Limit (music)

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In just intonation the limit is the highest prime number, that may appear in the ratios used to create a specific just tuning.

For instance, in a 5 limit tuning only ratios using 2, 3, 5, and all of their respective multiples, would be used. However, intervals are usually referred to within one octave, so that 10/2 is usually called 5/2.

While in general people use more notes in their scales when tuning to a higher limit, such as Harry Partch's 43 note 11-limit scale, in actuality even a 3 limit scale can create an infinite amount of pitches. For instance a series of perfect fifths may be built upon the starting pitch, 1, at the ratio 3:2, the highest allowed by 3-limit. Since 1 * 1.5^x ≠ 1 * 2^x the series would go on forever. This is not true in equal temperament, since the perfect fifth is approximated at 1.4983070768766815..., or 700 centss, and 1.4983070768766815¹² = 2⁷. This is exactly the issue which led to equal temperament.

Some people can tune an amazing number of intervals in just intonation by ear alone. Such an example of a person who exhibited this ability was the late composer Lou Harrison, who could tune up to 7-limit but expressed the desire to eventually reach 13.

Links:

Just Intonation Explained

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