1.
Classical music
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Classical music is art music produced or rooted in the traditions of Western music, including both liturgical and secular music. The central norms of this tradition became codified between 1550 and 1900, which is known as the common-practice period, Western staff notation is used by composers to indicate to the performer the pitches, tempo, meter and rhythms for a piece of music. This can leave less room for such as improvisation and ad libitum ornamentation. The term classical music did not appear until the early 19th century, the earliest reference to classical music recorded by the Oxford English Dictionary is from about 1836. This score typically determines details of rhythm, pitch, and, the written quality of the music has enabled a high level of complexity within them, J. S. The use of written notation also preserves a record of the works, Musical notation enables 2000s-era performers to sing a choral work from the 1300s Renaissance era or a 1700s Baroque concerto with many of the features of the music being reproduced. That said, the score does not provide complete and exact instructions on how to perform a historical work, even if the tempo is written with an Italian instruction, we do not know exactly how fast the piece should be played. Bach was particularly noted for his complex improvisations, during the Classical era, the composer-performer Mozart was noted for his ability to improvise melodies in different styles. During the Classical era, some virtuoso soloists would improvise the cadenza sections of a concerto, during the Romantic era, Beethoven would improvise at the piano. The instruments currently used in most classical music were largely invented before the mid-19th century and they consist of the instruments found in an orchestra or in a concert band, together with several other solo instruments. The symphony orchestra is the most widely known medium for music and includes members of the string, woodwind, brass. The concert band consists of members of the woodwind, brass and it generally has a larger variety and number of woodwind and brass instruments than the orchestra but does not have a string section. However, many bands use a double bass. Many of the used to perform medieval music still exist. Medieval instruments included the flute, the recorder and plucked string instruments like the lute. As well, early versions of the organ, fiddle, Medieval instruments in Europe had most commonly been used singly, often self accompanied with a drone note, or occasionally in parts. From at least as early as the 13th century through the 15th century there was a division of instruments into haut, during the earlier medieval period, the vocal music from the liturgical genre, predominantly Gregorian chant, was monophonic, using a single, unaccompanied vocal melody line. Polyphonic vocal genres, which used multiple independent vocal melodies, began to develop during the medieval era, becoming prevalent by the later 13th
2.
Western culture
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The term also applies beyond Europe, to countries and cultures whose histories are strongly connected to Europe by immigration, colonization, or influence. For example, Western Culture includes countries in the Americas and Australasia, whose language, before the Cold War era, the traditional Western viewpoint identified Western Civilization with the Western Christian countries and culture. Ancient Greece is considered the birthplace of Western culture, with the worlds first democratic system of government and major advances in philosophy, science, Greece was followed by Rome, which made key contributions in law, government, engineering and political organization. European culture developed with a range of philosophy, medieval scholasticism, and mysticism. Rational thinking developed through an age of change and formation, with the experiments of the Enlightenment. More often an ideology is what will be used to categorize it as a Western society. There is some disagreement about what nations should or should not be included in the category, many parts of the Eastern Roman Empire are considered Western today but were Eastern in the past. Since the context is highly biased and context-dependent, there is no agreed definition what the West is and it is difficult to determine which individuals fit into which category and the East–West contrast is sometimes criticized as relativistic and arbitrary. Globalism has spread Western ideas so widely that almost all cultures are, to some extent. Stereotyped views of the West have been labeled Occidentalism, paralleling Orientalism—the term for the 19th-century stereotyped views of the East, as Europe discovered the wider world, old concepts adapted. The area that had formerly considered the Orient became the Near East, as the interests of the European powers interfered with Meiji Japan and Qing China for the first time. Thus, the Sino-Japanese War in 1894–1895 occurred in the Far East, the Greeks contrasted themselves to their Eastern neighbors, such as the Trojans in Iliad, setting an example for later contrasts between east and west. In the Middle Ages, the Near East provided a contrast to the West, concepts of what is the West arose out of legacies of the Western Roman Empire and the Eastern Roman Empire. Later, ideas of the west were formed by the concepts of Latin Christendom, Western culture is neither homogeneous nor unchanging. As with all cultures, it has evolved and gradually changed over time. Nevertheless, it is possible to follow the evolution and history of the West, and appreciate its similarities and differences, its borrowings from, and contributions to, other cultures of humanity. Nevertheless, the Greeks felt they were the most civilized and saw themselves as something between the wild barbarians of most of Europe and the soft, slavish Middle-Easterners. In the meantime, however, Greece, under Alexander, had become a capital of the East, the Celts also created some significant literature in the ancient world whenever they were given the opportunity
3.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
4.
Accidental (music)
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In music, an accidental is a note of a pitch that is not a member of the scale or mode indicated by the most recently applied key signature. In musical notation, the sharp, flat, and natural symbols, among others, in the measure where it appears, an accidental sign raises or lowers the immediately following note from its normal pitch, overriding sharps or flats in the key signature. A note is raised or lowered by a semitone, although microtonal music may use fractional accidental signs. One occasionally sees double sharps or flats, which raise or lower the note by a whole tone. Accidentals apply within the measure and octave in which appear, unless canceled by another accidental sign. If a note has an accidental and the note is repeated in a different octave within the same measure, the accidental does not apply to the same note of the different octave. The modern accidental signs derive from the two forms of the letter b used in Gregorian chant manuscripts to signify the two pitches of B, the only note that could be altered. The round b became the sign, while the square b diverged into the sharp. Sometimes the black keys on a keyboard are called accidentals or sharps. In most cases, a sharp raises the pitch of a note one semitone while a flat lowers it a semitone, a natural is used to cancel the effect of a flat or sharp. This system of accidentals operates in conjunction with the key signature, whose effect continues throughout an entire piece, an accidental can also be used to cancel a previous accidental or reinstate the flats or sharps of the key signature. Accidentals apply to subsequent notes on the staff position for the remainder of the measure where they occur, unless explicitly changed by another accidental. Once a barline is passed, the effect of the accidental ends, subsequent notes at the same staff position in the second or later bars are not affected by the accidental carried through with the tied note. Though this convention is still in use particularly in music, it may be cumbersome in music that features frequent accidentals. As a result, a system of note-for-note accidentals has been adopted. Accidentals are not repeated for repeated notes unless one or more different pitches intervene, if a sharp or flat pitch is followed directly by its natural form, a natural is used. Cautionary accidentals or naturals may be used to clarify ambiguities, because seven of the twelve notes of the chromatic equal-tempered scale are naturals this system can significantly reduce the number of naturals required in a notated passage. Thus, the effect of the accidental must be understood in relation to the meaning of the notes staff position
5.
Sharp (music)
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In music, sharp, dièse, or diesis means higher in pitch. More specifically, in musical notation, sharp means higher in pitch by a semitone, and has a sharp symbol, ♯. Sharp is contrasted with flat, which refers to a lowering of pitch, intonation may be flat, sharp, or both, successively or simultaneously. Under twelve-tone equal temperament, B♯, for instance, sounds the same as, or is equivalent to, C natural. In other tuning systems, such enharmonic equivalences in general do not exist, to allow extended just intonation, composer Ben Johnston uses a sharp to indicate a note is raised 70.6 cents, or a flat to indicate a note is lowered 70.6 cents. In tuning, sharp can also slightly higher in pitch. If two simultaneous notes are out of tune, the higher-pitched one is said to be sharp with respect to the other. Furthermore, the verb sharpen means raise the frequency of a note, double sharps also exist, these are denoted by the symbol and raise a note by two semitones, or one whole tone. They should not be confused with a ghost note, less often one will encounter other sharps. A half sharp raises a note by a quarter tone =50 cents, a sharp and a half raises a note by three quarter tones =150 cents and may be denoted. Although very uncommon, a triple sharp can sometimes be found and it raises a note by three semitones. The sharp symbol may be confused with the number sign, both signs have two sets of parallel double-lines. However, a correctly drawn sharp sign must have two slanted parallel lines which rise from left to right, to avoid being obscured by the staff lines, the number sign, in contrast, has two compulsory horizontal strokes in this place. In addition, while the sharp also always has two vertical lines, the number sign may or may not contain perfectly vertical lines. The order of sharps in key signature notation is F♯, C♯, G♯, D♯, A♯, E♯, B♯, each extra sharp being added successively in the sequence of major keys. Similarly the order of flats is based on the natural notes in reverse order, B♭, E♭, A♭, D♭, G♭, C♭, F♭, encountered in the following series of major keys. In the above progression, key C♯ may be conveniently written as the harmonically equivalent key D♭
6.
Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale, the smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones and they can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies, for example, any two notes an octave apart have a frequency ratio of 2,1. This means that successive increments of pitch by the same result in an exponential increase of frequency. For this reason, intervals are often measured in cents, a derived from the logarithm of the frequency ratio. In Western music theory, the most common naming scheme for intervals describes two properties of the interval, the quality and number, examples include the minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled, the importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. The size of an interval can be represented using two alternative and equivalently valid methods, each appropriate to a different context, frequency ratios or cents, the size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, Intervals with small-integer ratios are often called just intervals, or pure intervals. Most commonly, however, musical instruments are tuned using a different tuning system. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, for instance, an equal-tempered fifth has a frequency ratio of 2 7⁄12,1, approximately equal to 1.498,1, or 2.997,2. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems, the standard system for comparing interval sizes is with cents. The cent is a unit of measurement. If frequency is expressed in a scale, and along that scale the distance between a given frequency and its double is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament, a system in which all semitones have the same size. Hence, in 12-TET the cent can be defined as one hundredth of a semitone
7.
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
8.
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
9.
Just intonation
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In music, just intonation or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval, pure intervals are important in music because they correspond to the vibrational patterns found in physical objects which correlate to human perception. The two notes in any just interval are members of the harmonic series. Frequency ratios involving large integers such as 1024,729 are not generally said to be justly tuned, the Indian classical music system uses just intonation tuning as codified in the Natya Shastra. Various societies perceive pure intervals as pleasing or satisfying consonant and, conversely, however, various societies do not have these associations. Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch, however, except for doubling of frequencies, no other intervals are exact ratios of small integers. Each just interval differs a different amount from its analogous, equally tempered interval, justly tuned intervals can be written as either ratios, with a colon, or as fractions, with a solidus. For example, two tones, one at 300 hertz, and the other at 200 hertz are both multiples of 100 Hz and as members of the harmonic series built on 100 Hz. Thus 3,2, known as a fifth, may be defined as the musical interval between the second and third harmonics of any fundamental pitch. Just intonation An A-major scale, followed by three major triads, and then a progression of fifths in just intonation, equal temperament An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. By listening to the file, and then listening to this one, one might be able to hear the beating in this file. Equal temperament and just intonation compared A pair of major thirds, the first in each pair is in equal temperament, the second is in just intonation. Equal temperament and just intonation compared with square waveform A pair of major chords, the first is in equal temperament, the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just temperament between the two chords, in the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent, the square waveform makes the difference between equal and just temperaments more obvious. Harmonic intervals come naturally to horns, vibrating strings, and in human singing voices. Pythagorean tuning, perhaps the first tuning system to be theorized in the West, is a system in all tones can be found using powers of the ratio 3,2. It is easier to think of this system as a cycle of fifths
10.
Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has become the method of representing and comparing musical pitches. Like a decibels relation to intensity, a cent is a ratio between two close frequencies, for the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone spans 100 cents by definition, an octave—two notes that have a frequency ratio of 2, 1—spans twelve semitones and therefore 1200 cents.0005777895. For example, in just intonation the major third is represented by the frequency ratio 5,4, applying the formula at the top shows that this is about 386 cents. The equivalent interval on the piano would be 400 cents. The difference,14 cents, is about a seventh of a half step, as x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a linear function that is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1 +0. 0005946n instead of 2 n⁄1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50 and this error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes. It is difficult to establish how many cents are perceptible to humans, one author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians ability to recognize, as out-of-tune and it has also been established that increased tonal context enables listeners to judge pitch more accurately. Free, online web sites for self-testing are available, while intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords. When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch, normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents, iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone, centitone, and millitone, for example, Equal tempered perfect fifth =700 cents =175.6 savarts =583.3 millioctaves =350 centitones
11.
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
12.
Diminished second
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In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone. It is enharmonically equivalent to a perfect unison, thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C♭ immediately above, in particular, it may be regarded as the difference between a diatonic and chromatic semitone. For instance, the interval from B to C is a semitone, the interval from B to B♯ is a chromatic semitone, and their difference. Being diminished, it is considered a dissonant interval and this makes it a highly variable quantity between tuning systems. Hence for example C♯ is narrower than D♭ by a diminished second interval, in 12-tone equal temperament, the diminished second is identical to the unison, because both semitones have the same size. In 19-tone equal temperament, on the hand, it is identical to the chromatic semitone and is a respectable 63.16 cents wide. It shows a similar size in third-comma meantone, where it coincides with the greater diesis, the most commonly used meantone temperaments fall between these extremes, giving it an intermediate size. In Pythagorean tuning, however, the interval actually shows a descending direction, i. e. a ratio below unison, such is also the case in twelfth-comma meantone, although that diminished second is only a twelfth of the Pythagorean one. The table below summarizes the definitions of the second in the main tuning systems. In the column labeled Difference between semitones, m2 is the second, A1 is the augmented unison, and S1, S2, S3. Notice that for 5-limit tuning, 1/6-, 1/4-, and 1/3-comma meantone, List of musical intervals List of pitch intervals List of meantone intervals Comma
13.
Comma (music)
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In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. Within the same tuning system, two enharmonically equivalent notes may have a different frequency, and the interval between them is a comma. The interval between notes, the diesis, is an easily audible comma. Commas are often defined as the difference in size between two semitones, each meantone temperament tuning system produces a 12-tone scale characterized by two different kinds of semitones, and hence by a comma of unique size. The same is true for Pythagorean tuning, in just intonation, more than two kinds of semitones may be produced. Thus, a tuning system may be characterized by several different commas. For instance, a commonly used version of five-limit tuning produces a 12-tone scale with four kinds of semitones, the size of commas is commonly expressed and compared in terms of cents – 1/1200 fractions of an octave on a logarithmic scale. In the column labeled Difference between semitones, m2 is the second, A1 is the augmented unison, and S1, S2, S3. In the columns labeled Interval 1 and Interval 2, all intervals are presumed to be tuned in just intonation, notice that the Pythagorean comma and the syntonic comma are basic intervals that can be used as yardsticks to define some of the other commas. For instance, the difference between them is a small comma called schisma, a schisma is not audible in many contexts, as its size is narrower than the smallest audible difference between tones. Many other commas have been enumerated and named by microtonalists The syntonic comma has a role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds, in Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third and minor third were dissonant, and this prevented musicians from using triads and chords. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, for instance, if you decrease by a syntonic comma the frequency of E, C-E, and E-G become just. Since then, other tuning systems were developed, and the comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the belonging to the syntonic temperament continuum. More exactly, in these systems the diminished second is a descending inteval. For instance, the Pythagorean comma can be computed as the difference between a chromatic and a semitone, which is the opposite of a Pythagorean diminished second
14.
Quarter-comma meantone
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Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma, the purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be sonorous and just, later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude. In a meantone tuning, we have diatonic and chromatic semitones, in Pythagorean tuning, these correspond to the Pythagorean limma and the Pythagorean apotome, only now the apotome is larger. Two fifths up and an octave down make up a whole tone consisting of one diatonic, four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones. This large interval of a seventeenth contains 5 + + + =20 −3 =17 staff positions. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths,4 =8116 =8016 ⋅8180 =5 ⋅8180. In quarter-comma meantone temperament, where a just major third is required, by definition, however, a seventeenth of the same size must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as used in Pythagorean tuning, produce a slightly wider seventeenth. Notice that, in quarter-comma meantone, the seventeenth is 81/80 times narrower than in Pythagorean tuning and this difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. The difference between two sizes is a quarter of a syntonic comma, ≈701.955 −696.578 ≈5.377 ≈21.5064 cents. In sum, this system tunes the major thirds to the just ratio of 5,4, most of the tones in the ratio √5,2. This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth and it is this that gives the system its name of quarter-comma meantone. The whole chromatic scale, can be constructed by starting from a base note. This method is identical to Pythagorean tuning, except for the size of the fifth, the construction table below illustrates how the pitches of the notes are obtained with respect to D, in a D-based scale. As in Pythagorean tuning, this method generates 13 pitches, but A♭ and G♯ have almost the same frequency, the table above shows a D-based stack of fifths. Some authors prefer showing a C-based stack of fifths, ranging from A♭ to G♯, the only difference is that the construction table shows intervals from C, rather than from D
15.
Enharmonic
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Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in equal temperament, the notes C♯. Namely, they are the key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony. In other words, if two notes have the pitch but are represented by different letter names and accidentals, they are enharmonic. Enharmonic intervals are intervals with the sound that are spelled differently…, of course. Enharmonic equivalence is peculiar to post-tonal theory, much music since at least the 18th century, however, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent. Some key signatures have an enharmonic equivalent that represents a scale identical in sound, the number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is equivalent to the key of C♭ major with 7 flats. Keys past 7 sharps or flats exist only theoretically and not in practice, the enharmonic keys are six pairs, three major and three minor, B major/C♭ major, G♯ minor/A♭ minor, F♯ major/G♭ major, D♯ minor/E♭ minor, C♯ major/D♭ major and A♯ minor/B♭ minor. There are practically no works composed in keys that require double sharps or double flats in the key signature, in practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings. Enharmonic equivalents can also used to improve the readability of a line of music, for example, a sequence of notes is more easily read as ascending or descending if the noteheads are on different positions on the staff. Doing so may reduce the number of accidentals that must be used. Thus, in the key of B♭ major, the sequence B♭-B♮-B♭ is more easily using the enharmonic spelling C♭ instead of B♮. For example the intervals of a sixth on C, on B♯. The most common enharmonic intervals are the fourth and diminished fifth, or tritone. Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals. In principle, the musical use of the word enharmonic to mean identical tones is correct only in equal temperament. In other tuning systems, however, enharmonic associations can be perceived by listeners, in Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2
16.
Enharmonically equivalent
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Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in equal temperament, the notes C♯. Namely, they are the key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony. In other words, if two notes have the pitch but are represented by different letter names and accidentals, they are enharmonic. Enharmonic intervals are intervals with the sound that are spelled differently…, of course. Enharmonic equivalence is peculiar to post-tonal theory, much music since at least the 18th century, however, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent. Some key signatures have an enharmonic equivalent that represents a scale identical in sound, the number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is equivalent to the key of C♭ major with 7 flats. Keys past 7 sharps or flats exist only theoretically and not in practice, the enharmonic keys are six pairs, three major and three minor, B major/C♭ major, G♯ minor/A♭ minor, F♯ major/G♭ major, D♯ minor/E♭ minor, C♯ major/D♭ major and A♯ minor/B♭ minor. There are practically no works composed in keys that require double sharps or double flats in the key signature, in practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings. Enharmonic equivalents can also used to improve the readability of a line of music, for example, a sequence of notes is more easily read as ascending or descending if the noteheads are on different positions on the staff. Doing so may reduce the number of accidentals that must be used. Thus, in the key of B♭ major, the sequence B♭-B♮-B♭ is more easily using the enharmonic spelling C♭ instead of B♮. For example the intervals of a sixth on C, on B♯. The most common enharmonic intervals are the fourth and diminished fifth, or tritone. Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals. In principle, the musical use of the word enharmonic to mean identical tones is correct only in equal temperament. In other tuning systems, however, enharmonic associations can be perceived by listeners, in Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2
17.
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
18.
Diaschisma
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The diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds. It can be represented by the ratio 2048,2025 and is about 19.5 cents, the use of the name diaschisma for this interval is due to Helmholtz, earlier Rameau had called that interval a diminished comma or comma minor. A diaschisma is the difference between a schisma and a comma, as well as the difference between the greater chromatic semitone and the just minor second. Medieval theorists Boethius and Tinctoris described the diaschisma as one-half of the Pythagorean minor second, or 256/243, the diaschisma may be approximated by 89/88,19.56 cents. Tempering out the diaschisma, in the meaning of the term. However, it is possible to improve the tuning a good deal over that of 12-et and still temper out the diaschisma, the equal temperaments with 22,34 and 46 notes all temper it out
19.
Schisma
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In music, the schisma is the interval between a Pythagorean comma and a syntonic comma and equals 32805,32768, which is 1.9537 cents. Schisma is a Greek word meaning a split whose musical sense was introduced by Boethius at the beginning of the 6th century in the 3rd book of his De institutione musica, Boethius was also the first to define diaschisma. Andreas Werckmeister defined the grad as the root of the Pythagorean comma. This value,1.955 cents, may be approximated by the ratio 886,885 and this interval is also sometimes called a schisma. Curiously, 21/12 51/7 appears very close to 4,3, thats because the difference between a grad and a schisma is so small. So, a rational version of equal temperament may be realized by flattening the fifth by a schisma rather than a grad, a fact first noted by Johann Kirnberger. Twelve of these Kirnberger fifths of 16384,10935 exceed seven octaves, and therefore fail to close, by the interval of 2161 3−84 5−12. Tempering out the schisma leads to schismatic temperament, as used by Descartes, a schisma added to a perfect fourth =27,20, a schisma subtracted from a perfect fifth =40,27, and a major sixth plus a schisma =27,16. By this definition is a schisma is what is known as the syntonic comma, septimal-Comma, Tonalsoft, Encyclopedia of Microtonal Music Theory
20.
Pythagorean comma
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It is equal to the frequency ratio 531441,524288, or approximately 23.46 cents, roughly a quarter of a semitone. The comma which musical temperaments often refer to tempering is the Pythagorean comma, the diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides therefore with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma, equal to about −23.46 cents. As described in the introduction, the Pythagorean comma may be derived in multiple ways, Difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B♯, or D♭, Difference between Pythagorean apotome and Pythagorean limma. Difference between twelve just perfect fifths and seven octaves, Difference between three Pythagorean ditones and one octave. A just perfect fifth has a ratio of 3/2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to an initial note. Apotome and limma are the two kinds of semitones defined in Pythagorean tuning, namely, the apotome is the chromatic semitone, or augmented unison, while the limma is the diatonic semitone, or minor second. A ditone is a formed by two major tones. In Pythagorean tuning, a major tone has a size of about 203.9 cents, the 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change, * The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are used, because the enharmonic scales with five accidentals are treated as equivalent. This interval has serious implications for the various tuning schemes of the scale, because in Western music,12 perfect fifths. Equal temperament, today the most common tuning used in the West, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma. Take the just fifth to the power, then subtract seven octaves. He concludes that raising this number by six whole tones yields a value G which is larger than that yielded by raising it by an octave and this much smaller interval was later named Mercators comma
21.
Pythagorean tuning
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Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio 3,2, which is 702 cents wide. Hence, it is a system of tuning in which the frequency ratios of all intervals are based on the ratio 3,2. This ratio, also known as the perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear. As Novalis put it, The musical proportions seem to me to be particularly correct natural proportions, the so-called Pythagorean tuning was used by musicians up to the beginning of the 16th century. The Pythagorean system would appear to be ideal because of the purity of the fifths, Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3,2, the next simplest ratio after 2,1. Since notes differing in frequency by a factor of 2 are given the same name, the purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it. This interval is called the basic octave. For instance, the A is tuned such that its frequency equals 3,2 times the frequency of D—if D is tuned to a frequency of 288 Hz, then A is tuned to 432 Hz. Similarly, the E above A is tuned such that its frequency equals 3,2 times the frequency of A, or 9,4 times the frequency of D—with A at 432 Hz, this puts E at 648 Hz. Since this E is outside the basic octave, it is usual to halve its frequency to move it within the basic octave. Therefore, E is tuned to 324 Hz, a 9,8 above D, the B at 3,2 above that E is tuned to the ratio 27,16 and so on. This frequency is doubled to bring it into the basic octave. When extending this tuning however, a problem arises, no stack of 3,2 intervals will fit exactly into any stack of 2,1 intervals. For instance a stack such as this, obtained by adding one note to the stack shown above A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ will be similar. More exactly, it will be about a quarter of a semitone larger, thus, A♭ and G♯, when brought into the basic octave, will not coincide as expected. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a Pythagorean comma, to get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E♭ to G♯. This, as shown above, the remaining interval is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as one is known as a wolf interval
22.
Philolaus
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Philolaus was a Greek Pythagorean and Presocratic philosopher. He argued that at the foundation of everything is the part played by the limiting and limitless and he is also credited with originating the theory that the Earth was not the center of the universe. According to August Böckh, who cites Nicomachus, Philolaus was the successor of Pythagoras, Philolaus is variously reported as being born in either Croton, or Tarentum, or Metapontum — all part of Magna Graecia. It is most likely that he came from Croton and he may have fled the second burning of the Pythagorean meeting-place around 454 BCE, after which he migrated to Greece. According to Platos Phaedo, he was the instructor of Simmias and Cebes at Thebes, around the time the Phaedo takes place and this would make him a contemporary of Socrates, and agrees with the statement that Philolaus and Democritus were contemporaries. The various reports about his life are scattered among the writings of later writers and are of dubious value in reconstructing his life. He apparently lived for time at Heraclea, where he was the pupil of Aresas. Diogenes Laërtius is the authority for the claim that Plato, shortly after the death of Socrates. The pupils of Philolaus were said to have included Xenophilus, Phanto, Echecrates, Diocles, Diogenes Laërtius speaks of Philolaus composing one book, but elsewhere he speaks of three books, as do Aulus Gellius and Iamblichus. It might have been one treatise divided into three books, Plato is said to have procured a copy of his book from which, it was later claimed, Plato composed much of his Timaeus. One of the works of Philolaus was called On Nature, which seems to be the work which Stobaeus calls On the World. Other writers refer to a work entitled Bacchae, which may have another name for the same work. However, it has been mentioned that Proclus describes the Bacchae as a book for teaching theology by means of mathematics, the second book appears to have been an exposition of the nature of numbers, which in the Pythagorean theory are the essence and source of all things. Additionally Charles Peter Mason noted Pythagoras and his earliest successors do not appear to have committed any of their doctrines to writing, but amid the different and inconsistent accounts of the matter, the first publication of the Pythagorean doctrines is pretty uniformly attributed to Philolaus. Other versions of the story represent Plato as purchasing it himself from Philolaus or his relatives when in Sicily, out of the materials which he derived from these books Plato is said to have composed his Timaeus. L. The story of Platos purchase of books from Philolaus was probably invented to authenticate the three forged treatises of Pythagoras. Burkerts arguments, supported by study, have led to a consensus that some 11 fragments are genuine. Fragments 1, 6a and 13 are identified as coming from the book On Nature by ancient sources, robert Scoon explained Philolaus universe in 1922 Philolaus is trying to show how the ordered universe that we know came into its present condition
23.
Jean-Philippe Rameau
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Jean-Philippe Rameau was one of the most important French composers and music theorists of the Baroque era. He replaced Jean-Baptiste Lully as the dominant composer of French opera and is considered the leading French composer for the harpsichord of his time. He was almost 50 before he embarked on the career on which his reputation chiefly rests today. His debut, Hippolyte et Aricie, caused a stir and was fiercely attacked by the supporters of Lullys style of music for its revolutionary use of harmony. Rameaus music had gone out of fashion by the end of the 18th century, today, he enjoys renewed appreciation with performances and recordings of his music ever more frequent. The details of Rameaus life are obscure, especially concerning his first forty years. He was a man, and even his wife knew nothing of his early life. Rameaus early years are particularly obscure and he was born on 25 September 1683 in Dijon, and baptised the same day. His father, Jean, worked as an organist in churches around Dijon. The couple had children, of whom Jean-Philippe was the seventh. Rameau was taught music before he could read or write and he was educated at the Jesuit college at Godrans, but he was not a good pupil and disrupted classes with his singing, later claiming that his passion for opera had begun at the age of twelve. Initially intended for the law, Rameau decided he wanted to be a musician, and his father sent him to Italy, on his return, he worked as a violinist in travelling companies and then as an organist in provincial cathedrals before moving to Paris for the first time. Here, in 1706, he published his earliest known compositions, the works that make up his first book of Pièces de clavecin. In 1709, he moved back to Dijon to take over his fathers job as organist in the main church, the contract was for six years, but Rameau left before then and took up similar posts in Lyon and Clermont. During this period, he composed motets for church performance as well as secular cantatas, in 1722, he returned to Paris for good, and here he published his most important work of music theory, Traité de lharmonie. This soon won him a reputation, and it was followed in 1726 by his Nouveau système de musique théorique. In 1724 and 1729, he published two more collections of harpsichord pieces. Rameau took his first tentative steps into composing music when the writer Alexis Piron asked him to provide songs for his popular comic plays written for the Paris Fairs
24.
Syntonic comma
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The comma is referred to as a comma of Didymus because it is the amount by which Didymus corrected the Pythagorean major third to a just major third. Namely,81,64 ÷5,4 =81,80, the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has a size of 3,2, a just major third has a size of 5,4, and one of them plus two octaves is equal to 5,1. The difference between these is the syntonic comma, namely,81,16 ÷5,1 =81,80. The difference between one octave plus a justly tuned minor third, and three justly tuned perfect fourths, namely,12,5 ÷64,27 =81,80. The difference between the two kinds of major second which occur in 5-limit tuning, major tone and minor tone, namely,9,8 ÷10,9 =81,80. The difference between a Pythagorean major sixth and a justly tuned or pure major sixth, namely,27,16 ÷5,3 =81,80. On a piano keyboard a stack of four fifths is exactly equal to two octaves plus a major third, in other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves, fifths, and thirds, however, the ratio between their frequencies, as explained above, is a syntonic comma. Pythagorean tuning uses justly tuned fifths as well, but uses the complex ratio of 81,64 for major thirds. Quarter-comma meantone uses justly tuned major thirds, but flattens each of the fifths by a quarter of a syntonic comma and this is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments. Mathematically, by Størmers theorem,81,80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5,4, and a number is one whose prime factors are limited to 2,3. Thus, although smaller intervals can be described within 5-limit tunings, the syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds, in Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third and minor third were dissonant, and this prevented musicians from using triads and chords, in late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a comma, C-E. But the fifth C-G stays consonant, since only E has been flattened, since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them
25.
19 equal temperament
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In music,19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a ratio of 19√2, or 63.16 cents. 19-edo is the tuning of the temperament in which the tempered perfect fifth is equal to 694.737 cents. Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory, the greater diesis, the ratio of four minor thirds to an octave was almost exactly a nineteenth of an octave. Interest in such a system goes back to the 16th century. Costeley understood and desired the circulating aspect of this tuning, in 1577, music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1⁄3-comma meantone, in which the fifth is of size 694.786 cents, the fifth of 19-edo is 694.737, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error. Salinas suggested tuning nineteen tones to the octave to this tuning, in the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50-edo. The composer Joel Mandelbaum wrote his Ph. D. thesis on the properties of the 19-edo tuning, Mandelbaum and Joseph Yasser have written music with 19-edo. Easley Blackwood has stated that 19-edo makes possible a substantial enrichment of the tonal repertoire, here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series, the difference column measures in cents the distance from an exact fit to these ratios. For reference, the difference from the fifth in the widely used 12 equal temperament is 1.955 cents. Beta scale Elaine Walker Levy, Kenneth J. Costeleys Chromatic Chanson, Annales Musicologues, Moyen-Age et Renaissance, Tome III, howe, Hubert S. Jr. 19-Tone Theory and Applications, Aaron Copland School of Music at Queens College. Sethares, William A. Tunings for 19 Tone Equal Tempered Guitar, Experimental Musical Instruments, Vol. VI, hair, Bailey, Morrison, Pearson and Parncutt, Rehearsing Microtonal Music, Grappling with Performance and Intonational Problems, Microtonalism. ZiaSpace. com - 19tet downloadable mp3s by Elaine Walker of Zia, the Music of Jeff Harrington, Parnasse. com. Jeff Harrington is a composer who has several pieces for piano in the 19-TET tuning. Chris Vaisvil, GR-20 Hexaphonic 19-ET Guitar Improvisation Arto Juhani Heino, Artone 19 Guitar Design, naming the 19 note scale Parvatic
26.
Septimal diesis
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In music, septimal diesis is an interval with the ratio of 49,48 play, which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone and it may also be the ratio 36,35, or 48.77 cents. Play In 12 equal temperament this interval is not tempered out and this makes the diesis a semitone, about three times its correct size. It is not tempered out however by 22-ET or 31-ET
27.
Slendro
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Slendro is a pentatonic scale, Play the older of the two most common scales used in Indonesian gamelan music, the other being pélog. From one region of Indonesia to another the slendro scale often varies widely, the amount of variation also varies from region to region. For example, slendro in Central Java varies much less from gamelan to gamelan than it does in Bali, the five pitches of the Javanese version are roughly equally spaced within the octave. As in pelog, although the intervals vary from one gamelan to the next and it is thought that this contributes to the very busy and shimmering sound of gamelan ensembles. In the religious ceremonies that contain gamelan, these beats are meant to give the listener a feeling of a gods presence or a stepping stone to a meditative state. For the instruments that do not need fixed pitches and the voice, the Sundanese musicologist/teacher Raden Machjar Angga Koesoemadinata identified 17 vocal pitches used in slendro. These microtonal adjustments bear some similarity to Indian śruti, in Java, the notes of the slendro scale can be designated in different ways, one common way is the use of numbers often called by their names in Javanese, especially in a shortened form. An older set uses names derived from parts of the body, notice that both systems have the same designations for 5 and 6. There is no 4, possibly this is because it appears as a tone in pelog and is not used when modulating between the systems. The name barang is also used for 1 in slendro. In Bali, the scale starts on the note named ding, for experienced participants in gamelan music, the pelog and slendro scales each have a particular feeling, related to the rituals and circumstances in which the scale is used. For example, in Bali, slendro is felt to have a sad sound because it is used as the tuning of gamelan angklung, the connotation also depends on the pathet used. There are three slendro pathet used in Javanese gamelan, nem, sanga, and manyura and that is the order in which they appear in a wayang performance, which historically used only slendro pathet. Consequently, they have the implications of where they appear in the evening, the origin of the slendro scale is unknown. However the name slendro is derived from Sailendra, the ancient dynasty of Medang Kingdom in Central Java, the slendro scale is thought to be brought to Srivijaya by Mahayana Buddhist from Gandhara of India, via Nalanda and Srivijaya from there to Java and Bali. It is similar to scales used in Indian and Chinese music as well as areas of Asia. A salendro scale is found in the neighboring musical ensemble of Kulintang. This is very difficult if not impossible to determine, even within Indonesia it is difficult to determine the evolution of scales
28.
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
29.
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
30.
31 equal temperament
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In music,31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Play Each step represents a ratio of 31√2, or 38.71 cents. 31-ET is a good approximation of quarter-comma meantone temperament. More generally, it is a diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents. In 1666, Lemme Rossi first proposed an equal temperament of this order, shortly thereafter, having discovered it independently, scientist Christiaan Huygens wrote about it also. Huygens not only realized this, he went farther and noted that 31-ET provides an excellent approximation of septimal, the composer Joel Mandelbaum used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series. The tuning has poor matches to both the 9,8 and 10,9 intervals, however, it has a match for the average of the two. Practically it is close to quarter-comma meantone. This tuning can be considered a meantone temperament, many chords of 31-ET are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad, which might be written C–E–G, C–D–G or C–F–G, and the Orwell tetrad, usual chords like the major chord is rendered nicely in 31-ET because the third and the fifth are very well approximated. Also, it is possible to play chords and supermajor chords. It is also possible to render nicely the harmonic seventh chord, for example on C with C–E–G–A♯. The seventh here is different from stacking a fifth and a minor third and this difference cannot be made in 12-ET
31.
Ditone
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In music, a ditone is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded, the Pythagorean ditone is the major third in Pythagorean tuning, which has an interval ratio of 81,64, which is 407.82 cents. The Pythagorean ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma, because it is a comma wider than a perfect major third of 5,4, it is called a comma-redundant interval. Play The major third that appears commonly in the system is properly known as the Pythagorean ditone. This is the interval that is sharp, at 408c. It may also be thought of as four justly tuned fifths minus two octaves, the prime factorization of the 81,64 ditone is 3^4/2^6. In Didymuss diatonic and Ptolemys syntonic tunings, the ditone is a just major third with a ratio of 5,4, made up of two unequal tones—a major and a tone of 9,8 and 10,9. The difference between the two systems is that Didymus places the minor tone below the major, whereas Ptolemy does the opposite, in meantone temperaments, the major tone and minor tone are replaced by a mean tone which is somewhere in between the two. Two of these make a ditone or major third. This major third is exactly the just major third in quarter-comma meantone and this is the source of the name, the note exactly halfway between the bounding tones of the major third is called the mean tone. Modern writers occasionally use the word ditone to describe the interval of a third in equal temperament. For example, In modern acoustics, the equal-tempered semitone has 100 cents, the tone 200 cents, the ditone or major third 400 cents, the perfect fourth 500 cents, and so on
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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
33.
Aulos
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An aulos or tibia was an ancient Greek wind instrument, depicted often in art and also attested by archaeology. An aulete was the musician who performed on an aulos, the ancient Roman equivalent was the tibicen, from the Latin tibia, pipe, aulos. There were several kinds of aulos, single or double, the most common variety was a reed instrument. Archeological finds, surviving iconography and other evidence indicate that it was double-reeded, like the modern oboe, a single pipe without a reed was called the monaulos. A single pipe held horizontally, as the flute, was the plagiaulos. A pipe with a bag to allow for sound, that is a bagpipe, was the askaulos. Like the Great Highland Bagpipe, the aulos has been used for martial music and it was the standard accompaniment of the passionate elegiac poetry. It also accompanied physical activities such as wrestling matches, the broad jump, plato associates it with the ecstatic cults of Dionysus and the Korybantes, banning it from his Republic but reintroducing it in Laws. It appears that some variants of the instrument were loud, shrill, sometimes a second strap was used over the top of the head to prevent the phorbeiá from slipping down. Aulos players are depicted with puffed cheeks. The playing technique almost certainly made use of breathing, very much like the Sardinian launeddas and Armenian duduk. Nevertheless, such musicians could achieve fame, the Romano-Greek writer Lucian discusses aulos playing in his dialogue Harmonides, in which Alexander the Greats aulete Timotheus discusses fame with his pupil Harmonides. Timotheus advises him to impress the experts within his profession rather than seek popular approval in big public venues, if leading musicians admire him, popular approval will follow. However, Lucian reports that Harmonides died from excessive blowing during practicing, in myth, Marsyas the satyr was supposed to have invented the aulos, or else picked it up after Athena had thrown it away because it caused her cheeks to puff out and ruined her beauty. But Apollo and his lyre beat Marsyas and his aulos, and since the pure lord of Delphis mind worked in different ways from Marsyass, he celebrated his victory by stringing his opponent up from a tree and flaying him alive. King Midas was cursed with donkeys ears for judging Apollo as the lesser player, Marsyass blood and the tears of the Muses formed the river Marsyas in Asia Minor. This tale was a warning against committing the sin of hubris, or overweening pride, some of this is a result of 19th century AD classical interpretation, i. e. Apollo versus Dionysus, or Reason opposed to Madness. In the temple to Apollo at Delphi, there was also a shrine to Dionysus, and his Maenads are shown on drinking cups playing the aulos, so a modern interpretation can be a little more complicated than just simple duality
34.
Hermann von Helmholtz
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Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of institutions, the Helmholtz Association, is named after him. In physics, he is known for his theories on the conservation of energy, work in electrodynamics, chemical thermodynamics, Helmholtzs work is influenced by the philosophy of Johann Gottlieb Fichte and Immanuel Kant. He tried to trace their theories in empirical matters like physiology, as a young man, Helmholtz was interested in natural science, but his father wanted him to study medicine at the Charité because there was financial support for medical students. Trained primarily in physiology, Helmholtz wrote on other topics, ranging from theoretical physics, to the age of the Earth. Helmholtzs first academic position was as a teacher of Anatomy at the Academy of Arts in Berlin in 1848 and he then moved to take a post of associate professor of physiology at the Prussian University of Königsberg, where he was appointed in 1849. In 1855 he accepted a professorship of anatomy and physiology at the University of Bonn. He was not particularly happy in Bonn, however, and three later he transferred to the University of Heidelberg, in Baden, where he served as professor of physiology. In 1871 he accepted his final university position, as professor of physics at the University of Berlin and his first important scientific achievement, an 1847 treatise on the conservation of energy, was written in the context of his medical studies and philosophical background. He discovered the principle of conservation of energy while studying muscle metabolism and he tried to demonstrate that no energy is lost in muscle movement, motivated by the implication that there were no vital forces necessary to move a muscle. This was a rejection of the tradition of Naturphilosophie which was at that time a dominant philosophical paradigm in German physiology. He published his theories in his book Über die Erhaltung der Kraft, in the 1850s and 60s, building on the publications of William Thomson, Helmholtz and William Rankine popularized the idea of the heat death of the universe. In fluid dynamics, Helmholtz made several contributions, including Helmholtzs theorems for vortex dynamics in inviscid fluids, Helmholtz was a pioneer in the scientific study of human vision and audition. He coined the term psychophysics, to capture the distinction between the measurement of physical stimuli and their effect on human perception. The physical sound needs to be increased exponentially in order for equal steps to seem linear, Helmholtz paved the way in experimental studies on the relationship between the physical energy and its appreciation, with the goal in mind to develop psychophysical laws. The sensory physiology of Helmholtz was the basis of the work of Wilhelm Wundt, a student of Helmholtz and he, more explicitly than Helmholtz, described his research as a form of empirical philosophy and as a study of the mind as something separate. Helmholtz had, in his early repudiation of Naturphilosophie, stressed the importance of materialism, in 1851, Helmholtz revolutionized the field of ophthalmology with the invention of the ophthalmoscope, an instrument used to examine the inside of the human eye. Helmholtzs interests at that time were focused on the physiology of the senses
35.
Alexander John Ellis
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Alexander John Ellis, FRS was an English mathematician and philologist, who also influenced the field of musicology. He changed his name from his fathers name Sharpe to his mothers maiden name Ellis in 1825 and he is buried in Kensal Green Cemetery, London. He was born Alexander John Sharpe in Hoxton, Middlesex to a wealthy family and his father James Birch Sharpe was a notable artist and physician, who was later appointed Esquire of Windlesham. His mother Ann Ellis was from a background, but it is not known how her family made its fortune. Alexanders brother James Birch Sharpe junior, died at the Battle of Inkerman and his other brother William Henry Sharpe served with the Lancashire Fusiliers after moving north with his family to Cumberland, due to military work. Alexander was educated at Shrewsbury School, Eton College and Trinity College, Cambridge, initially trained in mathematics and the classics, he became a well-known phonetician of his time. Through his work in phonetics, he became interested in vocal pitch and by extension, in musical pitch as well as speech. Ellis is noted for translating and extensively annotating Hermann von Helmholtzs On the Sensations of Tone, the second edition of this translation, published in 1885, contains an appendix which summarises Ellis own work on related matters. In his writings on musical pitch and scales, Ellis elaborates his notion and notation of cents for musical intervals and this concept became especially influential in Comparative musicology, a predecessor of ethnomusicology. Analyzing the scales of various European musical traditions, Ellis also showed that the diversity of systems cannot be explained by a single physical law. In part V of his work On Early English Pronunciation, he applied the Dialect Test across Britain and he distinguished forty-two different dialects in England and the Scottish Lowlands. This was one of the first works to apply phonetics to English speech and he was acknowledged by George Bernard Shaw as the prototype of Professor Henry Higgins of Pygmalion. He was elected in June 1864 as a Fellow of the Royal Society, elliss son Tristram James Ellis trained as an engineer, but later became a noted painter of the Middle East. Ellis developed two phonetic alphabets, phonotype, which used many new letters, and palæotype, which replaced many of these with turned letters, small caps, and italics. Two of his novel letters survived, ⟨ʃ⟩ and ⟨ʒ⟩ were passed on to Sweets Romic alphabet,1885, On the Musical Scales of Various Nations, Journal of the Society of Arts 33, p.485. (Link is to a HTML transcription 1890, English Dialects – Their Sounds and Homes Chisholm, Hugh, ed. Ellis, dictionary of National Biography,1901 supplement. London, Smith, Elder & Co. M. K
36.
Semitone
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A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, for example, C is adjacent to C♯, the interval between them is a semitone. In music theory, a distinction is made between a diatonic semitone, or minor second and a semitone or augmented unison. In twelve-tone equal temperament all semitones are equal in size, in other tuning systems, semitone refers to a family of intervals that may vary both in size and name. In quarter-comma meantone, seven of them are diatonic, and 117.1 cents wide, while the five are chromatic. 12-tone scales tuned in just intonation typically define three or four kinds of semitones, for instance, Asymmetric five-limit tuning yields chromatic semitones with ratios 25,24 and 135,128, and diatonic semitones with ratios 16,15 and 27,25. The condition of having semitones is called hemitonia, that of having no semitones is anhemitonia, a musical scale or chord containing semitones is called hemitonic, one without semitones is anhemitonic. The minor second occurs in the scale, between the third and fourth degree, and between the seventh and eighth degree. It is also called the diatonic semitone because it occurs between steps in the diatonic scale, the minor second is abbreviated m2. Its inversion is the major seventh, listen to a minor second in equal temperament. Here, middle C is followed by D♭, which is a tone 100 cents sharper than C, melodically, this interval is very frequently used, and is of particular importance in cadences. In the perfect and deceptive cadences it appears as a resolution of the leading-tone to the tonic, in the plagal cadence, it appears as the falling of the subdominant to the mediant. It also occurs in many forms of the cadence, wherever the tonic falls to the leading-tone. Harmonically, the interval usually occurs as some form of dissonance or a tone that is not part of the functional harmony. It may also appear in inversions of a seventh chord. In unusual situations, the second can add a great deal of character to the music. For instance, Frédéric Chopins Étude Op.25, No.5 opens with a melody accompanied by a line that plays fleeting minor seconds and these are used to humorous and whimsical effect, which contrasts with its more lyrical middle section. This eccentric dissonance has earned the piece its nickname, the wrong note étude and this kind of usage of the minor second appears in many other works of the Romantic period, such as Modest Mussorgskys Ballet of the Unhatched Chicks
37.
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
38.
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
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Major and minor
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In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval. Major and minor are frequently referred to in the titles of classical compositions, with regard to intervals, the words major and minor just mean large and small, so a major third is a wider interval, and a minor third a relatively narrow one. The intervals of the second, third, sixth, and seventh may be major or minor, the other uses of major and minor, in general, refer to musical structures containing major thirds or minor thirds. A major scale is one whose third degree is a third above the tonic. A major chord or major triad, similarly, contains a third above the root. In Western music, a chord, in comparison, sounds darker than a major chord. The hallmark that distinguishes major keys from minor is whether the scale degree is major or minor. This alteration in the third degree greatly changes the mood of the music, the minor scale can be described in two different ways. Minor keys are said to have a more interesting, possibly sadder sound than plain major scales. The minor mode, with its sixth and seventh degrees, offers nine notes, in C, C-D-E♭-F-G-A♭-A♮-B♭-B♮, over the major modes seven, in C. Harry Partch considers minor as, the faculty of ratios. The minor key and scale are considered less justifiable than the major, with Paul Hindemith calling it a clouding of major. The relative minor of a key has the same key signature and starts down a minor third, for example. Similarly the relative major of a minor key starts up a third, for example. This is in contrast with, for instance, transposition, transposition is done by moving all intervals up or down a certain constant interval, and does change key, but does not change mode, which requires the alteration of intervals. By analogy, the scales are given stereotypically masculine qualities. German uses the word Tongeschlecht for tonality, and the words Dur for major, major and minor chords may each be found in both the major and minor scales, constructed on different degrees of each. In the natural scale, all scale degrees are the same as the relative major
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