Just intonation in the context of "Terry Riley"

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (Play), 1.5, and may be approximated by an equal tempered perfect fifth (Play) which is 2 (about 1.498). If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.

Ratios, rather than direct frequency measurements, allow musicians to work with relative pitch measurements applicable to many instruments in an intuitive manner, whereas one rarely has the frequencies of fixed pitched instruments memorized and rarely has the capabilities to measure the changes of adjustable pitch instruments (electronic tuner). Ratios have an inverse relationship to string length, for example stopping a string at two-thirds (2:3) its length produces a pitch one and one-half (3:2) that of the open string (not to be confused with inversion).

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2·3·5 = 15/8.

Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths (plus one octave, which can be removed by multiplying by 1/2, i.e., 2). Powers of 5 represent intervals of major thirds (plus two octaves, removable by multiplying by 1/4, i.e., 2). Thus, 5-limit tunings are constructed entirely from stacking of three basic purely-tuned intervals (octaves, thirds and fifths). Since the perception of consonance seems related to low numbers in the harmonic series, and 5-limit tuning relies on the three lowest primes, 5-limit tuning should be capable of producing very consonant harmonies. Hence, 5-limit tuning is considered a method for obtaining just intonation.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Traditionally, there are two most common commata; the syntonic comma (80:81), "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma (524288:531441, approximately 73:74), "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Pitches separated by either comma are considered the same note because conventional notation does not distinguish Pythagorean intervals from 5-limit intervals. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53 TET, the harmonic seventh B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.

James Tenney (August 10, 1934 – August 24, 2006) was an American composer, music theorist, and pianist. He made significant early musical contributions to plunderphonics, sound synthesis, algorithmic composition, process music, spectral music, and microtonal tuning systems including extended just intonation. His theoretical writings variously concern musical form, texture, timbre, consonance and dissonance, and harmonic perception.

Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century composers in the West to work systematically with microtonal scales, alongside Lou Harrison. He built his own instruments in these tunings on which to play his compositions, and described the method behind his theory and practice in his book Genesis of a Music (1947).

Partch composed with scales dividing the octave into 43 unequal tones derived from the natural harmonic series; these scales allowed for more tones of smaller intervals than in standard Western tuning, which uses twelve equal intervals to the octave. To play his music, Partch built many unique instruments, with such names as the Chromelodeon, the Quadrangularis Reversum, and the Zymo-Xyl. Partch described his music as "corporeal" (emphasizing its physical/visceral elements), and distinguished it from abstract music, which he perceived as the dominant trend in Western music since the time of J.S. Bach, whose seminal book of preludes and fugues called The Well-tempered Clavier (in German, Das wohltemperierte Klavier) is often cited as the pivot point beyond which older mean-tone and ancient just intonation tunings were abandoned (in the late-18th century) and the then-future of Western Classical (and popular) instruments were (and most are still) based, for exploitation of all 24 theoretical key signatures. Partch's earliest compositions were small-scale pieces to be intoned with simple folkloric-like string instrumental backing; his later works were large-scale (like a fusion of theater and music decidedly related to but quite apart from Wagnerian opera), they were integrated theater productions in which he expected each of the performers to sing, dance, speak, and play instruments in a "corporeal apotheosis". Ancient Greek theatre and Japanese Noh and kabuki heavily influenced Harry Partch's music theatre.

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.

In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12 tone equal temperament, 12 TET or 12 ET, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2, (${\textstyle {\sqrt[{12}]{2}}}$ ≈ 1.05946). That resulting smallest interval, ⁠1/12⁠ the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12 TET.

The Theatre of Eternal Music (later sometimes called The Dream Syndicate) was an avant-garde musical group formed by La Monte Young in New York City in 1962. The first group (1962–1964) of performers consisted of La Monte Young, Marian Zazeela, Angus MacLise, and Billy Name. From 1964 to 1966, it consisted of La Monte Young (voice, saxophone), Marian Zazeela (voice, lighting), John Cale (viola), and Tony Conrad (violin), with sometimes also Terry Riley (voice). Since 1966, Theatre of Eternal Music has seen many permutations and has included Garrett List, Jon Gibson, Jon Hassell, Rhys Chatham, Alex Dea, Terry Jennings, and many others, including some members of the various 1960s groups. The group's self-described "dream music" explored drones and pure harmonic intervals, employing sustained tones and electric amplification in lengthy, all-night performances.

Archival recordings of the group's influential mid-1960s performances remain in La Monte Young's archive. None have ever seen official release following a dispute over compositional credit between Young and the pair of Conrad and Cale. Nonetheless, a 1965 bootleg recording removed from the archive by Young's first archivist, Arnold Dreyblatt, was controversially released in 2000 by Table of the Elements in CD as Day of Niagara. Other bootlegs of Theatre of Eternal Music have appeared online via file-sharing sites.

The Well-Tuned Piano is an ongoing improvisatory solo piano work begun in 1964 by La Monte Young. Young has never considered the composition or performance of this piece finished, and he has performed it differently several times since its debut in 1974. The composition requires a piano tuned in just intonation. A 1987 performance of the piece was released on DVD in 2000.

A typical performance lasts five to six hours. and is performed within the context of Marian Zazeela's light art installation The Magenta Lights. The Guardian described it as "one of the great achievements of 20th-century music."

Shri Camel is an album by experimental music and classical minimalism pioneer Terry Riley. Riley began composing the work in 1975 on commission from West Germany's Radio Bremen, and performed an early version of the work in Bremen in May 1976. In 1978 Riley recorded a different version of the piece, separated into four suites, at CBS Studios in San Francisco as the final part of a three album deal with CBS; however, CBS did not release the recording until 1980.

For the studio recording, Riley performed the work solo on a modified Yamaha YC-45D combo organ tuned in just intonation and augmented with studio digital delay.