1.
Polychord
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In music and music theory, a bichord or polychord consists of two or more chords, one on top of the other. In shorthand they are written with the top chord above a line, the use of polychords may suggest bitonality or polytonality. Examples may be found in Igor Stravinskys Petrushka, p.15 and they may also be found in the song Point of No Return from Andrew Lloyd Webbers The Phantom of the Opera, including chords such as E♭m over Fm. Extended chords contain more than one triad, and so can be regarded as a type of polychord, For example G7♯11♭9 is formed from G major, examples of extended chords include the Elektra chord. When one or both of the chords in a polychord are not chords in some exclusive sense according to some preferred chord theory or other, secundal Tertian Quartal Upper structure Chordioid
2.
Pulley
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A pulley is a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable, supporting shell is referred to as a block. A pulley may also be called a sheave or drum and may have a groove or grooves between two flanges around its circumference. The drive element of a system can be a rope, cable, belt. Hero of Alexandria identified the pulley as one of six simple machines used to lift weights, pulleys are assembled to form a block and tackle in order to provide mechanical advantage to apply large forces. Pulleys are also assembled as part of belt and chain drives in order to power from one rotating shaft to another. A set of pulleys assembled so that they rotate independently on the axle from a block. Two blocks with an attached to one of the blocks. A block and tackle is assembled so one block is attached to fixed mounting point, the ideal mechanical advantage of the block and tackle is equal to the number of parts of the rope that support the moving block. This system is included in the list of simple machines identified by Renaissance scientists, if the rope and pulley system does not dissipate or store energy, then its mechanical advantage is the number of parts of the rope that act on the load. This can be shown as follows, consider the set of pulleys that form the moving block and the parts of the rope that support this block. If there are p of these parts of the supporting the load W. This means the force on the rope is T=W/p. Thus, the block and tackle reduces the force by the factor p. The simplest theory of operation for a pulley system assumes that the pulleys and lines are weightless, and it is also assumed that the lines do not stretch. In equilibrium, the forces on the block must sum to zero. In addition the tension in the rope must be the same for each of its parts and this means that the two parts of the rope supporting the moving block must each support half the load. These are different types of systems, Fixed, A fixed pulley has an axle mounted in bearings attached to a supporting structure. A fixed pulley changes the direction of the force on a rope or belt that moves along its circumference, mechanical advantage is gained by combining a fixed pulley with a movable pulley or another fixed pulley of a different diameter
3.
Musical instrument
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A musical instrument is an instrument created or adapted to make musical sounds. In principle, any object that produces sound can be a musical instrument—it is through purpose that the object becomes a musical instrument, the history of musical instruments dates to the beginnings of human culture. Early musical instruments may have used for ritual, such as a trumpet to signal success on the hunt. Cultures eventually developed composition and performance of melodies for entertainment, Musical instruments evolved in step with changing applications. The date and origin of the first device considered an instrument is disputed. The oldest object that some refer to as a musical instrument. Some consensus dates early flutes to about 37,000 years ago, many early musical instruments were made from animal skins, bone, wood, and other non-durable materials. Musical instruments developed independently in many populated regions of the world, however, contact among civilizations caused rapid spread and adaptation of most instruments in places far from their origin. By the Middle Ages, instruments from Mesopotamia were in maritime Southeast Asia, development in the Americas occurred at a slower pace, but cultures of North, Central, and South America shared musical instruments. By 1400, musical instrument development slowed in areas and was dominated by the Occident. Musical instrument classification is a discipline in its own right, Instruments can be classified by their effective range, their material composition, their size, etc. However, the most common method, Hornbostel-Sachs, uses the means by which they produce sound. The academic study of instruments is called organology. Once humans moved from making sounds with their bodies—for example, by using objects to create music from sounds. Primitive instruments were designed to emulate natural sounds, and their purpose was ritual rather than entertainment. The concept of melody and the pursuit of musical composition were unknown to early players of musical instruments. A player sounding a flute to signal the start of a hunt does so without thought of the notion of making music. Musical instruments are constructed in an array of styles and shapes
4.
Laboratory
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A laboratory is a facility that provides controlled conditions in which scientific or technological research, experiments, and measurement may be performed. Laboratories used for scientific research take many forms because of the requirements of specialists in the various fields of science. A physics laboratory might contain a particle accelerator or vacuum chamber, a chemist or biologist might use a wet laboratory, while a psychologists laboratory might be a room with one-way mirrors and hidden cameras in which to observe behavior. In some laboratories, such as commonly used by computer scientists. Scientists in other fields will use other types of laboratories. Engineers use laboratories as well to design, build, and test technological devices, scientific laboratories can be found as research and learning spaces in schools and universities, industry, government, or military facilities, and even aboard ships and spacecraft. Early instances of laboratories recorded in English involved alchemy and the preparation of medicines, larger or more sophisticated equipment is generally called a scientific instrument. Both laboratory equipment and scientific instruments are increasingly being designed and shared using open hardware principles, open source labs use open source scientific hardware. The title of laboratory is used for certain other facilities where the processes or equipment used are similar to those in scientific laboratories. In many labs, though, hazards are present, in laboratories where dangerous conditions might exist, safety precautions are important. Rules exist to minimize the risk, and safety equipment is used to protect the lab user from injury or to assist in responding to an emergency. This standard is referred to as the Laboratory Standard. Under this standard, a laboratory is required to produce a Chemical Hygiene Plan which addresses the specific hazards found in its location, the CHP must be reviewed annually. Many schools and businesses employ safety, health, and environmental specialists, such as a Chemical Hygiene Officer to develop, manage, and evaluate their CHP. Additionally, third party review is used to provide an objective outside view which provides a fresh look at areas. An important element of such audits is the review of regulatory compliance, training is critical to the ongoing safe operation of the laboratory facility. Educators, staff and management must be engaged in working to reduce the likelihood of accidents, injuries, efforts are made to ensure laboratory safety videos are both relevant and engaging. The dictionary definition of laboratory at Wiktionary Media related to Laboratory at Wikimedia Commons Nobel Laureates Interactive 360° Laboratories
5.
Chord (music)
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A chord, in music, is any harmonic set of pitches consisting of two or more notes that are heard as if sounding simultaneously. In everyday use by musical ensembles such as bands and orchestras, however, the notes of a chord do not have to be played together at the same time, arpeggios and broken chords may, for many practical and theoretical purposes, also constitute chords. Other chords with more than three notes include added tone chords, extended chords and tone clusters, which are used in classical music, jazz. Triads commonly found in the Western classical tradition are major and minor chords, the descriptions major, minor, augmented, and diminished are referred to collectively as chordal quality. Chords are also classified by their root note—for instance, a C major triad consists of the pitch classes C, E. While most chords have at least three notes, power chords, which are used in rock music, particularly in hard rock. An ordered series of chords is called a chord progression, one example of a widely used chord progression in Western traditional music and blues is the 12 bar blues progression, the simplest versions of which include tonic, subdominant and dominant chords. To describe this, Western music theory has developed the practice of numbering chords using Roman numerals which represent the number of steps up from the tonic note of the scale. Common ways of notating or representing chords in Western music include Roman numerals, figured bass, macro symbols, the chords in a song or piece are also given names which refer to their function. The chord built on the first note of a scale is called the tonic chord. The chord built on the note of a major scale is called the subdominant chord. The chord built on the degree of the major scale is called the dominant chord. There are names for the built on every note of the major scale. Chords can be played on instruments, including piano, pipe organ, guitar. Chords can also be performed when multiple musicians play together in an ensemble or when multiple singers sing in a choir. The English word chord derives from Middle English cord, a shortening of accord in the sense of agreement and later. A sequence of chords is known as a progression or harmonic progression. These are frequently used in Western music, a chord progression aims for a definite goal of establishing a tonality founded on a key, root or tonic chord
6.
Musical bow
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The musical bow is a simple string musical instrument part of a number of South African cultures, also found in other places in the world through the result of slave trade. It consists of a flexible, usually wooden, stick 1.5 to 10 feet long and it can be played with the hands or a wooden stick or branch. Often, it is a normal archery bow used for music, types of bow are different mouth-resonated string bow, earth-resonated string bow, gourd-resonated string bow, bridged string bow, spiked fiddle, and bowed trough fiddle. Although the bow is now thought of as a weapon, it is not clear whether it was used in this way originally. A cave painting in the Trois Frères cave of southern France, dated to around 13,000 BCE, displays a bow being used as a musical instrument, Musical bows are still used in a number of cultures today. It can be found as far south as Swaziland, and as far east as eastern Africa, Madagascar, the usual way to make the bow sound is to pluck the string, although sometimes a subsidiary bow is used to scrape the string, much as on a violin. The Onavillu of Kerala sounds when struck with a thin stick, unlike string instruments used in classical music, however, they do not have a built-in resonator, although resonators may be made to work with the bow in a number of ways. The most usual type of resonator consists of an attached to the back of the string bearer. The bow may also be stood in a pit or gourd on the ground and this last method allows the size of the resonator to be varied as the instrument is played, thus allowing a melody to be heard consisting of the notes resonating in the players mouth. As well as various forms of resonators, the bow is frequently played without a resonator at all. In Africa the musical bow is usually played by a solo performer, in Capoeira, the berimbau is played as part of the roda, a musical group standing in a circle, in the centre of which the Capoeiristas perform or play. The Appalachian mouth-bow can be played amplified in old-time music jams, due to the nature of their construction and playing, musical bows are quiet instruments, therefore needing a resonator to resound. The resonator can either be a gourd or the players mouth Musical bows are the instruments of the Nguni and Sotho people. Historians believe that many of the musical came from Khoisan peoples. Although there are differences between musical bows, all of them share two things, a resonator, and at least two fundamental notes. The strongest notes are the fundamentals, the deepest notes produced by the string, there are at least two fundamental notes produced by all musical bows, an open and a closed. In Xhosa they are called vu and ba respectively and these two notes can already be on the string, if it is divided or stopped by a string attached to the gourd, as in the case of umakhweyana, xitende, berimbau, hungu, etc. The pitch difference between a vu and a ba is usually about a whole tone, in certain places in can be closer to a semitone or closer to a minor third
7.
Fingerboard
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The fingerboard is an important component of most stringed instruments. It is a thin, long strip of material, usually wood, the strings run over the fingerboard, between the nut and bridge. To play the instrument, a musician presses strings down to the fingerboard to change the vibrating length and this is called stopping the strings. Depending on the instrument and the style of music, the musician may pluck, on some instruments, notes can be sounded by the fretting hand alone, such as with hammer ons, an electric guitar technique. The word fingerboard in other languages sometimes occurs in musical directions, in particular, the direction sul tasto for bowed string instruments to play with the bow above the fingerboard. This reduces the prominence of upper harmonics, giving a more ethereal tone, a fingerboard may be fretted, having raised strips of hard material perpendicular to the strings, which the player presses the strings against to stop the strings. On modern guitars, frets are made of metal. Frets let the player stop the string consistently in the same place, as well, frets do not dampen string vibrations as much as fingers alone on an unfretted fingerboard. Frets may be fixed, as on a guitar or mandolin, or movable, fingerboards may also be unfretted, as they usually are on bowed instruments, where damping by the finger is of little consequence because of the sustained stimulation of the strings by the bow. Unfretted fingerboards allow a more control over subtle changes in pitch than fretted boards. Fingerboards may also be, though uncommon, a hybrid of these two, such a construction is seen on the sitar, where arched frets attach at the edges of a smooth fingerboard, unfrettable strings run inside the frets, while frettable ones run outside. A double dot or some other variation marks the 12th fret, variations on the standard dot shape can make a guitar more distinctive. Position markers are sometimes made luminescent to make more visible on stage. Position markers are sometimes repeated on the edge of the fingerboard for easy viewing. Over time, strings wear frets down, which can cause buzzing, fixing this occasionally requires replacing the frets—but more often they just need dressing. In fret dressing, a luthier levels and polishes the frets, stainless steel guitar frets may never need dressing, because of the density of the material. Not having frets carefully and properly aligned with the fingerboard can cause severe intonation issues, the ultimate way of determining the source of a buzz and detuning problem is to measure the levelness of the frets. A straightedge positioned on the neck in the lie of one of the strings should show nearly level frets, on bowed string instruments, the fingerboard is usually made of ebony, rosewood or other hardwood
8.
Fingering (music)
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In music, fingering, or on stringed instruments stopping, is the choice of which fingers and hand positions to use when playing certain musical instruments. Fingering typically changes throughout a piece, the challenge of choosing good fingering for a piece is to make the hand movements as comfortable as possible without changing hand position too often, a substitute fingering is an alternative to the indicated fingering, not to be confused with finger substitution. Depending on the instrument, not all the fingers may be used, for example, saxophonists do not use the right thumb and string instruments only use the fingers. Fingering applies to the rotary and piston valves employed on many brass instruments, the trombone, a fully chromatic brass instrument without valves, employs equivalent numbered notation for slide positions rather than fingering. On keyboard instruments all digits are used and the thumb is considered to be a finger, so there are numbers from 1 to five. The numbers are related to the fingers themselves, not to the position on the keyboard. There are only few publications about piano fingering and it is mentioned by Carl Philipp Emanuel Bach in his book Versuch über die wahre Art, das Clavier zu spielen where he dedicated several paragraphs to this topic. The British pianist Tobias Matthay wrote a small book Principles of Fingering, in 1971 Julien Musafia published his book The Art of Fingering in Piano Playing. The book includes musical examples mostly from the Beethovens Violin and Piano Sonatas, in 2012 Rami Bar-Niv published his book The Art of Piano Fingering -- Traditional, Advanced, and Innovative. The book teaches the craft of piano fingering using music examples, photos and diagrams, exercises, and injury-free techniques. On string instruments fingers are numbered from 1 to 4, beginning with the finger, the thumb not being counted because it does not normally play on a string. In those cases on string instruments where the thumb is used, guitar music indicates thumb, occasionally used to finger bass notes on the low E string, with a T. Position may be indicated through ordinal numbers or Roman numerals, a string may also be indicated through Roman numerals, often I-IV, or by its open-string note. A change in positions is referred to as a shift, guitar music indicates position with Roman numerals and string designations with circled numbers. The classical guitar also has a notation system for the plucking hand, known as pima, abbreviations of Spanish. It is usually notated in scores where a passage is particularly difficult. Otherwise, plucking-hand fingering is generally left to the discretion of the guitarist, several alternate fingerings may exist for any given pitch. Simple flutes as well as bagpipe chanters have open holes which are closed by the pads of the players fingertips, some such instruments use simple keywork to extend the players reach for one or two notes
9.
String (music)
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A string is the vibrating element that produces sound in string instruments such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a material that a musical instrument holds under tension so that they can vibrate freely. Wound strings have a core of one material, with an overwinding of other materials and this is to make the string vibrate at the desired pitch, while maintaining a low profile and sufficient flexibility for playability. This enabled stringed instruments to be made with less thick bass strings, on string instruments that the player plucks or bows directly, this enabled instrument makers to use thinner strings for the lowest-pitched strings, which made the lower-pitch strings easier to play. The end of the string that mounts to the tuning mechanism is usually plain. Depending on the instrument, the other, fixed end may have either a plain, loop. When a ball or loop is used with a guitar, this ensures that the string stays fixed in the bridge of the guitar, when a ball or loop is used with a violin-family instrument, this keeps the string end fixed in the tailpiece. Fender Bullet strings have a cylinder for more stable tuning on guitars equipped with synchronized tremolo systems. Strings for some instruments may be wrapped with silk at the ends to protect the string, the color and pattern of the silk often identifies attributes of the string, such as manufacturer, size, intended pitch, etc. There are several varieties of wound strings available, the simplest wound strings are roundwound—with round wire wrapped in a tight spiral around either a round or hexagonal core. Such strings are usually simple to manufacture and the least expensive and they have several drawbacks, however, Roundwound strings have a bumpy surface profile that produce friction on the players fingertips. This causes squeaking sounds when the fingers slide over the strings. Roundwound strings higher friction surface profile may hasten fingerboard and fret wear, when the core is round, the winding is less secure and may rotate freely around the core, especially if the winding is damaged after use. Flatwound strings also have either a round or hex core, however, the winding wire has a rounded square cross-section that has a shallower profile when tightly wound. This makes for more playing, and decreased wear for frets. Squeaking sounds due to fingers sliding along the strings are also decreased significantly, flatwound strings also have a longer playable life because of smaller grooves for dirt and oil to build up in. On the other hand, flatwound strings sound less bright than roundwounds, flatwounds also usually cost more than roundwounds because of less demand, less production, and higher overhead costs. Manufacturing is also difficult, as precise alignment of the flat sides of the winding must be maintained
10.
Bridge (instrument)
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Most stringed instruments produce their sound through the application of energy to the strings, which sets them into vibratory motion. The strings alone, however, produce only a faint sound because they displace only a volume of air as they vibrate. Consequently, the sound of the strings alone requires impedance matching to the air by transmitting their vibrations to a larger surface area capable of displacing larger volumes of air. This calls for an arrangement that allows for the strings to vibrate freely, a bridge is the customary means by which this is accomplished. Typically, the bridge is perpendicular to the strings and larger surface with the tension of the strings pressing down on the bridge and thus on the larger surface beneath it. That larger, more acoustically responsive surface may be coupled to a sound chamber—an enclosure such as the body of a guitar or violin—that provides resonance that helps amplify the sound, alternatively, a bridge may consist of multiple parts. One common form is a bridge that incorporates a separate bearing surface on which the strings rest and this is often of a material harder than the bridge itself, such as bone, ivory, high-density plastic, or metal. A classical guitar saddle sits loosely in the bridge and is kept there by string tension. The saddle has shallow grooves in it, at least for the treble strings, yet another type of multi-part bridge is common on instruments whose sound plate is curved rather than flat. Instruments of this type, such as guitars and mandolins, often have a bridge with a base. On classical and flat-top guitars the bridge is glued to the top, a bridge held on to the top by string tension, as in archtop guitars, is known as a floating bridge, and requires a separate tailpiece to anchor the strings. The bridge must transfer vibration to the board or other amplifying surface. As the strings are set in motion, it does this by bending to and this causes the sounding board to vibrate at the same frequency as the string producing a wave-like motion and an audible sound. Bridges are designed to hold the strings at a height above the fingerboard of the instrument. Bridge height may be fixed or alterable, in addition to supporting the strings and transmitting their vibrations, the bridge also controls the spacing between strings with shallow grooves cut in the bridge or its saddle. The strings sit in grooves, thus are held in their proper lateral position. The nut, at the end of the instrument from the bridge or tailpiece. Bridges for electric guitars can be divided into two groups, vibrato and non-vibrato
11.
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
12.
Graduation (instrument)
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Graduation is the marking of an instrument with indicators of a measurement. Linear graduation of a scale occurs on a straight instrument, the graduation can identify linear measures, such as inches or millimetres on a rule. They can also be such as logarithmic or other transcendental scales. Volumetric graduations, for instance on a cup, can vary in scale due to the containers conical or abstract shape. Circular graduations of a scale occur on an arc or limb of an instrument. In some cases, non-circular curves are graduated in instruments, a typical circular arc graduation is the division into angular measurements, such as degrees, minutes and seconds. There can also be non-uniform graduations such as logarithmic or other scales, graduations can be placed on an instrument by etching, scribing or engraving, painting, printing or other means. For durability and accuracy, etched or scribed marks are usually preferable to surface coatings such as paints, markings can be a combination of both physical marks such as a scribed line and a paint or other marking material. For example, it is common for black ink or paint to fill the grooves cut in a scribed rule, inexpensive plastic devices can be molded and painted or molded with two or more colours of plastic used. Some rather high-quality devices can be manufactured with plastic and reveal high-precision graduations, graduations traditionally have been scribed into an instrument by hand with a sharp, hard tool. Later developments in such as dividing engines allowed the process to automated with greater precision. Modern devices can be stamped, cut on a machine or with a CNC machine. In the case of stamping, the master has the precision built into itself, similarly, molding of plastic can be as precise as the mold process. With proper concern for such effects as thermal expansion or contraction and shrinkage, the precision can be very high
13.
Ruler
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A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing, engineering and building to measure distances or to rule straight lines. The ruler is a straightedge which may also contain calibrated lines to measure distances, rulers have long been made from different materials and in a wide range of sizes. Plastics have also used since they were invented, they can be molded with length markings instead of being scribed. Metal is used for more durable rulers for use in the workshop,12 inches or 30 cm in length is useful for a ruler to be kept on a desk to help in drawing. Shorter rulers are convenient for keeping in a pocket, longer rulers, e. g.18 inches are necessary in some cases. Rigid wooden or plastic yardsticks,1 yard long, and meter sticks,1 meter long, are also used, classically, long measuring rods were used for larger projects, now superseded by tape measure or laser rangefinders. Desk rulers are used for three purposes, to measure, to aid in drawing straight lines and as a straight guide for cutting and scoring with a blade. Practical rulers have distance markings along their edges, a line gauge is a type of ruler used in the printing industry. These may be made from a variety of materials, typically metal or clear plastic, units of measurement on a basic line gauge usually include inches, agate, picas, and points. More detailed line gauges may contain sample widths of lines, samples of common type in several point sizes, measuring instruments similar in function to rulers are made portable by folding or retracting into a coil when not in use. When extended for use, they are straight, like a ruler, the illustrations on this page show a 2-meter carpenters rule, which folds down to a length of 24 cm to easily fit in a pocket, and a 5-meter-long tape, which retracts into a small housing. A flexible length measuring instrument which is not necessarily straight in use is the tailors fabric tape measure and it is used to measure around a solid body, e. g. a persons waist measurement, as well as linear measurement, e. g. inside leg. It is rolled up when not in use, taking up little space, a contraction rule is made having larger divisions than standard measures to allow for shrinkage of a metal casting. They may also be known as a shrinkage or shrink rule, a ruler software program can be used to measure pixels on a computer screen or mobile phone. These programs are known as screen rulers. In geometry, a ruler without any marks on it may be used only for drawing lines between points. A straightedge is used to help draw accurate graphs and tables. A ruler and compass construction refers to using an unmarked ruler
14.
Musical note
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In music, the term note has three primary meanings, A sign used in musical notation to represent the relative duration and pitch of a sound, A pitched sound itself. Notes are the blocks of much written music, discretizations of musical phenomena that facilitate performance, comprehension. In the former case, one note to refer to a specific musical event, in the latter. Two notes with fundamental frequencies in an equal to any integer power of two are perceived as very similar. Because of that, all notes with these kinds of relations can be grouped under the pitch class. However, within the English-speaking and Dutch-speaking world, pitch classes are represented by the first seven letters of the Latin alphabet. A few European countries, including Germany, adopt an almost identical notation, the eighth note, or octave, is given the same name as the first, but has double its frequency. The name octave is also used to indicate the span between a note and another with double frequency, for example, the now-standard tuning pitch for most Western music,440 Hz, is named a′ or A4. There are two systems to define each note and octave, the Helmholtz pitch notation and the scientific pitch notation. Letter names are modified by the accidentals, a sharp ♯ raises a note by a semitone or half-step, and a flat ♭ lowers it by the same amount. In modern tuning a half step has a ratio of 12√2. The accidentals are written after the name, so, for example, F♯ represents F-sharp, B♭ is B-flat. Additional accidentals are the double-sharp, raising the frequency by two semitones, and double-flat, lowering it by that amount, in musical notation, accidentals are placed before the note symbols. Systematic alterations to the seven lettered pitches in the scale can be indicated by placing the symbols in the key signature, explicitly noted accidentals can be used to override this effect for the remainder of a bar. A special accidental, the natural symbol ♮, is used to indicate an unmodified pitch, effects of key signature and local accidentals do not accumulate. If the key signature indicates G♯, a flat before a G makes it G♭, though often this type of rare accidental is expressed as a natural. Likewise, a sharp sign on a key signature with a single sharp ♯ indicates only a double sharp. Assuming enharmonicity, many accidentals will create equivalences between pitches that are written differently, for instance, raising the note B to B♯ is equal to the note C
15.
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
16.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
17.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
18.
Course (music)
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A course, on a stringed musical instrument, is two or more adjacent strings that are closely spaced relative to the other strings, and typically played as a single string. The strings in each course are tuned in unison or an octave. Course may also refer to a single string normally played on its own on an instrument with other multi-string courses, an instrument with at least one course is referred to as coursed, while one whose strings are all played individually is uncoursed. Eventually, however, they came to be employed to alter the characteristics of instruments as well. A typical piano, for example, contains courses with one, all members of the mandolin family, except some versions of the lowest-pitched, have courses each of two or three strings, most commonly eight strings in four courses. The exception is some varieties of mando-bass, which have four individual strings, the baroque guitar has five courses, with the bottom four always double strung, and the top string either double or single. There are also several patterns of modern ten-string guitar but all have ten single strings, the twelve-string guitar has twelve strings, in six courses. The lowest three courses are tuned at octaves, with the primary string uppermost and the octave below it. The G course is either unison or at octaves, on some electric twelve-string guitars, and most notably the Rickenbacker 360/12, the octave strings are below the primary strings. The national instrument of Columbia, the tiple Colombiano, has four courses of three strings each and its higher-pitched relative, the tiple requinto, is similarly triple-strung. The American tiple, a smaller instrument loosely derived from the Columbian tiple, the electric twelve-string bass has twelve strings in four triple courses. Basses have also built with six double courses and other configs
19.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
20.
Pitch (music)
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Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major attribute of musical tones, along with duration, loudness. Pitch may be quantified as a frequency, but pitch is not a purely objective physical property, Pitch is an auditory sensation in which a listener assigns musical tones to relative positions on a musical scale based primarily on their perception of the frequency of vibration. Pitch is closely related to frequency, but the two are not equivalent, frequency is an objective, scientific attribute that can be measured. Pitch is each persons subjective perception of a wave, which cannot be directly measured. However, this not necessarily mean that most people wont agree on which notes are higher and lower. Sound waves themselves do not have pitch, but their oscillations can be measured to obtain a frequency and it takes a sentient mind to map the internal quality of pitch. However, pitches are usually associated with, and thus quantified as frequencies in cycles per second, or hertz, by comparing sounds with pure tones, Complex and aperiodic sound waves can often be assigned a pitch by this method. According to the American National Standards Institute, pitch is the attribute of sound according to which sounds can be ordered on a scale from low to high. That is, high pitch means very rapid oscillation, and low pitch corresponds to slower oscillation, despite that, the idiom relating vertical height to sound pitch is shared by most languages. At least in English, it is just one of many deep conceptual metaphors that involve up/down, the exact etymological history of the musical sense of high and low pitch is still unclear. There is evidence that humans do actually perceive that the source of a sound is slightly higher or lower in vertical space when the frequency is increased or reduced. The pitch of tones can be ambiguous, meaning that two or more different pitches can be perceived, depending upon the observer. In a situation like this, the percept at 200 Hz is commonly referred to as the missing fundamental, Pitch depends to a lesser degree on the sound pressure level of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases, for instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder, theories of pitch perception try to explain how the physical sound and specific physiology of the auditory system work together to yield the experience of pitch. In general, pitch perception theories can be divided into place coding, place theory holds that the perception of pitch is determined by the place of maximum excitation on the basilar membrane. However, a purely place-based theory cannot account for the accuracy of pitch perception in the low, temporal theories offer an alternative that appeals to the temporal structure of action potentials, mostly the phase-locking and mode-locking of action potentials to frequencies in a stimulus
21.
Mersenne's laws
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Mersennes laws are laws describing the frequency of oscillation of a stretched string or monochord, useful in musical tuning and musical instrument construction. The equation was first proposed by French mathematician and music theorist Marin Mersenne in his 1637 work Traité de lharmonie universelle. Mersennes laws govern the construction and operation of string instruments, pianos, harps, lower strings are thicker, thus having a greater mass per unit length. Higher-pitched strings typically are thinner, have higher tension, and may be shorter and this result does not differ substantially from Galileos, yet it is rightly known as Mersennes law, because Mersenne physically proved their truth through experiments. Mersenne investigated and refined these relationships by experiment but did not himself originate them, though his theories are correct, his measurements are not very exact, and his calculations where greatly improved by Joseph Sauveur through the use of acoustic beats and metronomes. These laws are derived from Mersennes equation 22, f 0 = ν λ =12 L F μ. The formula for the frequency is, f 0 =12 L F μ. Beat Cycloid Long-string instrument Mersenne conjectures Overtone Standing wave
22.
Pizzicato
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Pizzicato is a playing technique that involves plucking the strings of a string instrument. The exact technique varies depending on the type of stringed instrument. On bowed string instruments it is a method of playing by plucking the strings with the fingers and this produces a very different sound from bowing, short and percussive rather than sustained. On a keyboard string instrument, such as the piano, pizzicato may be employed as one of the variety of techniques involving direct manipulation of the strings known collectively as string piano. On the guitar, it is a form of plucking. For details of technique, see palm mute. When a string is struck or plucked, as with pizzicato and this complex timbre is called inharmonicity. The inharmonicity of a string depends on its characteristics, such as tension, composition, diameter. The first recognised use of pizzicato in classical music is found in Tobias Humes Captain Humes Poeticall Musicke, wherein he instructs the viola da gamba player to use pizzicato. Another early use is found in Claudio Monteverdis Il combattimento di Tancredi e Clorinda, later, in 1756, Leopold Mozart in his Versuch einer gründlichen Violinschule instructs the player to use the index finger of the right hand. This has remained the most usual way to execute a pizzicato, the bow is held in the hand at the same time unless there is enough time to put it down and pick it up again between bowed passages. In jazz and bluegrass, and the few popular music styles which use double bass, in classical double bass playing, pizzicato are often performed with the bow being held in the hand, as such, the string is usually only plucked with a single finger. In contrast, in jazz, bluegrass, and other non-Classical styles, in classical music, however, string instruments are most usually played with the bow, and composers give specific indications to play pizzicato where required. Pieces in classical music that are played entirely pizzicato include, J. S. Bach, johann Strauss II, Neue Pizzicato Polka. 4 Benjamin Britten, the movement of the Simple Symphony Leroy Anderson. He also included pizzicato in the movement of Winter from The Four Seasons. In music notation, a composer will normally indicate the performer should use pizzicato with the abbreviation pizz, a return to bowing is indicated by the Italian term arco. If a string player has to play pizzicato for a period of time
23.
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
24.
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
25.
Tetractys
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As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was associated with planetary motions. The first four numbers symbolize the musica universalis and the Cosmos as, the four rows add up to ten, which was unity of a higher order. The Tetractys symbolizes the four classical elements—fire, air, water, bless us, divine number, thou who generated gods and men. O holy, holy Tetractys, thou that containest the root, as a portion of the secret religion, initiates were required to swear a secret oath by the Tetractys. They then served as novices for a period of silence lasting three years. It is said that the Pythagorean musical system was based on the Tetractys as the rows can be read as the ratios of 4,3,3,2,2,1, forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths, pure fifths, and the ratios of the unison 1,1. Note that the diapason,2,1, and the diapason plus diapente,3,1, are consonant intervals according to the tetractys of the decad, the Tetractys is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity, there are some who believe that the tetractys and its mysteries influenced the early Kabbalah. A Hebrew tetractys has the letters of the Tetragrammaton inscribed on the ten positions of the tetractys, from right to left. It has been argued that the Kabbalistic Tree of Life, with its ten spheres of emanation, is in some way connected to the tetractys, but its form is not that of a triangle. The occultist Dion Fortune writes, The point is assigned to Kether, the line to Chokmah, in a Tarot reading, the various positions of the tetractys provide a representation for forecasting future events by signifying according to various occult disciplines, such as Alchemy. Below is only a variation for interpretation. The first row of a single position represents the Premise of the reading, forming a foundation for understanding all the other cards, the second row of two positions represents the cosmos and the individual and their relationship. The Light Card to the right represents the influence of the leading the individual to an action. The Dark Card to the left represents the reaction of the cosmos to the actions of the individual, the third row of three positions represents three kinds of decisions an individual must make. The Creator Card is rightmost, representing new decisions and directions that may be made, the Sustainer Card is in the middle, representing decisions to keep balance, and things that should not change
26.
Multiplication table
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In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an part of elementary arithmetic around the world. Many educators believe it is necessary to memorize the table up to 9 ×9, the oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60, the oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during Chinas Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras and it is also called the Table of Pythagoras in many languages, sometimes in English. In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a table up to 99 ×99. Leslie also recommended that young pupils memorize the table up to 25 ×25. The illustration below shows a table up to 12 ×12, there is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below, Figure 1 is used for multiples of 1,3,7, Figure 2 is used for the multiples of 2,4,6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, as you would start on the number you are multiplying, when you multiply by 0, you stay on 0. For example, to all the multiples of 7, Look at the 7 in the first picture. The next number in the direction of the arrow is 4, so think of the next number after 7 that ends with 4, which is 14. The next number in the direction of the arrow is 1, so think of the next number after 14 that ends with 1, which is 21. After coming to the top of column, start with the bottom of the next column. So think of the number after 21 that ends with 8. Proceed in the way until the last number,3. Next, use the 0 at the bottom, then, start again with the 7. This time it will correspond to 77, tables can also define binary operations on groups, fields, rings, and other algebraic systems
27.
Least common multiple
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Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm as 0 for all a, the LCM is the lowest common denominator that must be determined before fractions can be added, subtracted or compared. The LCM of more than two integers is also well-defined, it is the smallest positive integer that is divisible by each of them, a multiple of a number is the product of that number and an integer. For example,10 is a multiple of 5 because 5 ×2 =10, because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle,10 is the least common multiple of −5, in this article we will denote the least common multiple of two integers a and b as lcm. The programming language J uses a*. b What is the LCM of 4 and 6. Multiples of 4 are,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76. and the multiples of 6 are,6,12,18,24,30,36,42,48,54,60,66,72. Common multiples of 4 and 6 are simply the numbers that are in both lists,12,24,36,48,60,72. So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12. For instance,221 +16 =442 +742 =1142 where the denominator 42 was used because it is the least common multiple of 21 and 6. This formula is valid when exactly one of a and b is 0. However, if both a and b are 0, this formula would cause division by zero, lcm =0 is a special case, there are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, lcm =21 ⋅6 gcd =21 ⋅6 gcd =21 ⋅63 =1263 =42. Because gcd is a divisor of both a and b, it is efficient to compute the LCM by dividing before multiplying. This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results. Because gcd is a divisor of both a and b, the division is guaranteed to yield an integer, so the result can be stored in an integer. Done this way, the previous becomes, lcm =21 gcd ⋅6 =21 gcd ⋅6 =213 ⋅6 =7 ⋅6 =42. The unique factorization theorem says that every integer greater than 1 can be written in only one way as a product of prime numbers
28.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
29.
Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
30.
Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion, a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the forms a gradation. Consonance is associated with sweetness, pleasantness and acceptability and dissonance with harshness, unpleasantness, as Hindemith stressed, The two concepts have never been completely explained, and for a thousand years the definitions have varied. The opposition can be made in different contexts, In acoustics or psychophysiology, in modern times, it usually is based on the perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds. In music, even if the opposition often is founded on the preceding, objective distinction, it often is subjective, conventional, cultural. A major second would be considered dissonant if it occurred in a J. S, Bach prelude from the 1700s, however, the same interval may sound consonant in the context of a Claude Debussy piece from the early 1900s or an atonal contemporary piece. For this reason, consonance and dissonance have been considered particularly in the case of Western polyphonic music, most historical definitions of consonance and dissonance since about the 16th century have stressed their pleasant/unpleasant, or agreeable/disagreeable character. In addition, the oppositions pleasant/unpleasant or agreeable/disagreeable evidence a confusion between the concepts of dissonance and of noise and these include, Frequency ratios, with ratios of lower simple numbers being more consonant than those that are higher. Many of these definitions do not require exact integer tunings, only approximation, coincidence of partials, with consonance being a greater coincidence of partials. By this definition, consonance is dependent not only on the width of the interval between two notes, but also on the spectral distribution and thus sound quality of the notes. Thus, a note and the note one octave higher are highly consonant because the partials of the note are also partials of the lower note. Although Helmholtzs work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings, fusion, perception of unity or tonal fusion between two notes. A stable tone combination is a consonance, consonances are points of arrival, rest, an unstable tone combination is a dissonance, its tension demands an onward motion to a stable chord. Thus dissonant chords are active, traditionally they have been considered harsh and have expressed pain, grief, in Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance. Both consonance and dissonance are words applied to harmony, chords, and intervals and, by extension, to melody, tonality, and even rhythm and metre. Nevertheless, the ideas of dissonance, consonance, and resolution exist in some form in all musical traditions that have a concept of melody, harmony. Dissonance being the complement of consonance it may be defined, as above, as non-coincidence of partials, lack of fusion or pattern matching, or as complexity. For many musicians and composers, the ideas of dissonance
31.
Mandolin
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A mandolin is a musical instrument in the lute family and is usually plucked with a plectrum or pick. It commonly has four courses of doubled metal strings tuned in unison, although five, the courses are normally tuned in a succession of perfect fifths. It is the member of a family that includes the mandola, octave mandolin, mandocello. There are many styles of mandolin, but three are common, the Neapolitan or round-backed mandolin, the mandolin and the flat-backed mandolin. The round-back has a bottom, constructed of strips of wood. The carved-top or arch-top mandolin has a shallower, arched back. The flat-backed mandolin uses thin sheets of wood for the body, each style of instrument has its own sound quality and is associated with particular forms of music. Neapolitan mandolins feature prominently in European classical music and traditional music, carved-top instruments are common in American folk music and bluegrass music. Flat-backed instruments are used in Irish, British and Brazilian folk music. Some modern Brazilian instruments feature a fifth course tuned a fifth lower than the standard fourth course. There has also been a type and an instrument with sixteen-strings. Much of mandolin development revolved around the soundboard, pre-mandolin instruments were quiet instruments, strung with as many as six courses of gut strings, and were plucked with the fingers or with a quill. However, modern instruments are louder—using four courses of metal strings, the modern soundboard is designed to withstand the pressure of metal strings that would break earlier instruments. The soundboard comes in many shapes—but generally round or teardrop-shaped, sometimes with scrolls or other projections, there is usually one or more sound holes in the soundboard, either round, oval, or shaped like a calligraphic f. A round or oval sound hole may be covered or bordered with decorative rosettes or purfling, Mandolins evolved from the lute family in Italy during the 17th and 18th centuries, and the deep bowled mandolin, produced particularly in Naples, became common in the 19th century. Dating to around c.13,000 BC, a painting in the Trois Frères cave in France depicts what some believe is a musical bow. From the musical bow, families of stringed instruments developed, since each string played a note, adding strings added new notes, creating bow harps, harps. In turn, this led to being able to play dyads and chords, another innovation occurred when the bow harp was straightened out and a bridge used to lift the strings off the stick-neck, creating the lute
32.
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.
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
34.
Microtonal music
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Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are also called microintervals. It may also be extended to any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. Microtonal music can refer to any music containing microtones, therefore it is important to comprehend what a microtone is, the words microtone and microtonal were coined before 1912 by Maud MacCarthy Mann in order to avoid the misnomer quarter tone when speaking of the srutis of Indian music. Prior to this time the quarter tone was used, confusingly, not only for an interval actually half the size of a semitone. 1998, Wallon 1980,13, Whitfield 1989,13, microinterval is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers. In English, the two terms microtone and microinterval are synonymous, microtone is also sometimes used to refer to individual notes, microtonal pitches added to and distinct from the familiar twelve notes of the chromatic scale, as enharmonic microtones, for example. In English the word microtonality is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the blues scales. It was used earlier by W. McNaught with reference to developments in modernism in a 1939 record review of the Columbia History of Music. The term microinterval is used alongside microtone by American musicologist Margo Schulter in her articles on medieval music, the term microtonal music usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues, many microtonal equal divisions of the octave have been proposed, usually in order to achieve approximation to the intervals of just intonation. Terminology other than microtonal has been used or proposed by some theorists, in 1914, A. H. Fox Strangways objected that heterotone would be a better name for śruti than the usual translation microtone. Modern Indian researchers yet write, microtonal intervals called shrutis, a similar term, subchromatic, has been used recently by theorist Marek Žabka. Ivor Darreg proposed the term xenharmonic, another Russian authors use more international adjective microtonal and rendered it in Russian as микротоновый, but not microtonality. However are used микротональность and микротоника also, some authors writing in French have adopted the term micro-intervallique to describe such music. Italian musicologist Luca Conti dedicated two his monographs to microtonalismo, which is the term in Italian, and also in Spanish. The analogous English form, microtonalism, is found occasionally instead of microtonality. At the time when serialism and neoclassicism were still incipient a third movement emerged, the term macrotonal has also been used for musical form. The Hellenic civilizations of ancient Greece left fragmentary records of their music—e. g, ancient Greek intervals were of many different sizes, including microtones
35.
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
36.
Harry Partch
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Harry Partch was an American composer, music theorist, and creator of musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century composers in the West to work systematically with microtonal scales. He built custom-made instruments in these tunings on which to play his compositions, to play his music, Partch built a large number of unique instruments, with such names as the Chromelodeon, the Quadrangularis Reversum, and the Zymo-Xyl. Partch described his music as corporeal, and distinguished it from abstract music, ancient Greek theatre and Japanese Noh and kabuki heavily influenced his music theatre. Encouraged by his mother, Partch learned several instruments at a young age, by fourteen, he was composing, and in particular took to setting dramatic situations. He dropped out of the University of Southern Californias School of Music in 1922 over dissatisfaction with the quality of his teachers, in 1930, he burned all his previous compositions in a rejection of the European concert tradition. Partch frequently moved around the US, early in his career, he was a transient worker, and sometimes a hobo, later he depended on grants, university appointments, and record sales to support himself. In 1970, supporters created the Harry Partch Foundation to administer Partchs music, Partch was born on June 24,1901, in Oakland, California. His parents were Virgil Franklin Partch and Jennie, the Presbyterian couple were missionaries, and served in China from 1888 to 1893, and again from 1895 to 1900, when they fled the Boxer Rebellion. Partch moved with his family to Arizona for his mothers health and his father worked for the Immigration Service there, and they settled in the small town of Benson. It was still the Wild West there in the twentieth century. Nearby, there were native Yaqui people, whose music he heard and his mother sang to him in Mandarin Chinese, and he heard and sang songs in Spanish and the Yaqui language. His mother encouraged her children to music, and he learned the mandolin, violin, piano, reed organ. His mother taught him to read music, the family moved to Albuquerque, New Mexico, in 1913, where Partch seriously studied the piano. He had work playing keyboards for silent films while he was in high school, by 14, he was composing for the piano. He early found an interest in writing music for dramatic situations, and often cited the lost composition Death, Partch graduated from high school in 1919. The family moved to Los Angeles in 1919 following the death of Partchs father, there, his mother was killed in a trolley accident in 1920. He enrolled in the University of Southern Californias School of Music in 1920 and he moved to San Francisco and studied books on music in the libraries there and continued to compose
37.
Tony Conrad
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Anthony Schmalz Tony Conrad was an American avant-garde video artist, experimental filmmaker, musician, composer, sound artist, teacher, and writer. Active in a variety of media since the early 1960s, he was a pioneer of both film and drone music. Conrad was born in Concord, New Hampshire to Mary Elizabeth Parfitt, but raised in Baldwin, Maryland and Northern Virginia. His father worked with Everett Warner during World War II in designing dazzle camouflage for the US Navy, Conrad graduated from Harvard University in 1962 with a degree in Mathematics. While studying at Harvard, Conrad was exposed to the ideas of John Cage, after graduating, Conrad went to Copenhagen to see a friend, who was a research mathematician working on a computer project at the Physics Chemist Institute. That computer was the computer in Denmark at the time. Conrad worked on it in machine language during the summer, which helped build his computer skills and he also did work as a computer programmer for a year when he came back to the United States. After working as a programmer, Conrad got into the experimental music scene in New York City. At the time, film was institutionally unattached, which drew Conrad towards the community of New York filmmakers, in 1966, he made his first film, The Flicker, said to be a landmark in structural filmmaking. The film consists of completely black and completely white images. When the film was first screened several viewers in the audience became physically ill, Conrad wished to generalize the whole technology of film. He approached the film by considering the relationship between the subjective conditions of the flicker, and its relation to narrative and storytelling. Conrads concept came from an attempt at pushing the framework of film. He wanted to make a film that would last fifty years, Conrad began to work in video and performance in the 1970s as a professor at Antioch College in Ohio, where he overlapped with the filmmaker Paul Sharits. In 1976, Conrad joined the faculty at the Center for Media studies at the University at Buffalo. While in Buffalo, Conrad was part of a scene that included Sharits, as well as Hollis Frampton, Steina and Woody Vasulka, Peter Weibel, James Blue, and Gerald OGrady. Their practices in film, video, performance, and other forms were documented in the 2008 book Buffalo Heads, Media Study, Media Practice, Media Pioneers, 1973–1990, edited by Vasulka, in the mid-1970s, Conrad began performing film. With Sukiyaki Film he decided that the film should be prepared immediately before viewing, Sukiyaki was chosen as the paradigm for the work because it is a dish often cooked immediately before eating, in front of the diners
38.
Glenn Branca
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Glenn Branca is an American avant-garde composer and guitarist known for his use of volume, alternative guitar tunings, repetition, droning, and the harmonic series. Branca received a 2009 Foundation for Contemporary Arts Grants to Artists Award, born in Harrisburg, Pennsylvania, Branca started playing the guitar at age 15. He also created a number of sound art collage pieces for his own amusement. After attending York College in 1966–1967, he started the cover band The Crystal Ship with Al Whiteside. In the early 1970s, Branca studied theater at Emerson College in Boston, in 1973, he moved from Boston to London with his then girlfriend Meg English. After moving back to Boston in 1974, he met John Rehberger, while there, he began experimenting with sound as the founder of an experimental theater group called Bastard Theatre in 1975. Working out of a loft on Massachusetts Avenue they wrote and produced the music/theater piece Anthropophagoi for a two-week run, in 1976, The Bastard Theatres second production was What Actually Happened at a new loft in Central Square, Cambridge and later at The Boston Arts Group. Considering the unconventional and sometimes confrontational nature of the productions, the shows still received interested reviews from the Phoenix, all music for Bastard Theatre productions were original compositions by Branca or Rehberger and were performed live by the actor/musicians. In 1976 Branca moved to New York City to continue in experimental theater and he encountered the N. Dodo Band and watched their rehearsals in Chelsea, hoping to use the space for a theater production. Branca spent time with one of its members, Jeffrey Lohn, the two began forming a theater group when Branca decided he wanted to form a band, which he called the Static and later Theoretical Girls. Branca put up posters to recruit members, and after seeing one of the posters, lohns girlfriend Margaret De Wys joined the band as its bassist, and they borrowed drummer Mike Anthol from the N. Dodo Band. Artist Dan Graham booked the band at Franklin Furnace for its first performance, the group reformed in 1977 with Wharton Tiers as its drummer. In 1982, Branca launched his own label, Neutral Records, releasing Y Pants LP. In the early 1980s, he released his first album under his own name, in the same year, he composed several medium-length compositions for electric guitar ensembles, including The Ascension and Indeterminate Activity of Resultant Masses. The Ascension appeared on his second same titled album in 1981. 3, he began to compose for the harmonic series. In this project, Branca was initially influenced by the writings of Dane Rudhyar, Hermann von Helmholtz and he also built several electrically amplified instruments of his own invention, expanding his ensemble beyond the guitar. A few of these instruments were third bridge zithers he called harmonic guitars, in the early 1990s, David Baratier attempted to document Brancas teaching style in They Walked in Line
39.
Yuri Landman
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Yuri Landman started as a comic book artist and made his debut in the comics field in 1997 with Je Mag Alles Met Me Doen. In the follow-up, released in 1998, Het Verdiende Loon, for the second title he received the 1998 Breda Prize, an award for rising new comic artists in the Netherlands. Since then he has published no other comic books, together with Cees van Appeldoorn, he formed the lo-fi band Zoppo playing bass and prepared guitar in 1997. After 2 albums and several 7” singles, Landman left the band in 2000, Landman then formed the noise band Avec Aisance with drummer/producer Valentijn Höllander and released a CD, Vivre dans l’aisance in 2004. After quitting Avec-A in 2006, he focused mainly on instrument building, Landman is musically untrained and cannot play chords. While with Avec-A, Landman began creating and building several experimental string instruments, including electric zithers, electric Cymbalum, in the period between 2000-2005, Landman created 9 prototype instruments. In 2006 he changed his focus and stopped to perform. The Moodswinger was the first instrument Landman made for the band Liars, after the Moodswinger, he started making more instruments for other bands as well. From November 2006 to January 2007 Landman finished 2 copies of The Moonlander, the Springtime is an experimental electric guitar with seven strings and three outputs. The first prototype of this instrument, created in 2008, was made for guitar player Laura-Mary Carter of Blood Red Shoes, afterwards he also made copies for Lou Barlow and dEUS Mauro Pawlowski. For John Schmersal of Enon he built the Twister guitar, a version of the Springtime. In 2009 he finished instruments for The Dodos, Liam Finn, HEALTH, Micachu, for The Dodos and Finn he created electric 24 string drum guitars called the Tafelberg and for Andrews an electric 17 string harp guitar called the Burner guitar. He also started to again after a Perpignan Festival hosted by Vincent Moon. Meanwhile, he continued to build instruments for such as These Are Powers, Women. For his own musical career he develops a 25-meter electric long-string instrument, the Moodswinger led to a research on harmonics theory. He published a clarifying 3rd Bridge diagram related to this subject in 2012, besides his output as an inventor and noise musician, he starts to focus on music education, and participatory art. At first Landman started giving lectures at venues, festivals. He published an extensive 8 chapter guide on how to prepare a guitar and his lectures and presentations with his instruments lead to a request in 2009 for a practical building workshop
40.
Tuning peg
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A tuning peg is used to hold a string in the pegbox of a stringed instrument. It may be made of ebony, rosewood, boxwood or other material, some tuning pegs are ornamented with shell, metal, or plastic inlays, beads or rings. Turning the peg tightens or loosens the string, changing the pitch produced when the string is played, a pegbox is the part of certain stringed musical instruments that houses the tuning pegs. Friction pegs are most often used on violin family instruments They are also used on instruments, such as the Bulgarian gadulka. A properly working peg will turn easily and hold reliably, that is, modern pegs for violin and viola have conical shafts, turned to a 1,30 taper, changing in diameter by 1 mm over a distance of 30 mm. The taper allows the peg to turn more easily pulled out slightly. Peg dope is a used to coat the bearing surfaces of the tuning pegs of string instruments. Manufactured varieties are sold in either a small stick, a block. A commonly used home expedient treatment involves soap and chalk, in varying proportion depending on whether the peg tends to slip or stick, Peg dope serves two different purposes. It both lubricates the peg shaft so it easily in the pegbox and provides friction to keep the pegs from slipping with the force of string tension. Tuning pegs that are fitted and properly doped will both turn smoothly throughout an entire rotation and hold firmly wherever the player wishes. Without the proper amount of friction to hold the peg in place, string instruments with pegs that are slipping can be tuned briefly, but will be out of tune within minutes as soon as the peg slips again. With too much friction, adjusting the tuning at all is impossible, if the pegs or their holes are not perfectly round, or if the bearing surfaces of the pegs are indented from wear, peg dope will not remedy the resulting problems. Pegs for double bass and guitar family instruments are usually geared, machine head Peg shapers and pegbox reamers in various tapers
41.
Nut (string instrument)
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A nut, on a stringed musical instrument, is a small piece of hard material that supports the strings at the end closest to the headstock or scroll. The nut marks one end of the length of each open string, sets the spacing of the strings across the neck. Along with the bridge, the nut defines the vibrating lengths of the open strings, the nut may be made of ebony, ivory, cow bone, brass, Corian or plastic, and is usually notched or grooved for the strings. The grooves are designed to lead the string from the fingerboard to the headstock or pegbox in a smooth curve, bowed string instruments in particular benefit from an application of soft pencil graphite in the notches of the nut, to preserve the delicate flat windings of their strings. The word may have come from the German Nut, meaning groove or slot, in French, the nut is known as a fr, sillet, which, like German, can also translate to mean saddle. The Italian term, capo tasto, is the origin of the capo, not all string instruments have nuts as described, The nuts on some instruments are notched deeply enough that they are just string spacers. These instruments use a zero fret—a fret at the beginning of the scale where a normal nut would be that provides the correct string clearance, the zero fret is often found on less expensive instruments, as it is easier to set up an instrument this way. However, a zero fret also makes the sound of the open string similar to fretted notes, a conventional nut can make open strings sound slightly different—and for this reason some high-end instruments use a zero fret. String slots in a nut without a zero fret must be cut to the depth to make string height correct. Strings that are too low at the nut can buzz against the frets, some fretted instruments have a compensated nut. This type of nut provides better intonation across the instrument, the principle, given that strings are different thicknesses and have different tensions, the temperament of each fret is not 100% accurate for an equal temperament instrument. This is especially evident on the first few frets of an electric guitar, many guitar players notice how open position chords never sound in tune with each other. A compensated nut aims to correct this, by staggering the starting position of each string according to thickness, while not a complete solution such as a true temperament fretboard, there is a noticeable difference in tuning within chords. Many guitar companies, such as Music Man, and ESP include compensated nuts as standard on most of their instruments, another type is a locking nut. This nut—usually used in conjunction with a vibrato system such as a Floyd Rose or Kahler—clamps the strings against the nut. This improves tuning stability when using the vibrato bar, a drawback however, is that the locking nut must be loosened using an Allen wrench to tune outside the range of the fine tuners on the bridge. The erhu does not use a hard nut to define the length of the open string, but rather a qiān jin, a loop of string, or, less commonly. Some guitars have a rolling nut, in this design, made popular by the Fender company, the strings sit on roller bearings instead of nut slots
42.
Resonator
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The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones, another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency. A cavity resonator is one in which waves exist in a space inside the device. Acoustic cavity resonators, in sound is produced by air vibrating in a cavity with one opening, are known as Helmholtz resonators. A physical system can have as many resonant frequencies as it has degrees of freedom, systems with one degree of freedom, such as a mass on a spring, pendulums, balance wheels, and LC tuned circuits have one resonant frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers can have two resonant frequencies, a crystal lattice composed of N atoms bound together can have N resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to the next becomes significant, the vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next. The term resonator is most often used for an object in which vibrations travel as waves, at an approximately constant velocity, bouncing back. The material of the resonator, through which the waves flow, therefore, they can have millions of resonant frequencies, although only a few may be used in practical resonators. The oppositely moving waves interfere with other, and at its resonant frequencies reinforce each other to create a pattern of standing waves in the resonator. If the distance between the sides is d, the length of a trip is 2 d. To cause resonance, the phase of a sinusoidal wave after a round trip must be equal to the initial phase so the waves self-reinforce. The above analysis assumes the medium inside the resonator is homogeneous, so the waves travel at a constant speed, and they are then called overtones instead of harmonics. There may be several such series of resonant frequencies in a single resonator, an electrical circuit composed of discrete components can act as a resonator when both an inductor and capacitor are included. Oscillations are limited by the inclusion of resistance, either via a specific resistor component, such resonant circuits are also called RLC circuits after the circuit symbols for the components. A distributed-parameter resonator has capacitance, inductance, and resistance that cannot be isolated into separate lumped capacitors, inductors, an example of this, much used in filtering, is the helical resonator. A single layer coil that is used as a secondary or tertiary winding in a Tesla coil or magnifying transmitter is also a distributed resonator
43.
Endpin
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The endpin or spike is the component of a cello or double bass that makes contact with the floor to support the instruments weight. It is made of metal, carbon fiber, or, occasionally, wood, and is extensible from the bottom of the instrument, most bass clarinets and contrabassoons also have a similar fixture. Endpins are usually tipped with a point to stick into the floor, also, some endpins have a secondary extension for tall musicians. The endpin also may have notches cut in it, allowing it to have extra holding strength at these points and this design is particularly common on string basses and beginning-level cellos. The former often require these because of their greater weight, left-hand pressure on a cello fingerboard, acting against the fulcrum of the players chest and/or knees, may cause the endpin to slip forward on the floor. To prevent this slippage, objects known as endpin stoppers, pinstops, donuts, black holes, endpin anchors, endpin holders, one type of endpin stopper is placed between the endpin and the floor to add surface area and enhance friction, and stands alone. With this sort, the base must be out of a non-slippery material like rubber. One very common type consists of a disc surrounding a circular cup to hold the endpins tip. A different sort of endpin stopper uses the musicians chair as an anchor, t-shaped wooden stoppers are anchored by placing the top of the T behind the chair legs. Straight plank stoppers use one or two straps with loops at the end which are anchored around the chair legs, basses do not always require stoppers, being heavier and usually played in a more vertical orientation than a cello. However, bassists often use rockstops when sitting on a stool or when playing on high-glossed floors or uneven surfaces, pointed endpins can cause extensive damage, especially to tile and wooden flooring. Many music rooms bear evidence of this in a myriad of holes or chips. Here, rubber tips and/or stoppers are beneficial, on carpet, the damage is less extensive. The bare tip is thus most effective in outdoor conditions, carpeted areas, however, the sharper the endpin, the more likely it is to go through the standard rubber tip
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Guqin
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The guqin is a plucked seven-string Chinese musical instrument of the zither family. It is sometimes referred to by the Chinese as the father of Chinese music or the instrument of the sages, the guqin is not to be confused with the guzheng, another Chinese long zither also without frets, but with moveable bridges under each string. The prefix gu- was later added for clarification, thus, the instrument is called guqin today. It can also be called qixian-qin, because Robert Hans van Guliks book about the qin is called The Lore of the Chinese Lute, the guqin is sometimes inaccurately called a lute. Other incorrect classifications, mainly from music compact discs, include harp or table-harp, the guqin is a very quiet instrument, with a range of about four octaves, and its open strings are tuned in the bass register. Its lowest pitch is two octaves below middle C, or the lowest note on the cello. Sounds are produced by plucking open strings, stopped strings, the use of glissando—sliding tones—gives it a sound reminiscent of a pizzicato cello, fretless double bass or a slide guitar. The qin is also capable of a lot of harmonics, of which 91 are most commonly used and indicated by the dotted positions, by tradition the qin originally had five strings, but ancient qin-like instruments with 10 or more strings have been found. The modern form has been standardized for about two millennia, legend has it that the qin, the most revered of all Chinese musical instruments, has a history of about 5,000 years. This legend states that the figures of Chinas pre-history — Fuxi, Shennong and Huang Di. Nearly all qin books and tablature collections published prior to the twentieth century state this as the origins of the qin. It is mentioned in Chinese writings dating back nearly 3,000 years, the exact origins of the qin is still a very much continuing subject of debate over the past few decades. It is the longest excerpt included on the disc, in 2003, guqin music was proclaimed as one of the Masterpieces of the Oral and Intangible Heritage of Humanity by UNESCO. There are a number of ancient sources that discuss qin lore, qin theory, some of these books are available inserted into certain qinpu. The detail can be very concise to extremely detailed and thorough, some are mostly philosophical or artistic musings, others are scientific and technical. As with any other tradition, there are differences in ideals. Therefore, there exist different schools and societies which transmit these different ideas, many qin schools known as qin pai developed over the centuries. Such schools generally formed around areas where qin activity was greatest, some schools have come and gone, and some have offshoots
45.
Koto (instrument)
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The koto is a traditional Japanese stringed musical instrument derived from the Chinese zheng, and similar to the Mongolian yatga, the Korean gayageum, and the Vietnamese đàn tranh. The koto is the instrument of Japan. Koto are about 180 centimetres length, and made from kiri wood and they have 13 strings that are usually strung over 13 movable bridges along the width of the instrument. There is also a 17-string koto variant, players can adjust the string pitches by moving the white bridges before playing. To play the instrument, the strings are plucked using three finger picks, otherwise known as plectra, the character for koto is 箏, although 琴 is often used. However, 琴 usually refers to another instrument, the kin. 箏, however, many times the character 箏 is used in titles, while 琴 is used in telling the number of kotos used. The ancestor of the koto was the Chinese zheng and was first introduced to Japan from China in the 7th and 8th century, the first known version had five strings, which eventually increased to seven strings. This particular instrument is known throughout Asia but in different forms, the Japanese koto, which is a relative to the Chinese zheng, the Korean gayageum, and this variety of instrument came in two basic forms, a zither that had bridges and zithers without bridges. When the koto was first imported to Japan, the native word koto was a term for any. Over time the definition of koto could not describe the variety of these stringed instruments. The azumagoto or yamatogoto was called the wagon, the kin no koto was called the kin, the modern koto originates from the gakusō used in Japanese court music. It was a popular instrument among the wealthy, the instrument koto was considered a romantic one, some literary and historical records indicate that solo pieces for koto existed centuries before sōkyoku, the music of the solo koto genre, was established. According to Japanese literature, the koto was used as imagery, in one part of The Tales of Genji, Genji falls deeply in love with a mysterious woman, who he has never seen before, after he hears her playing the koto from a distance. Perhaps the most important influence on the development of koto was Yatsuhashi Kengyo and he was a gifted blind musician from Kyoto who changed the limited selection of six songs to a brand new style of koto music which he called kumi uta. Yatsuhashi changed the Tsukushi goto tunings, which were based on gagaku ways of tuning, and with this change, Yatsuhashi Kengyo is now known as the Father of Modern Koto. A smaller influence in the evolution of the koto is found in the inspiration of a woman named Keiko Nosaka, Keiko Nosaka, felt confined by playing a koto with just 13 strings, so she created new versions of the instrument with 20 or more strings. Japanese developments in bridgeless zithers include the one-stringed koto and two-stringed koto, around the 1920s, Goro Morita created a new version of the two-stringed koto, on this koto, one would push down buttons above the metal strings like the western autoharp. It was named the taishōgoto after the Taishō period, at the beginning of the Meiji Period, western music was introduced to Japan
Play (help·info). Half of this length will produce a pitch two octaves higher than the original—four times the initial frequency (4:1)—and so on. Standard diatonic Pythagorean tuning (Ptolemy's Diatonic Ditonic) is easily derived starting from superparticular ratios, (n+1)/n, constructed from the first four counting numbers, the tetractys, measured out on a monochord.[citation needed] The mathematics involved include the multiplication table, least common multiples, and prime and composite numbers.[4]
"Monochord". Encyclopædia Britannica. 18 (11th ed.). 1911.
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