1.
Quadrangularis Reversum
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One of Partchs scales has 43 tones to the octave. To play this music, he built a number of unique instruments, with names such as the Chromelodeon, the Quadrangularis Reversum. Partch called himself a philosophic music-man seduced into carpentry, the path towards Partchs use of a large number of unique instruments was a gradual one. Partch began in the 1920s using traditional instruments, and wrote a quartet in just intonation. He had his first specialized instrument built for him in 1930—the Adapted Viola and he re-tuned the reeds of several reed organs and labeled the keys with a color code. The first one was called the Ptolemy, in tribute to the ancient music theorist Claudius Ptolemaeus, whose musical scales included ratios of the 11-limit, the others were called Chromelodeons, a portmanteau of chrome and melodeon. Most of Partchs works used the instruments he created exclusively, some works made use of unaltered standard instruments such as oboe, clarinet, or cello, and Revelation in the Courtyard Park used an unaltered small wind band. In 1991, Dean Drummond became the custodian of the original Harry Partch instrument collection until his death in 2013, in 1999 the instruments began a residency at Montclair State University in Montclair, New Jersey which lasted until November 2014 when they moved to University of Washington in Seattle. They are currently under the care of Charles Corey, Partch built two steel-string guitars, the six-string Adapted Guitar I, and the ten-string Adapted Guitar II. Partch first started experimenting with such instruments in 1934, and the Adapted Guitar I first appeared in Barstow in 1941, in 1945, he began using amplification for both instruments. The Adapted Viola is earliest extant instrument created by Partch and it is made from a cello fingerboard neck attached to the body of a viola. It also includes a bridge which allows triple stops, i. e. full triads. Partch hammered brads into the fingerboard to aid with finding fingerings, the Adapted Viola was constructed in New Orleans with the assistance of an Edward Benton, a local violin maker. Partch originally called it the Monophone, but by 1933 it had known as the Adapted Viola. Partch tuned the Adapted Viola an octave below the violin and the brads inserted into the instrument fingerboard are placed at ratios commonly used in Partchs works. In playing the instrument, Partch called for a one-finger technique, specifically, he desired that notes should be approached by gliding from tone to tone, The finger may start slowly on its move, increase speed, and hit the next ratio exactly. It may move very fast from the first ratio, and then slowly and insinuatingly into the next—so slowly, sometimes. Or, all this may be reversed, what the bow is doing meanwhile, in its capacity of providing an infinitude of nuance, is supremely important
2.
Harry Partch
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Harry Partch was an American composer, music theorist, and creator of musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century composers in the West to work systematically with microtonal scales. He built custom-made instruments in these tunings on which to play his compositions, to play his music, Partch built a large number of unique instruments, with such names as the Chromelodeon, the Quadrangularis Reversum, and the Zymo-Xyl. Partch described his music as corporeal, and distinguished it from abstract music, ancient Greek theatre and Japanese Noh and kabuki heavily influenced his music theatre. Encouraged by his mother, Partch learned several instruments at a young age, by fourteen, he was composing, and in particular took to setting dramatic situations. He dropped out of the University of Southern Californias School of Music in 1922 over dissatisfaction with the quality of his teachers, in 1930, he burned all his previous compositions in a rejection of the European concert tradition. Partch frequently moved around the US, early in his career, he was a transient worker, and sometimes a hobo, later he depended on grants, university appointments, and record sales to support himself. In 1970, supporters created the Harry Partch Foundation to administer Partchs music, Partch was born on June 24,1901, in Oakland, California. His parents were Virgil Franklin Partch and Jennie, the Presbyterian couple were missionaries, and served in China from 1888 to 1893, and again from 1895 to 1900, when they fled the Boxer Rebellion. Partch moved with his family to Arizona for his mothers health and his father worked for the Immigration Service there, and they settled in the small town of Benson. It was still the Wild West there in the twentieth century. Nearby, there were native Yaqui people, whose music he heard and his mother sang to him in Mandarin Chinese, and he heard and sang songs in Spanish and the Yaqui language. His mother encouraged her children to music, and he learned the mandolin, violin, piano, reed organ. His mother taught him to read music, the family moved to Albuquerque, New Mexico, in 1913, where Partch seriously studied the piano. He had work playing keyboards for silent films while he was in high school, by 14, he was composing for the piano. He early found an interest in writing music for dramatic situations, and often cited the lost composition Death, Partch graduated from high school in 1919. The family moved to Los Angeles in 1919 following the death of Partchs father, there, his mother was killed in a trolley accident in 1920. He enrolled in the University of Southern Californias School of Music in 1920 and he moved to San Francisco and studied books on music in the libraries there and continued to compose
3.
Music theory
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Music theory is the study of the practices and possibilities of music. The term is used in three ways in music, though all three are interrelated. The first is what is otherwise called rudiments, currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, Theory in this sense is treated as the necessary preliminary to the study of harmony, counterpoint, and form. The second is the study of writings about music from ancient times onwards, Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. However, this medieval discipline became the basis for tuning systems in later centuries, Music theory as a practical discipline encompasses the methods and concepts composers and other musicians use in creating music. The development, preservation, and transmission of music theory in this sense may be found in oral and written music-making traditions, musical instruments, and other artifacts. In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia and ancient China, have also considered music theory in more formal ways such as written treatises, in modern academia, music theory is a subfield of musicology, the wider study of musical cultures and history. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a person who researches, teaches, or writes articles about music theory is a music theorist. University study, typically to the M. A. or Ph. D level, is required to teach as a music theorist in a US or Canadian university. Methods of analysis include mathematics, graphic analysis, and especially analysis enabled by Western music notation, comparative, descriptive, statistical, and other methods are also used. See for instance Paleolithic flutes, Gǔdí, and Anasazi flute, several surviving Sumerian and Akkadian clay tablets include musical information of a theoretical nature, mainly lists of intervals and tunings. The scholar Sam Mirelman reports that the earliest of these dates from before 1500 BCE. Further, All the Mesopotamian texts are united by the use of a terminology for music that, much of Chinese music history and theory remains unclear. The earliest texts about Chinese music theory are inscribed on the stone and they include more than 2800 words describing theories and practices of music pitches of the time. The bells produce two intertwined pentatonic scales three tones apart with additional pitches completing the chromatic scale, Chinese theory starts from numbers, the main musical numbers being twelve, five and eight. Twelve refers to the number of pitches on which the scales can be constructed, the Lüshi chunqiu from about 239 BCE recalls the legend of Ling Lun. On order of the Yellow Emperor, Ling Lun collected twelve bamboo lengths with thick, blowing on one of these like a pipe, he found its sound agreeable and named it huangzhong, the Yellow Bell
4.
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
5.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
6.
Otonality
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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
7.
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
8.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
9.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
10.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
11.
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
12.
Equivalence class
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In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the equivalence class if. Formally, given a set S and an equivalence relation ~ on S and it may be proven from the defining properties of equivalence relations that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. If X is the set of all cars, and ~ is the relation has the same color as. X/~ could be identified with the set of all car colors. Let X be the set of all rectangles in a plane, for each positive real number A there will be an equivalence class of all the rectangles that have area A. Consider the modulo 2 equivalence relation on the set Z of integers, x ~ y if and this relation gives rise to exactly two equivalence classes, one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the element of Z/~. Let X be the set of ordered pairs of integers with b not zero, the same construction can be generalized to the field of fractions of any integral domain. In this situation, each equivalence class determines a point at infinity, the equivalence class of an element a is denoted and is defined as the set = of elements that are related to a by ~. An alternative notation R can be used to denote the class of the element a. This is said to be the R-equivalence class of a, the set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R. The surjective map x ↦ from X onto X/R, which each element to its equivalence class, is called the canonical surjection or the canonical projection map. When an element is chosen in each class, this defines an injective map called a section. If this section is denoted by s, one has = c for every equivalence class c, the element s is called a representative of c. Any element of a class may be chosen as a representative of the class, sometimes, there is a section that is more natural than the other ones
13.
Octave
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In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are written using eight notes. For example, the C major scale is typically written C D E F G A B C, two notes separated by an octave have the same letter name and are of the same pitch class. Three commonly cited examples of melodies featuring the perfect octave as their opening interval are Singin in the Rain, Somewhere Over the Rainbow, the interval between the first and second harmonics of the harmonic series is an octave. The octave has occasionally referred to as a diapason. To emphasize that it is one of the intervals, the octave is designated P8. The octave above or below a note is sometimes abbreviated 8a or 8va, 8va bassa. For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, the ratio of frequencies of two notes an octave apart is therefore 2,1. Further octaves of a note occur at 2n times the frequency of that note, such as 2,4,8,16, etc. and the reciprocal of that series. For example,55 Hz and 440 Hz are one and two away from 110 Hz because they are 1⁄2 and 4 times the frequency, respectively. After the unison, the octave is the simplest interval in music, the human ear tends to hear both notes as being essentially the same, due to closely related harmonics. Notes separated by a ring together, adding a pleasing sound to music. For this reason, notes an octave apart are given the note name in the Western system of music notation—the name of a note an octave above A is also A. The conceptualization of pitch as having two dimensions, pitch height and pitch class, inherently include octave circularity, thus all C♯s, or all 1s, in any octave are part of the same pitch class. Octave equivalency is a part of most advanced cultures, but is far from universal in primitive. The languages in which the oldest extant written documents on tuning are written, leon Crickmore recently proposed that The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week
14.
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
15.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
16.
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
17.
Diamond marimba
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Harry Partch was an American composer, music theorist, and creator of musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century composers in the West to work systematically with microtonal scales. He built custom-made instruments in these tunings on which to play his compositions, to play his music, Partch built a large number of unique instruments, with such names as the Chromelodeon, the Quadrangularis Reversum, and the Zymo-Xyl. Partch described his music as corporeal, and distinguished it from abstract music, ancient Greek theatre and Japanese Noh and kabuki heavily influenced his music theatre. Encouraged by his mother, Partch learned several instruments at a young age, by fourteen, he was composing, and in particular took to setting dramatic situations. He dropped out of the University of Southern Californias School of Music in 1922 over dissatisfaction with the quality of his teachers, in 1930, he burned all his previous compositions in a rejection of the European concert tradition. Partch frequently moved around the US, early in his career, he was a transient worker, and sometimes a hobo, later he depended on grants, university appointments, and record sales to support himself. In 1970, supporters created the Harry Partch Foundation to administer Partchs music, Partch was born on June 24,1901, in Oakland, California. His parents were Virgil Franklin Partch and Jennie, the Presbyterian couple were missionaries, and served in China from 1888 to 1893, and again from 1895 to 1900, when they fled the Boxer Rebellion. Partch moved with his family to Arizona for his mothers health and his father worked for the Immigration Service there, and they settled in the small town of Benson. It was still the Wild West there in the twentieth century. Nearby, there were native Yaqui people, whose music he heard and his mother sang to him in Mandarin Chinese, and he heard and sang songs in Spanish and the Yaqui language. His mother encouraged her children to music, and he learned the mandolin, violin, piano, reed organ. His mother taught him to read music, the family moved to Albuquerque, New Mexico, in 1913, where Partch seriously studied the piano. He had work playing keyboards for silent films while he was in high school, by 14, he was composing for the piano. He early found an interest in writing music for dramatic situations, and often cited the lost composition Death, Partch graduated from high school in 1919. The family moved to Los Angeles in 1919 following the death of Partchs father, there, his mother was killed in a trolley accident in 1920. He enrolled in the University of Southern Californias School of Music in 1920 and he moved to San Francisco and studied books on music in the libraries there and continued to compose
18.
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
19.
Perfect fifth
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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3,2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five notes in a diatonic scale. The perfect fifth spans seven semitones, while the diminished fifth spans six, for example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. Play The perfect fifth may be derived from the series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and it occurs above the root of all major and minor chords and their extensions. Until the late 19th century, it was referred to by one of its Greek names. Its inversion is the perfect fourth, the octave of the fifth is the twelfth. The term perfect identifies the perfect fifth as belonging to the group of perfect intervals, so called because of their simple pitch relationships and their high degree of consonance. However, when using correct enharmonic spelling, the fifth in Pythagorean tuning or meantone temperament is actually not a perfect fifth. The perfect unison has a pitch ratio 1,1, the perfect octave 2,1, the perfect fourth 4,3, within this definition, other intervals may also be called perfect, for example a perfect third or a perfect major sixth. In terms of semitones, these are equivalent to the tritone, the justly tuned pitch ratio of a perfect fifth is 3,2, meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a violin is tuned, if adjacent strings are adjusted to the ratio of 3,2, the result is a smooth and consonant sound. Keyboard instruments such as the piano normally use a version of the perfect fifth. In 12-tone equal temperament, the frequencies of the perfect fifth are in the ratio 7 or approximately 1.498307. An equally tempered fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents. Kepler explored musical tuning in terms of ratios, and defined a lower imperfect fifth as a 40,27 pitch ratio. His lower perfect fifth ratio of 1.4815 is much more imperfect than the equal temperament tuning of 1.498, the perfect fifth is a basic element in the construction of major and minor triads, and their extensions
20.
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
21.
Just minor third
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In the music theory of Western culture, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the third as encompassing three staff positions. The minor third is one of two commonly occurring thirds and it is called minor because it is the smaller of the two, the major third spans an additional semitone. For example, the interval from A to C is a minor third, diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically, notable examples of ascending minor thirds include the opening two notes of Greensleeves and of Light My Fire. The minor third may be derived from the series as the interval between the fifth and sixth harmonics, or from the 19th harmonic. The minor third is used to express sadness in music. It is also a quartal tertian interval, as opposed to the major thirds quintality, the minor third is also obtainable in reference to a fundamental note from the undertone series, while the major third is obtainable as such from the overtone series. The minor scale is so named because of the presence of this interval between its tonic and mediant scale degrees, minor chords too take their name from the presence of this interval built on the chords root. A minor third, in just intonation, corresponds to a ratio of 6,5 or 315.64 cents. In an equal tempered tuning, a third is equal to three semitones, a ratio of 21/4,1, or 300 cents,15.64 cents narrower than the 6,5 ratio. If a minor third is tuned in accordance with the fundamental of the series, the result is a ratio of 19,16. The 12-TET minor third more closely approximates the 19-limit minor third 16,19 Play with only 2.49 cents error. Other pitch ratios are given related names, the minor third with ratio 7,6. The minor third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, instruments in A – most commonly the A clarinet, sound a minor third lower than the written pitch. In music theory, a semiditone is the interval 32,27 and it is the minor third in Pythagorean tuning. The 32,27 Pythagorean minor third arises in the C major scale between D and F, Play It can be thought of as two octaves minus three justly tuned fifths. It is narrower than a justly tuned minor third by a syntonic comma, musical tuning List of meantone intervals Pythagorean interval
22.
Unison
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In music, unison is two or more musical parts sounding the same pitch or at an octave interval, usually at the same time. Rhythmic patterns which are homorhythmic are also called unison, two pitches that are the same or two that move as one. Unison or perfect unison may refer to the interval formed by a tone and its duplication, for example C–C, as differentiated from the second, C–D, in the unison the two pitches have the ratio of 1,1 or 0 half steps and zero cents. This is because a pair of tones in unison come from different locations or can have different colors, voices with different colors have, as sound waves, different waveforms. These waveforms have the fundamental frequency but differ in the amplitudes of their higher harmonics. The unison is considered the most consonant interval while the near unison is considered the most dissonant, the unison is also the easiest interval to tune. The unison is abbreviated as P1, a point is the beginning of a line, although, it is not itself a line. But a line is not composed of points, since a point has no length, width, or depth that can be extended, or joined to another point. So a unison is only the beginning of consonance or interval, it is neither consonance nor interval, several singers singing a melody together. In orchestral music unison can mean the simultaneous playing of a note by different instruments, either at the pitch, or in a different octave, for example, cello. Typically a section string player plays unison with the rest of the section, thus, in the divisi first violins the outside players might play the top note of the chord, while the inside seated players play the middle note, and the second violins play the bottom note. At the point where the first violins no longer play divisi, when several people sing together, as in a chorus, the simplest way for them to sing is to sing in one voice, in unison. If there is an instrument accompanying them, then the instrument must play the notes being sung by the singers. Otherwise the instrument is considered a voice and there is no unison. If there is no instrument, then the singing is said to be a cappella, Music in which all the notes sung are in unison is called monophonic. From this sense can be derived another, figurative, sense, if people do something in unison it means they do it simultaneously, in tandem. Related terms are univocal and unanimous, monophony could also conceivably include more than one voice which do not sing in unison but whose pitches move in parallel, always maintaining the same interval of an octave. A pair of notes sung one or a multiple of an octave apart are almost in unison, when there are two or more voices singing different notes, this is called part singing
23.
Minor sixth
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In classical music from Western culture, a sixth is a musical interval encompassing six staff positions, and the minor sixth is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two, the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a sixth, as the note F lies eight semitones above A. Diminished and augmented sixths span the same number of staff positions, in equal temperament, the minor sixth is enharmonically equivalent to the augmented fifth. It occurs in first inversion major and dominant seventh chords and second inversion minor chords, the ratios of both major and minor sixths are corresponding numbers of the Fibonacci sequence,5 and 8 for a minor sixth and 3 and 5 for a major. The 11,7 undecimal minor sixth is 782.49 cents, in Pythagorean tuning, the minor sixth is the ratio 128,81, or 792.18 cents. In just intonation, the sixth is classed as a consonance of the 5-limit. Any note will appear in major scales from any of its minor sixth major scale notes. Musical tuning List of meantone intervals Sixth chord Subminor sixth
24.
Major sixth
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In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two, the major sixth spans nine semitones. Its smaller counterpart, the sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth and its a sixth because it encompasses six note letter names and six staff positions. Its a major sixth, not a sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of letter names and staff positions. A commonly cited example of a melody featuring the major sixth as its opening is My Bonnie Lies Over the Ocean. The major sixth is one of the consonances of common practice music, along with the unison, octave, perfect fifth, major and minor thirds, minor sixth, in just intonation, the major sixth is classed as a consonance of the 5-limit. Assuming close-position voicings for the examples, the major sixth occurs in a first inversion minor triad, a second inversion major triad. It also occurs in the second and third inversions of a dominant seventh chord, the septimal major sixth is approximated in 53 tone equal temperament by an interval of 41 steps or 928 cents. Many intervals in a various tuning systems qualify to be called major sixth, the following examples are sorted by increasing width. In just intonation, the most common major sixth is the ratio of 5,3. In 12-tone equal temperament, a sixth is equal to nine semitones, exactly 900 cents. The 27,16 Pythagorean major sixth arises in the C Pythagorean major scale between F and D, as well as between C and A, G and E, and D and B. Play The septimal major sixth is approximated in 53-tone equal temperament by an interval of 41 steps, giving a frequency ratio of the root of 2 over 1. Another just major sixth is the 12,7 septimal major sixth or supermajor sixth of approximately 933 cents, the nineteenth subharmonic is a major sixth, A = 32/19 =902.49 cents. Musical tuning list of meantone intervals sixth chord Duckworth, William, in Sound and Light, La Monte Young, Marian Zazeela, edited by William Duckworth and Richard Fleming, p.167. Lewisburg, Bucknell University Press, London and Cranbury, NJ, Associated University Presses
25.
Perfect fourth
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In classical music from Western culture, a fourth is a musical interval encompassing four staff positions, and the perfect fourth is a fourth spanning five semitones. For example, the interval from C to the next F is a perfect fourth, as the note F lies five semitones above C. Diminished and augmented fourths span the same number of staff positions, the perfect fourth may be derived from the harmonic series as the interval between the third and fourth harmonics. The term perfect identifies this interval as belonging to the group of perfect intervals, so called because they are neither major nor minor, up until the late 19th century, the perfect fourth was often called by its Greek name, diatessaron. Its most common occurrence is between the fifth and upper root of all major and minor triads and their extensions. A perfect fourth in just intonation corresponds to a ratio of 4,3, or about 498 cents, while in equal temperament a perfect fourth is equal to five semitones. A helpful way to recognize a fourth is to hum the starting of the Bridal Chorus from Wagners Lohengrin. Other examples are the first two notes of the Christmas carol Hark, the Herald Angels Sing or El Cóndor Pasa, and, for a descending perfect fourth, the second and third notes of O Come All Ye Faithful. The perfect fourth is a perfect interval like the unison, octave, and perfect fifth, in common practice harmony, however, it is considered a stylistic dissonance in certain contexts, namely in two-voice textures and whenever it appears above the bass. Conventionally, adjacent strings of the bass and of the bass guitar are a perfect fourth apart when unstopped, as are all pairs. Sets of tom-tom drums are also tuned in perfect fourths. The 4,3 just perfect fourth arises in the C major scale between G and C, play The use of perfect fourths and fifths to sound in parallel with and to thicken the melodic line was prevalent in music prior to the European polyphonic music of the Middle Ages. In the 13th century, the fourth and fifth together were the concordantiae mediae after the unison and octave, in the 15th century the fourth came to be regarded as dissonant on its own, and was first classed as a dissonance by Johannes Tinctoris in his Terminorum musicae diffinitorium. In practice, however, it continued to be used as a consonance when supported by the interval of a third or fifth in a lower voice. Modern acoustic theory supports the medieval interpretation insofar as the intervals of unison, octave, the octave has the ratio of 2,1, for example the interval between a at A440 and a at 880 Hz, giving the ratio 880,440, or 2,1. The fifth has a ratio of 3,2, and its complement has the ratio of 3,4, ancient and medieval music theorists appear to have been familiar with these ratios, see for example their experiments on the Monochord. In early western polyphony, these simpler intervals were generally preferred, however, in its development between the 12th and 16th centuries, In the earliest stages, these simple intervals occur so frequently that they appear to be the favourite sound of composers. Later, the more complex intervals move gradually from the margins to the centre of musical interest
26.
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
27.
Septimal tritone
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A septimal tritone is a tritone that involves the factor seven. There are two that are inverses, the lesser septimal tritone is the musical interval with ratio 7,5. The greater septimal tritone, is an interval with ratio 10,7 and they are also known as the sub-fifth and super-fourth, or subminor fifth and supermajor fourth, respectively. The 7,5 interval is equal to a 6,5 minor third plus a 7,6 subminor third. The 10,7 interval is equal to a 5,4 major third plus a 8,7 supermajor second, the difference between these two is the septimal sixth tone Play. 12 equal temperament and 22 equal temperament do not distinguish between these tritones,19 equal temperament does distinguish them but doesnt match them closely,31 equal temperament and 41 equal temperament both distinguish between and closely match them. The lesser septimal tritone is the most consonant tritone when measured by combination tones, harmonic entropy, and period length. Depending on the temperament used, the tritone, defined as three tones, may be identified as either a lesser septimal tritone, a greater septimal tritone, neither
28.
Septimal minor third
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In music, the septimal minor third play, also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5, in 24-tone equal temperament five quarter tones approximate the septimal minor third at 250 cents. It has a darker but generally pleasing character when compared to the 6/5 third, a triad formed by using it in place of the minor third is called a septimal minor or subminor triad play. In the meantone era the interval made its appearance as the minor third in remote keys. Composer Ben Johnston uses a small 7 as an accidental to indicate a note is lowered 49 cents, the position of this note also appears on the scale of the Moodswinger. Yuri Landman indicated the positions of his instrument in a color dotted series. The septimal minor third position is cyan blue as well as the knotted positions of the seventh harmonic. Twelve-tone equal temperament, as used in Western music, does not provide a good approximation for this interval. 19-TET, 22-TET, 31-TET, 41-TET, 53-TET, and 72-TET each offer successively better matches to this interval, several non-Western and just intonation tunings, such as the 43-tone scale developed by Harry Partch, do feature the septimal minor third. Depending on the timbre of the pitches, humans sometimes perceive this root pitch even if it is not played, the phenomenon of hearing this root pitch is evident in the following sound file, which uses a pure sine wave. For comparison, the pitch is played after the interval has been played
29.
Subminor and supermajor
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In music, a subminor interval is an interval that is noticeably wider than a diminished interval but noticeably narrower than a minor interval. It is found in between a minor and diminished interval, thus making it below, or subminor to, the minor interval, a supermajor interval is a musical interval that is noticeably wider than a major interval but noticeably narrower than an augmented interval. It is found in between a major and augmented interval, thus making it above, or supermajor to, the major interval. The inversion of an interval is a subminor interval, and there are four major and four minor intervals, allowing for eight supermajor and subminor intervals. Traditionally, supermajor and superminor, the given to certain thirds found in the justly intoned scale with a natural or subminor seventh. Thus, a second is intermediate between a minor second and a diminished second. An example of such an interval is the ratio 26,25, another example is the ratio 28,27, or 62.96 cents. A supermajor seventh is an intermediate between a major seventh and an augmented seventh. It is the inverse of a subminor second, examples of such an interval is the ratio 25,13, or 1132.10 cents, the ratio 27,14, or 1137.04 cents, and 35,18, or 1151.23 cents. A subminor third is in between a third and a diminished third. An example of such an interval is the ratio 7,6, or 266.87 cents, the minor third. Another example is the ratio 13,11, or 289.21 cents, a supermajor sixth is noticeably wider than a major sixth but noticeably narrower than an augmented sixth, and may be a just interval of 12,7. In 24 equal temperament A = B, the septimal major sixth is an interval of 12,7 ratio, or about 933 cents. It is the inversion of the 7,6 subminor third, a subminor sixth or septimal sixth is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth, enharmonically equivalent to the major fifth. The sub-minor sixth is an interval of a 14,9 ratio or alternately 11,7, the 21st subharmonic is 729.22 cents. Play A supermajor third is in between a third and an augmented third, enharmonically equivalent to the minor fourth. An example of such an interval is the ratio 9,7, or 435.08 cents, another example is the ratio 50,39, or 430.14 cents. A subminor seventh is an interval between a seventh and a diminished seventh
30.
Septimal whole tone
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In music, the septimal whole tone, septimal major second, or supermajor second play is the musical interval exactly or approximately equal to an 8/7 ratio of frequencies. It is about 231 cents wide in just intonation, although 24 equal temperament does not match this interval particularly well, its nearest representation is at 250 cents, approximately 19 cents sharp. It can also be thought of as the inversion of the 7/4 interval. No close approximation to this exists in the standard 12 equal temperament used in most modern western music. 26 equal temperament matches this interval almost perfectly with an error of only 0.4 cents,31 equal temperament, which has much more accurate fifths and major thirds, approximates 8/7 with a slightly higher error of 1.1 cents
31.
Modulatory space
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The spaces described in this article are pitch class spaces which model the relationships between pitch classes in some musical system. These models are often graphs, groups or lattices, closely related to pitch class space is pitch space, which represents pitches rather than pitch classes, and chordal space, which models relationships between chords. The simplest pitch space model is the real line, to create circular pitch class space we identify or glue together pitches p and p +12. The result is a continuous, circular pitch class space that mathematicians call Z/12Z, other models of pitch class space, such as the circle of fifths, attempt to describe the special relationship between pitch classes related by perfect fifth. In equal temperament, twelve successive fifths equate to seven octaves exactly and we say that the pitch class of the fifth generates – or is a generator of – the space of twelve pitch classes. By drawing a line between two classes when they differ by a generator, we can depict the circle of generators as a cycle graph. We may now define a graph with n vertices on which the group acts, the result is a graph of genus one, which is to say, a graph with a donut or torus shape. Such a graph is called a toroidal graph and we may generalize immediately to any number of relatively prime factors, producing graphs can be drawn in a regular manner on an n-torus. A linear temperament is a temperament of rank two generated by the octave and another interval, commonly called the generator. The most familiar example by far is meantone temperament, whose generator is a flattened, the pitch classes of any linear temperament can be represented as lying along an infinite chain of generators, in meantone for instance this would be -F-C-G-D-A- etc. This defines a linear modulatory space, a temperament of rank two which is not linear has one generator which is a fraction of an octave, called the period. We may represent the space of such a temperament as n chains of generators in a circle. Here n is the number of periods in an octave, for example, diaschismic temperament is the temperament which tempers out the diaschisma, or 2048/2025. It can be represented as two chains of slightly sharp fifths a half-octave apart, which can be depicted as two chains perpendicular to a circle and at side of it. The fifths are along the axis, and the major thirds point off to the right at an angle of sixty degrees. Another sixty degrees gives us the axis of major sixths, pointing off to the left, the non-unison elements of the 5-limit tonality diamond, 3/2, 5/4, 5/3, 4/3, 8/5, 6/5 are now arranged in a regular hexagon around 1. The triads are the triangles of this lattice, with the upwards-pointing triangles being major triads. When two lattice points are as close as possible, a distance apart, then and only then are they separated by a consonant interval
32.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
33.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
34.
Erv Wilson
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Ervin Wilson was a Mexican/American music theorist. Ervin Wilson was born in a area of northwest Chihuahua, Mexico. His mother taught him to play the organ and to read musical notation. He began to compose at an age, but immediately discovered that some of the sounds he was hearing mentally could not be reproduced by the conventional intervals of the organ. As a teenager he began to read books on Indian music, while he was in the Air Force in Japan, a chance meeting with a total stranger introduced him to musical harmonics, which changed the course of his life and work. Influenced by the work of Joseph Yasser, Wilson began to think of the scale as a living process—like a crystal or plant. He has been mentor to many composers and instrument builders, despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design. Among his developments are Moments of Symmetry, Combination Product Sets, Golden Horograms, scales based on recurrence relations and he cites Augusto Novaro and Joseph Yasser as influences. Wilson built instruments and explored the resources of 31 and 41 equal divisions of the octave and he supported the work of Harry Partch, helped build the Quadrangularis Reversum, and provided diagrams for Partchs book Genesis of a Music. The goal of his research with scales is to make them accessible to the composer. I sculpt in the architecture of the scale, other people come along and animate it. Finnamore The Sonic Sky - site about the musical realm of Erv Wilson, and the vast universe of his musical structures
35.
Farey sequence
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Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends with the value 1, denoted by the fraction 1⁄1. A Farey sequence is called a Farey series, which is not strictly correct. — Beiler Chapter XVI Farey sequences are named after the British geologist John Farey, Farey conjectured, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Fareys letter was read by Cauchy, who provided a proof in his Exercices de mathématique, in fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy. Thus it was an accident that linked Fareys name with these sequences. This is an example of Stiglers law of eponymy, the Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn−1 and also contains an additional fraction for each number that is less than n, thus F6 consists of F5 together with the fractions 1/6 and 5/6. The middle term of a Farey sequence Fn is always 1/2, from this, we can relate the lengths of Fn and Fn−1 using Eulers totient function φ, | F n | = | F n −1 | + φ. Using the fact that |F1| =2, we can derive an expression for the length of Fn, the asymptotic behaviour of |Fn| is, | F n | ∼3 n 2 π2. The index I n = k of a fraction a k, n in the Farey sequence F n = is simply the position that a k, n occupies in the sequence. This is of relevance as it is used in an alternative formulation of the Riemann hypothesis. Various useful properties follow, I n =0, I n =1, I n = /2, I n = | F n | −1, I n = | F n | −1 − I n. Fractions which are neighbouring terms in any Farey sequence are known as a Farey pair and have the following properties, if a/b and c/d are neighbours in a Farey sequence, with a/b < c/d, then their difference c/d − a/b is equal to 1/bd. Since c d − a b = b c − a d b d, thus 1/3 and 2/5 are neighbours in F5, and their difference is 1/15. If b c − a d =1 for positive integers a, b, c and d with a < b and c < d then a/b, thus the first term to appear between 1/3 and 2/5 is 3/8, which appears in F8. The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 and 1, Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions — in one the final term is 1, in the other the term is greater than 1. Farey sequences are useful to find rational approximations of irrational numbers
36.
Euler's totient function
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In number theory, Eulers totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ or ϕ and it can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd is equal to 1. The integers k of this form are referred to as totatives of n. For example, the totatives of n =9 are the six numbers 1,2,4,5,7 and 8. They are all relatively prime to 9, but the three numbers in this range,3,6, and 9 are not, because gcd = gcd =3. As another example, φ =1 since for n =1 the only integer in the range from 1 to n is 1 itself, Eulers totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ = φφ. This function gives the order of the group of integers modulo n. It also plays a key role in the definition of the RSA encryption system, leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it, he wrote πD for the multitude of less than D. This definition varies from the current definition for the totient function at D =1 but is otherwise the same, the now-standard notation φ comes from Gausss 1801 treatise Disquisitiones Arithmeticae. Although Gauss didnt use parentheses around the argument and wrote φA, thus, it is often called Eulers phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, so it is referred to as Eulers totient function. Jordans totient is a generalization of Eulers, the cototient of n is defined as n − φ. It counts the number of positive integers less than or equal to n that have at least one factor in common with n. There are several formulas for computing φ and it states φ = n ∏ p ∣ n, where the product is over the distinct prime numbers dividing n. The proof of Eulers product formula depends on two important facts and this means that if gcd =1, then φ = φ φ. If p is prime and k ≥1, then φ = p k − p k −1 = p k −1 = p k, proof, since p is a prime number the only possible values of gcd are 1, p, p2
37.
Coprime integers
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
38.
Yuri Landman
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Yuri Landman started as a comic book artist and made his debut in the comics field in 1997 with Je Mag Alles Met Me Doen. In the follow-up, released in 1998, Het Verdiende Loon, for the second title he received the 1998 Breda Prize, an award for rising new comic artists in the Netherlands. Since then he has published no other comic books, together with Cees van Appeldoorn, he formed the lo-fi band Zoppo playing bass and prepared guitar in 1997. After 2 albums and several 7” singles, Landman left the band in 2000, Landman then formed the noise band Avec Aisance with drummer/producer Valentijn Höllander and released a CD, Vivre dans l’aisance in 2004. After quitting Avec-A in 2006, he focused mainly on instrument building, Landman is musically untrained and cannot play chords. While with Avec-A, Landman began creating and building several experimental string instruments, including electric zithers, electric Cymbalum, in the period between 2000-2005, Landman created 9 prototype instruments. In 2006 he changed his focus and stopped to perform. The Moodswinger was the first instrument Landman made for the band Liars, after the Moodswinger, he started making more instruments for other bands as well. From November 2006 to January 2007 Landman finished 2 copies of The Moonlander, the Springtime is an experimental electric guitar with seven strings and three outputs. The first prototype of this instrument, created in 2008, was made for guitar player Laura-Mary Carter of Blood Red Shoes, afterwards he also made copies for Lou Barlow and dEUS Mauro Pawlowski. For John Schmersal of Enon he built the Twister guitar, a version of the Springtime. In 2009 he finished instruments for The Dodos, Liam Finn, HEALTH, Micachu, for The Dodos and Finn he created electric 24 string drum guitars called the Tafelberg and for Andrews an electric 17 string harp guitar called the Burner guitar. He also started to again after a Perpignan Festival hosted by Vincent Moon. Meanwhile, he continued to build instruments for such as These Are Powers, Women. For his own musical career he develops a 25-meter electric long-string instrument, the Moodswinger led to a research on harmonics theory. He published a clarifying 3rd Bridge diagram related to this subject in 2012, besides his output as an inventor and noise musician, he starts to focus on music education, and participatory art. At first Landman started giving lectures at venues, festivals. He published an extensive 8 chapter guide on how to prepare a guitar and his lectures and presentations with his instruments lead to a request in 2009 for a practical building workshop
39.
Harmonic series (music)
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A harmonic series is the sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency. Pitched musical instruments are based on an approximate harmonic oscillator such as a string or a column of air. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves, interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency. The musical timbre of a tone from such an instrument is determined by the relative strengths of each harmonic. A complex tone can be described as a combination of many simple periodic waves or partials, each with its own frequency of vibration, amplitude, a partial is any of the sine waves of which a complex tone is composed. A harmonic is any member of the series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The fundamental is also considered a harmonic because it is 1 times itself, a harmonic partial is any real partial component of a complex tone that matches an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic, Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds that are rich in inharmonic partials, an overtone is any partial except the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no special meaning other than to exclude the fundamental. It is the relative strengths of the different overtones that gives an instrument its particular timbre, some electronic instruments, such as synthesizers, can play a pure frequency with no overtones. Synthesizers can also combine pure frequencies into more complex tones, such as to other instruments. Certain flutes and ocarinas are very nearly without overtones, in most pitched musical instruments, the fundamental is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality, the fact that a string is fixed at each end means that the longest allowed wavelength on the string is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2,3,4,5,6, etc. times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies, the harmonic series is an arithmetic series. In terms of frequency, the difference between consecutive harmonics is therefore constant and equal to the fundamental, but because human ears respond to sound nonlinearly, higher harmonics are perceived as closer together than lower ones
40.
Moodswinger
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The Moodswinger is a twelve-string electric zither with an additional third bridge designed by Yuri Landman. The rod which functions as the third bridge divides the strings into two sections to cause an overtone multiphonic sound, one of the copies of the instrument is part of the collection of the Musical Instrument Museum in Phoenix, Arizona. In March 2006 Landman was contacted by the noise band Liars to make an instrument for them, after 6 months he finished two copies of The Moodswinger, an electric 12-string 3rd-bridge overtone koto, one for guitarist/drummer Aaron Hemphill and one for himself. Although it closely resembles a guitar, it is actually a zither. The pickup and electronics are built into the neck instead of in the body like usual electric guitars, in 2008 the Moodswinger II was released as a serial product. Jessie Stein of The Luyas owns a copy, in 2009 Landman created a derivative version of the instrument called the Home Swinger, for workshops at festivals, where participants built their own copy within four hours. In 2010 the Musical Instrument Museum in Phoenix included a Moodswinger as well as a Home Swinger in their collection as two of the pieces of the Dutch section of musical instrument inventions, the 3rd bridge divides the strings into two segments with different pitches. Depending on where the string is played, a bell-like harmonic second tone is created, the string resonates more or less when the back side is struck, depending on the position of the 3rd bridge along the string. This can be explained by resonance and microtonality. At harmonic nodal positions, the string more than at other positions. For instance, dividing the string 1/3 + 2/3 creates a clear overtone, the color dotted scale indicates the simple-number ratios up to the 7 limit On these positions just intoned harmonic dyads occur. The Moodswinger is focused on a playing technique. A mathematical scale is added to specify 23 harmonic positions on the strings, because the instrument has 12 strings, tuned in a circle of fourths, it is always possible to play every note of the equal tempered scale. However some positions have a + or - indication, because the equal tempered scale is not a perfect well-tempered scale, the tuning of this instrument is a circle of fourths, E-A-D-G-C-F-A#-D#-G#-C#-F#-B, arranged in 3 clusters of 4 strings each to make the field of strings better readable. Because of this tuning all five neighbouring strings form a harmonic pentatonic scale and this allows a very easy fingerpicking technique without picking false notes, if the right key is chosen. On the Home Swinger the same color system occurs, the sound of a 3rd-bridged string is a combination of 3 tones. A soft-sounding attack tone of the string part hit at the body side, the diagram below shows the tone combinations of the overtone and the low tone of the counterpart. The attack tone is in most positions exact the same note as the overtone, exceptions are 3/4, 3/5, 3/7 and 5/7
41.
Lattice (music)
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In musical tuning, a lattice is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern, each point on the lattice corresponds to a ratio. The lattice can be two-, three-, or n-dimensional, with each corresponding to a different prime-number partial or chroma. The points in a lattice represent pitch classes, and the connectors in a lattice represent the intervals between them, repeatedly adding the same vector moves you further in the same direction. Lattices in just intonation are theoretically infinite, however, lattices are sometimes also used to notate limited subsets that are particularly interesting. Examples of musical lattices include the Tonnetz of Euler and Hugo Riemann, musical intervals in just intonation are related to those in equal tuning by Adriaan Fokkers Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv Wilson, the limit is the highest prime number used in the ratios that define the intervals used by a tuning. Here is a template he used to generate what he called a “Euler“ lattice after where he drew his inspiration, each prime harmonic has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure. Another feature of the template is worth pointing out, harmonically generated intervals will always appear above the fundamental and the subharmonics below. With a 9-limit system the direction will be both above and to the right, leaving the other quadrant for more complex ratios with the opposite with the subharmonic and this makes it quite easy to understand what is being represented. Erv Wilson would commonly use 10-squares-to-the-inch graph paper and that way, he had room to notate both ratios and often the scale degree, which explains why he didnt use a template where all the numbers where divided by 2. The scale degree always followed a period or dot to separate it from the ratios, numerous examples appear throughout the Wilson Archives Tonality diamond Johnston, Ben. Rational Structure in Music, Maximum Clarity and Other Writings on Music, edited by Bob Gilmore
42.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
43.
Genesis of a Music
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Genesis of a Music is a book first published in 1949 by microtonal composer Harry Partch. He then goes on to explain his tuning theory based on just intonation, the ensemble of instruments of his own invention. The book has been influential to succeeding generations of microtonal composers, including Lou Harrison, Ben Johnston. A ratio thus always belongs to two tonalities, an Otonality in accordance with its Odentity, and an Utonality in accordance with its Udentity, Partch admirer Ben Johnston uses 31-limit or higher extended just intonation. See, Emancipation of the dissonance Genesis of a Music, Monophony, the relation of its music to historic and contemporary trends, its philosophy, concepts, and its application to musical instruments, University of Wisconsin Press,362 pp.1949. ASIN B0007DM7I8 Genesis Of A Music, An Account Of A Creative Work, Its Roots, And Its Fulfillments, Second Edition, Da Capo Press, Paperback,544 pp.1979
44.
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
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West Coast School
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The West Coast School are composers and compositional style associated with the West Coast of the United States, specifically California. Henry Cowell is considered, the father of West Coast experimentalism, rather than Orient/Occident, composer Lou Harrison argues for a Pacific/Atlantic conception, where the West Coast is necessarily part of the Pacific, and thus associated with Asia more than Europe. Other influences and interests include the use of tonality, just intonation, techniques employed by composers of the West Coast School include found and percussion instruments, such as the Indonesian gamelan. Harrison cites, new instruments and new tunings, composers considered to be part of the West Coast School include Henry Cowell, John Cage, Lou Harrison, and Harry Partch. Those four composers were each ultramodernist, Californian, and gay, all of which would have contributed to marginalization
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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
![[1]](http://www.hypercustom.nl/utonaldiagram.jpg){kind=link}
