1.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
2.
Reel-to-reel audio tape recording
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Reel-to-reel or open-reel audio tape recording is the form of magnetic tape audio recording in which the recording medium is held on a reel, rather than being securely contained within a cassette. Reel-to-reel systems use a tape that is 1⁄4 inches in width and this compares to 0.15 inches wide and 1.875 inches per second for a cassette. By writing out the audio signal across more tape, reel-to-reel systems offer much higher fidelity. In spite of the tapes, less convenient use and generally higher cost media. Studer, Stellavox and Denon still produced reel to reel tape recorders in the 1990s, but as of 2014, only Nagra, Otari, originally, this format had no name, since all forms of magnetic tape recorders used it. The earliest machines produced distortion during the process which German engineers significantly reduced during the Nazi era by applying a bias signal to the tape. In 1939 one machine was found to make consistently better recordings than other ostensibly identical models, and when it was taken apart a minor flaw was noticed. It was introducing an AC signal to the tape, and this was adapted to new models using a high-frequency AC bias that has remained a part of audio tape recording to this day. American audio engineer Jack Mullin was a member of the U. S. Army Signal Corps during World War II and he acquired two Magnetophon recorders and 50 reels of I. G. Farben recording tape and shipped them home, over the next two years, he worked to develop the machines for commercial use, hoping to interest the Hollywood film studios in using magnetic tape for movie soundtrack recording. Crosby invested $50,000 in an electronics company, Ampex. Ampex went on to develop the first practical videotape recorders in the early 1950s to pre-record Crosbys TV shows, inexpensive reel-to-reel tape recorders were widely used for voice recording in the home and in schools before the Philips compact cassette, introduced in 1963, gradually took over. Cassettes eventually displaced reel-to-reel recorders for consumer use, however, the narrow tracks and slow recording speeds used in cassettes compromised fidelity. Columbia House carried pre-recorded reel-to-reel tapes from 1960 to 1984, even today, some artists of all genres prefer analog tapes musical, natural and especially warm sound. Due to harmonic distortion, bass can thicken up, creating a fuller-sounding mix, in addition, high end can be slightly compressed, which is more natural to the human ear. It is common for artists to record to digital and re-record the tracks to analog reels for this effect of natural sound. In addition to all of these attributes of tape, tape saturation is a form of distortion that many rock, blues. For the first time, audio could be manipulated as a physical entity, Tape editing is performed simply by cutting the tape at the required point, and rejoining it to another section of tape using adhesive tape, or sometimes glue
3.
Phonograph
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The phonograph is a device invented in 1877 for the mechanical recording and reproduction of sound. In its later forms it is called a gramophone. To recreate the sound, the surface is rotated while a playback stylus traces the groove and is therefore vibrated by it. In later electric phonographs, the motions of the stylus are converted into an electrical signal by a transducer. The phonograph was invented in 1877 by Thomas Edison, while other inventors had produced devices that could record sounds, Edisons phonograph was the first to be able to reproduce the recorded sound. His phonograph originally recorded sound onto a sheet wrapped around a rotating cylinder. A stylus responding to sound vibrations produced an up and down or hill-and-dale groove in the foil, in the 1890s, Emile Berliner initiated the transition from phonograph cylinders to flat discs with a spiral groove running from the periphery to near the center. Later improvements through the years included modifications to the turntable and its system, the stylus or needle. The disc phonograph record was the dominant audio recording format throughout most of the 20th century, from the mid-1980s on, phonograph use on a standard record player declined sharply because of the rise of the cassette tape, compact disc and other digital recording formats. Records are still a favorite format for some audiophiles and DJs, vinyl records are still used by some DJs and musicians in their concert performances. Musicians continue to release their recordings on vinyl records, the original recordings of musicians are sometimes re-issued on vinyl. Usage of terminology is not uniform across the English-speaking world, in more modern usage, the playback device is often called a turntable, record player, or record changer. When used in conjunction with a mixer as part of a DJ setup, the term phonograph was derived from the Greek words φωνή and γραφή. The similar related terms gramophone and graphophone have similar root meanings, the roots were already familiar from existing 19th-century words such as photograph, telegraph, and telephone. In British English, gramophone may refer to any sound-reproducing machine using disc records, the term phonograph was usually restricted to machines that used cylinder records. Gramophone generally referred to a wind-up machine, after the introduction of the softer vinyl records, 33 1⁄3-rpm LPs and 45-rpm single or two-song records, and EPs, the common name became record player or turntable. Often the home record player was part of a system that included a radio and, later, from about 1960, such a system began to be described as a hi-fi or a stereo. In American English, phonograph, properly specific to machines made by Edison, was used in a generic sense as early as the 1890s to include cylinder-playing machines made by others
4.
The Well-Tempered Clavier
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The Well-Tempered Clavier, BWV 846–893, is a collection of two series of Preludes and Fugues in all 24 major and minor keys, composed for solo keyboard by Johann Sebastian Bach. In the German of Bachs time Clavier was a name indicating a variety of keyboard instruments, most typically a harpsichord or clavichord –. The modern German spelling for the collection is Das wohltemperierte Klavier, some 20 years later Bach compiled a second book of the same kind, which became known as The Well-Tempered Clavier, Part Two. Modern editions usually refer to both parts as The Well-Tempered Clavier, Book I and The Well-Tempered Clavier, Book II, the collection is generally regarded as being among the most influential works in the history of Western classical music. Each set contains twenty-four pairs of preludes and fugues, the first pair is in C major, the second in C minor, the third in C♯ major, the fourth in C♯ minor, and so on. The rising chromatic pattern continues until every key has been represented, the first set was compiled in 1722 during Bachs appointment in Köthen, the second followed 20 years later in 1742 while he was in Leipzig. The C♯ major prelude and fugue in book one was originally in C major – Bach added a key signature of seven sharps, although the Well-Tempered Clavier was the first collection of fully worked keyboard pieces in all 24 keys, similar ideas had occurred earlier. His contemporary Johann Heinrich Kittel also composed a cycle of 12 organ preludes in successive keys, Fischer was published in 1702 and reissued 1715. It is a set of 20 prelude-fugue pairs in ten major and nine minor keys, Bach knew the collection and borrowed some of the themes from Fischer for the Well-Tempered Clavier. Finally, a lost collection by Johann Pachelbel, Fugen und Praeambuln über die gewöhnlichsten Tonos figuratos and it was later shown that this was the work of a composer who was not even born in 1689, Bernhard Christian Weber. It was in written in 1745–50, and in imitation of Bachs example. Bachs title suggests that he had written for a tuning system in which all keys sounded in tune. The opposing system in Bachs day was meantone temperament in which keys with many accidentals sound out of tune, Bach would have been familiar with different tuning systems, and in particular as an organist would have played instruments tuned to a meantone system. There is debate whether Bach meant a range of temperaments, perhaps even altered slightly in practice from piece to piece. During much of the 20th century it was assumed that Bach wanted equal temperament, internal evidence for this may be seen in the fact that in Book 1 Bach paired the E♭ minor prelude with its enharmonic key of D♯ minor for the fugue. This represents an equation of the most tonally remote enharmonic keys where the flat, any performance of this pair would have required both of these enharmonic keys to sound identically tuned, thus implying equal temperament in the one pair, as the entire work implies as a whole. However, research has continued into various unequal systems contemporary with Bachs career, accounts of Bachs own tuning practice are few and inexact. The three most cited sources are Forkel, Bachs first biographer, Friedrich Wilhelm Marpurg, who received information from Bachs sons and pupils, and Johann Kirnberger, one of those pupils
5.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
6.
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
7.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
8.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
9.
Simon Stevin
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Simon Stevin, sometimes called Stevinus, was a Flemish/Dutch/Netherlandish mathematician, physicist and engineer. He was active in a great areas of science and engineering. Very little is known with certainty about Stevins life and what we know is mostly inferred from other recorded facts, the exact birth date and the date and place of his death are uncertain. It is assumed he was born in Bruges since he enrolled at Leiden University under the name Simon Stevinus Brugensis and his name is usually written as Stevin, but some documents regarding his father use the spelling Stevijn. This is a normal spelling shift in 16th century Dutch and he was born around the year 1548 to unmarried parents, Anthonis Stevin and Catelyne van der Poort. His father is believed to have been a son of a mayor of Veurne. While Simons father was not mentioned in the book of burghers, many other Stevins were later mentioned in the Poorterboeken. Simon Stevins mother Cathelijne was the daughter of a family from Ypres. Her father Hubert was a poorter of Bruges, Simons mother Cathelijne later married Joost Sayon who was involved in the carpet and silk trade and a member of the schuttersgilde Sint-Sebastiaan. Through her marriage Cathelijne became a member of a family of Calvinists and it is believed that Stevin grew up in a relatively affluent environment and enjoyed a good education. He was likely educated at a Latin school in his hometown, Stevin left Bruges in 1571 apparently without a particular destination. Stevin was most likely a Calvinist since a Catholic would likely not have risen to the position of trust he later occupied with Maurice, Prince of Orange and it is assumed that he left Bruges to escape the religious persecution of Protestants by the Spanish rulers. Based on references in his work Wisconstighe Ghedaechtenissen, it has been inferred that he must have moved first to Antwerp where he began his career as a merchants clerk. Some biographers mention that he travelled to Prussia, Poland, Denmark, Norway and Sweden and other parts of Northern Europe and it is possible that he completed these travels over a longer period of time. In 1577 Simon Stevin returned to Bruges and was appointed city clerk by the aldermen of Bruges and he worked in the office of Jan de Brune of the Brugse Vrije, the castellany of Bruges. Why he had returned to Bruges in 1577 is not clear and it may have been related to the political events of that period. Bruges was the scene of religious conflict. Catholics and Calvinists alternately controlled the government of the city and they usually opposed each other but would occasionally collaborate in order to counteract the dictates of King Philip II of Spain
10.
Root of unity
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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in branches of mathematics, and are especially important in number theory, the theory of group characters. In field theory and ring theory the notion of root of unity also applies to any ring with an identity element. Any algebraically closed field has exactly n nth roots of unity if n is not divisible by the characteristic of the field, an nth root of unity, where n is a positive integer, is a number z satisfying the equation z n =1. Without further specification, the roots of unity are complex numbers, however the defining equation of roots of unity is meaningful over any field F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either numbers, if the characteristic of F is 0, or, otherwise. Conversely, every element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details, an nth root of unity is primitive if it is not a kth root of unity for some smaller k, z k ≠1. Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n. In fact, if z1 =1 then z is a primitive first root of unity, otherwise if z2 =1 then z is a second root of unity. And, as z is a root of unity, one finds a first a such that za =1. If z is an nth root of unity and a ≡ b then za = zb, Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. Any integer power of an nth root of unity is also an nth root of unity, n = z k n = k =1 k =1. In particular, the reciprocal of an nth root of unity is its complex conjugate, let z be a primitive nth root of unity. Zn−1, zn = z0 =1 are all distinct, assume the contrary, that za = zb where 1 ≤ a < b ≤ n. But 0 < b − a < n, which contradicts z being primitive. Since an nth-degree polynomial equation can only have n distinct roots, from the preceding, it follows that if z is a primitive nth root of unity, z a = z b ⟺ a ≡ b. If z is not primitive there is only one implication, a ≡ b ⟹ z a = z b