1.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
2.
Musical temperament
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In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system, tempering is the process of altering the size of an interval by making it narrower or wider than pure. The development of well temperament allowed fixed-pitch instruments to play well in all of the keys. The famous Well-Tempered Clavier by Johann Sebastian Bach takes full advantage of this breakthrough, however, while unpleasant intervals were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies, allowing intervals varying from the highest consonance to highly dissonant, for instance,660 Hz /440 Hz constitutes a fifth, and 880 Hz /440 Hz an octave. Such intervals have a stability, or purity to their sound, if, for example, two sound signals with frequencies that vary just by 0. When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency, the waveform of such a tone is characterized by a shape that is complex compared to a simple waveform, but remains periodic. When two tones depart from exact integer ratios, the shape waveform becomes erratic—a phenomenon that may be described as destabilization, as the composite waveform becomes more erratic, the consonance of the interval also changes. Tempering an interval involves the use of such minor adjustments to enable musical possibilities that are impractical using just intonation. Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning, Pythagorean tuning was a system of just intonation that tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the practice until then. The major third of Pythagorean tuning differed from a just major third by an amount known as syntonic comma, with the correct amount of tempering, the syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this system with a slight beating. Pythagorean tuning also had a problem, which meantone temperament does not solve, which is the problem of modulation. A series of 12 just fifths as in Pythagorean tuning does not return to the pitch, but rather differs by a Pythagorean comma. In meantone temperament, this effect is more pronounced. The use of 53 equal temperament provides a solution for the Pythagorean tuning, when building an instrument, this can be very impractical. Well temperament is the given to a variety of different systems of temperament that were employed to solve this problem, in which some keys are more in tune than others
3.
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
4.
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
5.
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
6.
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
7.
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
8.
Just major third
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In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two, the major third spans four semitones, the third three. The major third may be derived from the series as the interval between the fourth and fifth harmonics. The major scale is so named because of the presence of this interval between its tonic and mediant scale degrees, the major chord also takes its name from the presence of this interval built on the chords root. The older concept of a made a dissonantly wide major third with the ratio 81,64. The septimal major third is 9,7, the major third is 14,11. A helpful way to recognize a third is to hum the first two notes of Kumbaya or of When the Saints Go Marching In. A descending major third is heard at the starts of Goodnight, Ladies and Swing Low, in equal temperament three major thirds in a row are equal to an octave. This is sometimes called the circle of thirds, in just intonation, however, three 5,4 major thirds are less than an octave. For example, three 5,4 major thirds from C is B♯, the difference between this just-tuned B♯ and C, like that between G♯ and A♭, is called a diesis, about 41 cents. The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, a diminished fourth is enharmonically equivalent to a major third. For example, B–D♯ is a third, but if the same pitches are spelled B and E♭. B–E♭ occurs in the C harmonic minor scale, the major third is used in guitar tunings. For the standard tuning, only the interval between the 3rd and 2nd strings is a third, each of the intervals between the other pairs of consecutive strings is a perfect fourth. In an alternative tuning, the tuning, each of the intervals are major thirds. Decade, compound just major third Ear training List of meantone intervals Doubling the cube, 21/3 = 3√2
9.
Five-limit tuning
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Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths, powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals, hence, 5-limit tuning is considered a method for obtaining just intonation. If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes extending indefinitely in two directions, however, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical to have the number of pitches in each octave. In that case, the system can also be thought of as an octave-repeating scale of a certain number of pitches per octave. For example, if we have a 5-limit tuning system where the note is C256, then fC =256 Hz. There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a frequency ratio of 5/4, or using fifths one may go up four factors of 3 and down six factors of 2, reaching 81/64. The prominent notes of a scale are tuned so that their frequencies form ratios of relatively small integers. Here the row headed Natural expresses all these ratios using a common list of natural numbers. In other words, the lowest occurrence of this one-octave scale shape within the series is as a subset of 8 of the 25 harmonics found in the octave from harmonics 24 to 48 inclusive. The three major thirds are correct, and three of the thirds are as expected, but D to F is a semiditone or Pythagorean minor third. As a consequence, we obtain a scale in which EGB and ACE are just minor triads, but the DFA triad doesnt have the minor shape or sound we might expect, being. Furthermore, the BDF triad is not the diminished triad that we would get by stacking two 6,5 minor thirds, being instead, Another way to do it is as follows. The three major thirds are still 5,4, and three of the thirds are still 6,5 with the fourth being 32,27. FAC and CEG still form just major triads, but GBD is now, there are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else. It is evidently not possible to get all seven diatonic triads in the configuration for major, for minor and that demonstrates the need for increasing the numbers of pitches to execute the desired harmonies in tune
10.
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
11.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
12.
Harmonic seventh
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The harmonic seventh interval play, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7,4 ratio. This is somewhat narrower than and is, particularly sweet, sweeter in quality than a minor seventh. The harmonic seventh may be derived from the series as the interval between the seventh harmonic and the fourth harmonic. Composer Ben Johnston uses a small 7 as an accidental to indicate a note is lowered 49 cents, thus, in C major, the seventh partial, or harmonic seventh, is notated as ♭ B with 7 written above the flat. Instead, Hagerman and Sundberg found that tuning of major and minor third intervals in barbershop lies between just and equal temperament, the harmonic seventh differs from the augmented sixth by 224/225, or about one-third of a comma. The harmonic seventh note is one third of a semitone flatter than an equal tempered minor seventh. When this flatter seventh is used, the dominant seventh chords need to resolve down a fifth is weak or non-existent and this chord is often used on the tonic and functions as a fully resolved final chord. The twenty-first harmonic is the seventh of the dominant, and would then arise in chains of secondary dominants in styles using harmonic sevenths. The Tonal Phoenix, A Study of Tonal Progression Through the Prime Numbers Three, Five, & Sundberg, J. Fundamental frequency adjustment in barbershop singing. Journal of Research in Singing,4, 3-17
13.
Tritone
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In music theory, the tritone is strictly defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B, according to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale, a tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave, for instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half, in classical music, the tritone is a harmonic and melodic dissonance and is important in the study of musical harmony. The tritone can be used to avoid traditional tonality, Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality. Contrarily, the found in the dominant seventh chord helps establish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music, the condition of having tritones is called tritonia, that of having no tritones is atritonia. A musical scale or chord containing tritones is called tritonic, one without tritones is atritonic, since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each starting from a different pitch and spanning six semitones. According to a complex but widely used naming convention, six of them are classified as augmented fourths, under that convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions. The augmented fourth and diminished fifth are defined as the produced by widening the perfect fourth. They both span six semitones, and they are the inverse of each other, meaning that their sum is equal to one perfect octave. In twelve-tone equal temperament, the most commonly used tuning system, in most other tuning systems, they are not equivalent, and neither is exactly equal to half an octave. Any augmented fourth can be decomposed into three whole tones, for instance, the interval F–B is an augmented fourth and can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. It is not possible to decompose a diminished fifth into three adjacent whole tones, the reason is that a whole tone is a major second, and according to a rule explained elsewhere, the composition of three seconds is always a fourth. To obtain a fifth, it is necessary to add another second, for instance, using the notes of the C major scale, the diminished fifth B–F can be decomposed into the four adjacent intervals B–C, C–D, D–E, and E–F. Using the notes of a scale, B–F may be also decomposed into the four adjacent intervals B–C♯, C♯–D♯, D♯–E♯. Notice that the diminished second is formed by two enharmonically equivalent notes
14.
Septimal tritone
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A septimal tritone is a tritone that involves the factor seven. There are two that are inverses, the lesser septimal tritone is the musical interval with ratio 7,5. The greater septimal tritone, is an interval with ratio 10,7 and they are also known as the sub-fifth and super-fourth, or subminor fifth and supermajor fourth, respectively. The 7,5 interval is equal to a 6,5 minor third plus a 7,6 subminor third. The 10,7 interval is equal to a 5,4 major third plus a 8,7 supermajor second, the difference between these two is the septimal sixth tone Play. 12 equal temperament and 22 equal temperament do not distinguish between these tritones,19 equal temperament does distinguish them but doesnt match them closely,31 equal temperament and 41 equal temperament both distinguish between and closely match them. The lesser septimal tritone is the most consonant tritone when measured by combination tones, harmonic entropy, and period length. Depending on the temperament used, the tritone, defined as three tones, may be identified as either a lesser septimal tritone, a greater septimal tritone, neither
15.
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
16.
Septimal major third
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It is equal to 435 cents, sharper than a just major third by the septimal quarter tone. In 24-TET the septimal major third is approximated by 9 quarter tones, both 24 and 19 equal temperament map the septimal major third and the septimal narrow fourth to the same interval. The septimal major third has a brassy sound which is much less sweet than a pure major third. Together with the root 1,1 and the fifth of 3,2, it makes up the septimal major triad. However, in terms of the series, this is a utonal rather than otonal chord, being an inverted 6,7,9, i. e. a 9⁄9, 9⁄7. The septimal major triad can also be represented by the ratio 14,18,21, the septimal major triad contains an interval of a septimal minor third between its third and fifth. Similarly, the major third is the interval between the third and the fifth of the septimal minor triad. In the early meantone era the interval made its appearance as the major third in remote keys
17.
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
18.
Major third
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In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two, the major third spans four semitones, the third three. The major third may be derived from the series as the interval between the fourth and fifth harmonics. The major scale is so named because of the presence of this interval between its tonic and mediant scale degrees, the major chord also takes its name from the presence of this interval built on the chords root. The older concept of a made a dissonantly wide major third with the ratio 81,64. The septimal major third is 9,7, the major third is 14,11. A helpful way to recognize a third is to hum the first two notes of Kumbaya or of When the Saints Go Marching In. A descending major third is heard at the starts of Goodnight, Ladies and Swing Low, in equal temperament three major thirds in a row are equal to an octave. This is sometimes called the circle of thirds, in just intonation, however, three 5,4 major thirds are less than an octave. For example, three 5,4 major thirds from C is B♯, the difference between this just-tuned B♯ and C, like that between G♯ and A♭, is called a diesis, about 41 cents. The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, a diminished fourth is enharmonically equivalent to a major third. For example, B–D♯ is a third, but if the same pitches are spelled B and E♭. B–E♭ occurs in the C harmonic minor scale, the major third is used in guitar tunings. For the standard tuning, only the interval between the 3rd and 2nd strings is a third, each of the intervals between the other pairs of consecutive strings is a perfect fourth. In an alternative tuning, the tuning, each of the intervals are major thirds. Decade, compound just major third Ear training List of meantone intervals Doubling the cube, 21/3 = 3√2
19.
Neutral third
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A neutral third is a musical interval wider than a minor third play but narrower than a major third play, named by Jan Pieter Land in 1880, the name has been taken over by Alois Hába. Three distinct intervals may be termed neutral thirds, The undecimal neutral third has a ratio of 11,9 between the frequencies of the two tones, or about 347.41 cents play. A tridecimal neutral third play has a ratio of 16,13 between the frequencies of the two tones, or about 359.47 cents and this is the largest neutral third, and occurs infrequently in music, as little music utilizes the 13th harmonic. An equal-tempered neutral third play is characterized by a difference in 350 cents between the two tones, a wider than the 11,9 ratio, and exactly half of an equal-tempered perfect fifth. These intervals are all within about 12 cents and are difficult for most people to distinguish by ear, neutral thirds are roughly a quarter tone sharp from 12 equal temperament minor thirds and a quarter tone flat from 12-ET major thirds. In just intonation, as well as in such as 31-ET, 41-ET, or 72-ET, which more closely approximate just intonation. A neutral third can be formed by stacking a neutral second together with a whole tone, zalzals wosta, a neutral third of 354.55 cents, may be constructed through the addition of a whole tone and a three quarter tone, 9/8 X 12/11 = 27/22. Based on its positioning in the series, the undecimal neutral third implies a root one whole tone below the lower of the two notes. A triad formed by two thirds is neither major nor minor, thus the neutral thirds triad is ambiguous. While it is not found in twelve tone equal temperament it is found in such as the quarter tone scale Play. Infants experiment with singing, and a few studies of individual infants singing found that neutral thirds regularly arise in their improvisations, the neutral third has been used by a number of modern composers, including Charles Ives, James Tenney, and Gayle Young. The equal-tempered neutral third may be found in the tone scale. Undecimal neutral thirds appear in traditional Georgian music, neutral thirds are also found in American folk music. Blue notes on the note of a scale can be seen as a variant of a neutral third with the tonic. Similarly the blue note on the note of the scale can be seen as a neutral third with the dominant. Unlike most classical music, blue notes do not have exact values, two steps of seven-tone equal temperament form an interval of 342.8571 cents, which is within 5 cents of 347.4079 for the undecimal neutral third. This is an equal temperament in reasonably common use, at least in the form of seven equal. Close approximations to the neutral third appear in 53-ET and 72-ET
20.
Minor third
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In the music theory of Western culture, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the third as encompassing three staff positions. The minor third is one of two commonly occurring thirds and it is called minor because it is the smaller of the two, the major third spans an additional semitone. For example, the interval from A to C is a minor third, diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically, notable examples of ascending minor thirds include the opening two notes of Greensleeves and of Light My Fire. The minor third may be derived from the series as the interval between the fifth and sixth harmonics, or from the 19th harmonic. The minor third is used to express sadness in music. It is also a quartal tertian interval, as opposed to the major thirds quintality, the minor third is also obtainable in reference to a fundamental note from the undertone series, while the major third is obtainable as such from the overtone series. The minor scale is so named because of the presence of this interval between its tonic and mediant scale degrees, minor chords too take their name from the presence of this interval built on the chords root. A minor third, in just intonation, corresponds to a ratio of 6,5 or 315.64 cents. In an equal tempered tuning, a third is equal to three semitones, a ratio of 21/4,1, or 300 cents,15.64 cents narrower than the 6,5 ratio. If a minor third is tuned in accordance with the fundamental of the series, the result is a ratio of 19,16. The 12-TET minor third more closely approximates the 19-limit minor third 16,19 Play with only 2.49 cents error. Other pitch ratios are given related names, the minor third with ratio 7,6. The minor third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, instruments in A – most commonly the A clarinet, sound a minor third lower than the written pitch. In music theory, a semiditone is the interval 32,27 and it is the minor third in Pythagorean tuning. The 32,27 Pythagorean minor third arises in the C major scale between D and F, Play It can be thought of as two octaves minus three justly tuned fifths. It is narrower than a justly tuned minor third by a syntonic comma, musical tuning List of meantone intervals Pythagorean interval
21.
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
22.
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
23.
Major second
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In Western music theory, a major second is a second spanning two semitones. A second is an interval encompassing two adjacent staff positions. For example, the interval from C to D is a major second, diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones. The major second is the interval that occurs between the first and second degrees of a scale, the tonic and the supertonic. On a musical keyboard, a second is the interval between two keys separated by one key, counting white and black keys alike. On a guitar string, it is the interval separated by two frets, in moveable-do solfège, it is the interval between do and re. It is considered a step, as opposed to larger intervals called skips. Intervals composed of two semitones, such as the second and the diminished third, are also called tones, whole tones. One source says step is chiefly US. The preferred usage has been argued since the 19th century, Mr. M. in teaching the Diatonic scale calls a tone a step, and a semitone a half step, now, who ever heard of a step in music, or in sound. The largest ones are called major tones or greater tones, the smallest are called minor tones or lesser tones and their size differs by exactly one syntonic comma. Some equal temperaments, such as 15-ET and 22-ET, also distinguish between a greater and a lesser tone, the major second was historically considered one of the most dissonant intervals of the diatonic scale, although much 20th-century music saw it reimagined as a consonance. It is common in different musical systems, including Arabic music, Turkish music and music of the Balkans. It occurs in both diatonic and pentatonic scales, listen to a major second in equal temperament. Here, middle C is followed by D, which is a tone 200 cents sharper than C, the difference in size between a major tone and a minor tone is equal to one syntonic comma. The major tone is the 9,8 interval play, and it is an approximation thereof in other tuning systems, the major tone may be derived from the harmonic series as the interval between the eighth and ninth harmonics. The minor tone may be derived from the series as the interval between the ninth and tenth harmonics. The 10,9 minor tone arises in the C major scale between D and e and G and A, and is a sharper dissonance than 9,8
24.
Neutral interval
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In music theory, a neutral interval is an interval that is neither a major nor minor, but instead in between. For example, in equal temperament, a third is 400 cents, a minor third is 300 cents. A neutral interval inverts to a neutral interval, for example, the inverse of a neutral third is a neutral sixth. Roughly, neutral intervals are a tone sharp from minor intervals. In just intonation, as well as in such as 31-ET, 41-ET, or 72-ET, which more closely approximate just intonation. The lesser undecimal neutral second may be derived from the series as the interval between the eleventh and twelfth harmonics. The greater undecimal neutral second may be derived from the series as the interval between the tenth and eleventh harmonics. The equal-tempered neutral second may be found in the tone scale. Because neutral seconds are essentially a semitone plus a quarter-tone, they may be considered three-quarter tones, approximations to the 12,11 and 11,10 neutral seconds can be found in a number of equally tempered tuning systems. 11,10 is very closely matched by 22-ET, whereas 12,11 is matched by 24-ET, 31-ET and 41-ET, 72-ET matches both intervals closely and is also the smallest widely used equal temperament that uniquely matches both intervals. Tuning systems that temper out the comma of 121,120 do not distinguish between the two intervals, 17-ET has a neutral second between 12,11 and 13,12, and a neutral third between 16,13 and 11,9. A neutral seventh is an interval wider than a minor seventh play. Four distinct intervals may be termed neutral sevenths, A septimal neutral seventh play has a ratio of 64,35 or about 1045 cents, the just undecimal neutral seventh has a ratio of 11,6 between the frequencies of the two tones, or about 1049 cents play. Alternately,13,7 or about 1071.7 cents, a tridecimal neutral seventh play has a ratio of 24,13 between the frequencies of the two tones, or about 1061 cents. This is the largest neutral seventh, and occurs infrequently in music, an equal-tempered neutral seventh play is characterized by a difference in 1050 cents between the two tones, a hair larger than the 11,6 ratio, and exactly half of an equal-tempered major thirteenth. These intervals are all within about 12 cents of each other and are difficult for most people to distinguish, a neutral seventh can be formed by stacking a neutral third together with a perfect fifth. Based on its positioning in the series, the undecimal neutral third implies a root one perfect fifth below the lower of the two notes. Major fourth and minor fifth Subminor and supermajor List of pitch intervals Microtonal music
25.
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
26.
Augmented unison
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In other words, it is a unison where one note has been altered by a half-step, such as B♭ and B♮ or C♮ and C♯. The interval is described as a chromatic semitone. The term, in its French form unisson superflu, appears to have coined by Jean-Philippe Rameau in 1722. Historically, this interval, like the tritone, is described as being mi contra fa, in 12-tone equal temperament, it is the enharmonic equivalent of a diatonic semitone or minor second, although in other tunings the diatonic semitone is a different interval. The term diminished unison or diminished prime is found occasionally. It is found once in Rameaus writings, for example, as well as subsequent French, German, other sources reject the possibility or utility of the diminished unison on the grounds that any alteration to the unison increases its size, thus augmenting rather than diminishing it. The term is sometimes justified as an interval, and also in terms of violin double-stopping technique on analogy to parallel intervals found on other strings. Some theoreticians make a distinction for this form of the unison, stating it is only valid as a melodic interval. False relation List of musical intervals List of pitch intervals
27.
Diesis
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For instance, an octave spans from C to C, and three justly tuned major thirds span from C to B♯. The difference between C-C and C-B♯ is the diesis, notice that this coincides with the interval between B♯ and C, also called a diminished second. The above-mentioned 128,125 comma is also known as the lesser diesis, as shown in the picture, in the quarter-comma meantone tuning system, the diminished second coincides with the diesis. In any tuning system, the deviation of an octave from three major thirds, however large that is, is referred to as a diminished second. The diminished second is an interval between pairs of enharmonically equivalent notes, for instance the interval between E and F♭, as mentioned above, the term diesis most commonly refers to the diminished second in quarter-comma meantone temperament. Less frequently and less strictly, the term is also used to refer to a diminished second of any size. In third-comma meantone, the second is typically denoted as a greater diesis. In quarter-comma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the value of 128,125. Notice that 128,125 is larger than a unison and this means that, for instance, C is sharper than B♯. In eleventh-comma meantone, the second is within 1/716 of a cent above unison. The word diesis has also used to describe a large number of intervals, of varying sizes. Philolaus used it to describe the interval now usually called a limma, other theorists have used it for various other intervals.57 cents. Being larger, this diesis was termed greater while the 128,125 diesis was termed lesser, the small diesis Play is 3125,3072 or approximately 29.61 cents. The septimal diesis is an interval with the ratio of 49,48 play and it is about 35.70 cents wide. The undecimal diesis is equal to 45,44 or about 38.91 cents, closely approximated by 31 equal temperaments 38.71 cent interval
28.
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
29.
Kleisma
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In music theory and tuning, the kleisma, or semicomma majeur, is a minute and barely perceptible comma type interval important to musical temperaments. It is the difference between six justly tuned minor thirds and one justly tuned tritave or perfect twelfth and it is equal to a frequency ratio of 15625/15552 = 2−6 3−556, or approximately 8.1 cents. It can be defined as the difference between five justly tuned minor thirds and one justly tuned major tenth. The interval was named by Shohé Tanaka after the Greek for closure and it is also tempered out by 19 equal temperament and 72 equal temperament, but it is not tempered out in 12 equal temperament. Namely, in 12 equal temperament the difference between six minor thirds and one perfect twelfth is not a comma, but a semitone, the same is true for the difference between five minor thirds and one major tenth. The interval was described but not used by Rameau in 1726
30.
Regular temperament
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Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, when only two generators are needed, with one of them the octave, this is called linear temperament. The best-known example of a linear temperaments is meantone temperament, where the generating intervals are given in terms of a slightly flattened fifth. Other linear temperaments include the schismatic temperament of Hermann von Helmholtz, if the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. In mathematical terminology, the products of these define a free abelian group. The number of independent generators is the rank of an abelian group, the rank-one tuning systems are equal temperaments, all of which can be spanned with only a single generator. A rank-two temperament has two generators, hence, meantone is a rank-2 temperament. In studying regular temperaments, it can be useful to regard the temperament as having a map from p-limit just intonation to the set of tempered intervals. To properly classify a temperaments dimensionality one must determine how many of the generators are independent. Another way of considering this problem is that the rank of a temperament should be the rank of its image under this map, other methods of linear and multilinear algebra can be applied to the map. For instance, a maps kernel consists of p-limit intervals called commas, a. Milne, W. A. Sethares, and J. Enharmonic instruments and music, 1470-1900
31.
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
32.
Schismatic temperament
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A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805,32768 to a unison. It is also called the temperament, Helmholtz temperament, or quasi-Pythagorean temperament. In Pythagorean tuning all notes are tuned as a number of perfect fifths, the major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E+, to differ from the just major third, E♮, C — G — D — A+ — E+ Elliss skhismic temperament instead uses the note eight fifths below C, F♭--, the Pythagorean diminished fourth or schismatic major third. Though spelled incorrectly for a third, this note is only 1.95 cents flat of E♮. As Ellis puts it, the Fifths should be perfect and the Skhisma should be disregarded. E+ ≈ F♭-- F♭-- — C♭-- — G♭-- — D♭-- — A♭- — E♭- — B♭- — F — C In his eighth-schisma Helmholtzian temperament the note eight fifths below C is also used as the third above C. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or tempered, thus the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, the major Thirds are taken perfect, E♮ ≈ F♭-- E♮ — ≈ C♭-- — ≈ G♭-- — ≈ D♭-- — ≈ A♭- — ≈ E♭- — ≈ B♭- — ≈ F — C Compare Pythagorean vs. Skhismic. In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much more accurate perfect fifths. This places them well outside the span of a diatonic scale. Various equal temperaments lead to schismatic tunings which can be described in the same terms, dividing the octave by 53 provides an approximately 1/29-schisma temperament, by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named,171, produces very accurate septimal intervals, the −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7,4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7,4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning, historically significant is the eighth-schisma tuning of Hermann von Helmholtz and Norwegian composer Eivind Groven. Helmholtz had a special Physharmonica with 24 tones to the octave, a 1/9-schisma tuning has also been proposed by Eduardo Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning. Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the medieval period. This was not temperament but merely 12-tone Pythagorean tuning, schismic, Tonalsoft - Encyclopedia of microtonal music theory
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Septimal kleisma
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In music, the ratio 225/224 is called the septimal kleisma. It is a minute comma type interval of approximately 7.7 cents, factoring it into primes gives 2−53252 7−1, which can be rewritten 2−12. That says that it is the amount that two thirds of 5/4 and a septimal major third, or supermajor third, of 9/7 exceeds the octave. The existence of such a chord, which might be termed the septimal kleisma augmented triad, is a significant feature of a tuning system, the septimal kleisma can also be viewed as the difference between the diatonic semitone and the septimal diatonic semitone
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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
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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
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Meantone temperament
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Meantone temperament is a musical temperament, which is a system of musical tuning. Quarter-comma meantone is the best known type of meantone temperament, Meantone can receive the following equivalent definitions, The meantone is the mean between the major whole tone and the minor whole tone, i. e. the geometric mean of 9,8 and 10,9. The meantone is the mean of the just major third, i. e. the square root of 5,4, all meantone temperaments are linear temperaments, distinguished by the width of its generator in cents, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a portion of this tuning continuum. In Figure 1, the tuning ranges of 5-limit, 7-limit, and 11-limit tunings are shown. This last ratio was termed R by American composer, pianist and theoretician Easley Blackwood, if we multiply by 1200, we have the size of fifth in cents. In these terms, some historically notable meantone tunings are listed below, the relationship between the first two columns is exact, while that between them and the third is closely approximate. Equal temperaments useful as meantone tunings include 19-ET, 50-ET, 31-ET, 43-ET, the farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by tempering the timbre to match the tuning. A whole number of just perfect fifths will never add up to a number of octaves. If a stacked-up whole number of perfect fifths is to close with the octave, then one of the fifths must have a different width than all of the others. For example, to make the 12-note chromatic scale in Pythagorean tuning close at the octave, one fifth must be out of tune by the Pythagorean comma, wolf intervals are an artifact of keyboard design. This can be shown most easily using a keyboard, such as that shown in Figure 2. On an isomorphic keyboard, any musical interval has the same shape wherever it appears. On the keyboard shown in Figure 2, from any given note, there are no wolf intervals within the note-span of this keyboard. The problem is at the edge, on the note E♯, the note thats a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown. Because there is no B♯ button, when playing an E♯ power chord, one must choose some other note, such as C, even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes. When the perfect fifth is exactly 700 cents wide then the tuning is identical to the familiar 12-tone equal temperament and this appears in the table above when R =2,1. Because of the forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments
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Dominant seventh chord
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In music theory, a dominant seventh chord, or major minor seventh chord, is a chord composed of a root, major third, perfect fifth, and minor seventh. It can be viewed as a major triad with an additional minor seventh. When using popular-music symbols, it is denoted by adding a superscript 7 after the letter designating the chord root, the dominant seventh is found almost as often as the dominant triad. In Roman numerals it is represented as V7, the chord can be represented by the integer notation. Of all the chords, perhaps the most important is the dominant seventh. It was the first seventh chord to appear regularly in classical music, the name comes from the fact that it occurs naturally in the seventh chord built upon the dominant of a given major diatonic scale. Take for example the C major scale, The note G is the dominant degree of C major—its fifth note, the note F is a minor seventh from G, and it is also called the dominant seventh with respect to G. The function of the dominant seventh chord is to drive to or resolve to the note or chord. The demand of the V7 for resolution is, to our ears, the dominant seventh is, in fact, the central propulsive force in our music, it is unambiguous and unequivocal. This basic dominant seventh chord is useful to composers because it both a major triad and the interval of a tritone. The major triad confers a very strong sound, the tritone is created by the co-occurrence of the third degree and seventh degree. In a diatonic context, the third of the chord is the leading-tone of the scale, the seventh of the chord acts as an upper leading-tone to the third of the scale. Because of this usage, it also quickly became an easy way to trick the listeners ear with a deceptive cadence. The dominant seventh may work as part of a progression, preceded by the supertonic. S. A. and Loggins. Chuck Berrys Rock And Roll Music uses the dominant seventh on I, IV, the dominant seventh is enharmonically equivalent to the German sixth, causing the chords to be spelled enharmonically, for example the German sixth G♭–B♭–D♭–E and the dominant seventh F♯–A♯–C♯–E. The dominant seventh is used to approximate a Harmonic seventh chord. Others include 20,25,30,36 Play, found on I, renaissance composers decided in terms of intervals rather than chords, however, certain dissonant sonorities suggest that the dominant seventh chord occurred with some frequency. Monteverdi and other early baroque composers begin to treat the V7 as a chord as part of the introduction of functional harmony, the V7 was in constant use during the classical period, with similar treatment to that of the baroque
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Otonality and Utonality
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Otonality and utonality are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone, respectively. An otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, for example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore composed of members of a harmonic series, similarly, the ratios of a utonality share the same numerator. 7/4, 7/5, 7/6, and 1/1 form a utonality, every utonality is therefore composed of members of a subharmonic series. An otonality corresponds to a series of frequencies, or lengths of a vibrating string. Brass instruments naturally produce otonalities, and indeed otonalities are inherent in the harmonics of a fundamental tone. Tuvan Khoomei singers produce otonalities with their vocal tracts, Utonality is the opposite, corresponding to a subharmonic series of frequencies, or an arithmetic series of wavelengths. The arithmetical proportion may be considered as a demonstration of utonality, microtonalists define a just intonation chord as otonal if its odd limit increases on being inverted, utonal if its odd limit decreases, and ambitonal if its odd limit is unchanged. The chord is not inverted in the sense, where C E G becomes E G C or G C E. Instead. A chords odd limit is the largest odd limit of each of the numbers in the chords extended ratio, for example, the major triad 4,5,6 has an odd limit of 5. Its inverse 10,12,15 has an odd limit of 15, Partch said that his 1931 coinage of otonality and utonality was, hastened, by having read Henry Cowells discussion of undertones in New Musical Resources. The 5-limit otonality is simply a just major chord, and the 5-limit utonality is a just minor chord, thus otonality and utonality can be viewed as extensions of major and minor tonality respectively. This chord might be, for example, A♭-C-E♭-G♭ Play, standing alone, it has something of the sound of a dominant seventh, but considerably less dissonant. Utonal chords, while containing the same dyads and roughness as otonal chords, numerary nexus Scale of harmonics Tonality flux Otonality and ADO system at 96-EDO Utonality and EDL system at 96-EDO
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Regular diatonic tuning
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In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator. In the ordinary diatonic scales the Ts here are tones and the Ss are semitones which are half, but in the more general regular diatonic tunings, the two steps can be of any relation within the range between T=171.43 and T=240 cents. Note that regular diatonic tunings are not limited to the notes of the scale which defines them. As the semitones get larger, eventually the steps are all the size. Regular here is understood in the sense of a mapping from Pythagorean diatonic such that all the relationships are preserved. These scales however are not included as regular diatonic tunings, all regular diatonic tunings are also Linear temperaments, i. e. Regular temperaments with two generators, the octave and the tempered fifth. One can use the fourth as an alternative generator. Another moment of symmetry with two interval sizes. g, from E to F between notes five steps apart in the cycle. Here, the seven equal system is the limit as the chromatic semitone tends to zero, however, his range of recognizability is more restrictive than regular diatonic tuning. For instance, he requires the diatonic semitone to be at least 25 cents in size and this includes 1/3 comma meantone - achieves pure minor thirds 6/5, fifth is 694.786 cents. So for instance, a 1/8 schizma temperament will achieve a pure 8/5 in a chain of eight fifths. 53 equal temperament achieves an approximation to Schismatic temperament. At around 703. 4-705.0 cents, with fifths mildly tempered in the wide direction, at 705.882 cents, or tempered in the wide direction by 3.929 cents, the result is the diatonic scale in 17 tone equal. Beyond this point, the major and minor thirds approximate simple ratios of numbers with prime factors 2-3-7, such as the 9/7 or septimal major third. At the same time, the regular tones more and more closely approximate a large 8/7 tone and this septimal range extends out to around 711.111 cents or 27-ed2, or a bit further. This combination is necessary and sufficient to define a set of relationships among tonal intervals that is constant across the syntonic temperaments tuning range. Hence, it defines a constant mapping -- all across the tuning continuum -- between the notes at these tonal intervals, and the corresponding partials of a pseudo-harmonic timbre. Hence, the relationship between the temperament and its note-aligned timbres can be seen as a generalization of the special relationship between Just Intonation and the Harmonic Series
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Limit (music)
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In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony, roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant will the intervals of the tuning be perceived. A scale belonging to a prime limit has a distinctive hue that makes it aurally distinguishable from scales with other limits. Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher harmonics in its constructs, in medieval music, only chords made of octaves and perfect fifths were considered consonant. In the West, triadic harmony arose around the time of the Renaissance, the major and minor thirds of these triads invoke relationships among the first 5 harmonics. Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music, in conventional music theory pedagogy, these seventh chords are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5, for example, the dominant 7th chord in 12-ET approximates 4,5,6,7, while the major 7th chord approximates 8,10,12,15. In just intonation, intervals between pitches are drawn from the rational numbers, since Partch, two distinct formulations of the limit concept have emerged, odd limit and prime limit. Odd limit and prime limit n do not include the same even when n is an odd prime. For a positive odd number n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than n. In Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partchs theoretical prediction of the dissonance of intervals are very similar to those of theorists including Hermann von Helmholtz, William Sethares. An identity is each of the odd numbers below and including the limit in a tuning, for example, the identities included in 5-limit tuning are 1,3, and 5. The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number. Partch defines identity as one of the correlatives, major or minor, in a tonality, one of the odd-number ingredients, odentity and udentity are, short for Over-Identity, and, Under-Identity, respectively. An udentity is an identity of an utonality, for a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with numerator and denominator both n-smooth, given a prime number p, the subset of Q + consisting of those rational numbers x whose prime factorization has the form x = p 1 α1 p 2 α2. P r ≤ p forms a subgroup of and we say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup
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Music of Turkey
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Much of its modern popular music can trace its roots to the emergence in the early 1930s drive for Westernization. With the assimilation of immigrants from various regions the diversity of musical genres, Turkey has also seen documented folk music and recorded popular music produced in the ethnic styles of Greek, Armenian, Albanian, Polish, Azeri and Jewish communities, among others. Many Turkish cities and towns have vibrant local music scenes which, in turn, despite this however, western-style pop music lost popularity to arabesque in the late 70s and 80s, with even its greatest proponents Ajda Pekkan and Sezen Aksu falling in status. It became popular again by the beginning of the 1990s, as a result of an opening economy, with the support of Aksu, the resurging popularity of pop music gave rise to several international Turkish pop stars such as Tarkan and Sertab Erener. Ottoman court music has a large and varied system of modes or scales known as makams, a number of notation systems were used for transcribing classical music, the most dominant being the Hamparsum notation in use until the gradual introduction of western notation. Turkish classical music is taught in conservatories and social clubs, the most respected of which is Istanbuls Üsküdar Musiki Cemiyeti, a full fasıl concert would include four different instrumental forms and three vocal forms, including a light classical song, şarkı. A strictly classical fasıl remains is the same throughout, from the introductory taksim. Composers and Performers Other famous proponents of this genre include Sufi Dede Efendi, Prince Cantemir, Baba Hamparsum, Kemani Tatyos Efendi, Sultan Selim III, the most popular modern Turkish classical singer is Münir Nurettin Selçuk, who was the first to establish a lead singer position. Other performers include Bülent Ersoy, Zeki Müren, Müzeyyen Senar, from the makams of the royal courts to the melodies of the royal harems, a type of dance music emerged that was different from the oyun havası of fasıl music. In the Ottoman Empire, the harem was that part of a house set apart for the women of the family and it was a place in which non-family males were not allowed. Eunuchs guarded the sultans harems, which were large, including several hundred women who were wives and concubines. There, female dancers and musicians entertained the women living in the harem, belly dance was performed by women for women. This female dancer, known as a rakkase, hardly ever appeared in public and this type of harem music was taken out of the sultans private living quarters and to the public by male street entertainers and hired dancers of the Ottoman Empire, the male rakkas. These dancers performed publicly for wedding celebrations, feasts, festivals, modern oriental dance in Turkey is derived from this tradition of the Ottoman rakkas. However, Çiftetelli is now a form of music, with names of songs that describe their local origins. Dancers are also known for their use of finger cymbals as instruments. Romani are known throughout Turkey for their musicianship and their urban music brought echoes of classical Turkish music to the public via the meyhane or taverna. This type of music with food and alcoholic beverages is often associated with the underclass of Turkish society