1.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
2.
Music
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Music is an art form and cultural activity whose medium is sound organized in time. The common elements of music are pitch, rhythm, dynamics, different styles or types of music may emphasize, de-emphasize or omit some of these elements. The word derives from Greek μουσική, Ancient Greek and Indian philosophers defined music as tones ordered horizontally as melodies and vertically as harmonies. Common sayings such as the harmony of the spheres and it is music to my ears point to the notion that music is often ordered and pleasant to listen to. However, 20th-century composer John Cage thought that any sound can be music, saying, for example, There is no noise, the creation, performance, significance, and even the definition of music vary according to culture and social context. There are many types of music, including music, traditional music, art music, music written for religious ceremonies. For example, it can be hard to draw the line between some early 1980s hard rock and heavy metal, within the arts, music may be classified as a performing art, a fine art or as an auditory art. People may make music as a hobby, like a teen playing cello in a youth orchestra, the word derives from Greek μουσική. According to the Online Etymological Dictionary, the music is derived from mid-13c. Musike, from Old French musique and directly from Latin musica the art of music and this is derived from the. Greek mousike of the Muses, from fem. of mousikos pertaining to the Muses, from Mousa Muse. In classical Greece, any art in which the Muses presided, Music is composed and performed for many purposes, ranging from aesthetic pleasure, religious or ceremonial purposes, or as an entertainment product for the marketplace. With the advent of recording, records of popular songs. Some music lovers create mix tapes of their songs, which serve as a self-portrait. An environment consisting solely of what is most ardently loved, amateur musicians can compose or perform music for their own pleasure, and derive their income elsewhere. Professional musicians sometimes work as freelancers or session musicians, seeking contracts and engagements in a variety of settings, There are often many links between amateur and professional musicians. Beginning amateur musicians take lessons with professional musicians, in community settings, advanced amateur musicians perform with professional musicians in a variety of ensembles such as community concert bands and community orchestras. However, there are many cases where a live performance in front of an audience is also recorded and distributed. Live concert recordings are popular in classical music and in popular music forms such as rock, where illegally taped live concerts are prized by music lovers
3.
Musical notation
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Types and methods of notation have varied between cultures and throughout history, and much information about ancient music notation is fragmentary. Although many ancient cultures used symbols to represent melodies and rhythms, none of them were particularly comprehensive, the seeds of what would eventually become modern western notation were sown in medieval Europe, starting with the Catholic churchs goal for ecclesiastical uniformity. The church began notating plainchant melodies so that the same chants could be used throughout the church, Music notation developed in the Renaissance and Baroque music eras. The introduction of figured bass notation in the Baroque era marked the beginning of the first compositions based around chord progressions, in the classical period and the Romantic music era, notation continued to develop as new musical instrument technologies were developed. Music notation has been adapted to many kinds of music, including music, popular music. The earliest form of notation can be found in a cuneiform tablet that was created at Nippur, in Sumer. The tablet represents fragmentary instructions for performing music, that the music was composed in harmonies of thirds, a tablet from about 1250 BC shows a more developed form of notation. Although the interpretation of the system is still controversial, it is clear that the notation indicates the names of strings on a lyre. Although they are fragmentary, these represent the earliest notated melodies found anywhere in the world. The notation consists of symbols placed above text syllables, an example of a complete composition is the Seikilos epitaph, which has been variously dated between the 2nd century BC to the 1st century AD. Three hymns by Mesomedes of Crete exist in manuscript, the Delphic Hymns, dated to the 2nd century BC, also use this notation, but they are not completely preserved. Ancient Greek notation appears to have out of use around the time of the Decline of the Roman Empire. Byzantine music has survived as music for court ceremonies, including vocal religious music. It is not known if it is based on the monodic modal singing, unlike Western notation Byzantine neumes always indicate modal steps in relation to a clef or modal key. These step symbols themselves, or better phonic neumes, resemble brush strokes and are colloquially called gántzoi in Modern Greek, Notes as pitch classes or modal keys are represented in written form only between these neumes. In modern notation they simply serve as a reminder and modal and tempo directions have been added. In Papadic notation medial signatures usually meant a change into another echos. With exception of vú and zō they do correspond to Western solmization syllables as re, mi, fa, sol, la, si
4.
Akkadian
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Akkadian is an extinct East Semitic language that was spoken in ancient Mesopotamia. The earliest attested Semitic language, it used the writing system, which was originally used to write the unrelated Ancient Sumerian. The language was named after the city of Akkad, a centre of Mesopotamian civilization during the Akkadian Empire. The mutual influence between Sumerian and Akkadian had led scholars to describe the languages as a sprachbund, Akkadian proper names were first attested in Sumerian texts from around the mid 3rd-millennium BC. From the second half of the third millennium BC, texts written in Akkadian begin to appear. By the second millennium BC, two variant forms of the language were in use in Assyria and Babylonia, known as Assyrian and Babylonian respectively, for centuries, Akkadian was the native language in Mesopotamian nations such as Assyria and Babylonia. However, it began to decline during the Neo-Assyrian Empire around the 8th century BC, by the Hellenistic period, the language was largely confined to scholars and priests working in temples in Assyria and Babylonia. The last known Akkadian cuneiform document dates from the 1st century AD, Akkadian belongs with the other Semitic languages in the Near Eastern branch of the Afroasiatic languages, a family native to East Africa, which then spread to West, Northwest and Northeast Africa. Within the Near Eastern Semitic languages, Akkadian forms an East Semitic subgroup and this novel word order is due to the influence of the Sumerian substratum, which has an SOV order. Additionally Akkadian is the only Semitic language to use the prepositions ina and ana, other Semitic languages like Arabic and Aramaic have the prepositions bi/bə and li/lə. The origin of the Akkadian spatial prepositions is unknown, in contrast to most other Semitic languages, Akkadian has only one non-sibilant fricative, ḫ. Akkadian lost both the glottal and pharyngeal fricatives, which are characteristic of the other Semitic languages, until the Old Babylonian period, the Akkadian sibilants were exclusively affricated. Old Akkadian is preserved on clay tablets dating back to c.2500 BC and it was written using cuneiform, a script adopted from the Sumerians using wedge-shaped symbols pressed in wet clay. As employed by Akkadian scribes, the cuneiform script could represent either Sumerian logograms, Sumerian syllables, Akkadian syllables. For this reason, the sign AN can on the one hand be a logogram for the word ilum, additionally, this sign was used as a determinative for divine names. Another peculiarity of Akkadian cuneiform is that many signs do not have a phonetic value. Certain signs, such as AḪ, do not distinguish between the different vowel qualities, nor is there any coordination in the other direction, the syllable -ša-, for example, is rendered by the sign ŠA, but also by the sign NĪĜ. Both of these are used for the same syllable in the same text
5.
Cuneiform script
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Cuneiform script, one of the earliest systems of writing, was invented by the Sumerians. It is distinguished by its wedge-shaped marks on clay tablets, made by means of a blunt reed for a stylus, the name cuneiform itself simply means wedge shaped. Emerging in Sumer in the fourth millennium BC, cuneiform writing began as a system of pictograms. In the third millennium, the pictorial representations became simplified and more abstract as the number of characters in use grew smaller, the system consists of a combination of logophonetic, consonantal alphabetic and syllabic signs. Cuneiform writing was replaced by the Phoenician alphabet during the Neo-Assyrian Empire. By the second century CE, the script had become extinct, between half a million and two million cuneiform tablets are estimated to have been excavated in modern times, of which only approximately 30,000 –100,000 have been read or published. Most of these have lain in these collections for a century without being translated, studied or published, the cuneiform writing system was in use for more than three millennia, through several stages of development, from the 34th century BC down to the second century CE. Ultimately, it was replaced by alphabetic writing in the course of the Roman era. It had to be deciphered as an unknown writing system in 19th-century Assyriology. Successful completion of its deciphering is dated to 1857, the cuneiform script was developed from pictographic proto-writing in the late 4th millennium BC. Mesopotamias proto-literate period spans roughly the 35th to 32nd centuries, the first documents unequivocally written in Sumerian date to the 31st century at Jemdet Nasr. Originally, pictographs were either drawn on clay tablets in vertical columns with a reed stylus or incised in stone. This early style lacked the characteristic shape of the strokes. Proper names continued to be written in purely logographic fashion. The earliest known Sumerian king whose name appears on contemporary cuneiform tablets is Enmebaragesi of Kish, from about 2900 BC, many pictographs began to lose their original function, and a given sign could have various meanings depending on context. The sign inventory was reduced from some 1,500 signs to some 600 signs, determinative signs were re-introduced to avoid ambiguity. Cuneiform writing proper thus arises from the more primitive system of pictographs at about that time, by adjusting the relative position of the tablet to the stylus, the writer could use a single tool to make a variety of impressions. Cuneiform tablets could be fired in kilns to provide a permanent record, many of the clay tablets found by archaeologists were preserved because they were fired when attacking armies burned the building in which they were kept
6.
Lyre
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The lyre is a string instrument known for its use in Greek classical antiquity and later periods. The lyre is similar in appearance to a small harp but with distinct differences, the word comes via Latin from the Greek, the earliest reference to the word is the Mycenaean Greek ru-ra-ta-e, meaning lyrists and written in the Linear B script. The lyres of Ur, excavated in ancient Mesopotamia, date to 2500 BC, the earliest picture of a lyre with seven strings appears in the famous sarcophagus of Hagia Triada. The sarcophagus was used during the Mycenaean occupation of Crete, the recitations of the Ancient Greeks were accompanied by lyre playing. The lyre of classical antiquity was ordinarily played by being strummed with a plectrum, the fingers of the free hand silenced the unwanted strings in the chord. However, later lyres were played with a bow in Europe, one example from Wales that has been resurrected recently is the crwth. In organology, lyres are defined as yoke lutes, being lutes in which the strings are attached to a yoke which lies in the plane as the sound-table. A classical lyre has a body or sound-chest, which. Extending from this sound-chest are two raised arms, which are hollow, and are curved both outward and forward. They are connected near the top by a crossbar or yoke, an additional crossbar, fixed to the sound-chest, makes the bridge which transmits the vibrations of the strings. They were stretched between the yoke and bridge, or to a tailpiece below the bridge, according to ancient Greek mythology, the young god Hermes stole a herd of sacred cows from Apollo. In order not to be followed, he made shoes for the cows which forced them to walk backwards, Apollo, following the trails, could not follow where the cows were going. Along the way, Hermes slaughtered one of the cows and offered all, from the entrails and a tortoise/turtle shell, he created the Lyre. Apollo, figuring out it was Hermes who had his cows, Apollo was furious, but after hearing the sound of the lyre, his anger faded. Apollo offered to trade the herd of cattle for the lyre, hence, the creation of the lyre is attributed to Hermes. Other sources credit it to Apollo himself, locales in southern Europe, western Asia, or north Africa have been proposed as the historic birthplace of the genus. The instrument is played in north-eastern parts of Africa. Some of the cultures using and developing the lyre were the Aeolian and Ionian Greek colonies on the coasts of Asia bordering the Lydian empire, some mythic masters like Musaeus, and Thamyris were believed to have been born in Thrace, another place of extensive Greek colonization
7.
Brain
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The brain is an organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. The brain is located in the head, usually close to the organs for senses such as vision. The brain is the most complex organ in a vertebrates body, in a human, the cerebral cortex contains approximately 15–33 billion neurons, each connected by synapses to several thousand other neurons. Physiologically, the function of the brain is to exert centralized control over the other organs of the body, the brain acts on the rest of the body both by generating patterns of muscle activity and by driving the secretion of chemicals called hormones. This centralized control allows rapid and coordinated responses to changes in the environment, the operations of individual brain cells are now understood in considerable detail but the way they cooperate in ensembles of millions is yet to be solved. This article compares the properties of brains across the range of animal species. It deals with the human brain insofar as it shares the properties of other brains, the ways in which the human brain differs from other brains are covered in the human brain article. Several topics that might be covered here are instead covered there because more can be said about them in a human context. The most important is brain disease and the effects of brain damage, the shape and size of the brain varies greatly between species, and identifying common features is often difficult. Nevertheless, there are a number of principles of architecture that apply across a wide range of species. Some aspects of structure are common to almost the entire range of animal species, others distinguish advanced brains from more primitive ones. The simplest way to gain information about brain anatomy is by visual inspection, Brain tissue in its natural state is too soft to work with, but it can be hardened by immersion in alcohol or other fixatives, and then sliced apart for examination of the interior. Visually, the interior of the consists of areas of so-called grey matter, with a dark color, separated by areas of white matter. Further information can be gained by staining slices of tissue with a variety of chemicals that bring out areas where specific types of molecules are present in high concentrations. It is also possible to examine the microstructure of brain tissue using a microscope, the brains of all species are composed primarily of two broad classes of cells, neurons and glial cells. Glial cells come in types, and perform a number of critical functions, including structural support, metabolic support, insulation. Neurons, however, are considered the most important cells in the brain. The property that makes neurons unique is their ability to send signals to target cells over long distances
8.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
9.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
10.
Sheet music
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Sheet music is a handwritten or printed form of music notation that uses modern musical symbols to indicate the pitches, rhythms and/or chords of a song or instrumental musical piece. The first printed sheet music made with a press was made in 1473. Sheet music is the form in which Western classical music is notated so that it can be learned and performed by solo singers or instrumentalists or musical ensembles. Many forms of traditional and popular Western music are commonly learned by singers and musicians by ear, Score is a common alternative term for sheet music, and there are several types of scores, as discussed below. Sheet music from the 20th and 21st century typically indicates the title of the song or composition on a page or cover, or on the top of the first page. If the song or piece is from a movie, Broadway musical, or opera, if the songwriter or composer is known, her or his name is typically indicated along with the title. Black market sheet music, such as illegal jazz fake books may or may not indicate the songwriter or composer, the type of musical notation varies a great deal by genre or style of music. In most classical music, the melody and accompaniment parts are notated on the lines of a staff using round note heads, the lyrics, if present, are written near the melody notes. However, music from the Baroque music era or earlier eras may have neither a tempo marking nor a dynamic indication. The singers and musicians of that era were expected to know what tempo and loudness to play or sing a song or piece due to their musical experience. In the contemporary music era, and in some cases before, composers often used their native language for tempo indications. These conventions of music notation, and in particular the use of English tempo instructions, are also used for sheet music versions of 20th. Popular music songs often indicate both the tempo and genre, slow blues or uptempo rock, pop songs often contain chord names above the staff using letter names, so that an acoustic guitarist or piano player can improvise a chordal accompaniment. In other styles of music, different musical notation methods may be used, in jazz, while most professional performers can read classical-style notation, many jazz tunes are notated using chord charts, which indicate the chord progression of a song and its form. Like popular music songs, jazz tunes often indicate both the tempo and genre, slow blues or fast bop, professional country music session musicians typically use music notated in the Nashville Number System, which indicates the chord progression using numbers. Chord charts using letter names, numbers, or Roman numerals are widely used for notating music by blues, R&B, rock music. Many guitar players and electric bass players learn songs and note tunes using tablature, tab is widely used rock music and heavy metal guitarists. Singers in many music styles learn a song using only a lyrics sheet
11.
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
12.
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
