1.
Musical temperament
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In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system, tempering is the process of altering the size of an interval by making it narrower or wider than pure. The development of well temperament allowed fixed-pitch instruments to play well in all of the keys. The famous Well-Tempered Clavier by Johann Sebastian Bach takes full advantage of this breakthrough, however, while unpleasant intervals were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies, allowing intervals varying from the highest consonance to highly dissonant, for instance,660 Hz /440 Hz constitutes a fifth, and 880 Hz /440 Hz an octave. Such intervals have a stability, or purity to their sound, if, for example, two sound signals with frequencies that vary just by 0. When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency, the waveform of such a tone is characterized by a shape that is complex compared to a simple waveform, but remains periodic. When two tones depart from exact integer ratios, the shape waveform becomes erratic—a phenomenon that may be described as destabilization, as the composite waveform becomes more erratic, the consonance of the interval also changes. Tempering an interval involves the use of such minor adjustments to enable musical possibilities that are impractical using just intonation. Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning, Pythagorean tuning was a system of just intonation that tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the practice until then. The major third of Pythagorean tuning differed from a just major third by an amount known as syntonic comma, with the correct amount of tempering, the syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this system with a slight beating. Pythagorean tuning also had a problem, which meantone temperament does not solve, which is the problem of modulation. A series of 12 just fifths as in Pythagorean tuning does not return to the pitch, but rather differs by a Pythagorean comma. In meantone temperament, this effect is more pronounced. The use of 53 equal temperament provides a solution for the Pythagorean tuning, when building an instrument, this can be very impractical. Well temperament is the given to a variety of different systems of temperament that were employed to solve this problem, in which some keys are more in tune than others
2.
Limit (music)
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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, roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant will the intervals of the tuning be perceived. A scale belonging to a prime limit has a distinctive hue that makes it aurally distinguishable from scales with other limits. 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 perfect fifths were considered consonant. In the West, triadic harmony arose around the time of the Renaissance, 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 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 7th chord in 12-ET approximates 4,5,6,7, while the major 7th 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 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, since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partchs theoretical prediction of the dissonance of intervals are very similar to those of theorists including Hermann von Helmholtz, William Sethares. An identity is each of the odd numbers below and including the limit in a tuning, for example, the identities included in 5-limit tuning are 1,3, and 5. 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, odentity and udentity are, short for Over-Identity, and, Under-Identity, respectively. An udentity is an identity of an utonality, for a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with numerator and denominator both n-smooth, given a prime number p, the subset of Q + consisting of those rational numbers x whose prime factorization has the form x = p 1 α1 p 2 α2. P r ≤ p forms a subgroup of and we say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup
3.
Meantone temperament
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Meantone temperament is a musical temperament, which is a system of musical tuning. Quarter-comma meantone is the best known type of meantone temperament, Meantone can receive the following equivalent definitions, The meantone is the mean between the major whole tone and the minor whole tone, i. e. the geometric mean of 9,8 and 10,9. The meantone is the mean of the just major third, i. e. the square root of 5,4, all meantone temperaments are linear temperaments, distinguished by the width of its generator in cents, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a portion of this tuning continuum. In Figure 1, the tuning ranges of 5-limit, 7-limit, and 11-limit tunings are shown. This last ratio was termed R by American composer, pianist and theoretician Easley Blackwood, if we multiply by 1200, we have the size of fifth in cents. In these terms, some historically notable meantone tunings are listed below, the relationship between the first two columns is exact, while that between them and the third is closely approximate. Equal temperaments useful as meantone tunings include 19-ET, 50-ET, 31-ET, 43-ET, the farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by tempering the timbre to match the tuning. A whole number of just perfect fifths will never add up to a number of octaves. If a stacked-up whole number of perfect fifths is to close with the octave, then one of the fifths must have a different width than all of the others. For example, to make the 12-note chromatic scale in Pythagorean tuning close at the octave, one fifth must be out of tune by the Pythagorean comma, wolf intervals are an artifact of keyboard design. This can be shown most easily using a keyboard, such as that shown in Figure 2. On an isomorphic keyboard, any musical interval has the same shape wherever it appears. On the keyboard shown in Figure 2, from any given note, there are no wolf intervals within the note-span of this keyboard. The problem is at the edge, on the note E♯, the note thats a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown. Because there is no B♯ button, when playing an E♯ power chord, one must choose some other note, such as C, even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes. When the perfect fifth is exactly 700 cents wide then the tuning is identical to the familiar 12-tone equal temperament and this appears in the table above when R =2,1. Because of the forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments
4.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
5.
19 equal temperament
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In music,19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a ratio of 19√2, or 63.16 cents. 19-edo is the tuning of the temperament in which the tempered perfect fifth is equal to 694.737 cents. Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory, the greater diesis, the ratio of four minor thirds to an octave was almost exactly a nineteenth of an octave. Interest in such a system goes back to the 16th century. Costeley understood and desired the circulating aspect of this tuning, in 1577, music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1⁄3-comma meantone, in which the fifth is of size 694.786 cents, the fifth of 19-edo is 694.737, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error. Salinas suggested tuning nineteen tones to the octave to this tuning, in the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50-edo. The composer Joel Mandelbaum wrote his Ph. D. thesis on the properties of the 19-edo tuning, Mandelbaum and Joseph Yasser have written music with 19-edo. Easley Blackwood has stated that 19-edo makes possible a substantial enrichment of the tonal repertoire, here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series, the difference column measures in cents the distance from an exact fit to these ratios. For reference, the difference from the fifth in the widely used 12 equal temperament is 1.955 cents. Beta scale Elaine Walker Levy, Kenneth J. Costeleys Chromatic Chanson, Annales Musicologues, Moyen-Age et Renaissance, Tome III, howe, Hubert S. Jr. 19-Tone Theory and Applications, Aaron Copland School of Music at Queens College. Sethares, William A. Tunings for 19 Tone Equal Tempered Guitar, Experimental Musical Instruments, Vol. VI, hair, Bailey, Morrison, Pearson and Parncutt, Rehearsing Microtonal Music, Grappling with Performance and Intonational Problems, Microtonalism. ZiaSpace. com - 19tet downloadable mp3s by Elaine Walker of Zia, the Music of Jeff Harrington, Parnasse. com. Jeff Harrington is a composer who has several pieces for piano in the 19-TET tuning. Chris Vaisvil, GR-20 Hexaphonic 19-ET Guitar Improvisation Arto Juhani Heino, Artone 19 Guitar Design, naming the 19 note scale Parvatic
6.
31 equal temperament
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In music,31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Play Each step represents a ratio of 31√2, or 38.71 cents. 31-ET is a good approximation of quarter-comma meantone temperament. More generally, it is a diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents. In 1666, Lemme Rossi first proposed an equal temperament of this order, shortly thereafter, having discovered it independently, scientist Christiaan Huygens wrote about it also. Huygens not only realized this, he went farther and noted that 31-ET provides an excellent approximation of septimal, the composer Joel Mandelbaum used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series. The tuning has poor matches to both the 9,8 and 10,9 intervals, however, it has a match for the average of the two. Practically it is close to quarter-comma meantone. This tuning can be considered a meantone temperament, many chords of 31-ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad, which might be written C–E–G, C–D–G or C–F–G, and the Orwell tetrad, usual chords like the major chord is rendered nicely in 31-ET because the third and the fifth are very well approximated. Also, it is possible to play chords and supermajor chords. It is also possible to render nicely the harmonic seventh chord, for example on C with C–E–G–A♯. The seventh here is different from stacking a fifth and a minor third and this difference cannot be made in 12-ET
7.
7-limit tuning
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7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven, the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example,50,49 is a 7-limit interval, for example, the greater just minor seventh,9,5 Play is a 5-limit ratio, the harmonic seventh has the ratio 7,4 and is thus a septimal interval. Similarly, the chromatic semitone,21,20, is a septimal interval as 21÷7=3. The harmonic seventh is used in the seventh chord and music. Compositions with septimal tunings include La Monte Youngs The Well-Tuned Piano,4, and Lou Harrisons Incidental Music for Corneilles Cinna. The Great Highland Bagpipe is tuned to a ten-note seven-limit scale,1,1,9,8,5,4,4,3,27,20,3,2,5,3,7,4,16,9,9,5. In the 2nd century Ptolemy described the septimal intervals, 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7. Those considering 7 to be dissonant include Gioseffo Zarlino, René Descartes, Jean-Philippe Rameau, Hermann von Helmholtz, A. J. von Öttingen, Hugo Riemann, Colin Brown, the 7-limit tonality diamond, This diamond contains four identities. Similarly, the 2,3,5,7 pitch lattice contains four identities and thus 3-4 axes, laMonte Young created a lattice containing only identities 3 and 7, thus requiring only two axes, for The Well-Tuned Piano. It is possible to approximate 7-limit music using equal temperament, for example 31-ET
8.
Major third
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In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two, the major third spans four semitones, the third three. The major third may be derived from the series as the interval between the fourth and fifth harmonics. The major scale is so named because of the presence of this interval between its tonic and mediant scale degrees, the major chord also takes its name from the presence of this interval built on the chords root. The older concept of a made a dissonantly wide major third with the ratio 81,64. The septimal major third is 9,7, the major third is 14,11. A helpful way to recognize a third is to hum the first two notes of Kumbaya or of When the Saints Go Marching In. A descending major third is heard at the starts of Goodnight, Ladies and Swing Low, in equal temperament three major thirds in a row are equal to an octave. This is sometimes called the circle of thirds, in just intonation, however, three 5,4 major thirds are less than an octave. For example, three 5,4 major thirds from C is B♯, the difference between this just-tuned B♯ and C, like that between G♯ and A♭, is called a diesis, about 41 cents. The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, a diminished fourth is enharmonically equivalent to a major third. For example, B–D♯ is a third, but if the same pitches are spelled B and E♭. B–E♭ occurs in the C harmonic minor scale, the major third is used in guitar tunings. For the standard tuning, only the interval between the 3rd and 2nd strings is a third, each of the intervals between the other pairs of consecutive strings is a perfect fourth. In an alternative tuning, the tuning, each of the intervals are major thirds. Decade, compound just major third Ear training List of meantone intervals Doubling the cube, 21/3 = 3√2
9.
Harmonic seventh
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The harmonic seventh interval play, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7,4 ratio. This is somewhat narrower than and is, particularly sweet, sweeter in quality than a minor seventh. The harmonic seventh may be derived from the series as the interval between the seventh harmonic and the fourth harmonic. Composer Ben Johnston uses a small 7 as an accidental to indicate a note is lowered 49 cents, thus, in C major, the seventh partial, or harmonic seventh, is notated as ♭ B with 7 written above the flat. Instead, Hagerman and Sundberg found that tuning of major and minor third intervals in barbershop lies between just and equal temperament, the harmonic seventh differs from the augmented sixth by 224/225, or about one-third of a comma. The harmonic seventh note is one third of a semitone flatter than an equal tempered minor seventh. When this flatter seventh is used, the dominant seventh chords need to resolve down a fifth is weak or non-existent and this chord is often used on the tonic and functions as a fully resolved final chord. The twenty-first harmonic is the seventh of the dominant, and would then arise in chains of secondary dominants in styles using harmonic sevenths. The Tonal Phoenix, A Study of Tonal Progression Through the Prime Numbers Three, Five, & Sundberg, J. Fundamental frequency adjustment in barbershop singing. Journal of Research in Singing,4, 3-17
10.
Quarter-comma meantone
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Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma, the purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be sonorous and just, later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude. In a meantone tuning, we have diatonic and chromatic semitones, in Pythagorean tuning, these correspond to the Pythagorean limma and the Pythagorean apotome, only now the apotome is larger. Two fifths up and an octave down make up a whole tone consisting of one diatonic, four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones. This large interval of a seventeenth contains 5 + + + =20 −3 =17 staff positions. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths,4 =8116 =8016 ⋅8180 =5 ⋅8180. In quarter-comma meantone temperament, where a just major third is required, by definition, however, a seventeenth of the same size must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as used in Pythagorean tuning, produce a slightly wider seventeenth. Notice that, in quarter-comma meantone, the seventeenth is 81/80 times narrower than in Pythagorean tuning and this difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. The difference between two sizes is a quarter of a syntonic comma, ≈701.955 −696.578 ≈5.377 ≈21.5064 cents. In sum, this system tunes the major thirds to the just ratio of 5,4, most of the tones in the ratio √5,2. This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth and it is this that gives the system its name of quarter-comma meantone. The whole chromatic scale, can be constructed by starting from a base note. This method is identical to Pythagorean tuning, except for the size of the fifth, the construction table below illustrates how the pitches of the notes are obtained with respect to D, in a D-based scale. As in Pythagorean tuning, this method generates 13 pitches, but A♭ and G♯ have almost the same frequency, the table above shows a D-based stack of fifths. Some authors prefer showing a C-based stack of fifths, ranging from A♭ to G♯, the only difference is that the construction table shows intervals from C, rather than from D
11.
Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has become the method of representing and comparing musical pitches. Like a decibels relation to intensity, a cent is a ratio between two close frequencies, for the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone spans 100 cents by definition, an octave—two notes that have a frequency ratio of 2, 1—spans twelve semitones and therefore 1200 cents.0005777895. For example, in just intonation the major third is represented by the frequency ratio 5,4, applying the formula at the top shows that this is about 386 cents. The equivalent interval on the piano would be 400 cents. The difference,14 cents, is about a seventh of a half step, as x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a linear function that is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1 +0. 0005946n instead of 2 n⁄1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50 and this error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes. It is difficult to establish how many cents are perceptible to humans, one author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians ability to recognize, as out-of-tune and it has also been established that increased tonal context enables listeners to judge pitch more accurately. Free, online web sites for self-testing are available, while intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords. When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch, normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents, iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone, centitone, and millitone, for example, Equal tempered perfect fifth =700 cents =175.6 savarts =583.3 millioctaves =350 centitones
12.
Syntonic comma
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The comma is referred to as a comma of Didymus because it is the amount by which Didymus corrected the Pythagorean major third to a just major third. Namely,81,64 ÷5,4 =81,80, the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has a size of 3,2, a just major third has a size of 5,4, and one of them plus two octaves is equal to 5,1. The difference between these is the syntonic comma, namely,81,16 ÷5,1 =81,80. The difference between one octave plus a justly tuned minor third, and three justly tuned perfect fourths, namely,12,5 ÷64,27 =81,80. The difference between the two kinds of major second which occur in 5-limit tuning, major tone and minor tone, namely,9,8 ÷10,9 =81,80. The difference between a Pythagorean major sixth and a justly tuned or pure major sixth, namely,27,16 ÷5,3 =81,80. On a piano keyboard a stack of four fifths is exactly equal to two octaves plus a major third, in other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves, fifths, and thirds, however, the ratio between their frequencies, as explained above, is a syntonic comma. Pythagorean tuning uses justly tuned fifths as well, but uses the complex ratio of 81,64 for major thirds. Quarter-comma meantone uses justly tuned major thirds, but flattens each of the fifths by a quarter of a syntonic comma and this is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments. Mathematically, by Størmers theorem,81,80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5,4, and a number is one whose prime factors are limited to 2,3. Thus, although smaller intervals can be described within 5-limit tunings, the syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds, in Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third and minor third were dissonant, and this prevented musicians from using triads and chords, in late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a comma, C-E. But the fifth C-G stays consonant, since only E has been flattened, since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them
13.
Septimal semicomma
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In music, the septimal semicomma, a seven-limit semicomma, is the ratio 126/125 and is equal to approximately 13.79 cents. It is also called the septimal comma and the starling comma after its use in starling temperament. This comma is important to certain tuning systems, such as septimal meantone temperament, a diminished seventh chord consisting of three minor thirds and a subminor third making up an octave is possible in such systems. This characteristic feature of these systems is known as the septimal semicomma diminished seventh chord. It is tempered out in 19 equal temperament and 31 equal temperament, but not in 22 equal temperament,34 equal temperament,41 equal temperament, or 53 equal temperament
14.
Septimal kleisma
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In music, the ratio 225/224 is called the septimal kleisma. It is a minute comma type interval of approximately 7.7 cents, factoring it into primes gives 2−53252 7−1, which can be rewritten 2−12. That says that it is the amount that two thirds of 5/4 and a septimal major third, or supermajor third, of 9/7 exceeds the octave. The existence of such a chord, which might be termed the septimal kleisma augmented triad, is a significant feature of a tuning system, the septimal kleisma can also be viewed as the difference between the diatonic semitone and the septimal diatonic semitone
15.
Otonality and Utonality
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Otonality and utonality are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone, respectively. An otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, for example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore composed of members of a harmonic series, similarly, the ratios of a utonality share the same numerator. 7/4, 7/5, 7/6, and 1/1 form a utonality, every utonality is therefore composed of members of a subharmonic series. An otonality corresponds to a series of frequencies, or lengths of a vibrating string. Brass instruments naturally produce otonalities, and indeed otonalities are inherent in the harmonics of a fundamental tone. Tuvan Khoomei singers produce otonalities with their vocal tracts, Utonality is the opposite, corresponding to a subharmonic series of frequencies, or an arithmetic series of wavelengths. The arithmetical proportion may be considered as a demonstration of utonality, microtonalists define a just intonation chord as otonal if its odd limit increases on being inverted, utonal if its odd limit decreases, and ambitonal if its odd limit is unchanged. The chord is not inverted in the sense, where C E G becomes E G C or G C E. Instead. A chords odd limit is the largest odd limit of each of the numbers in the chords extended ratio, for example, the major triad 4,5,6 has an odd limit of 5. Its inverse 10,12,15 has an odd limit of 15, Partch said that his 1931 coinage of otonality and utonality was, hastened, by having read Henry Cowells discussion of undertones in New Musical Resources. The 5-limit otonality is simply a just major chord, and the 5-limit utonality is a just minor chord, thus otonality and utonality can be viewed as extensions of major and minor tonality respectively. This chord might be, for example, A♭-C-E♭-G♭ Play, standing alone, it has something of the sound of a dominant seventh, but considerably less dissonant. Utonal chords, while containing the same dyads and roughness as otonal chords, numerary nexus Scale of harmonics Tonality flux Otonality and ADO system at 96-EDO Utonality and EDL system at 96-EDO
16.
Augmented sixth chord
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In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, further developed in the Baroque, conventionally used with a predominant function, the three more common types of augmented sixth chords are usually called Italian sixth, French sixth, and German sixth. The augmented sixth interval is typically between the degree of the minor scale and the raised fourth degree. With standard voice leading, the chord is followed directly or indirectly by some form of the dominant chord and this tendency to resolve outwards to 5 is why the interval is spelled as an augmented sixth, rather than enharmonically as a minor seventh. Although augmented sixth chords are common in the minor mode. However, this may be used as the derivation of the augmented sixth chord, for example, F–A♭–C is a minor triad. F♯–A♭–C is a diminished triad. Note that it is equivalent to G♭–A♭–C, the tritone substitute resolving to G. Its inversion, A♭–C–F♯, is the Italian augmented sixth chord resolving to G, there are three main types of augmented sixth chords, commonly known as Italian sixth, French sixth, and German sixth. Though each is named after a European nationality, theorists disagree on their origins and have struggled for centuries to define their roots. According to Kosta and Payne, the two terms are similar to the Italian sixth, which, has no historical authenticity- simply a convenient. The Italian sixth is derived from iv6 with an altered fourth scale degree, ♯4, ♭6–1–♯4, A♭–C–F♯ in C major and this is the only augmented sixth chord comprising just three distinct notes, in four-part writing, the tonic pitch is doubled. The Italian sixth is enharmonically equivalent to a dominant seventh. The French sixth is similar to the Italian, but with a tone,2, ♭6–1–2–♯4, A♭–C–D–F♯ in C major. The notes of the French sixth chord are all contained within the whole tone scale. This chord has the notes as a dominant seventh flat five chord. The German sixth is also like the Italian, but with an added tone ♭3, ♭6–1–♭3–♯4, A♭–C–E♭–F♯ in C major and C minor. In Classical music, however, it appears in much the same places as the other variants and it appears frequently in the works of Beethoven, and in ragtime music
17.
Tristan chord
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The Tristan chord is a chord made up of the notes F, B, D♯, and G♯. More generally, it can be any chord that consists of these intervals, augmented fourth, augmented sixth. It is so named as it is heard in the phrase of Richard Wagners opera Tristan und Isolde as part of the leitmotif relating to Tristan. The notes of the Tristan chord are not unusual, they could be respelled enharmonically to form a common half-diminished seventh chord, what distinguishes the chord is its unusual relationship to the implied key of its surroundings. This motif also appears in measures 6,10, and 12, several times later in the work, much has been written about the Tristan chords possible harmonic functions or voice leading, and the motif has been interpreted in various ways. For instance, Arnold Schering traces the development of the Tristan chord through ten intermediate steps, beginning with the Phrygian cadence. Martin Vogel points out the chord in earlier works by Guillaume de Machaut, Carlo Gesualdo, Bach, Mozart, Beethoven, or Louis Spohr as in the following example from Beethovens Piano Sonata No. 18, tempo allegro, The chord is found in works by Fryderyk Chopin, from as early as 1828, in the Sonata in C minor. It is only in late works where tonal ambiguities similar to Wagners arise, as in the Prelude in A minor, Op.28,2, and the posthumously published Mazurka in F minor, Op.68, No.4. The Tristan chords significance is in its away from traditional tonal harmony. With this chord, Wagner actually provoked the sound or structure of musical harmony to become more predominant than its function, a notion which was soon explored by Debussy and others. In the words of Robert Erickson, The Tristan chord is, among other things, an identifiable sound, although at the same time enharmonically sounding like the half-diminished chord F-A♭-C♭-E♭, it can also be interpreted as the suspended altered subdominant II, B-D♯-F-G♯. I have never been able to understand how the idea that Tristan could be made the prototype of an atonality grounded in destruction of all tension could possibly have gained credence. After this, it easy to convince naive readers that such an aggregation escapes classification in terms of harmony textbooks. Nattiez, distinguishes between functional and nonfunctional analyses of the chord and this interpretation of the chord is confirmed by its subsequent appearances in the Preludes first period, the IV6 chord remains constant, notes foreign to that chord vary. Nonfunctional analyses are based on structure, and are characterized as vertical characterizations or linear analyses and this ascent by minor third is mirrored by the descending line, a descent by minor third, making the D♯, like A♯, an appoggiatura. This makes the chord a diminished seventh, thus in the soprano motif, the G♯ and the A♯ are heard as appoggiaturas, as the F and D♯ in the initial motif. The chord is thus a chord with added sixth on the fourth degree
18.
Enharmonic
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Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in equal temperament, the notes C♯. Namely, they are the key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony. In other words, if two notes have the pitch but are represented by different letter names and accidentals, they are enharmonic. Enharmonic intervals are intervals with the sound that are spelled differently…, of course. Enharmonic equivalence is peculiar to post-tonal theory, much music since at least the 18th century, however, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent. Some key signatures have an enharmonic equivalent that represents a scale identical in sound, the number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is equivalent to the key of C♭ major with 7 flats. Keys past 7 sharps or flats exist only theoretically and not in practice, the enharmonic keys are six pairs, three major and three minor, B major/C♭ major, G♯ minor/A♭ minor, F♯ major/G♭ major, D♯ minor/E♭ minor, C♯ major/D♭ major and A♯ minor/B♭ minor. There are practically no works composed in keys that require double sharps or double flats in the key signature, in practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings. Enharmonic equivalents can also used to improve the readability of a line of music, for example, a sequence of notes is more easily read as ascending or descending if the noteheads are on different positions on the staff. Doing so may reduce the number of accidentals that must be used. Thus, in the key of B♭ major, the sequence B♭-B♮-B♭ is more easily using the enharmonic spelling C♭ instead of B♮. For example the intervals of a sixth on C, on B♯. The most common enharmonic intervals are the fourth and diminished fifth, or tritone. Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals. In principle, the musical use of the word enharmonic to mean identical tones is correct only in equal temperament. In other tuning systems, however, enharmonic associations can be perceived by listeners, in Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2
19.
Musical tuning
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In music, there are two common meanings for tuning, Tuning practice, the act of tuning an instrument or voice. Tuning systems, the systems of pitches used to tune an instrument. Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones, Tuning is usually based on a fixed reference, such as A =440 Hz. Out of tune refers to a pitch/tone that is too high or too low in relation to a given reference pitch. While an instrument might be in relative to its own range of notes. Some instruments become out of tune with damage or time and must be readjusted or repaired, different methods of sound production require different methods of adjustment, Tuning to a pitch with ones voice is called matching pitch and is the most basic skill learned in ear training. Turning pegs to increase or decrease the tension on strings so as to control the pitch, instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs, while others such as the violin can be tuned manually. Modifying the length or width of the tube of an instrument, brass instrument, pipe, bell. The sounds of instruments such as cymbals are inharmonic—they have irregular overtones not conforming to the harmonic series. Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other, a tuning fork or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano is used. Symphony orchestras and concert bands tend to tune to an A or a B♭, respectively, interference beats are used to objectively measure the accuracy of tuning. As the two approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected, for other intervals, this is dependent on the tuning system being used. Harmonics may be used to facilitate tuning of strings that are not themselves tuned to the unison, for example, lightly touching the highest string of a cello at the middle while bowing produces the same pitch as doing the same a third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the fifth between the fundamentals of the two strings. In music, the open string refers to the fundamental note of the unstopped. The strings of a guitar are tuned to fourths, as are the strings of the bass guitar. Violin, viola, and cello strings are tuned to fifths, however, non-standard tunings exist to change the sound of the instrument or create other playing options
20.
Pitch (music)
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Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major attribute of musical tones, along with duration, loudness. Pitch may be quantified as a frequency, but pitch is not a purely objective physical property, Pitch is an auditory sensation in which a listener assigns musical tones to relative positions on a musical scale based primarily on their perception of the frequency of vibration. Pitch is closely related to frequency, but the two are not equivalent, frequency is an objective, scientific attribute that can be measured. Pitch is each persons subjective perception of a wave, which cannot be directly measured. However, this not necessarily mean that most people wont agree on which notes are higher and lower. Sound waves themselves do not have pitch, but their oscillations can be measured to obtain a frequency and it takes a sentient mind to map the internal quality of pitch. However, pitches are usually associated with, and thus quantified as frequencies in cycles per second, or hertz, by comparing sounds with pure tones, Complex and aperiodic sound waves can often be assigned a pitch by this method. According to the American National Standards Institute, pitch is the attribute of sound according to which sounds can be ordered on a scale from low to high. That is, high pitch means very rapid oscillation, and low pitch corresponds to slower oscillation, despite that, the idiom relating vertical height to sound pitch is shared by most languages. At least in English, it is just one of many deep conceptual metaphors that involve up/down, the exact etymological history of the musical sense of high and low pitch is still unclear. There is evidence that humans do actually perceive that the source of a sound is slightly higher or lower in vertical space when the frequency is increased or reduced. The pitch of tones can be ambiguous, meaning that two or more different pitches can be perceived, depending upon the observer. In a situation like this, the percept at 200 Hz is commonly referred to as the missing fundamental, Pitch depends to a lesser degree on the sound pressure level of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases, for instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder, theories of pitch perception try to explain how the physical sound and specific physiology of the auditory system work together to yield the experience of pitch. In general, pitch perception theories can be divided into place coding, place theory holds that the perception of pitch is determined by the place of maximum excitation on the basilar membrane. However, a purely place-based theory cannot account for the accuracy of pitch perception in the low, temporal theories offer an alternative that appeals to the temporal structure of action potentials, mostly the phase-locking and mode-locking of action potentials to frequencies in a stimulus
21.
Millioctave
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The millioctave is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave, from this it follows that one millioctave is equal to the ratio 21/1000, the 1000th root of 2, or approximately 1.0006934. A millioctave is exactly 1.2 cents, the millioctave was introduced by the German physicist Arthur von Oettingen in his book Das duale Harmoniesystem. The invention goes back to John Herschel, who proposed a division of the octave into 1000 parts, compared to the cent, the millioctave has not been as popular. It is, however, occasionally used by authors who wish to avoid the association between the cent and equal temperament. However, it has criticized that it introduces a bias for the less familiar 10-tone equal temperament. Cent Savart Musical tuning Logarithm Degree Chiliagon Logarithmic Interval Measures
22.
Savart
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The savart /səˈvɑːr/ is a unit of measurement for musical pitch intervals. One savart is equal to one thousandth of a decade,3.9863 cents, musically, in just intonation, the interval of a decade is precisely a just major twenty-fourth, or, in other words, three octaves and a just major third. Today the savart has largely replaced by the cent and the millioctave. The savart is practically the same as the earlier heptameride, one seventh of a meride, one tenth of an heptameride is a decameride and a hundredth of an heptameride is a jot. The number of savarts in an octave is 1000 times the logarithm of 2. Sometimes this is rounded to 300, which makes the unit more useful for equal temperament, Sauveur proposed the méride, eptaméride, and decaméride. In English these are meride, heptameride, and decameride respectively, the octave is divided into 43 merides, the meride is divided into seven heptamerides, and the heptameride is divided into ten decamerides. There are thus 43 ×7 =301 heptamerides in an octave. The attraction of this scheme to Sauveur was that log10 is very close to.301 and this is equivalent to assuming 1000 heptamerides in a decade rather than 301 in an octave, the same as Savarts definition. The unit was given the name savart sometime in the 20th century, a disadvantage of this scheme is that there are not an exact number of heptamerides/savarts in an equal tempered semitone. For this reason Alexander Wood used a definition of the savart, with 300 savarts in an octave. A related unit is the jot, of which there are 30103 in an octave, the jot is defined in a similar way to the savart, but has a more accurate rounding of log10 because more digits are used. There are approximately 100 jots in a savart, the unit was first described by Augustus de Morgan which he called an atom. The name jot was coined by John Curwen at the suggestion of Hermann von Helmholtz
23.
Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale, the smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones and they can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies, for example, any two notes an octave apart have a frequency ratio of 2,1. This means that successive increments of pitch by the same result in an exponential increase of frequency. For this reason, intervals are often measured in cents, a derived from the logarithm of the frequency ratio. In Western music theory, the most common naming scheme for intervals describes two properties of the interval, the quality and number, examples include the minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled, the importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. The size of an interval can be represented using two alternative and equivalently valid methods, each appropriate to a different context, frequency ratios or cents, the size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, Intervals with small-integer ratios are often called just intervals, or pure intervals. Most commonly, however, musical instruments are tuned using a different tuning system. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, for instance, an equal-tempered fifth has a frequency ratio of 2 7⁄12,1, approximately equal to 1.498,1, or 2.997,2. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems, the standard system for comparing interval sizes is with cents. The cent is a unit of measurement. If frequency is expressed in a scale, and along that scale the distance between a given frequency and its double is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament, a system in which all semitones have the same size. Hence, in 12-TET the cent can be defined as one hundredth of a semitone
24.
Interval ratio
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In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3,2,1.5, if the A above middle C is 440 Hz, the perfect fifth above it would be E, at 660 Hz, while the equal tempered E5 is 659.255 Hz. Ratios have a relationship to string length, for example stopping a string at two-thirds its length produces a pitch one. Intervals may be ranked by relative consonance and dissonance, as such ratios with lower integers are generally more consonant than intervals with higher integers. For example,2,1,4,3,9,8,65536,59049, consonance and dissonance may more subtly be defined by limit, wherein the ratios whose limit, which includes its integer multiples, is lower are generally more consonant. For example, the 3-limit 128,81 and the 7-limit 14,9, despite having larger integers 128,81 is less dissonant than 14,9, as according to limit theory. For ease of comparison intervals may also be measured in cents, for example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents. Frequency ratios are used to describe intervals in both Western and non-Western music. When a musical instrument is tuned using a just intonation tuning system, intervals with small-integer ratios are often called just intervals, or pure intervals. To most people, just intervals sound consonant, i. e. pleasant, although the size of equally tuned intervals is typically similar to that of just intervals, in most cases it cannot be expressed by small-integer ratios. For instance, a tempered perfect fifth has a frequency ratio of about 1.4983,1. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems
25.
Pitch class
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In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e. g. the pitch class C consists of the Cs in all octaves. The pitch class C stands for all possible Cs, in whatever octave position, important to musical set theory, a pitch class is, all pitches related to each other by octave, enharmonic equivalence, or both. Psychologists refer to the quality of a pitch as its chroma, a chroma is an attribute of pitches, just like hue is an attribute of color. A pitch class is a set of all pitches that share the same chroma, because octave-related pitches belong to the same class, when an octave is reached, the numbers begin again at zero. Indeed, the mapping from pitch to real numbers defined in this forms the basis of the MIDI Tuning Standard. To represent pitch classes, we need to identify or glue together all pitches belonging to the same pitch class—i. e, all numbers p and p +12. The result is a quotient group that musicians call pitch class space. Points in this space can be labelled using real numbers in the range 0 ≤ x <12. These numbers provide numerical alternatives to the names of elementary music theory,0 = C,1 = C♯/D♭,2 = D,2.5 = D,3 = D♯/E♭. In this system, pitch classes represented by integers are classes of equal temperament. In music, integer notation is the translation of pitch classes and/or interval classes into whole numbers, thus if C =0, then C♯ =1. A♯ =10, B =11, with 10 and 11 substituted by t and e in some sources and this allows the most economical presentation of information regarding post-tonal materials. In the integer model of pitch, all classes and intervals between pitch classes are designated using the numbers 0 through 11. It is not used to music for performance, but is a common analytical and compositional tool when working with chromatic music, including twelve tone, serial. The C above this is not 12, but 0 again, thus arithmetic modulo 12 is used to represent octave equivalence. One advantage of this system is that it ignores the spelling of notes according to their diatonic functionality, there are a few disadvantages with integer notation. First, theorists have used the same integers to indicate elements of different tuning systems. Thus, the numbers 0,1,2,5, are used to notate pitch classes in 6-tone equal temperament
26.
Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion, a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the forms a gradation. Consonance is associated with sweetness, pleasantness and acceptability and dissonance with harshness, unpleasantness, as Hindemith stressed, The two concepts have never been completely explained, and for a thousand years the definitions have varied. The opposition can be made in different contexts, In acoustics or psychophysiology, in modern times, it usually is based on the perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds. In music, even if the opposition often is founded on the preceding, objective distinction, it often is subjective, conventional, cultural. A major second would be considered dissonant if it occurred in a J. S, Bach prelude from the 1700s, however, the same interval may sound consonant in the context of a Claude Debussy piece from the early 1900s or an atonal contemporary piece. For this reason, consonance and dissonance have been considered particularly in the case of Western polyphonic music, most historical definitions of consonance and dissonance since about the 16th century have stressed their pleasant/unpleasant, or agreeable/disagreeable character. In addition, the oppositions pleasant/unpleasant or agreeable/disagreeable evidence a confusion between the concepts of dissonance and of noise and these include, Frequency ratios, with ratios of lower simple numbers being more consonant than those that are higher. Many of these definitions do not require exact integer tunings, only approximation, coincidence of partials, with consonance being a greater coincidence of partials. By this definition, consonance is dependent not only on the width of the interval between two notes, but also on the spectral distribution and thus sound quality of the notes. Thus, a note and the note one octave higher are highly consonant because the partials of the note are also partials of the lower note. Although Helmholtzs work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings, fusion, perception of unity or tonal fusion between two notes. A stable tone combination is a consonance, consonances are points of arrival, rest, an unstable tone combination is a dissonance, its tension demands an onward motion to a stable chord. Thus dissonant chords are active, traditionally they have been considered harsh and have expressed pain, grief, in Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance. Both consonance and dissonance are words applied to harmony, chords, and intervals and, by extension, to melody, tonality, and even rhythm and metre. Nevertheless, the ideas of dissonance, consonance, and resolution exist in some form in all musical traditions that have a concept of melody, harmony. Dissonance being the complement of consonance it may be defined, as above, as non-coincidence of partials, lack of fusion or pattern matching, or as complexity. For many musicians and composers, the ideas of dissonance
27.
Microtonal music
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Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are also called microintervals. It may also be extended to any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. Microtonal music can refer to any music containing microtones, therefore it is important to comprehend what a microtone is, the words microtone and microtonal were coined before 1912 by Maud MacCarthy Mann in order to avoid the misnomer quarter tone when speaking of the srutis of Indian music. Prior to this time the quarter tone was used, confusingly, not only for an interval actually half the size of a semitone. 1998, Wallon 1980,13, Whitfield 1989,13, microinterval is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers. In English, the two terms microtone and microinterval are synonymous, microtone is also sometimes used to refer to individual notes, microtonal pitches added to and distinct from the familiar twelve notes of the chromatic scale, as enharmonic microtones, for example. In English the word microtonality is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the blues scales. It was used earlier by W. McNaught with reference to developments in modernism in a 1939 record review of the Columbia History of Music. The term microinterval is used alongside microtone by American musicologist Margo Schulter in her articles on medieval music, the term microtonal music usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues, many microtonal equal divisions of the octave have been proposed, usually in order to achieve approximation to the intervals of just intonation. Terminology other than microtonal has been used or proposed by some theorists, in 1914, A. H. Fox Strangways objected that heterotone would be a better name for śruti than the usual translation microtone. Modern Indian researchers yet write, microtonal intervals called shrutis, a similar term, subchromatic, has been used recently by theorist Marek Žabka. Ivor Darreg proposed the term xenharmonic, another Russian authors use more international adjective microtonal and rendered it in Russian as микротоновый, but not microtonality. However are used микротональность and микротоника also, some authors writing in French have adopted the term micro-intervallique to describe such music. Italian musicologist Luca Conti dedicated two his monographs to microtonalismo, which is the term in Italian, and also in Spanish. The analogous English form, microtonalism, is found occasionally instead of microtonality. At the time when serialism and neoclassicism were still incipient a third movement emerged, the term macrotonal has also been used for musical form. The Hellenic civilizations of ancient Greece left fragmentary records of their music—e. g, ancient Greek intervals were of many different sizes, including microtones
28.
Just intonation
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In music, just intonation or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval, pure intervals are important in music because they correspond to the vibrational patterns found in physical objects which correlate to human perception. The two notes in any just interval are members of the harmonic series. Frequency ratios involving large integers such as 1024,729 are not generally said to be justly tuned, the Indian classical music system uses just intonation tuning as codified in the Natya Shastra. Various societies perceive pure intervals as pleasing or satisfying consonant and, conversely, however, various societies do not have these associations. Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch, however, except for doubling of frequencies, no other intervals are exact ratios of small integers. Each just interval differs a different amount from its analogous, equally tempered interval, justly tuned intervals can be written as either ratios, with a colon, or as fractions, with a solidus. For example, two tones, one at 300 hertz, and the other at 200 hertz are both multiples of 100 Hz and as members of the harmonic series built on 100 Hz. Thus 3,2, known as a fifth, may be defined as the musical interval between the second and third harmonics of any fundamental pitch. Just intonation An A-major scale, followed by three major triads, and then a progression of fifths in just intonation, equal temperament An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. By listening to the file, and then listening to this one, one might be able to hear the beating in this file. Equal temperament and just intonation compared A pair of major thirds, the first in each pair is in equal temperament, the second is in just intonation. Equal temperament and just intonation compared with square waveform A pair of major chords, the first is in equal temperament, the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just temperament between the two chords, in the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent, the square waveform makes the difference between equal and just temperaments more obvious. Harmonic intervals come naturally to horns, vibrating strings, and in human singing voices. Pythagorean tuning, perhaps the first tuning system to be theorized in the West, is a system in all tones can be found using powers of the ratio 3,2. It is easier to think of this system as a cycle of fifths
29.
Harmonic scale
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The Harmonic scale is a super-just musical scale allowing extended just intonation, beyond 5-limit to the 19th harmonic, and free modulation through the use of synthesizers. Transpositions and tuning tables are controlled by the hand on the appropriate note on a one-octave keyboard. For example, if the scale is tuned to a fundamental of C, then harmonics 16–32 are as follows, Some harmonics are not included,23,25,29. The 21st is a seventh above G, but not a great interval above C. Play diatonic scale It was invented by Wendy Carlos and used on three pieces on her album Beauty in the Beast, Just Imaginings, Thats Just It, versions of the scale have also been used by Ezra Sims and Frans Richter Herf. Though described by Carlos as containing 144 distinct pitches to the octave, technically there should then be duplicates and thus 57 pitches. For example, a fifth above G is the major tone above C
30.
Harry Partch's 43-tone scale
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The 43-tone scale is a just intonation scale with 43 pitches in each octave, invented and used by Harry Partch. Almost all of Partchs music is written in the 43-tone scale, Partch chose the 11 limit as the basis of his music, because the 11th harmonic is the first that is utterly foreign to Western ears. Here are all the ratios within the octave with odd factors up to and including 11, note that the inversion of every interval is also present, so the set is symmetric about the octave. There are two reasons why the 11-limit ratios by themselves would not make a good scale, first, the scale only contains a complete set of chords based on one tonic pitch. Second, it contains large gaps, between the tonic and the two pitches to either side, as well as other places. Both problems can be solved by filling in the gaps with multiple-number ratios, together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale. Erv Wilson who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables, a constant structure giving one the property of anytime a ratio appears it will be subtended by the same amount of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible, the 43-tone scale was published in Genesis of a Music, and is sometimes known as the Genesis scale, or Partchs pure scale. Besides the 11-limit diamond, he also published 5- and 13-limit diamonds, erv Wilson who did the original drawings in Partchs Genesis of a Music has made a series of diagrams of Partchs diamond as well as others like Diamonds
31.
Hexany
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In music theory, the hexany is a six-note just intonation structure, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube. The notes are arranged so that edge of the octahedron joins together notes that make a consonant dyad. This makes a musical geometry with the form of the octahedron. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series, the points represent musical notes and the three notes that make each of the triangular faces represent musical triads. Its constructed by taking four musical intervals, one of which can optionally be the unison, and then combining them in pairs, in all possible ways. So for instance if you start with 1/1, 3/1, 5/1 and 7/1 then combine them in pairs you get 1*3, 1*5, 1*7, 3*5, 3*7, 5*7 and those are the notes of the 1,3,5,7 hexany. The notes are often octave shifted to them all within the same octave, which has no effect on interval relations. The 1,3,5,7 hexany is found within any 3D cubic lattice of musical pitches, and so within the three factor Euler–Fokker genus based on a cube. If none of the used to construct it is the unity then you need to go into four dimensions. An example of this is the 3,5,7,11 hexany, the result is still a three dimensional figure, the octahedron, with vertices 3*5, 3*7, 3*11, 5*7, 5*11, 7*11. However when you embed it in the four factor Euler Fokker genus and then represent this in 4D and you can have 3D cross sections of a 4D shape much as you can obtain a triangle as a 2D cross section of a normal 3D cube. The Hexany is the invention of Erv Wilson and represents one of the simplest structures found in his Combination Product Sets, the numbers of vertices of his combination sets when set out as subdivisions of a Euler-Fokker genus follow the numbers in Pascals triangle. IN this construction, the hexany is the cross section of the four factor genus. Hexany is the name that Erv Wilson gave to the six notes in the 2-out-of-4 combination product set 4 CPS) and this shows the three dimensional version of the hexany. Here is another diagram showing how the hexany can be found in the three factor Euler Fokker genus, the hexany is the figure containing both the triangles shown as well as the connecting lines between them. Note - in this 3D construction it is not visually a perfect octahedron - it is somewhat squashed, but that is an inessential difference as the interval relationships are the same. See also figure 2 of Kraig Gradys paper, in its most general form the hexany is embedded in a four factor Euler–Fokker genus, or geometrically, a hypercube, also called a tesseract. The four dimensions of the hypercube are often tuned to distinct primes to achieve a hexany with maximally consonant triads, a single step in each dimension corresponds to multiplying the frequency by that prime
32.
Five-limit tuning
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Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths, powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals, hence, 5-limit tuning is considered a method for obtaining just intonation. If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes extending indefinitely in two directions, however, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical to have the number of pitches in each octave. In that case, the system can also be thought of as an octave-repeating scale of a certain number of pitches per octave. For example, if we have a 5-limit tuning system where the note is C256, then fC =256 Hz. There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a frequency ratio of 5/4, or using fifths one may go up four factors of 3 and down six factors of 2, reaching 81/64. The prominent notes of a scale are tuned so that their frequencies form ratios of relatively small integers. Here the row headed Natural expresses all these ratios using a common list of natural numbers. In other words, the lowest occurrence of this one-octave scale shape within the series is as a subset of 8 of the 25 harmonics found in the octave from harmonics 24 to 48 inclusive. The three major thirds are correct, and three of the thirds are as expected, but D to F is a semiditone or Pythagorean minor third. As a consequence, we obtain a scale in which EGB and ACE are just minor triads, but the DFA triad doesnt have the minor shape or sound we might expect, being. Furthermore, the BDF triad is not the diminished triad that we would get by stacking two 6,5 minor thirds, being instead, Another way to do it is as follows. The three major thirds are still 5,4, and three of the thirds are still 6,5 with the fourth being 32,27. FAC and CEG still form just major triads, but GBD is now, there are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else. It is evidently not possible to get all seven diatonic triads in the configuration for major, for minor and that demonstrates the need for increasing the numbers of pitches to execute the desired harmonies in tune
33.
Ptolemy's intense diatonic scale
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It is also supported by Giuseppe Tartini. It is produced through a tetrachord consisting of a tone, lesser tone. This is called Ptolemys intense diatonic tetrachord, as opposed to Ptolemys soft diatonic tetrachord, formed by 21/20, 10/9, in comparison to Pythagorean tuning, while both provide just perfect fourths and fifths, the Ptolemaic provides just thirds which are smoother and more easily tuned. Note that D-F is a Pythagorean minor third, D-A is a fifth, F-D is a Pythagorean major sixth. All of these differ from their just counterparts by a syntonic comma and this scale may also be considered as derived from the major chord, and the major chords on top and bottom of it, FAC-CEG-GBD
34.
Pythagorean tuning
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Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio 3,2, which is 702 cents wide. Hence, it is a system of tuning in which the frequency ratios of all intervals are based on the ratio 3,2. This ratio, also known as the perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear. As Novalis put it, The musical proportions seem to me to be particularly correct natural proportions, the so-called Pythagorean tuning was used by musicians up to the beginning of the 16th century. The Pythagorean system would appear to be ideal because of the purity of the fifths, Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3,2, the next simplest ratio after 2,1. Since notes differing in frequency by a factor of 2 are given the same name, the purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it. This interval is called the basic octave. For instance, the A is tuned such that its frequency equals 3,2 times the frequency of D—if D is tuned to a frequency of 288 Hz, then A is tuned to 432 Hz. Similarly, the E above A is tuned such that its frequency equals 3,2 times the frequency of A, or 9,4 times the frequency of D—with A at 432 Hz, this puts E at 648 Hz. Since this E is outside the basic octave, it is usual to halve its frequency to move it within the basic octave. Therefore, E is tuned to 324 Hz, a 9,8 above D, the B at 3,2 above that E is tuned to the ratio 27,16 and so on. This frequency is doubled to bring it into the basic octave. When extending this tuning however, a problem arises, no stack of 3,2 intervals will fit exactly into any stack of 2,1 intervals. For instance a stack such as this, obtained by adding one note to the stack shown above A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ will be similar. More exactly, it will be about a quarter of a semitone larger, thus, A♭ and G♯, when brought into the basic octave, will not coincide as expected. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a Pythagorean comma, to get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E♭ to G♯. This, as shown above, the remaining interval is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as one is known as a wolf interval
35.
Scale of harmonics
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The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale, most fret positions appearing on Non-Western string instruments are equal to positions of this scale. Unexpectedly, these positions are actually the corresponding undertones of the overtones from the harmonic series. The distance from the nut to the fret is a number lower than the distance from the fret to the bridge. On the guqin, the end of the dotted scale is a mirror image of the right end. The instrument is played with flageolet tones as well as pressing the strings on the wood, the flageolets appear on the harmonic positions of the overtone series, therefore these positions are marked as the musical scale of this instrument. The flageolet positions also represent the harmonic consonant relation of the string part with the open string. The guqin has one anomaly in its scale, the guqin scale represents the first six harmonics and the eighth harmonic. The seventh harmonic is left out, however this tone is still consonant related to the open string and has a lesser consonant relation to all other harmonic positions. A Vietnamese monochord, called the Đàn bầu, also functions with the scale of harmonics, on this instrument only the right half of the scale is present up to the limit of the first seven overtones. The dots are on the string lengths 1/2, 1/3, 1/4, 1/5, 1/6, partchs tone selection otonality from his utonality and otonality concept are the complement pitches of the overtones. For instance, the frequency ratio 5,4 is equal to 4/5th of the length and 4/5 is the complement of 1/5. Groven used the seljefløyte as basis for his research, the flute uses only the upper harmonic scale. The scale is present on the Moodswinger. Although this functions quite differently than a Guqin, oddly enough the scale occurs on this instrument while it is not played in a just intonation tuning, arithmetic progression Harmonic spectrum Otonality and Utonality Partch, Harry. Genesis Of A Music, An Account Of A Creative Work, Its Roots, article about the overtoning positions and their relation to musical scales
36.
Tonality diamond
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In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality. Equivalently, the diamond may be considered as a set of pitch classes, the tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Friedrich Meyer, the tonality diamond is now most associated with Harry Partch, Partch arranged the elements of the tonality diamond in the shape of a rhombus, and subdivided into 2/4 smaller rhombuses. Along the upper side of the rhombus are placed the odd numbers from 1 to n. These intervals are then arranged in ascending order, along the lower left side are placed the corresponding reciprocals,1 to 1/n, also reduced to the octave. These are placed in descending order, at all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition, diagonals sloping in one direction form Otonalities and the diagonals in the other direction form Utonalities. One of Partchs instruments, the marimba, is arranged according to the tonality diamond. Harry Partch used the 11-limit tonality diamond, but flipped it 90 degrees, the five- and seven-limit tonality diamonds exhibit a highly regular geometry within the modulatory space, meaning all non-unison elements of the diamond are only one unit from the unison. The five-limit diamond then becomes a regular hexagon surrounding the unison, further examples of lattices of diamonds ranging from the triadic to the ogdoadic diamond have been realised by Erv Wilson where each interval is given its own unique direction. Three properties of the tonality diamond and the contained, All ratios between neighboring ratios are superparticular ratios, those with a difference of 1 between numerator and denominator. Ratios with relatively lower numbers have more space between them than ratios with higher numbers, the system, including the ratios between ratios, is symmetrical within the octave when measured in cents not in ratios. For example, The ratio between 6/5 and 5/4 is 25/24, the ratios with relatively low numbers 4/3 and 3/2 are 203.91 cents apart, while the ratios with relatively high numbers 6/5 and 5/4 are 70.67 cents apart. The ratio between the lowest and 2nd lowest and the highest and 2nd highest ratios are the same, from this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to 2 π2 n 2. The first few values are the important ones, and the fact that the size of the diamond grows as the square of the size of the odd limit tells us that it becomes large fairly quickly. Yuri Landman rewrites Partchs diamond to clarify its relationship to string lengths. Landman flips the ratios and takes the complement string part to them easier to understand. In Partchs ratios, the over number corresponds to the amount of divisions of a vibrating string
37.
Whole tone scale
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In music, a whole tone scale is a scale in which each note is separated from its neighbours by the interval of a whole tone. This effect is especially emphasised by the fact that triads built on such scale tones are augmented, indeed, one can play all six tones of a whole tone scale simply with two augmented triads whose roots are a major second apart. Since they are symmetrical, whole tone scales do not give an impression of the tonic or tonality. The composer Olivier Messiaen called the tone scale his first mode of limited transposition. The composer and music theorist George Perle calls the whole tone scale interval cycle 2, since there are only two possible whole tone scale positions, it is either C20 or C21. For this reason, the tone scale is also maximally even. Due to this symmetry, the hexachord consisting of the scale is not distinct under inversion or more than one transposition. Use of the whole tone scale can be traced at least as far back as Johann Sebastian Bach. The concluding chorale movement of his cantata O Ewigkeit du Donnerwort BWV60, opens with four notes from the tone scale, Mozart also used the scale in his Musical Joke, for strings. Further examples can be found in the works of Rimsky-Korsakov, the sea kings music in Sadko and also in Scheherezade, colles names as the childhood of the whole-tone scale the music of Berlioz and Schubert in France and then Russians Glinka and Dargomyzhsky. The sense of mystery and ambiguity here even extends to the title of the piece, though the whole-tone scale is prominent in much of his music after 1905 when he encountered Debussy, it serves simply to fit the motifs over augmented chords. The same motifs return from the whole-tone to the scale without emphasizing the contrast. The first of Alban Bergs Seven Early Songs opens with a whole-tone passage both in the accompaniment and in the vocal line that enters a bar later. Berg also quotes the Bach chorale setting referred to above in his Violin Concerto. The last four notes of the 12-tone row Berg used are B, C♯, E♭ and F, an early instance of the use of the scale in jazz writing can be found in Don Redman’s “Chant of the Weed”. In 1958, Gil Evans recorded an arrangement that gives striking coloration to passages featuring whole-tone harmonies, however, these are only the most overt examples of the use of this scale in jazz. Art Tatum and Thelonious Monk are two pianists who used the tone scale extensively and creatively. Thelonious Monks Four in One and Trinkle-Tinkle are fine examples of this, a prominent example of the whole tone scale that made its way into pop music are bars 3 and 4 of the opening of Stevie Wonders 1972 song You are the Sunshine of my Life
38.
15 equal temperament
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In music,15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps. Each step represents a ratio of 15√2, or 80 cents. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro scale in Indonesian gamelan,15 equal temperament is not a meantone system. Guitars have been constructed for 15-ET tuning, the American musician Wendy Carlos used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell. Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar, Blackwood believes that 15 equal temperament, is likely to bring about a considerable enrichment of both classical and popular repertoire in a variety of styles. Here are the sizes of some common intervals in 15-ET, 15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The perfect fifth is more out of tune than in 12-ET, 19-ET, or 22-ET, and the third in 15-ET is the same as the major third in 12-ET. Unlike 12-ET and 19-ET, 15-ET matches the 11,8 and 16,11 ratios, 15-ET also has a neutral second and septimal whole tone. To construct a third, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals. Ivor Darreg, 15-TONE SCALE SYSTEM, Sonic-Arts. org, brewt, Fifteen note equal temperament tutorial
39.
17 equal temperament
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In music,17 tone equal temperament is the tempered scale derived by dividing the octave into 17 equal steps. Each step represents a ratio of 17√2, or 70.6 cents. Alexander J. Ellis refers to a tuning of seventeen tones based on fourths and fifths as the Arabic scale. This 17-tone system remained the primary theoretical system until the development of the tone scale. 17-ET is the tuning of the Regular diatonic tuning in which the tempered fifth is equal to 705.88 cents. 17-ET is where every other step in the 34-ET scale is included, conversely 34-ET is a subdivision of 17-ET. The 17-tone Puzzle — And the, microtonalismo Heptadecatonic System Applications Georg Hajdus 1992 ICMC paper on the 17-tone piano project ProyectoXVII Heptadecatonic System Applications project XVII - Peruvian
40.
Quarter tone
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A quarter tone play, is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide as a semitone, which is half a whole tone. Many composers are known for having written music including quarter tones or the quarter-tone scale —proposed by 19th-century music theorists Heinrich Richter in 1823, george Tucker, Ivan Alexandrovich Wyschnegradsky, and Iannis Xenakis. The term quarter tone can refer to a number of different intervals, for example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments. In the quarter tone scale, also called 24 tone equal temperament, the tone is 50 cents, or a frequency ratio of 24√2 or approximately 1.0293. In this scale the quarter tone is the smallest step, a semitone is thus made of two steps, and three steps make a three-quarter tone play or neutral second, half of a minor third. The 8-TET scale is composed of three-quarter tones, in just intonation the quarter tone can be represented by the septimal quarter tone,36,35, or by the undecimal quarter tone,33,32, approximately half the semitone of 16,15 or 25,24. The ratio of 36,35 is only 1.23 cents narrower than a 24-TET quarter tone and this just ratio is also the difference between a minor third and septimal minor third. Quarter tones and intervals close to also occur in a number of other equally tempered tuning systems. 22-TET contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET has an interval of 45.28 cents, 72-TET also has equally tempered quarter-tones, and indeed contains 3 quarter tone scales, since 72 is divisible by 24. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33,32, because many musical instruments manufactured today are designed for the 12-tone scale, not all are usable for playing quarter tones. Sometimes special playing techniques must be used, other instruments can be used to play quarter tones when using audio signal processing effects such as pitch shifting. Pairs of conventional instruments tuned a tone apart can be used to play some quarter tone music. Indeed, quarter-tone pianos have been built, which consist essentially of two stacked one above the other in a single case, one tuned a quarter tone higher than the other. Many Persian dastgah and Arabic maqamat contain intervals of three-quarter tone size, the quarter tone scale may be primarily a theoretical construct in Arabic music. Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, the Japanese multi-instrumentalist and experimental musical instrument builder Yuichi Onoue developed a 24-TET quarter tone tuning on his guitar. Several quarter-tone albums have been recorded by Jute Gyte, a one-man avantgarde black metal band from Missouri, the enharmonic genus of the Greek tetrachord consisted of a ditone or an approximate major third, and a semitone, which was divided into two microtones. Intervals matched particularly closely include the second, neutral third. The septimal minor third and septimal major third are approximated rather poorly, overall, 24-TET can be viewed as matching the 11th and 13th harmonics more closely than the 7th
41.
53 equal temperament
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In music,53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps. Play Each step represents a ratio of 21/53, or 22.6415 cents. 53-TET is a tuning of equal temperament in which the tempered fifth is 701.89 cents wide. Theoretical interest in this division goes back to antiquity, ching Fang, a Chinese music theorist, observed that a series of 53 just fifths is very nearly equal to 31 octaves. He calculated this difference with six-digit accuracy to be 177147 /176776, later the same observation was made by the mathematician and music theorist Nicholas Mercator, who calculated this value precisely as, which is known as Mercators comma. Mercators comma is of small value to begin with, but 53 equal temperament flattens each fifth by only 1/53 of that comma. Thus,53 equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning and this property of 53-TET may have been known earlier, Isaac Newtons unpublished manuscripts suggest that he had been aware of it as early as 1664–65. Because a distance of 31 steps in this scale is almost precisely equal to a just perfect fifth, however, 53-TET contains additional intervals that are very close to just intonation. For instance, the interval of 17 steps is also a major third, 53-TET is very good as an approximation to any interval in 5-limit just intonation. The matches to the just intervals involving the 7th harmonic are slightly less close, the 11th harmonic and intervals involving it are less closely matched, as illustrated with the undecimal neutral seconds and thirds in the table below. The 53-ET tuning equates to the unison, or tempers out, the intervals 32805/32768, known as the schisma, because it tempers these out, 53-ET can be used for both schismatic temperament, tempering out the schisma, and Hanson temperament, tempering out the kleisma. The interval of 7/4 is 4.8 cents sharp in 53-ET, and using it for 7-limit harmony means that the septimal kleisma, attempting to use standard notation, seven letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with 19-ET and 31-ET where there is little ambiguity, by not being meantone, it adds some problems that require more attention. Specifically, the third is different from a ditone, two tones, each of which is two fifths minus an octave. Likewise, the third is different from a semiditone. The fact that the comma is not tempered out means that notes. Instead, the triads are chords like C-F♭-G, where the major third is a diminished fourth. Likewise, the triads are chords like C-D♯-G
42.
Regular temperament
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Regular temperament is any tempered 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. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, when only two generators are needed, with one of them the octave, this is called linear temperament. The best-known example of a linear temperaments is meantone temperament, where the generating intervals are given in terms of a slightly flattened fifth. Other linear temperaments include the schismatic temperament of Hermann von Helmholtz, if the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. In mathematical terminology, the products of these define a free abelian group. The number of independent generators is the rank of an abelian group, the rank-one tuning systems are equal temperaments, all of which can be spanned with only a single generator. A rank-two temperament has two generators, hence, meantone is a rank-2 temperament. In studying regular temperaments, it can be useful to regard the temperament as having a map from p-limit just intonation to the set of tempered intervals. To properly classify a temperaments dimensionality one must determine how many of the generators are independent. Another way of considering this problem is that the rank of a temperament should be the rank of its image under this map, other methods of linear and multilinear algebra can be applied to the map. For instance, a maps kernel consists of p-limit intervals called commas, a. Milne, W. A. Sethares, and J. Enharmonic instruments and music, 1470-1900
♭
Play (help·info). Similarly, the tuning with pure 7:4 intervals has a fifth of size 696.88 cents 
, C-E♭-G♭-A, C-E♭-F♯-A or C-D♯-F♯-A. It also has a septimal augmented triad, which in various inversions can be C-E-G♯, C-E-A♭ or C-F♭-A♭. It has both a dominant seventh chord, C-E-G-B♭, and an otonal tetrad, C-E-G-A♯; the latter is familiar in common practice harmony under the name German sixth. It likewise has utonal tetrads, C-E♭-G-B
). The 11 is pure using this method if the fifth is of size 697.30 cents, very close to the fifth of 74 equal temperament. On the other hand, 13 meantone fourths up and two octaves down (e.g. C-G