Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name.p-Limit Tuning. Given a prime number p, the subset of Q + {displaystyle mathbb {Q} ^{+}} consisting of those rational numbers x whose prime factorization has the form In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name. Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher harmonics in its constructs (see emancipation of the dissonance). In medieval music, only chords made of octaves and perfect fifths (involving relationships among the first three harmonics) were considered consonant. In the West, triadic harmony arose (contenance angloise) around the time of the Renaissance, and triads quickly became the fundamental building blocks of Western music. The major and minor thirds of these triads invoke relationships among the first five harmonics. Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music. In conventional music theory pedagogy, these seventh chords are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the dominant seventh chord in 12-ET approximates 4:5:6:7, while the major seventh chord approximates 8:10:12:15. 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. 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. Partch's theoretical prediction of the sensory dissonance of intervals (his 'One-Footed Bride') are very similar to those of theorists including Hermann von Helmholtz, William Sethares, and Paul Erlich. See #Examples, below. 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: 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'.