In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series. In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... A ratio is a dimensionless, or unitless, quantity denoting an amount or magnitude of one quantity relative to another. ... The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... In music theory, the term interval describes the difference in pitch between two notes. ... Pitched musical instruments are usually based on a harmonic oscillator such as a string or a column of air. ...

Justly tuned intervals are usually written either as ratios, with a colon (for example, 3:2), or as fractions, with a slash (3/2). Sometimes a technical distinction is made between the two styles, but in general they are equivalent and interchangeable. There are other, less common notation methods. A cake divided into four equal quarters. ...

Although in theory two notes tuned in an arbitrary frequency ratio such as 1024:927 might be said to be justly tuned, in practice only ratios using quite small numbers tend to be called just; more complex ratios are often considered to be rational intonation but not necessarily just intonation. Intervals used are then capable of being more consonant. In music, a consonance (Latin consonare, sounding together) is a harmony, chord, or interval considered stable, as opposed to a dissonance, which is considered unstable. ...

Just intonation is usually compared to equal temperament, the tuning system that is by far the most common in the West, which arranges all notes at multiples of the same basic interval. This results in a tuning system where all intervals will sound equally out of tune in any key—all major thirds will have exactly the same character, for example—but the intervals themselves are detuned slightly. An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

Image File history File links Just_vs_equal. ... Software development stages In computer programming, development stage terminology expresses how the development of a piece of software has progressed and how much further development it may require. ...

The diatonic scale

It is possible to tune the familiar diatonic scale or chromatic scale in just intonation, in many ways, all of which make certain chords purely tuned and as consonant and stable as possible, and the other chords not accommodated sound considerably less stable. In music theory, a diatonic scale (from the Greek diatonikos, to stretch out; also known as the heptatonia prima; set form 7-35) is a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated. ... The chromatic scale is the scale that contains all twelve pitches of the Western tempered scale. ...

The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the notes G to D (perfect fifth) is 2/3, while that of G to C (perfect fourth) is 3/4. Three basic intervals can be used to construct any interval involving the prime numbers 2, 3 and 5 (known as 5-limit just intonation). In music theory, the major scale (or major mode) is one of the diatonic scales. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... The perfect fourth or diatessaron, abbreviated P4, is one of two musical intervals that span four diatonic scale degrees; the other being the augmented fourth, which is one semitone larger. ...

• s = 16:15 Semitone
• t = 10:9 Minor tone
• T = 9:8 Major tone

which combine to form A semitone (also known in the USA as a half step) is a musical interval. ... In harmony, the tonus is the ratio 9:8 between a pair of frequencies or, equivalently, the ratio 8:9 between a pair of wavelengths. ... In harmony, the tonus is the ratio 9:8 between a pair of frequencies or, equivalently, the ratio 8:9 between a pair of wavelengths. ...

• 6:5 = Ts (minor third)
• 5:4 = Tt (major third)
• 4:3 = Tts (perfect fourth)
• 3:2 = TTts (perfect fifth)
• 2:1 = TTTttss (octave)

A just diatonic scale may be derived as follows. Suppose we insist that the chords F-A-C, C-E-G, and G-B-D be just major triads (then A-C-E and E-G-B are just minor triads, but D-F-A is not). Generally speaking, a major chord is any chord which has a major third above its root, as opposed to a minor chord which has a minor third. ... Generally speaking, a minor chord is any chord which has a minor third above its root, as opposed to a major chord which has a major third. ...

Then we obtain this scale:

The major thirds are correct, and two minor thirds are right, but D-F is not.

Another way to do it is as follows. We can insist that the chords D-F-A, A-C-E, and E-G-B be just minor triads (then F-A-C and C-E-G are just major triads, but G-B-D is not). Generally speaking, a minor chord is any chord which has a minor third above its root, as opposed to a major chord which has a major third. ... Generally speaking, a major chord is any chord which has a major third above its root, as opposed to a minor chord which has a minor third. ...

Then we get the following scale:

The major thirds are correct, and two minor thirds are right, but B-D is not.

If we compare with the scale above, we see that six notes can be lined up, but one note, D, has changed its value. It is evidently not possible to get all six chords mentioned correct.

There are other possibilities; instead of lowering D, we can raise A. But this breaks something else.

The twelve tone scale

The most common way to define the familiar twelve tone scale in a just temperament is to base it just on compounding of second, third and fifth harmonics. The following table summarises the composition. Starting point in this diagram is C. Horizontally factors 3 are applied (repeated third harmonic) and vertically factors 5 (fifth harmonic). Then factors 2 are added to bring the tones within the same octave. At both ends of the chain, at F# and Gb, the scale does not quite close by a difference of about one percent (a semitone is about 6 %).

Indian scales

In Indian music, the just diatonic scale described above is used, though there are different possibilities for the 6th pitch (Dha), and further modifications may be made to all pitches excepting Sa, Ma, and Pa.^{[citation needed]} Timeline and Samples Genres Classical (Carnatic and Hindustani) - Rock - Pop - Hip hop Awards Bollywood Music Awards - Punjabi Music Awards Charts Festivals Sangeet Natak Akademi – Thyagaraja Aradhana – Cleveland Thyagaraja Aradhana Media Sruti, The Music Magazine National anthem Jana Gana Mana, also national song Vande Mataram Music of the states Andaman and...

Both possible scales appear problematic. The first has a problem because (27/16)/(5/4) = 27/20, which is a wolf interval, being uncomfortably close to the purer 4:3. However, because Indian music uses melodies over a drone dyad (usually 1/1 and 3/2), these two pitches (27/16 and 5/4) would seldom be heard sounding together. See sargam, swara, and sruti.^{[citation needed]} When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. ... It has been suggested that this article or section be merged into Swara. ... It has been suggested that Sargam be merged into this article or section. ... The śruti (Sanskrit thing heard, sound) is the smallest interval of the tuning system of Indian classical music. ...

The alternative, using 5/3 for Dha gives (5/3)/(5/4) = 4/3, and allows these notes to sound together in a consonant fashion, but then introduces another problem as (5/3)/(9/8) = 40/27, which is another wolf interval, this time close to 3/2. These wolf intervals are incompatible with much western music, but in Indian music they are largely irrelevant.^{[citation needed]}

Practical difficulties

Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for F:D, and still worse, a minor tone next to a fourth giving 40:27 for A:D. Moving D down to 10/9 alleviates these difficulties but creates new ones: G:D becomes 27:20, and B:G becomes 27:16. When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. ...

You can have more frets on a guitar to handle both A's, 9/8 with respect to G and 10/9 with respect to G so that C:A can be played as 6:5 while D:A can still be played as 3:2. 9/8 and 10/9 are less than 1/53 octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune chords such as C-E-G-A-D is left unresolved (for instance, A could be 4:3 below D (making it 9/8, if G is 1) or 4:3 above E (making it 10/9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). However the frets may be removed entirely—this, unfortunately, makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand—and the tuning of most complex chords in just intonation is generally ambiguous. The neck of a guitar showing the first four frets. ... This article or section does not adequately cite its references or sources. ...

For many instruments tuned in just intonation, you can't change keys without retuning your instrument. For instance, if you tune a piano to just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys impractical to impossible. In music theory, the key identifies the tonic triad, the chord, major or minor, which represents the final point of rest for a piece, or the focal point of a section. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another. ...

Synthesizers have proven a valuable tool for composers wanting to experiment with just intonation. Many commercial synthesizers provide the ability to use built-in just intonation scales or to program your own. Wendy Carlos used a system on her 1986 album Beauty in the Beast, where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between traditional equal temperament and just intonation. A synthesizer (or synthesiser) is an electronic musical instrument designed to produce electronically generated sound, using techniques such as additive, subtractive, FM, physical modelling synthesis, phase distortion, or Scanned synthesis. ... Wendy Carlos (November 14, 1939 in Pawtucket, Rhode Island) is an American composer and electronic musician. ... 1986 (MCMLXXXVI) was a common year starting on Wednesday of the Gregorian calendar. ... 1987 (MCMLXXXVII) was a common year starting on Thursday of the Gregorian calendar. ...

Singing

The human voice is the most pitch-flexible instrument in common use. Pitch can be varied with no restraints and adjusted in the midst of performance, without needing to retune (as even with the otherwise very flexible string instruments). Although the explicit use of just intonation fell out of favour concurrently with the increasing use of instrumental accompaniment (with its attendant constraints on pitch), most a cappella ensembles naturally tend toward just intonation because of the comfort of its stability. Barbershop quartets are a good example of this. A cappella music is vocal music or singing without instrumental accompaniment, or a piece intended to be performed in this way. ... The Dapper Dans, a barbershop quartet at Disneyworld Barbershop harmony, as codified during the barbershop revival era (1940s-present), is a style of a cappella, or unaccompanied vocal music characterized by consonant four-part chords for every melody note in a predominantly homophonic texture. ...

Two exemplary contemporary ensembles that meticulously tune their singing in accordance with just intonation (whenever indicated) are The Hilliard Ensemble and Orlando Consort. A self portrait by Nicholas Hilliard The Hilliard Ensemble is a British male vocal quartet devoted to the authentic performance of early music. ...

History

There were several other systems in use before equal temperament. Pythagorean tuning was perhaps the first to be theorized^[1], which is a system in which all tones can be found using the ratios 3:2 and 4:3. It is easier to think of this system as a cycle of fifths, but it must be noted that because a series of 12 fifths does not reach the same tone it began with, this system produces "wolf fifths" in the more distant keys (which were consequently unused). An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. ...

Another system that was popular for keyboards through the Renaissance was meantone temperament^[2]. In this system the simpler ratios of 3:2 and 4:3 were compromised in favour of exact 5:4 (major thirds) ratios. Specifically, the fifth (3:2) was slightly narrowed so that a series of four narrowed fifths would produce 5:4 exactly (at some octave transposition). Again, this system is not circular and produced some unplayable keys. (Some keyboards of the 18th century featured split keys differentiating sharp and flat notes to expand the range of usable keys.) Meantone temperament is a system of musical tuning. ...

The most common tuning today began as well temperament, which was replaced by the more rigorous equal temperament in the early 20th century. Well temperament largely abandoned just intonation by applying small changes to the intervals so that they became more homogenized and eliminated wolf intervals. In systems of well temperament, and there were many, the goal was to make all keys usable by compromising each of them slightly. Its development was necessary as composers moved toward expression through large harmonic changes (modulation), and required access to a wider realm of tonality. Bach's "Well-Tempered Clavier", a book of compositions in every key, is the most famous example, but the compositions of Chopin, for instance, rely much more on the devices of expression only allowed by well temperament^[3]. Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another. ... Places in which Bach resided throughout his life Johann Sebastian Bach (pronounced ) (21 March 1685 O.S. – 28 July 1750 N.S.) was a prolific German composer and organist whose sacred and secular works for choir, orchestra and solo instruments drew together the strands of the Baroque period and brought... Title-page of Das wohtemperierte Klavier A flat major (As-dur) fugue from the second part of Das wohtemperierte Klavier (manuscript) The Well-Tempered Clavier (Das wohltemperierte Klavier in German -- Klavier means piano, but the English word clavier (which means keyboard) looks more like the German title) consists of two... The only known photograph of Frédéric Chopin (commonly mistaken for a daguerreotype), believed to have been taken by Louis-Auguste Bisson in 1849 “Chopin” redirects here. ...

Equal temperament is essentially the most homogenized form of well temperament, in that it tunes an actual circle of fifths by narrowing each by the same amount. In equal temperament, every interval is the same as all other intervals of its type. There are no longer pure and "wolf" fifths, or even good and bad fifths, but simply fifths (or thirds, or seconds, et cetera). Equal temperament is not a form of just intonation.

Today, the dominance of repertoire composed under well tempered systems, the prominence of the piano in musical training, the lack of just-intonation capable instruments, and the fact that tuning is not normally a significant part of a musician's education have made equal temperament so prevalent that alternatives are not often discussed.

Despite the obstacles, many today find reasons to pursue just intonation. The purity and stability of its intervals are found quite beautiful by many, but this stability also allows extreme intonational precision as well. The practical study of just intonation can greatly increase one's analytical ability with respect to sound, and yield improvement to musicianship even in well temperament repertoire.

In practice it is very difficult to produce true equal temperament. There are instruments such as the piano where tuning is not dependent on the performer, but these instruments are a minority. The main problem with equal temperament is that its intervals must sound somewhat unstable, and thus the performer has to learn to suppress the more stable just intervals in favour of equal tempered ones. This is counterintuitive, and in small groups, notably string quartets, just intonation is often approached either by accident or design because it is much easier to find (and hear) a point of stability than a point of arbitrary instability.

Western composers

Most composers don't specify how instruments are to be tuned, although historically most have assumed one tuning system which was common in their time; in the 20th century most composers assumed equal temperament would be used. However, a few have specified just intonation systems for some or all of their compositions, including Glenn Branca, Wendy Carlos, Tony Conrad, Stuart Dempster, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Ben Johnston, Elodie Lauten, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Adam Silverman, James Tenney, Ernesto Rodrigues, Daniel James Wolf and La Monte Young. Eivind Groven is often considered a just intonation composer but just intonation purists will disagree. His tuning system was in fact schismatic temperament, which is indeed capable of far closer approximations to just intonation consonances than 12-note equal temperament or even meantone temperament, but still alters the pure ratios of just intonation slightly in order to achieve a simpler and more flexible system than true just intonation. An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... Glenn Branca (born October 6, 1948 in Harrisburg, Pennsylvania) is an avant-garde composer and guitarist. ... Wendy Carlos (November 14, 1939 in Pawtucket, Rhode Island) is an American composer and electronic musician. ... Tony Conrad (born Anthony S. Conrad in 1940) is an American avant-garde video artist, experimental filmmaker, musician/composer, sound artist, teacher and writer. ... Stuart Dempster (born 1936 in Berkeley, California) is a trombonist, didjeridu player, improvisor, composer, author of The Modern Trombone: A Definition of Its Idioms (1979), and on the faculty of the University of Washington. ... Arnold Dreyblatt (b. ... Kyle Gann (born November 21 1955) is a composer and music critic born in Dallas, Texas. ... Kraig Grady is a composer who uses microtonal just intonation. ... Lou Silver Harrison (May 14, 1917 - February 2, 2003) was an American composer. ... Ben Johnston Benjamin Burwell Johnston, Junior (born March 15, 1926 in Macon, Georgia) is a composer of contemporary music in the just intonation system. ... Elodie Lauten (b. ... Douglas (Harry) Leedy, born March 3rd, 1938 in Portland, Oregon is an American composer. ... Pauline Oliveros (born 1932 in Houston, Texas) is an accordionist and composer who currently resides in Kingston, New York. ... Harry Partch (June 24, 1901 – September 3, 1974) was an American composer. ... Robert Rich is an ambient musician and composer based in California, USA. With a discography spanning over twenty years, he is widely regarded as a figure whose sound has greatly influenced todays ambient, new age, and even IDM music. ... Terry Riley – (Portrait by Betty Freeman) Terry Riley (born 24 June 1935) is an American composer associated with the minimalist school. ... Adam Benjamin Silverman (born in 1973, Atlanta, GA) is a composer of contemporary classical music. ... James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ... Ernesto Rodrigues Ernesto Rodrigues (born in Lisbon, August 29, 1959) is a Portuguese composer, violinist, violist and electronic musician. ... Daniel James Wolf (born September 13, 1961 in Upland, California) is an American composer of serious music and a music scholar. ... La Monte Young (born October 14, 1935) is an American composer whose eccentric and often hard-to-find works have been included among the most important post World War II avant-garde or experimental music. ... Eivind Groven (October 8, 1901–February 8, 1977) was a Norwegian microtonal composer and music-theorist. ... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... Meantone temperament is a system of musical tuning. ...

Music written in just intonation is most often tonal but need not be; some music of Kraig Grady and Daniel James Wolf uses just intonation scales designed by Erv Wilson explicitly for a consonant form of atonality, and Ben Johnston's Sonata for Microtonal Piano (1964) uses serialism to achieve an atonal result. Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10/7, for example, would be permitted, but 11/7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is a double octave of 3, while 9 is a square of 3). Tonality is a system of writing music according to certain hierarchical pitch relationships around a key center or tonic. ... Ervin Wilson is an American music theorist whose work, outside of the academic community, is noted for the variety and originality of its ideas. ... In music, a consonance (Latin consonare, sounding together) is a harmony, chord, or interval considered stable, as opposed to a dissonance, which is considered unstable. ... Atonality describes music not conforming to the system of tonal hierarchies, which characterizes the sound of classical European music between the seventeenth and nineteenth centuries. ... Ben Johnston Benjamin Burwell Johnston, Junior (born March 15, 1926 in Macon, Georgia) is a composer of contemporary music in the just intonation system. ... Sonata for Microtonal Piano is a sonata for specifically microtonally tuned piano by Ben Johnston written in 1964 (see also just intonation). ... Serialism is a technique for composing music that uses sets to describe musical elements, and allows the composer manipulations of those sets to create music. ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... y=x², for all integer values of 1≤x≤25. ...

See also

• Equal temperament
• Mathematics of musical scales
• Meantone temperament
• Microtonal music
• Microtuner
• Musical tuning
• Pythagorean tuning
• Pythagorean intervals
• Semitone
• Well temperament
• List of meantone intervals
• Whole-tone scale
• Regular number

An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... A musical scale is a discrete set of pitches used in making or describing music. ... Meantone temperament is a system of musical tuning. ... Microtonal music is music using microtones — intervals of less than an equally spaced semitone, or as Charles Ives put it, the notes between the cracks of the piano. ... Microtuner is an electronic device or audio software endowed with microtuning capabilities specifically designed and used to modify the tuning of musical instruments (in particular synthesizers), hence allowing for microtonal scales, just intonation scales and tunings other than the twelve-tone equal temperament to be played (microtonal tuner). ... In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... The intervals of Pythagorean tuning are just intervals involving only powers of two and three. ... A semitone (also known in the USA as a half step) is a musical interval. ... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ... The following is a list of intervals of meantone temperament. ... In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole step. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

Instruments designed for just intonation

• 3rd Bridge
• Moodswinger
• Bagpipe
• Natural horn
• Trumpet marine

In 2006 luthier Yuri Landman built the Moodswinger, a 12 string overtone guitar for Aaron Hemphill of the noiseband Liars The 3rd bridge guitar is an electric prepared guitar with an additional 3rd bridge. ... In 2006 luthier Yuri Landman built the Moodswinger, a 12 string overtone zither for Aaron Hemphill of the noiseband Liars The Moodswinger is a custom made string instrument made by Yuri Landman. ... A bagpipe performer in Amsterdam. ... The musical instrument natural horn is the ancestor of the modern-day French horn differentiated by its lack of valves. ... The trumpet marine, also known as the tromba marina, is a stringed instrument. ...

References

1. ^ The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts dating to roughly . See: West, M.L.. The Babylonian Musical Notation and the Hurrian Melodic Texts, Music & Letters vol. 75 no. 2 (May 1994). pp. 161-179.
2. ^ Grout, Donald Jay and Claude Palisca. A History of Western Music, sixth edition. W.W. Norton & Company Inc., New York, 2000. p. 349. ISBN 0-393-97527-4
3. ^ Chopin would also write a set of compositions in every key, his 24 Preludes, which in contrast to Bach's Well-Tempered Clavier, made extensive use of the chromatic modulations characteristic of Romantic music. See: Ibid. p. 579.

The Preludes Op. ... The era of Romantic music is defined as the period of European classical music that runs roughly from the early 1800s to the first decade of the 20th century, as well as music written according to the norms and styles of that period. ...

External links

• The Wilson Archives
• Tonalsoft Encyclopedia of Tuning
• Just Intonation Explained by Kyle Gann
• Just Intonation Network
• The Chrysalis Foundation -- Just Intonation: Two Definitions
• Medieval Music and Arts Foundation
• Why does our musical scale have twelve notes?
• Why does Just Intonation sound so good?
• Art of the States: microtonal/just intonation works using just intonation by American composers
• Music Novatory - Just Intonation
• Dante Rosati's 21 Tone Just Intonation guitar

Kyle Gann (born November 21 1955) is a composer and music critic born in Dallas, Texas. ... In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... The 43-tone scale is a just intonation scale with 43 pitches in each octave invented and used by Harry Partch. ... Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ... In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. ... A quarter tone is an interval half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone. ... In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the scale derived by dividing the octave into 31 equally large steps. ... In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into fifty-three equally large steps. ... In music, 72 equal temperament, called 72-tet, 72-edo, or 72-et, is the scale derived by dividing the octave into twelfth-tones, or in other words 72 equally large steps. ... Meantone temperament is a system of musical tuning. ... Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. ... LucyTuning is a form of meantone temperament, in which the fifth is of size 600+300/π (= approximately 695. ... In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal... In music, schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ... In music, miracle temperament is a regular temperament invented by George Secor which has as a generator an interval, called the secor, which serves as both the 15/14 and 16/15 semitones. ... Well temperament (also circular or circulating temperament) is a type of tempered tuning described in twentieth-century music theory. ...