1.
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
2.
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
3.
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
4.
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
5.
Interval ratio
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In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3,2,1.5, if the A above middle C is 440 Hz, the perfect fifth above it would be E, at 660 Hz, while the equal tempered E5 is 659.255 Hz. Ratios have a relationship to string length, for example stopping a string at two-thirds its length produces a pitch one. Intervals may be ranked by relative consonance and dissonance, as such ratios with lower integers are generally more consonant than intervals with higher integers. For example,2,1,4,3,9,8,65536,59049, consonance and dissonance may more subtly be defined by limit, wherein the ratios whose limit, which includes its integer multiples, is lower are generally more consonant. For example, the 3-limit 128,81 and the 7-limit 14,9, despite having larger integers 128,81 is less dissonant than 14,9, as according to limit theory. For ease of comparison intervals may also be measured in cents, for example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents. Frequency ratios are used to describe intervals in both Western and non-Western music. When a musical instrument is tuned using a just intonation tuning system, intervals with small-integer ratios are often called just intervals, or pure intervals. To most people, just intervals sound consonant, i. e. pleasant, although the size of equally tuned intervals is typically similar to that of just intervals, in most cases it cannot be expressed by small-integer ratios. For instance, a tempered perfect fifth has a frequency ratio of about 1.4983,1. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems
6.
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
7.
Gioseffo Zarlino
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Gioseffo Zarlino was an Italian music theorist and composer of the Renaissance. He was possibly the most famous European music theorist between Aristoxenus and Rameau, and made a contribution to the theory of counterpoint as well as to musical tuning. Zarlino was born in Chioggia, near Venice and his early education was with the Franciscans, and he later joined the order himself. In 1536 he was a singer at Chioggia Cathedral, and by 1539 he not only became a deacon, in 1540 he was ordained, and in 1541 went to Venice to study with the famous contrapuntist and maestro di cappella of Saint Marks, Adrian Willaert. In 1565, on the resignation of Cipriano de Rore, Zarlino took over the post of maestro di cappella of St. Marks, one of the most prestigious positions in Italy. While he was a prolific composer, and his motets are polished and display a mastery of canonic counterpoint. Zarlino also described the 1/4-comma meantone and 1/3-comma meantone, considering all three temperaments to be usable and these are the precursors to the 50- 31- and 19-tone equal temperaments, respectively. In his Dimostrationi harmoniche of 1571, he revised the numbering of modes to emphasize C, Zarlino was the first to theorize the primacy of triad over interval as a means of structuring harmony. His exposition of just intonation based on proportions within the Senario and 8 is a departure from the previously established Pythagorean diatonic system as passed on by Boethius, zarlinos writings, primarily published by Francesco Franceschi, spread throughout Europe at the end of the 16th century. Zarlinos compositions are more conservative in idiom than those of many of his contemporaries and his madrigals avoid the homophonic textures commonly used by other composers, remaining polyphonic throughout, in the manner of his motets. His works were published between 1549 and 1567, and include 41 motets, mostly for five and six voices and his 10 motets on the Song of Songs used the text of Isidoro Chiaris translation of the Bible. GCD921406 Zarlino, Modulationes sex vocum, Singer Pur, OEHMS CLASSICS873 Article Gioseffo Zarlino, in The New Grove Dictionary of Music and Musicians, ISBN 1-56159-174-2 Gustave Reese, Music in the Renaissance. ISBN 0-393-09530-4 Gioseffo Zarlino, Istituzioni armoniche, tr. Oliver Strunk, in Source Readings in Music History
8.
Francisco de Salinas
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In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma meantone tunings. The 19th-century musicologist Alexander John Ellis maintained that Salinas really meant to characterize 1/6-comma meantone, Salinas was also interested in just intonation, and advocated a 5-limit just intonation scale of 24 notes he called instrumentum perfectum. Blind from the age of eleven, Salinas served as organist to the celebrated Duke of Alba and his own compositions for organ have been lost. The poet Fray Luis de Leon admired Salinas greatly, knew him personally, Salinas, Francisco de, De musica libri septem, Mathias Gastius, Salamanca,1577,1592. Salinas, in New Grove Dictionary of Music and Musicians, Macmillan Publishers, Francisco de Salinas on the Huygens-Fokker Foundation site
9.
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
10.
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
11.
Octave
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In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are written using eight notes. For example, the C major scale is typically written C D E F G A B C, two notes separated by an octave have the same letter name and are of the same pitch class. Three commonly cited examples of melodies featuring the perfect octave as their opening interval are Singin in the Rain, Somewhere Over the Rainbow, the interval between the first and second harmonics of the harmonic series is an octave. The octave has occasionally referred to as a diapason. To emphasize that it is one of the intervals, the octave is designated P8. The octave above or below a note is sometimes abbreviated 8a or 8va, 8va bassa. For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, the ratio of frequencies of two notes an octave apart is therefore 2,1. Further octaves of a note occur at 2n times the frequency of that note, such as 2,4,8,16, etc. and the reciprocal of that series. For example,55 Hz and 440 Hz are one and two away from 110 Hz because they are 1⁄2 and 4 times the frequency, respectively. After the unison, the octave is the simplest interval in music, the human ear tends to hear both notes as being essentially the same, due to closely related harmonics. Notes separated by a ring together, adding a pleasing sound to music. For this reason, notes an octave apart are given the note name in the Western system of music notation—the name of a note an octave above A is also A. The conceptualization of pitch as having two dimensions, pitch height and pitch class, inherently include octave circularity, thus all C♯s, or all 1s, in any octave are part of the same pitch class. Octave equivalency is a part of most advanced cultures, but is far from universal in primitive. The languages in which the oldest extant written documents on tuning are written, leon Crickmore recently proposed that The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week
12.
Staff (music)
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Appropriate music symbols, depending on the intended effect, are placed on the staff according to their corresponding pitch or function. Musical notes are placed by pitch, percussion notes are placed by instrument, the absolute pitch of each line of a non-percussive staff is indicated by the placement of a clef symbol at the appropriate vertical position on the left-hand side of the staff. For example, the treble clef, also known as the G clef, is placed on the second line, the lines and spaces are numbered from bottom to top, the bottom line is the first line and the top line is the fifth line. The musical staff is analogous to a graph of pitch with respect to time. Pitches of notes are given by their position on the staff. A time signature to the right of the clef indicates the relationship between timing counts and note symbols, while bar lines group notes on the staff into measures, staff is more common in American English, stave in British English. The plural is staves in either case, stave is, in fact, the vertical position of the notehead on the staff indicates which note to play, higher-pitched notes are marked higher on the staff. The notehead can be placed with the center of its notehead intersecting a line, notes outside the range of the staff are placed on or between ledger lines—lines the width of the note they need to hold—added above or below the staff. Exactly which staff positions represent which notes is determined by a clef placed at the beginning of the staff, the clef identifies a particular line as a specific note, and all other notes are determined relative to that line. For example, the treble clef puts the G above middle C on the second line, the interval between adjacent staff positions is one step in the diatonic scale. Once fixed by a clef, the represented by the positions on the staff can be modified by the key signature. A clefless staff may be used to represent a set of percussion sounds, a single vertical line drawn to the left of multiple staves creates a system, indicating that the music on all the staves is to be played simultaneously. A bracket is a vertical line joining staves, to show groupings of instruments that function as a unit. A brace is used to join multiple staves that represent a single instrument, such as a piano, organ, harp, or marimba. Sometimes, a bracket is used to show instruments grouped in pairs, such as the first and second oboes, or the first. In some cases, a brace is used for this instead of a bracket. When more than one system appears on a page, often two parallel diagonal strokes are placed on the left side of the score to separate them. Four-part SATB vocal settings, especially in hymnals, use a notation on a two-staff system with soprano and alto voices sharing the upper staff
13.
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
14.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
15.
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
16.
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
17.
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
18.
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
19.
Minor sixth
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In classical music from Western culture, a sixth is a musical interval encompassing six staff positions, and the minor sixth is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two, the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a sixth, as the note F lies eight semitones above A. Diminished and augmented sixths span the same number of staff positions, in equal temperament, the minor sixth is enharmonically equivalent to the augmented fifth. It occurs in first inversion major and dominant seventh chords and second inversion minor chords, the ratios of both major and minor sixths are corresponding numbers of the Fibonacci sequence,5 and 8 for a minor sixth and 3 and 5 for a major. The 11,7 undecimal minor sixth is 782.49 cents, in Pythagorean tuning, the minor sixth is the ratio 128,81, or 792.18 cents. In just intonation, the sixth is classed as a consonance of the 5-limit. Any note will appear in major scales from any of its minor sixth major scale notes. Musical tuning List of meantone intervals Sixth chord Subminor sixth
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.
Minor seventh
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In the music of Western culture, a seventh is a musical interval encompassing seven staff positions, and the minor seventh is one of two commonly occurring sevenths. It is qualified as minor because it is the smaller of the two, the minor seventh spans ten semitones, the major seventh eleven, minor seventh intervals are rarely featured in melodies but occur more often than major sevenths. The best-known example, in due to its frequent use in theory classes, is found between the first two words of the phrase Theres a place for us in the song Somewhere in West Side Story. Another well-known example occurs between the first two notes of the introduction to the theme music from Star Trek, The Original Series theme. The most common occurrence of the seventh is built on the root of the prevailing keys dominant triad. Consonance and dissonance are relative, depending on context, the seventh being defined as a dissonance requiring resolution to a consonance. In just intonation there is both a 16,9 small just minor seventh, also called Pythagorean small minor seventh, an interval close in frequency is the harmonic seventh. Minor seventh chord Musical tuning List of meantone intervals
22.
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
23.
Unison
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In music, unison is two or more musical parts sounding the same pitch or at an octave interval, usually at the same time. Rhythmic patterns which are homorhythmic are also called unison, two pitches that are the same or two that move as one. Unison or perfect unison may refer to the interval formed by a tone and its duplication, for example C–C, as differentiated from the second, C–D, in the unison the two pitches have the ratio of 1,1 or 0 half steps and zero cents. This is because a pair of tones in unison come from different locations or can have different colors, voices with different colors have, as sound waves, different waveforms. These waveforms have the fundamental frequency but differ in the amplitudes of their higher harmonics. The unison is considered the most consonant interval while the near unison is considered the most dissonant, the unison is also the easiest interval to tune. The unison is abbreviated as P1, a point is the beginning of a line, although, it is not itself a line. But a line is not composed of points, since a point has no length, width, or depth that can be extended, or joined to another point. So a unison is only the beginning of consonance or interval, it is neither consonance nor interval, several singers singing a melody together. In orchestral music unison can mean the simultaneous playing of a note by different instruments, either at the pitch, or in a different octave, for example, cello. Typically a section string player plays unison with the rest of the section, thus, in the divisi first violins the outside players might play the top note of the chord, while the inside seated players play the middle note, and the second violins play the bottom note. At the point where the first violins no longer play divisi, when several people sing together, as in a chorus, the simplest way for them to sing is to sing in one voice, in unison. If there is an instrument accompanying them, then the instrument must play the notes being sung by the singers. Otherwise the instrument is considered a voice and there is no unison. If there is no instrument, then the singing is said to be a cappella, Music in which all the notes sung are in unison is called monophonic. From this sense can be derived another, figurative, sense, if people do something in unison it means they do it simultaneously, in tandem. Related terms are univocal and unanimous, monophony could also conceivably include more than one voice which do not sing in unison but whose pitches move in parallel, always maintaining the same interval of an octave. A pair of notes sung one or a multiple of an octave apart are almost in unison, when there are two or more voices singing different notes, this is called part singing
24.
Major sixth
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In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two, the major sixth spans nine semitones. Its smaller counterpart, the sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth and its a sixth because it encompasses six note letter names and six staff positions. Its a major sixth, not a sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of letter names and staff positions. A commonly cited example of a melody featuring the major sixth as its opening is My Bonnie Lies Over the Ocean. The major sixth is one of the consonances of common practice music, along with the unison, octave, perfect fifth, major and minor thirds, minor sixth, in just intonation, the major sixth is classed as a consonance of the 5-limit. Assuming close-position voicings for the examples, the major sixth occurs in a first inversion minor triad, a second inversion major triad. It also occurs in the second and third inversions of a dominant seventh chord, the septimal major sixth is approximated in 53 tone equal temperament by an interval of 41 steps or 928 cents. Many intervals in a various tuning systems qualify to be called major sixth, the following examples are sorted by increasing width. In just intonation, the most common major sixth is the ratio of 5,3. In 12-tone equal temperament, a sixth is equal to nine semitones, exactly 900 cents. The 27,16 Pythagorean major sixth arises in the C Pythagorean major scale between F and D, as well as between C and A, G and E, and D and B. Play The septimal major sixth is approximated in 53-tone equal temperament by an interval of 41 steps, giving a frequency ratio of the root of 2 over 1. Another just major sixth is the 12,7 septimal major sixth or supermajor sixth of approximately 933 cents, the nineteenth subharmonic is a major sixth, A = 32/19 =902.49 cents. Musical tuning list of meantone intervals sixth chord Duckworth, William, in Sound and Light, La Monte Young, Marian Zazeela, edited by William Duckworth and Richard Fleming, p.167. Lewisburg, Bucknell University Press, London and Cranbury, NJ, Associated University Presses
25.
Major seventh
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In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions, and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two, the major seventh spans eleven semitones, its smaller counterpart being the minor seventh, spanning ten semitones. For example, the interval from C to B is a seventh, as the note B lies eleven semitones above C. Diminished and augmented sevenths span the same number of staff positions, the easiest way to locate and identify the major seventh is from the octave rather than the unison, and it is suggested that one sings the octave first. For example, the most commonly cited example of a melody featuring a major seventh is the seventh of the opening to Over the Rainbow. Not many songwriters begin a melody with a major seventh interval, however, two songs provide exceptions to this generalisation, Cole Porters I love you opens with a descending major seventh and Jesse Harriss Dont Know Why, starts with an ascending one. The major seventh occurs most commonly built on the root of major triads, resulting in the type also known as major seventh chord or major-major seventh chord, including I7. Major seven chords add jazziness to a musical passage, alone, a major seventh interval can sound ugly. In 24-tone equal temperament a supermajor seventh, semiaugmented seventh or, the small major seventh is a ratio of 9,5, now identified as a just minor seventh. 35,18, or 1151.23 cents, is the ratio of the septimal semi-diminished octave, the 15,8 just major seventh occurs arises in the extended C major scale between C & B and F & E. Play F & E The major seventh interval is considered one of the most dissonant intervals after its inversion the minor second, for this reason, its melodic use is infrequent in classical music. However, in the genial Gavotte from J. S, another is the closing duet from Verdis Aida, O terra addio. During the early 20th century, the seventh was used increasingly both as a melodic and a harmonic interval, particularly by composers of the Second Viennese School. Anton Weberns Variations for Piano, Op.27, opens with a major seventh, under equal temperament this interval is enharmonically equivalent to a diminished octave. The major seventh chord is very common in jazz, especially cool jazz. The major seventh chord consists of the first, third, fifth and seventh degrees of the major scale, in the key of C, it comprises the notes C E G and B. List of meantone intervals Major seventh chord Minor seventh Musical tuning
26.
Augmented fifth
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In classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a fifth, seven semitones wide. Being augmented, it is considered a dissonant interval and its inversion is the diminished fourth, and its enharmonic equivalent is the minor sixth. This was achieved by raising the seventh degree to match that of the unstable seventh degree of the major mode. A consequence of this was that the interval between the modes already lowered third degree and the newly raised seventh degree, previously a perfect fifth, had now been augmented by a semitone. Another result of practice was the appearance of the first augmented triads, built on the same degree. As music became increasingly chromatic, the fifth was used with correspondingly greater freedom. Near the end of the century the augmented fifth became commonly used in a dominant chord. This would create a dominant chord. The augmented fifth of the chord would then act as a tone to the third of the next chord. This augmented V chord would never precede a minor tonic chord since the fifth of the dominant chord is identical to the third of the tonic chord. In an equal tempered tuning, a fifth is equal to eight semitones. The 25,16 just augmented fifth arises in the C harmonic minor scale between E♭ and B, play The augmented fifth is a context-dependent dissonance. That is, when heard in certain contexts, such as described above. In other contexts, however, the same eight-semitone interval will simply be heard as its consonant enharmonic equivalent, the Pythagorean augmented fifth is the ratio 6561,4096, or about 815.64 cents
27.
Wolf interval
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In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a tuning system, widely used in the sixteenth and seventeenth centuries. More broadly, it is used to refer to similar intervals produced by other tuning systems. In mean-tone systems, this interval is usually from C♯ to A♭ or from G♯ to E♭, the eleven perfect fifths sound almost perfectly consonant. Conversely, the sixth is severely dissonant and seems to howl like a wolf. Since the diminished sixth is meant to be equivalent to a perfect fifth. Besides the above-mentioned quarter comma meantone, other tuning systems may produce severely dissonant diminished sixths. Conversely, in 12-tone equal temperament, which is currently the most commonly used tuning system, by extension, any interval which is perceived as severely dissonant and may be regarded as howling like a wolf may be called a wolf interval. In 12-tone scales, the value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a value of 700 − ε cents, as in quarter-comma meantone and most other meantone temperament tuning systems, the value of ε changes depending on the tuning system. In other tuning systems, eleven fifths may have a size of 700 + ε cents, if 11ε is very large, as in the quarter-comma meantone tuning system, the diminished sixth is regarded as a wolf fifth. In terms of ratios, the product of the fifths must be 128. We likewise find varied tunings for the thirds, three of these diminished fourths form major triads with perfect fifths, but one of them forms a major triad with the diminished sixth. If the diminished sixth is an interval, this triad is called the wolf major triad. Similarly, we obtain nine minor thirds of 300 ± 3ε cents, meantone tunings with slightly flatter fifths produce even closer approximations to the subminor and supermajor thirds and corresponding triads. These thirds therefore hardly deserve the appellation of wolf, and in fact historically have not been given that name, a fifth this flat can also be regarded as howling like a wolf. There are also now eight sharp and four major thirds. Five-limit tuning determines one diminished sixth of size 1024,675, whether this interval should be considered dissonant enough to be called a wolf fifth is a controversial matter
28.
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
29.
Johann Kirnberger
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Johann Philipp Kirnberger was a musician, composer, and music theorist. He was a student of Johann Sebastian Bach, according to Ingeborg Allihn, Kirnberger played a significant role in the intellectual and cultural exchange between Germany and Poland in the mid-18th century. Between 1741 and 1751 Kirnberger lived and worked in Poland for powerful magnates including Lubomirski, Poninski and he spent much time collecting Polish national dances and compiled them in his treatise Die Charaktere der Taenze. He became a violinist at the court of Frederick II of Prussia in 1751 and he was the music director to the Prussian Princess Anna Amalia from 1758 until his death. Bach, and sought to secure the publication of all of Bachs chorale settings, many of Bachs manuscripts have been preserved in Kirnbergers library. He is known primarily for his theoretical work Die Kunst des reinen Satzes in der Musik. The well-tempered tuning systems known as Kirnberger II and Kirnberger III are associated with his name, ein Richtiges Gefuehl von der Natuerlichen Bewegung, Johann Philipp Kirnberger als Sammler von Nationaltaenzen, Oschersleben Michaelstein
30.
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
31.
Septimal diesis
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In music, septimal diesis is an interval with the ratio of 49,48 play, which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone and it may also be the ratio 36,35, or 48.77 cents. Play In 12 equal temperament this interval is not tempered out and this makes the diesis a semitone, about three times its correct size. It is not tempered out however by 22-ET or 31-ET
32.
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
33.
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
34.
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
35.
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
36.
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
37.
Boldface
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In typography, emphasis is the exaggeration of words in a text with a font in a different style from the rest of the text—to emphasize them. It is the equivalent of prosodic stress in speech, the most common methods in Bold fall under the general technique of emphasis through a change or modification of font, italics, boldface and small caps. Other methods include the alteration of letter case and spacing as well as color, the human eye is very receptive to differences in brightness within a text body. Therefore, one can differentiate types of emphasis according to whether the emphasis changes the “blackness” of text. With one or the other of these techniques, words can be highlighted without making them out much from the rest of the text. This was used for marking passages that have a different context, such as words from languages, book titles. By contrast, a bold font weight makes text darker than the surrounding text, for example, printed dictionaries often use boldface for their keywords, and the names of entries can conventionally be marked in bold. Small capitals are used for emphasis, especially for the first line of a section, sometimes accompanied by or instead of a drop cap. If the text body is typeset in a typeface, it is also possible to highlight words by setting them in a sans serif face. It is still using some font superfamilies, which come with matching serif and sans-serif variants. In Japanese typography, due to the legibility of heavier Minchō type. Of these methods, italics, small capitals and capitalisation are oldest, with bold type, the house styles of many publishers in the United States use all caps text for, chapter and section headings, newspaper headlines, publication titles, warning messages, word of important meaning. Capitalization is used less commonly today by British publishers. All-uppercase letters are a form of emphasis where the medium lacks support for boldface, such as old typewriters, plain-text email, SMS. Culturally all-caps text has become an indication of shouting, for example when quoting speech and it was also once often used by American lawyers to indicate important points in a legal text. Another means of emphasis is to increase the spacing between the letters, rather than making them darker, but still achieving a distinction in blackness and this results in an effect reverse to boldface, the emphasized text becomes lighter than its environment. This is often used in typesetting and typewriter manuscripts. On typewriters a full space was used between the letters of a word and also one before and one after the word
38.
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
39.
Enharmonically equivalent
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Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in equal temperament, the notes C♯. Namely, they are the key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony. In other words, if two notes have the pitch but are represented by different letter names and accidentals, they are enharmonic. Enharmonic intervals are intervals with the sound that are spelled differently…, of course. Enharmonic equivalence is peculiar to post-tonal theory, much music since at least the 18th century, however, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent. Some key signatures have an enharmonic equivalent that represents a scale identical in sound, the number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is equivalent to the key of C♭ major with 7 flats. Keys past 7 sharps or flats exist only theoretically and not in practice, the enharmonic keys are six pairs, three major and three minor, B major/C♭ major, G♯ minor/A♭ minor, F♯ major/G♭ major, D♯ minor/E♭ minor, C♯ major/D♭ major and A♯ minor/B♭ minor. There are practically no works composed in keys that require double sharps or double flats in the key signature, in practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings. Enharmonic equivalents can also used to improve the readability of a line of music, for example, a sequence of notes is more easily read as ascending or descending if the noteheads are on different positions on the staff. Doing so may reduce the number of accidentals that must be used. Thus, in the key of B♭ major, the sequence B♭-B♮-B♭ is more easily using the enharmonic spelling C♭ instead of B♮. For example the intervals of a sixth on C, on B♯. The most common enharmonic intervals are the fourth and diminished fifth, or tritone. Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals. In principle, the musical use of the word enharmonic to mean identical tones is correct only in equal temperament. In other tuning systems, however, enharmonic associations can be perceived by listeners, in Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2
40.
Diminished sixth
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In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a sixth, eight semitones wide. Being diminished, it is considered a dissonant interval and its inversion is the augmented third, and its enharmonic equivalent is the perfect fifth. A severely dissonant diminished sixth is observed when the instrument is tuned using a Pythagorean or a meantone temperament tuning system, typically, this is the interval between G♯ and E♭. Since it seems to howl like a wolf, and since it is meant to be the equivalent to a fifth. Notice that a justly tuned fifth is the most consonant interval after the perfect unison and the perfect octave
41.
Diminished third
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In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a third, three semitones wide, and both the intervals from A♯ to C, and from A to C♭ are diminished thirds. Being diminished, it is considered a dissonant interval, in equal temperament a diminished third is enharmonic with the major second, both having a value of 200 cents. However, in tunings with fifths flatter than the 700 cents of equal temperament. In 19 equal temperament it is in fact equivalent to an augmented second. In 31 equal temperament it has a typical value of 232.3 cents. In a twelve-note keyboard tuned in a meantone tuning from E♭ to G♯, in superpythagorean tunings, the diminished third is narrower than the major second. In 22 equal temperament, the third is ~109 cents while the chromatic semitone is ~163 cents. Thus, 22-ET is a system in which both semitones are not in fact semitones, but the third is a semitone. For example, a German sixth chord E♭-G-B♭-C♯-E♭ exhibits a diminished third between C♯ and E♭ which complements the augmented sixth between E♭ and C♯, the just diminished third arises in the extended C major scale between F♯ and A♭, Play and between B and D♭
42.
Augmented second
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For instance, the interval from C to D is a major second, two semitones wide, and the interval from C to D♯ is an augmented second, spanning three semitones. Augmented seconds occur in many scales, most important the harmonic minor and they also occur in the various Gypsy scales. In harmonic minor scales, the second occurs between the sixth and seventh scale degrees. For example, in the scale of A harmonic minor, the notes F, an augmented second is enharmonically equivalent to a minor third in equal temperament, but is not the same interval in other meantone tunings. In any tuning close to quarter-comma meantone it will be close to the 7,6 ratio of the minor third. Hence the distinction is not, as thought and even taught, a purely formal and contextual one. The 75,64 just augmented second arises in the C harmonic minor scale between A♭ and B
43.
Diminished fourth
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In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a fourth, five semitones wide. Being diminished, it is considered a dissonant interval, a diminished fourth is enharmonically equivalent to a major third, that is, it spans the same number of semitones, and they are physically the same pitch in twelve-tone equal temperament. For example, B–D♯ is a third, but if the same pitches are spelled B and E♭, as occurs in the C harmonic minor scale. In other tunings, however, they are not necessarily identical, for example, in 31 equal temperament the diminished fourth is slightly wider than a major third, and is instead the same width as the septimal major third. The Pythagorean diminished fourth, also known as the major third, is closer to the just major third than the Pythagorean major third. The 32,25 just diminished fourth arises in the C harmonic minor scale between B and E♭