Limit (music).html



In music theory, limit can refer to a variety of methods used to characterize the harmonies found in a piece of music, genre of music, or by extension, the harmonies that can be made with a particular scale or class of scales. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony in a piece of music; hence the name.

Owing to the difficulty of defining the 'complexity' of a harmony, there are several different formulations of the limit concept (see below). However, they all attempt to characterize harmony using a one-dimensional measure on the natural numbers.

The harmonic series and the evolution of music

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. In medieval music, only chords made of octaves and pure fifths (involving relationships among the first 3 harmonics) were considered consonant. In the West, triadic harmony sprang up 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 5 harmonics.

Around the turn of the 20th century, tetrads debuted as the fundamental building blocks of African-American music. In conventional music theory pedagogy, these tetrads are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from relationships among harmonics > 5.

This story is suggestive of evolution through a regime of punctuated equilibria, wherein the predominant technology of each epoch (e.g. triads) almost completely replaces that of previous epochs (e.g. medieval open fourths and fifths), at least in the genre undergoing the revolution (plain triads are very seldom used in jazz). This can be seen as a justification for using upper bounds to describe harmonic complexity.

Odd-limit and prime-limit

There are two distinct types of limit in music theory literature: prime limit and odd limit. Not all authors are aware of the distinction.

In a just intonation tuning, intervals between pitches are drawn from the rational numbers. In an n-limit prime limit tuning, intervals between pitches are drawn from rational numbers that can be factored using prime numbers no greater than n, where n is prime. In an n-limit odd limit tuning, intervals between pitches are drawn from rational numbers which, after all factors of 2 are removed, have numerators and denominators no greater than n, where n is an odd whole number. Note that prime limit and odd limit do not cover the same scales even when n is an odd prime.

In just intonation, musical intervals are represented by frequency ratios. In Genesis of a Music, Harry Partch defined the complexity of a just intonation interval as proportional to the size of the numbers in its ratio (when the ratio is in lowest terms), modulo octaves. Since octaves are represented by factors of 2 in just intonation, the complexity of any interval smaller than an octave is measured simply by the largest odd number in its ratio. To distinguish it from other formulations of the limit concept, this is often called odd-limit. Partch further defined the limit of a scale as the odd-limit of the most complex interval in the scale. Paul Erlich and others have subsequently shown that modern psychoacoustics places Partch's scheme on sound footing.

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. Taking the Partch odd-limit of such scales often produces misleading results, so these musicians instead used the largest prime number required to factor the ratios of all intervals in a given scale. This subsequently became known as prime-limit. While prime-limit works well in the context of such scales, it does not produce a psychoacoustically valid consonance ranking on bare intervals, as odd-limit does.

Examples

Beyond just intonation

In musical temperament, the simple ratios of just intonation are mapped to nearby irrational numbers. This operation, if successful, does not change the relative harmonic complexities of the different intervals, but it does make the calculation of the harmonic limits more difficult. One must first decide which rational intervals are being approximated and then calculate the odd- or prime-limit. But since some chords (such as the diminished seventh chord in 12-ET) can be tuned in just intonation in several valid ways, this procedure is sometimes frustrated.

External links

• Just Intonation Explained
• Some music theory from Paul Erlich