Just intonation: Difference between revisions
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== Just Intonation explained == | == Just Intonation explained == | ||
Just Intonation (JI) describes [[ | Just Intonation (JI) describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[Wikipedia: Rational number|rational numbers]]) between the frequencies of pitches<ref>. Just Intonation is sometimes distinguished from ''rational intonation'', by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)<sup>2</sup>/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)<sup>9</sup> is less than a cent short of an octave, while (27/25)<sup>2</sup> is almost precisely 7/6, the septimal minor third.</ref> | ||
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 [[Wikipedia: 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 [[OverToneSeries|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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''1. The principle of "[[Harmonic_Limit|harmonic limits]]," 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|3-limit]], [[5-limit|5-limit]], [[7-limit|7-limit]], [[11-limit|11-limit]], [[13-limit|13-limit]].'' | ''1. The principle of "[[Harmonic_Limit|harmonic limits]]," 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|3-limit]], [[5-limit|5-limit]], [[7-limit|7-limit]], [[11-limit|11-limit]], [[13-limit|13-limit]].'' | ||
''2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [ | ''2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[Wikipedia: Harry Partch|Harry Partch]]'s [[Wikipedia: Tonality diamond|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.'' | ||
''3. Other theorists who, in contrast to the above, advocate the use of [ | ''3. Other theorists who, in contrast to the above, advocate the use of [[Wikipedia: Hexany|products sets]] of given arrays of prime numbers, such as [[Wikipedia: Erv Wilson|Ervin Wilson]], Robert Dussaut, and others.'' | ||
''4. [[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.'' | ''4. [[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.'' | ||