1.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b

2.
Second
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The second is the base unit of time in the International System of Units. It is qualitatively defined as the division of the hour by sixty. SI definition of second is the duration of 9192631770 periods of the corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. Seconds may be measured using a mechanical, electrical or an atomic clock, SI prefixes are combined with the word second to denote subdivisions of the second, e. g. the millisecond, the microsecond, and the nanosecond. Though SI prefixes may also be used to form multiples of the such as kilosecond. The second is also the unit of time in other systems of measurement, the centimetre–gram–second, metre–kilogram–second, metre–tonne–second. Absolute zero implies no movement, and therefore zero external radiation effects, the second thus defined is consistent with the ephemeris second, which was based on astronomical measurements. The realization of the second is described briefly in a special publication from the National Institute of Standards and Technology. 1 international second is equal to, 1⁄60 minute 1⁄3,600 hour 1⁄86,400 day 1⁄31,557,600 Julian year 1⁄, more generally, = 1⁄, the Hellenistic astronomers Hipparchus and Ptolemy subdivided the day into sixty parts. They also used an hour, simple fractions of an hour. No sexagesimal unit of the day was used as an independent unit of time. The modern second is subdivided using decimals - although the third remains in some languages. The earliest clocks to display seconds appeared during the last half of the 16th century, the second became accurately measurable with the development of mechanical clocks keeping mean time, as opposed to the apparent time displayed by sundials. The earliest spring-driven timepiece with a hand which marked seconds is an unsigned clock depicting Orpheus in the Fremersdorf collection. During the 3rd quarter of the 16th century, Taqi al-Din built a clock with marks every 1/5 minute, in 1579, Jost Bürgi built a clock for William of Hesse that marked seconds. In 1581, Tycho Brahe redesigned clocks that displayed minutes at his observatory so they also displayed seconds, however, they were not yet accurate enough for seconds. In 1587, Tycho complained that his four clocks disagreed by plus or minus four seconds, in 1670, London clockmaker William Clement added this seconds pendulum to the original pendulum clock of Christiaan Huygens. From 1670 to 1680, Clement made many improvements to his clock and this clock used an anchor escapement mechanism with a seconds pendulum to display seconds in a small subdial

3.
Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale, the smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones and they can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies, for example, any two notes an octave apart have a frequency ratio of 2,1. This means that successive increments of pitch by the same result in an exponential increase of frequency. For this reason, intervals are often measured in cents, a derived from the logarithm of the frequency ratio. In Western music theory, the most common naming scheme for intervals describes two properties of the interval, the quality and number, examples include the minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled, the importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. The size of an interval can be represented using two alternative and equivalently valid methods, each appropriate to a different context, frequency ratios or cents, the size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, Intervals with small-integer ratios are often called just intervals, or pure intervals. Most commonly, however, musical instruments are tuned using a different tuning system. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, for instance, an equal-tempered fifth has a frequency ratio of 2 7⁄12,1, approximately equal to 1.498,1, or 2.997,2. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems, the standard system for comparing interval sizes is with cents. The cent is a unit of measurement. If frequency is expressed in a scale, and along that scale the distance between a given frequency and its double is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament, a system in which all semitones have the same size. Hence, in 12-TET the cent can be defined as one hundredth of a semitone

4.
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

5.
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

6.
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