1.
Whole tone scale
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In music, a whole tone scale is a scale in which each note is separated from its neighbours by the interval of a whole tone. This effect is especially emphasised by the fact that triads built on such scale tones are augmented, indeed, one can play all six tones of a whole tone scale simply with two augmented triads whose roots are a major second apart. Since they are symmetrical, whole tone scales do not give an impression of the tonic or tonality. The composer Olivier Messiaen called the tone scale his first mode of limited transposition. The composer and music theorist George Perle calls the whole tone scale interval cycle 2, since there are only two possible whole tone scale positions, it is either C20 or C21. For this reason, the tone scale is also maximally even. Due to this symmetry, the hexachord consisting of the scale is not distinct under inversion or more than one transposition. Use of the whole tone scale can be traced at least as far back as Johann Sebastian Bach. The concluding chorale movement of his cantata O Ewigkeit du Donnerwort BWV60, opens with four notes from the tone scale, Mozart also used the scale in his Musical Joke, for strings. Further examples can be found in the works of Rimsky-Korsakov, the sea kings music in Sadko and also in Scheherezade, colles names as the childhood of the whole-tone scale the music of Berlioz and Schubert in France and then Russians Glinka and Dargomyzhsky. The sense of mystery and ambiguity here even extends to the title of the piece, though the whole-tone scale is prominent in much of his music after 1905 when he encountered Debussy, it serves simply to fit the motifs over augmented chords. The same motifs return from the whole-tone to the scale without emphasizing the contrast. The first of Alban Bergs Seven Early Songs opens with a whole-tone passage both in the accompaniment and in the vocal line that enters a bar later. Berg also quotes the Bach chorale setting referred to above in his Violin Concerto. The last four notes of the 12-tone row Berg used are B, C♯, E♭ and F, an early instance of the use of the scale in jazz writing can be found in Don Redman’s “Chant of the Weed”. In 1958, Gil Evans recorded an arrangement that gives striking coloration to passages featuring whole-tone harmonies, however, these are only the most overt examples of the use of this scale in jazz. Art Tatum and Thelonious Monk are two pianists who used the tone scale extensively and creatively. Thelonious Monks Four in One and Trinkle-Tinkle are fine examples of this, a prominent example of the whole tone scale that made its way into pop music are bars 3 and 4 of the opening of Stevie Wonders 1972 song You are the Sunshine of my Life
2.
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
3.
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♭
4.
Inversion (music)
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In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, the concept of inversion also plays a role in musical set theory. An interval is inverted by raising or lowering either of the notes using displacement of the octave so that retain their names. Under inversion, perfect intervals remain perfect, major intervals become minor and vice versa, augmented intervals become diminished, traditional interval names add together to make nine, seconds become sevenths and vice versa, thirds become sixes and vice versa, and fourths become fifths and vice versa. Thus a perfect fourth becomes a fifth, an augmented fourth becomes a diminished fifth, and a simple interval and its inversion. A chords inversion describes the relationship of its bass to the tones in the chord. For instance, a C major triad contains the tones C, E and G, its inversion is determined by which of these tones is the bottom note in the chord. The term inversion is used to categorically refer to the different possibilities. In texts that make this restriction, the position may be used instead to refer to all of the possibilities as a category. A root-position chord Play is sometimes known as the parent chord of its inversions. For example, C is the root of a C major triad and is in the bass when the triad is in position, the 3rd. The following chord is also a C major triad in root position, the rearrangement of the notes above the bass into different octaves and the doubling of notes, is known as voicing. In an inverted chord, the root is not in the bass, the inversions are numbered in the order their bass tones would appear in a closed root position chord. In the second inversion Play, the bass is G—the 5th of the triad—with the root and the 3rd above it, forming a 4th and a 6th above the bass of G, respectively. This inversion can be either consonant or dissonant, and analytical notation sometimes treats it differently depending on the harmonic, for more details, look at Second inversion Third inversions exist only for chords of four or more tones, such as 7th chords. In a third-inversion chord Play, the 7th of the chord is in the bass position, each numeral expresses the interval that results from the voices above it. For example, in root-position triad C-E-G, the intervals above bass note C are a 3rd and a 5th, giving the figures 5-3. If this triad were inverted, the figures 6-3 would apply, due to the intervals of third and sixth appearing above bass note E. Figured bass is similarly applied to 7th chords, certain arbitrary conventions of abbreviation exist in the use of figured bass
5.
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
6.
Interval class
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For example, the interval class between pitch classes 4 and 9 is 5 because 9 −4 =5 is less than 4 −9 = −5 ≡7. See modular arithmetic for more on modulo 12, the largest interval class is 6 since any greater interval n may be reduced to 12 − n. The concept of interval class accounts for octave, enharmonic, consider, for instance, the following passage, (To hear a MIDI realization, click the following,106 KB In the example above, all four labeled pitch-pairs, or dyads, share a common intervallic color. In atonal theory, this similarity is denoted by interval class—ic 5, tonal theory, however, classifies the four intervals differently, interval 1 as perfect fifth,2, perfect twelfth,3, diminished sixth, and 4, perfect fourth. The unordered pitch class interval i may be defined as i = the smaller of i ⟨ a, b ⟩ and i ⟨ b, a ⟩, where i <a, b> is an ordered pitch-class interval. While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris, prefer to use braces, pitch interval Similarity relation Morris, Robert. Class Notes for Atonal Music Theory, ISBN 0-300-04536-0 ISBN 0-300-04537-9 Solomons Set Theory Primer
7.
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
8.
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
9.
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
10.
Quarter tone
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A quarter tone play, is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide as a semitone, which is half a whole tone. Many composers are known for having written music including quarter tones or the quarter-tone scale —proposed by 19th-century music theorists Heinrich Richter in 1823, george Tucker, Ivan Alexandrovich Wyschnegradsky, and Iannis Xenakis. The term quarter tone can refer to a number of different intervals, for example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments. In the quarter tone scale, also called 24 tone equal temperament, the tone is 50 cents, or a frequency ratio of 24√2 or approximately 1.0293. In this scale the quarter tone is the smallest step, a semitone is thus made of two steps, and three steps make a three-quarter tone play or neutral second, half of a minor third. The 8-TET scale is composed of three-quarter tones, in just intonation the quarter tone can be represented by the septimal quarter tone,36,35, or by the undecimal quarter tone,33,32, approximately half the semitone of 16,15 or 25,24. The ratio of 36,35 is only 1.23 cents narrower than a 24-TET quarter tone and this just ratio is also the difference between a minor third and septimal minor third. Quarter tones and intervals close to also occur in a number of other equally tempered tuning systems. 22-TET contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET has an interval of 45.28 cents, 72-TET also has equally tempered quarter-tones, and indeed contains 3 quarter tone scales, since 72 is divisible by 24. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33,32, because many musical instruments manufactured today are designed for the 12-tone scale, not all are usable for playing quarter tones. Sometimes special playing techniques must be used, other instruments can be used to play quarter tones when using audio signal processing effects such as pitch shifting. Pairs of conventional instruments tuned a tone apart can be used to play some quarter tone music. Indeed, quarter-tone pianos have been built, which consist essentially of two stacked one above the other in a single case, one tuned a quarter tone higher than the other. Many Persian dastgah and Arabic maqamat contain intervals of three-quarter tone size, the quarter tone scale may be primarily a theoretical construct in Arabic music. Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, the Japanese multi-instrumentalist and experimental musical instrument builder Yuichi Onoue developed a 24-TET quarter tone tuning on his guitar. Several quarter-tone albums have been recorded by Jute Gyte, a one-man avantgarde black metal band from Missouri, the enharmonic genus of the Greek tetrachord consisted of a ditone or an approximate major third, and a semitone, which was divided into two microtones. Intervals matched particularly closely include the second, neutral third. The septimal minor third and septimal major third are approximated rather poorly, overall, 24-TET can be viewed as matching the 11th and 13th harmonics more closely than the 7th
Play (help·info)). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more details). For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions. Diminished, minor and augmented seconds are notated on adjacent staff positions as well, but consist of a different number of semitones (zero, one, and three).
