Just intonation: Difference between revisions
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'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] where [[pitch]]es are chosen in a way such that every [[interval]] can be expressed as a whole-number [[ratio]] of the [[frequencies]] of pitches. '''Just intervals''' naturally occur in the [[harmonic series]] as intervals between any two [[harmonic]]s of a fundamental tone produced with a harmonic [[timbre]]. For instance, an interval with a frequency ratio of [[3/2]] appears between the 2nd and 3rd harmonics. Just intonation is particularly efficient when used with harmonic instruments, because it allows the tuning and the timbre to reinforce each other. | '''Just intonation''' ('''JI''') is an approach to [[musical tuning]] where [[pitch]]es are chosen in a way such that every [[interval]] can be expressed as a whole-number [[ratio]] of the [[frequencies]] of pitches. '''Just intervals''' naturally occur in the [[harmonic series]] as intervals between any two [[harmonic]]s of a fundamental tone produced with a harmonic [[timbre]]. For instance, an interval with a frequency ratio of [[3/2]] appears between the 2nd and 3rd harmonics. Just intonation is particularly efficient when used with harmonic instruments, because it allows the tuning and the timbre to reinforce each other. | ||
In theory, there are infinitely many just intervals, because each possible | In theory, there are infinitely many just intervals, because each possible {{w|fraction}} corresponds to a just interval. In practice, however, additional constraints are used to reduce the number of intervals to a reasonable amount, but also in many cases to prioritize [[consonant]] intervals. Usual constraints include [[subgroup]]s of [[generator]]s (including [[prime limit]]s), common denominators or numerators (as used in [[primodality]]), and [[complexity]] limits (usually [[height]] limits). Multiple constraints can be applied at the same time as well, such as the intersection of a prime limit and an [[odd limit]]. | ||
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,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> 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]]. | 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,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> 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]]. | ||
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== Just intonation explained == | == Just intonation explained == | ||
If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and | If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and {{w|Pitch (music)|pitch}}. Kyle Gann's ''[http://www.kylegann.com/tuning.html Just Intonation Explained]'' is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[harmonic series]]. | ||
In languages other than English, the original conceptions of "just intonation" are more obviously retained in the terms used in those languages: German ''Reine Stimmung'' (pure, that is, beatless, tuning), Ukrainian ''Натуральний стрій'' and French ''gamme naturelle'' (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian ''intonazione naturale'' (natural intonation, once again intervals derived from harmonic series), and so on. | In languages other than English, the original conceptions of "just intonation" are more obviously retained in the terms used in those languages: German ''Reine Stimmung'' (pure, that is, beatless, tuning), Ukrainian ''Натуральний стрій'' and French ''gamme naturelle'' (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian ''intonazione naturale'' (natural intonation, once again intervals derived from harmonic series), and so on. | ||
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# ''The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].'' | # ''The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].'' | ||
# ''Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of | # ''Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of {{w|Harry Partch}}'s {{w|tonality diamond}}. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.'' | ||
# ''Other theorists who, in contrast to the above, advocate the use of [[combination product sets|products sets]] of given arrays of prime numbers, such as [[Ervin Wilson]], Robert Dussaut, and others.'' | # ''Other theorists who, in contrast to the above, advocate the use of [[combination product sets|products sets]] of given arrays of prime numbers, such as [[Ervin Wilson]], Robert Dussaut, and others.'' | ||
# ''[[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in just intonation.'' | # ''[[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in just intonation.'' | ||