coined by Ben Johnston, usually refers to higher prime limits,[1] such as the 7-limit, the 11-limit and the 13-limit. The practice of just intonation without any particular constraint is sometimes referred to as rational intonation (RI) or as free style JI.

The structure of just intonation has several implications on music composition. Wolf intervals and commas, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are comma pumps, which may cause the tonal center of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, adaptive just intonation and temperament.

Just intonation in use

Here are multiple ways in which musicians and theorists have used just intonation.

Free style JI
Lou Harrison used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music.

Harmonic limits and subgrops
Harmonic limits set a limit for the highest prime number in the factorization of any ratio used. Subgroups name a list of allowable prime numbers used.

Restrictions on the denominator or numerator
Some approaches restrict "the denominator to one or very few values"[2] (the harmonic series, isoharmonic chords, AFDOs/overtone scales, primodality, ringer scales), the "numerator to one or a very few values" (the subharmonic series, IFDOs/undertone scales), or both (tonality diamonds)

Mediants
The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.

Approximations/alterations of tempered tunings
These are detemperings, including NEJI systems.

Other approaches
Other approaches include Meru scales, titriadic scales, and product sets.

Instruments

  • The array mbira was designed by Bill Wesley as a versatile just intonation instrument, covering a 5 octave range.
  • Most of Harry Partch's instruments were designed to be for just intonation.

Music

Main article: Music in just intonation

Notation

There are various notation systems for just intonation.

See also

Todo: cleanup

References

  1. ↑ Sabat, Marc. On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation
  2. ↑ From Jacques Dudon, "Differential Coherence", 1/1 vol. 11, no. 2: p.1).

Further reading

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