just intonation

Vincenzo Galilei (father of Galileo Galilei) may have been the first person to advocate equal temperament (in a 1581 treatise). The first person known to introduce a mathematically accurate specification for equal temperament is probably Chu Tsai-Yu (朱載堉) in the Ming Dynasty, who published a theory of the temperament in 1584. Soon after, European mathematicians Simon Stevin (1585, inspired by V. Galilei) and Marin Mersenne ( ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

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"Well tempered" means that the 12 notes per octave of the standard keyboard are tuned in such a way that it is possible to play music in any major or minor key and it will not sound perceptibly out of tune. In most tuning systems used before 1700, one or more intervals on the 12-note keyboard were so far from any pure interval that they were unusable in harmony and were called a "wolf". The most used system immediately before Werckmeister was meantone in which one fifth (usually E♭–G♯) was nearly two commas wider than pure ...

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Well temperament, Well temperament - Origins, Well temperament - Forms of well temperament

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Human beings distinguish sounds on the basis of their frequency. Actually what really matters is the ratio between their frequencies. The natural scale is attributed to the Grecian philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies. After fixing the frequency of the first note — the C of the scale — the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the fol ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

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Five limit just intonation has a modulatory space based on the fact that its pitch classes can be represented by 3a 5b, where a and b are integers. It is therefore a free abelian group with the two generators 3 and 5, and can be represented in terms of a square lattice with fifths along the horizontal axis, and major thirds along the vertical axis. In many ways a more enlightening picture emerges if we represent it in terms of a hexagonal lattice instead; this is the Tonnetz of Hugo Riemann, discovered independen ...

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Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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Zarlino was born in Chioggia, near Venice. His early education was with the Franciscans, and he later joined the order himself. In 1536 he was a singer at Chioggia Cathedral, and by 1539 he not only became a deacon, but became principal organist. In 1540 he was ordained, and in 1541 went to Venice to study with the famous contrapuntist and maestro di cappella of Saint Mark's, Adrian Willaert. In 1565, on the resignation of Cipriano de Rore, Zarlino took over the post of maestro di cappella of St. Mark's, one of the most ...

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Gioseffo Zarlino, Gioseffo Zarlino - Life, Gioseffo Zarlino - Works and influence

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In diatonic or tonal theory intervals are labelled according to their diatonic function and according to the number of members or degrees they span in a diatonic scale. The interval number of a note from a given tonic note is the number of staff positions enclosed within the interval, as shown at right. Intervals larger than an octave are called compound intervals; for example, a tenth is known as a compound third. Intervals larger than a thirteenth are rarely spoken of, since going above this by stacking thirds would re ...

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Interval music, Interval music - Frequency ratios, Interval music - Interval number and quality, Interval music - Shorthand notation, Interval music - Enharmonic intervals, Interval music - Steps and skips, Interval music - Pitch class intervals, Interval music - Ordered and unordered pitch and pitch class intervals, Interval music - Generic and specific intervals, Interval music - Cents, Interval music - Comparison of different interval naming systems, Interval music - Consonant and dissonant intervals, Interval music - Inversion, Interval music - Interval roots, Interval music - Interval cycles, Interval music - Other intervals, Interval music - Sources

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The term meantone temperament is sometimes used to refer specifically to quarter-comma meantone. However, systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems. Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison. While the term meanto ...

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Meantone temperament, Meantone temperament - Meantone temperaments, Meantone temperament - Wolf intervals and extended meantones

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All musical tuning have advantages and disadvantages. Twelve tone equal temperament is the standard and most usual tuning system used in western music today because it gives the advantage of modulation to any key without dramatically going out of tune, as all keys are equally and slightly out of tune. However, just intonation provides the advantage of being entirely in tune, with at least some, and possible a great deal, loss in ease of modulation. Referring to 12-tet the composer Terry Riley, who has written music for both tuning systems, h ...

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Musical tuning, Musical tuning - Subjects in general, Musical tuning - Ways of tuning the twelve-note chromatic scale, Musical tuning - Tunings of other scale systems, Musical tuning - Comparisons and controversies between tunings

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Each note produced by a musical instrument is made of a number of distinct frequencies, measured in hertz (Hz). The lowest frequency is called the fundamental and the pitch produced by this frequency is used to name the note. For example, in western music, instruments are normally tuned to A = 440 Hz. However, the richness of the sound is produced by the combination of this fundamental with a series of harmonics and/or partials (also collectively called overtones). Most western instruments produce harmonic sounds, and these can ...

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Timbre, Timbre - Terms, Timbre - American Standards Association definition, Timbre - Attributes, Timbre - Spectra, Timbre - Envelope, Timbre - In music, Timbre - Sources

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Harrison was born in Portland, Oregon, but moved with his family to a number of locations around the San Francisco Bay area as a child. The diverse music which he was to exposed to there, including Cantonese opera, Native American music, Mexican music and jazz as well as classical music, was to have a major influence on him. He also heard recordings of Indonesian music early in life. Harrison took Henry Cowell's "Music of the Peoples of the World" course, and also studied counterpoint and composition with him. He later went to the Uni ...

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Lou Harrison, Lou Harrison - Biography, Lou Harrison - Harrison's music, Lou Harrison - Source

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The violin is usually held under the chin and supported by the left shoulder. The strings may be sounded by either plucking (pizzicato) with either hand, or more commonly, by drawing the hair of the bow across them (arco). Rarely, they may be struck with the bow stick (col legno). The left hand regulates the sounding length of the string by stopping it against the fingerboard with the fingertips, producing different pitches. With the left hand in one position, a continuous range of slightly more than two octaves may be sounded across the different strings. Pla ...

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Playing the violin, Playing the violin - Playing the violin, Playing the violin - Left Hand & Producing Pitch, Playing the violin - Right Hand & Tone Colour, Playing the violin - Mute, Playing the violin - Tuning

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Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245/243 (sometimes called the minor Bohlen-Pierce diesis) and 3125/3072 (sometimes called the major Bohlen-Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81/80 and 128/125 to a unison. One can produce a 7-limit linear temperament by tempering out both of these intervals; the resulting Bohlen-Pierce temperament no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact ...

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Bohlen-Pierce scale, Bohlen-Pierce scale - Bohlen-Pierce temperament, Bohlen-Pierce scale - External link

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Western common practice music usually cannot be played in just intonation, even when it is confined to a single key. This is because the supertonic chord, or ii-chord, which is the most important of the minor triads in a major key, serves to bridge between the dominant and subdominant, having a root at once a minor third below the root of the subdominant triad, and hence sharing two of its notes, and a fifth above the root of the dominant triad or dominant seventh chord. The problem becomes still worse when modulation, the key changes so imp ...

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Mathematics of musical scales, Mathematics of musical scales - Pythagorean tuning, Mathematics of musical scales - Just intonation, Mathematics of musical scales - Temperament, Mathematics of musical scales - Equal temperament, Mathematics of musical scales - Sound samples, Mathematics of musical scales - Source

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In diatonic or tonal theory intervals are labelled according to their diatonic function and according to the number of members or degrees they span in a diatonic scale. The interval number of a note from a given tonic note is the number of staff positions enclosed within the interval, as shown at right. Intervals larger than an octave are called compound intervals; for example, a tenth is known as a compound third. Intervals larger than a thirteenth are rarely spoken of (but see 8va for use of 15ma). ...

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Interval music, Interval music - Frequency ratios, Interval music - Interval number and quality, Interval music - Shorthand notation, Interval music - Enharmonic intervals, Interval music - Steps and skips, Interval music - Pitch class intervals, Interval music - Ordered and unordered pitch and pitch class intervals, Interval music - Generic and specific intervals, Interval music - Cents, Interval music - Comparison of different interval naming systems, Interval music - Consonant and dissonant intervals, Interval music - Inversion, Interval music - Interval roots, Interval music - Interval cycles, Interval music - Other intervals, Interval music - Sources

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The American hardcore punk band Black Flag (1976-86) made interesting vernacular use of microtonal intervals, via guitarist Greg Ginn, a free jazz aficionado also familiar with modern classical. (During their peak in the late '70s and early '80s, long before American punk was mainstream, the band was considered, not unwarrantedly, a thuggish and hostile street unit, although time has given their work a considerable measure of musical acclaim.) A worthwhile song is "Damaged II," from 1981's Damaged LP -- a live-in-studio recording in w ...

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Microtonal music, Microtonal music - Microtonalism in rock music, Microtonal music - Source

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If we divide the octave into n parts, where n = rs is the product of two relatively prime integers r and s, we may represent every element of the tone space as the product of a certain number of "r" generators times a certain number of "s" generators; in other words, as the direct sum of two cyclic groups of orders r and s. We may now define a graph with n verticies on which the group acts, by adding an edge between to pitch classes whenever they differ by either an "r" generator or an "s" generator. The result is a graph of genus one, which is to say, a graph with a ...

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Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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A temperament of rank two which is not linear has one generator which is a fraction of an octave, called the period. We may represent the modulatory space of a such a temperament as n chains of generators in a circle, forming a cylinder. Here n is the number of periods in an octave. For example, diaschismic temperament is the temperament which tempers out the diaschisma, or 2048/2025. It can be represented as two chains of slightly (3.25 to 3.55 cents) sharp fifths a half-octave apart, which can be depicted as two chains perpendicular ...

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Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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In similar fashion, we can define a modulatory space for seven-limit just intonation, by representing 3a 5b 7c in terms of a corresponding cubic lattice. Once again, however, a more enlightening picture emerges if we represent it instead in terms of the three-dimensional analog of the hexagonal lattice, a lattice called A3, which is equivalent to the face centered cubic lattice, or D3. Abstractly, it can be defined as the integer triples (a, b, c), associated to 3a 5bSee also:

Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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A basic and important example of a modulatory space is the circle of fifths. In equal temperament, twelve succesive fifths equate to seven octaves exactly, and hence in terms of pitch classes closes back to itself, forming a circle. Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve. If we divide the octave into n equal parts, and choose an integer m<n such that m and n are relatively prime, we may obtain similar circles, which all have the structure of finite cyc ...

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Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental theorem of arithmetic). Therefore, a chromatic scale in Pythagorean tuning must have one fifth that is out of tune by the Pythagorean comma, called a wolf fifth. Most meantone temperaments share this problem, except for the case where the fifth is exactly 700 cents (tempered by approximately 1/11 of a syntonic comma) and the meantone becomes the ...

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Meantone temperament, Meantone temperament - Meantone temperaments, Meantone temperament - Wolf intervals and extended meantones

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Violins are tuned by turning the pegs in the pegbox under the scroll, or by winding the fine tuner screws at the tailpiece. A violin always has pegs, but Fine Tuners (also called adjustors) are optional. These permit the tension of the string to be adjusted in very small increments by rotating a small knob more easily than by using the pegs. Fine tuners are usually recommended for younger players, fractional sized instruments, those using high tension or metal strings, or beginners. Adjustors are most useful with solid m ...

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Playing the violin, Playing the violin - Playing the violin, Playing the violin - Left Hand & Producing Pitch, Playing the violin - Right Hand & Tone Colour, Playing the violin - Mute, Playing the violin - Tuning

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