In
just intonation, intervals between pitches are drawn from the
rational numbers. Since Partch, two distinct formulations of the limit concept have emerged:
odd limit and
prime limit. Odd limit and prime limit
n do not include the same intervals even when
n is an odd prime.
Odd limit For a positive odd number
n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than
n. In
Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves. Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio.
Identity An
identity is each of the
odd numbers below and including the (odd) limit in a tuning. For example, the identities included in 5-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the
harmonic series and may thus be considered an identity: C C G C E G ... 2 4 6 8 10 12 ... According to Partch: "The number 9, though not a
prime, is nevertheless an identity in music, simply because it is an odd number." Partch defines "identity" as "one of the correlatives, '
major' or '
minor', in a
tonality; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".
Odentity and
udentity are short for
over-identity and
under-identity, respectively. According to music software producer Tonalsoft: "An udentity is an identity of an
utonality".
Prime limit For a
prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than
n. In other words, it is the set of rationals with numerator and denominator both
n-
smooth. {{quote|
p-Limit Tuning. Given a prime number
p, the subset of \mathbb{Q}^+ consisting of those rational numbers
x whose prime factorization has the form x=p_1^{\alpha_1} p_2^{\alpha_2}... p_r^{\alpha_r} with p_1,...,p_r \le p forms a subgroup of (\mathbb{Q}^+,\cdot). ... We say that a scale or system of tuning uses
p-limit tuning if all interval ratios between pitches lie in this subgroup.}} In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the
American gamelan school. Inspired by Indonesian
gamelan, musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer
Lou Harrison. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct
Fokker periodicity blocks. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale. Prime-limit tuning and intervals are often referred to using the term for the
numeral system based on the limit. For example, 7-limit tuning and intervals are called septimal, 11-limit is called undecimal, and so on. ==Examples==