Just intonation: Difference between revisions

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)²/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)⁹ is less than a cent short of an octave, while (27/25)² is almost precisely 7/6, the septimal minor third.</ref>.  

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.

Of course, a historical description of something as "natural" does not prove that something is "natural." Similarly labeling something "natural" without any ground, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call "Just Intonation" do claim a "natural" status, and Just Intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.

In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: "3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Where we to want a Just Intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.

There is more to it than this, of course, but the basic principles of Just Intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals was the concern, understanding and defining "Just" was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...

== Just Intonation in use ==

Look at [[Notation|notation systems]] for Just Intonation.  

=== Free Style Just ===

=== Constrained Just ===

# ''[[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.''

== Links ==

== Articles ==