1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Fundamental frequency
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The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as f0, in other contexts, it is more common to abbreviate it as f1, the first harmonic. All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. The period of a waveform is the T for which the equation is true. This means that this equation and a definition of the values over any interval of length T is all that is required to describe the waveform completely. Every waveform may be described using any multiple of this period, there exists a smallest period over which the function may be described completely and this period is the fundamental period. The fundamental frequency is defined as its reciprocal, f 0 =1 T Since the period is measured in units of time, when the time units are seconds, the frequency is in s −1, also known as Hertz. For a tube of length L with one end closed and the end open the wavelength of the fundamental harmonic is 4 L. If the ends of the tube are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2 L. By the same method as above, the frequency is found to be f 0 = v 2 L. At 20 °C the speed of sound in air is 343 m/s and this speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature. The fundamental may be created by vibration over the length of a string or air column. The fundamental is one of the harmonics, a harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The reason a fundamental is also considered a harmonic is because it is 1 times itself, the fundamental is the frequency at which the entire wave vibrates. Overtones are other components present at frequencies above the fundamental. All of the components that make up the total waveform, including the fundamental
3.
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
4.
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
5.
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
6.
Standing wave
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In physics, a standing wave – also known as a stationary wave – is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude. The locations at which the amplitude is minimum are called nodes, standing waves were first noticed by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container, franz Melde coined the term standing wave around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings. For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy, as an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots, standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. Many standing river waves are popular river surfing breaks, the effect is a series of nodes and anti-nodes at fixed points along the transmission line. The failure of the line to power at the standing wave frequency will usually result in attenuation distortion. In practice, losses in the line and other components mean that a perfect reflection. The result is a standing wave, which is a superposition of a standing wave. The degree to which the wave resembles either a standing wave or a pure traveling wave is measured by the standing wave ratio. Another example is standing waves in the ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, in one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. The reflected wave has to have the amplitude and frequency as the incoming wave. At locations which are multiples of a quarter wavelength x =. −3 λ2, − λ, − λ2,0, λ2, λ,3 λ2, called the nodes, the amplitude is always zero, whereas at locations which are odd multiples of a quarter wavelength x =. −5 λ4, −3 λ4, − λ4, λ4,3 λ4,5 λ4, called the anti-nodes, the amplitude is maximum, with a value of twice the amplitude of the original waves. The distance between two nodes or anti-nodes is λ/2. Standing waves can occur in two- or three-dimensional resonators
7.
Sound
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In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a transmission medium such as air or water. In physiology and psychology, sound is the reception of such waves, humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and below 20 Hz is infrasound, other animals have different hearing ranges. Acoustics is the science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound. A scientist who works in the field of acoustics is an acoustician, an audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound. Auditory sensation evoked by the oscillation described in, sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium, as the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, at an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave and this is intuitively obvious for a solid, and the same is true for liquids and gases. During propagation, waves can be reflected, refracted, or attenuated by the medium, the behavior of sound propagation is generally affected by three things, A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium, if the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the wave will be decreased by the speed of the wind. Medium viscosity determines the rate at which sound is attenuated, for many media, such as air or water, attenuation due to viscosity is negligible. When sound is moving through a medium that does not have constant physical properties, the mechanical vibrations that can be interpreted as sound can travel through all forms of matter, gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium, sound cannot travel through a vacuum. Sound is transmitted through gases, plasma, and liquids as longitudinal waves and it requires a medium to propagate
8.
Resonance
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In physics, resonance is a phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at a specific preferential frequency. Frequencies at which the amplitude is a relative maximum are known as the systems resonant frequencies or resonance frequencies. At resonant frequencies, small periodic driving forces have the ability to produce large amplitude oscillations, Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes. However, there are some losses from cycle to cycle, called damping, when damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies, resonant systems can be used to generate vibrations of a specific frequency, or pick out specific frequencies from a complex vibration containing many frequencies. Resonance occurs widely in nature, and is exploited in many manmade devices and it is the mechanism by which virtually all sine waves and vibrations are generated. Many sounds we hear, such as when hard objects of metal, glass, light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. A familiar example is a swing, which acts as a pendulum. Pushing a person in a swing in time with the interval of the swing makes the swing go higher and higher. This is because the energy the swing absorbs is maximized when the match the swings natural oscillations. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, avoiding resonance disasters is a major concern in every building, tower, and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies, the Taipei 101 building relies on a 660-tonne pendulum —a tuned mass damper—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur, buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. Clocks keep time by mechanical resonance in a wheel, pendulum. The cadence of runners has been hypothesized to be energetically favorable due to resonance between the energy stored in the lower limb and the mass of the runner. Acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations across the range of human hearing. Like mechanical resonance, acoustic resonance can result in failure of the object at resonance. The classic example of this is breaking a glass with sound at the precise resonant frequency of the glass
9.
Harmonic
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A harmonic is any member of the harmonic series, the divergent infinite series. Every term of the series after the first is the mean of the neighboring terms. The phrase harmonic mean likewise derives from music, the term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves, a harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the frequency, the sum of harmonics is also periodic at that frequency. On strings, harmonics that are bowed have a glassy, pure tone, harmonics may also be called overtones, partials or upper partials. In some music contexts, the harmonic, overtone and partial are used fairly interchangeably. Most acoustic instruments emit complex tones containing many individual partials, rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are long and thin. Wind instruments whose air column is open at one end, such as trumpets and clarinets. However they only produce partials matching the odd harmonics, at least in theory, the reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials, antique singing bowls are known for producing multiple harmonic partials or multiphonics. An overtone is any partial higher than the lowest partial in a compound tone, the relative strengths and frequency relationships of the component partials determine the timbre of an instrument. This chart demonstrates how the three types of names are counted, In many musical instruments, it is possible to play the upper harmonics without the note being present. In a simple case this has the effect of making the note go up in pitch by an octave, in some cases it also changes the timbre of the note. This is part of the method of obtaining higher notes in wind instruments. The extended technique of playing multiphonics also produces harmonics, on string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch
10.
Vibration
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Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem, the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable, for example, the motion of a fork, the reed in a woodwind instrument or harmonica. In many cases, however, vibration is undesirable, wasting energy, for example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the parts, uneven friction. Careful designs usually minimize unwanted vibrations, the studies of sound and vibration are closely related. Sound, or pressure waves, are generated by vibrating structures, hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a system is set in motion with an initial input. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork, the mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, the periodic input can be a harmonic or a non-harmonic disturbance. Damped vibration, When the energy of a system is gradually dissipated by friction and other resistances. The vibrations gradually reduce or change in frequency or intensity or cease, Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT is attached to the table of a shaker, Vibration testing is performed to examine the response of a device under test to a defined vibration environment. The measured response may be life, resonant frequencies or squeak. Squeak and rattle testing is performed with a type of quiet shaker that produces very low sound levels while under operation. For relatively low frequency forcing, servohydraulic shakers are used, for higher frequencies, electrodynamic shakers are used. Generally, one or more input or control points located on the DUT-side of a fixture is kept at a specified acceleration, other response points experience maximum vibration level or minimum vibration level
11.
Timbre
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The physical characteristics of sound that determine the perception of timbre include spectrum and envelope. Singers and instrumental musicians can change the timbre of the music they are singing/playing by using different singing or playing techniques, for example, a violinist can use different bowing styles or play on different parts of the string to obtain different timbres. On electric guitar and electric piano, performers can change the timbre using effects units, in simple terms, timbre is what makes a particular musical sound have a different sound from another, even when they have the same pitch and loudness. For instance, it is the difference in sound between a guitar and a playing the same note at the same volume. Experienced musicians are able to distinguish between different instruments of the type based on their varied timbres, even if those instruments are playing notes at the same pitch. Tone quality and tone color are synonyms for timbre, as well as the attributed to a single instrument. However, the texture can also refer to the type of music, such as multiple. The sound of an instrument may be described with words such as bright, dark, warm, harsh. There are also colors of noise, such as pink and white, in visual representations of sound, timbre corresponds to the shape of the image. The Acoustical Society of America Acoustical Terminology definition 12, Timbre has been called. the psychoacousticians multidimensional waste-basket category for everything that cannot be labeled pitch or loudness. Many commentators have attempted to decompose timbre into component attributes, the richness of a sound or note a musical instrument produces is sometimes described in terms of a sum of a number of distinct frequencies. The lowest frequency is called the frequency, and the pitch it produces is used to name the note. The dominant frequency is the frequency that is most heard, for example, the dominant frequency for the transverse flute is double the fundamental frequency. Other significant frequencies are called overtones of the frequency, which may include harmonics. Harmonics are whole number multiples of the frequency, such as ×2, ×3, ×4. There are also sometimes subharmonics at whole number divisions of the fundamental frequency, most instruments produce harmonic sounds, but many instruments produce partials and inharmonic tones, such as cymbals and other indefinite-pitched instruments. When the tuning note in an orchestra or concert band is played, each instrument in the orchestra or concert band produces a different combination of these frequencies, as well as harmonics and overtones. The sound waves of the different frequencies overlap and combine, the timbre of a sound is also greatly affected by the following aspects of its envelope, attack time and characteristics, decay, sustain, release and transients
12.
Sine wave
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A sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time is, y = A sin = A sin where, A = the amplitude, F = the ordinary frequency, the number of oscillations that occur each second of time. ω = 2πf, the frequency, the rate of change of the function argument in units of radians per second φ = the phase. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds, a negative value represents a delay, and a positive value represents an advance. The sine wave is important in physics because it retains its shape when added to another sine wave of the same frequency and arbitrary phase. It is the only periodic waveform that has this property and this property leads to its importance in Fourier analysis and makes it acoustically unique. The wavenumber is related to the frequency by. K = ω v =2 π f v =2 π λ where λ is the wavelength, f is the frequency, and v is the linear speed. This equation gives a wave for a single dimension, thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. This could, for example, be considered the value of a wave along a wire, in two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, and their product as a dot product. For more complex such as the height of a water wave in a pond after a stone has been dropped in. This wave pattern occurs often in nature, including wind waves, sound waves, a cosine wave is said to be sinusoidal, because cos = sin , which is also a sine wave with a phase-shift of π/2 radians. Because of this start, it is often said that the cosine function leads the sine function or the sine lags the cosine. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a frequency with no harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, on the other hand, if the sound contains aperiodic waves along with sine waves, then the sound will be perceived noisy as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier discovered that sinusoidal waves can be used as building blocks to describe and approximate any periodic waveform
13.
Phase (waves)
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Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its initial value. The graphic to the right shows how one cycle constitutes 360° of phase, the graphic also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57. 3°. Phase can also be an expression of relative displacement between two corresponding features of two waveforms having the same frequency, in sinusoidal functions or in waves phase has two different, but closely related, meanings. One is the angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the cycle that has elapsed relative to the origin. Phase shift is any change that occurs in the phase of one quantity and this symbol, φ is sometimes referred to as a phase shift or phase offset because it represents a shift from zero phase. For infinitely long sinusoids, a change in φ is the same as a shift in time, if x is delayed by 14 of its cycle, it becomes, x = A ⋅ cos = A ⋅ cos whose phase is now φ − π2. It has been shifted by π2 radians, Phase difference is the difference, expressed in degrees or time, between two waves having the same frequency and referenced to the same point in time. Two oscillators that have the frequency and no phase difference are said to be in phase. Two oscillators that have the frequency and different phases have a phase difference. The amount by which such oscillators are out of phase with each other can be expressed in degrees from 0° to 360°, if the phase difference is 180 degrees, then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then interference will occur. It is common for waves of electromagnetic, acoustic or other energy to become superposed in their transmission medium, when that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes, time is sometimes used to express position within the cycle of an oscillation. A phase difference is analogous to two athletes running around a track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time, but the time difference between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds, the difference is undefined
14.
Fourier analysis
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In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics, for example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis, in mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transformation and its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations. Each transform used for analysis has an inverse transform that can be used for synthesis. This wide applicability stems from many useful properties of the transforms, The transforms are linear operators and, the exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently, the discrete version of the Fourier transform can be evaluated quickly on computers using Fast Fourier Transform algorithms. In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum, the FT method is used to decode the measured signals and record the wavelength data. Fourier transformation is also useful as a representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation of small pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, a large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. One function is transformed into another, and the operation is reversible, evaluating this quantity for all values of f produces the frequency-domain function. Then s can be represented as a recombination of complex exponentials of all frequencies, s = ∫ − ∞ ∞ S ⋅ e 2 i π f t d f. The complex number, S, conveys both amplitude and phase of frequency f, see Fourier series for more information, including the historical development. The DTFT is the dual of the time-domain Fourier series. Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function and that is a cornerstone in the foundation of digital signal processing. Furthermore, under idealized conditions one can theoretically recover S and s exactly
15.
Inharmonicity
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In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency. Acoustically, a note perceived to have a distinct pitch in fact contains a variety of additional overtones. Many percussion instruments, such as cymbals, tam-tams, and chimes, create complex, any departure from this ideal harmonic series is known as inharmonicity. The less elastic the strings are, the more inharmonicity they exhibit, Music harmony and intonation depends strongly on the harmonicity of tones. An ideal, homogeneous, infinitesimally thin or infinitely flexible string or column of air has exactly harmonic modes of vibration. In any real musical instrument, the resonant body that produces the music tone—typically a string, wire, or column of air—deviates from this ideal and has some small or large amount of inharmonicity. For instance, a thick string behaves less as an ideal string and more like a cylinder. For this reason, a single tone played by a string instrument, brass instrument. The inharmonicity of a string depends on its characteristics, such as tension, stiffness. For instance, a string under low tension exhibits a high degree of inharmonicity. A wound string generally exhibits less inharmonicity than the equivalent solid string, in 1943, Schuck and Young were the first scientists to measure the spectral inharmonicity in piano tones. They found that the spectral partials in piano tones are progressively stretched—that is to say, in 1962, research by Harvey Fletcher and his collaborators indicated that the spectral inharmonicity is important for tones to sound piano-like. They proposed that inharmonicity is responsible for the warmth property common to real piano tones, according to their research synthesized piano tones sounded more natural when some inharmonicity was introduced. In general, electronic instruments that duplicate acoustic instruments must duplicate both the inharmonicity and the resulting stretched tuning of the original instruments. When pianos are tuned by piano tuners, the technician sometimes listens for the sound of beating when two notes are played together, and tunes to the point that minimizes roughness between tones. Piano tuners must deal with the inharmonicity of piano strings, which is present in different amounts in all of the ranges of the instrument, the result is that octaves are tuned slightly wider than the harmonic 2,1 ratio. Because of the problem of inharmonicity, electronic piano tuning devices used by piano technicians are not designed to tune according to a harmonic series. Rather, the use various means to duplicate the stretched octaves
16.
Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has become the method of representing and comparing musical pitches. Like a decibels relation to intensity, a cent is a ratio between two close frequencies, for the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone spans 100 cents by definition, an octave—two notes that have a frequency ratio of 2, 1—spans twelve semitones and therefore 1200 cents.0005777895. For example, in just intonation the major third is represented by the frequency ratio 5,4, applying the formula at the top shows that this is about 386 cents. The equivalent interval on the piano would be 400 cents. The difference,14 cents, is about a seventh of a half step, as x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a linear function that is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1 +0. 0005946n instead of 2 n⁄1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50 and this error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes. It is difficult to establish how many cents are perceptible to humans, one author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians ability to recognize, as out-of-tune and it has also been established that increased tonal context enables listeners to judge pitch more accurately. Free, online web sites for self-testing are available, while intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords. When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch, normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents, iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone, centitone, and millitone, for example, Equal tempered perfect fifth =700 cents =175.6 savarts =583.3 millioctaves =350 centitones
17.
Percussion
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A percussion instrument is a musical instrument that is sounded by being struck or scraped by a beater, struck, scraped or rubbed by hand, or struck against another similar instrument. The percussion family is believed to include the oldest musical instruments, the percussion section of an orchestra most commonly contains instruments such as timpani, snare drum, bass drum, cymbals, triangle and tambourine. However, the section can also contain non-percussive instruments, such as whistles and sirens, percussive techniques can also be applied to the human body, as in body percussion. On the other hand, keyboard instruments, such as the celesta, are not normally part of the percussion section, Percussion instruments may play not only rhythm, but also melody and harmony. Percussion is commonly referred to as the backbone or the heartbeat of an ensemble, often working in close collaboration with bass instruments. In jazz and other popular ensembles, the pianist, bassist, drummer. Most classical pieces written for full orchestra since the time of Haydn and Mozart are orchestrated to place emphasis on the strings, woodwinds, however, often at least one pair of timpani is included, though they rarely play continuously. Rather, they serve to provide additional accents when needed, in the 18th and 19th centuries, other percussion instruments have been used, again generally sparingly. The use of percussion instruments became more frequent in the 20th century classical music, in almost every style of music, percussion plays a pivotal role. In classic jazz, one almost immediately thinks of the rhythm of the hi-hats or the ride cymbal when the word swing is spoken. Because of the diversity of instruments, it is not uncommon to find large musical ensembles composed entirely of percussion. Rhythm, melody, and harmony are all represented in these ensembles, music for pitched percussion instruments can be notated on a staff with the same treble and bass clefs used by many non-percussive instruments. Music for percussive instruments without a pitch can be notated with a specialist rhythm or percussion-clef. The word percussion has evolved from Latin terms, percussio, as a noun in contemporary English it is described in Wiktionary as the collision of two bodies to produce a sound. Hornbostel–Sachs has no high-level section for percussion, Most percussion instruments are classified as idiophones and membranophones.1 Concussion idiophones or clappers, played in pairs and beaten against each other, such as zills and clapsticks. 111.2 Percussion idiophones, includes many percussion instruments played with the hand or by a mallet, such as the hang, gongs and the xylophone. 21 Struck drums, includes most types of drum, such as the timpani, snare drum, (Included in most drum sets or 412. Stringed instruments played with such as the hammered dulcimer
18.
Marimba
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The marimba is a percussion instrument consisting of a set of wooden bars struck with mallets to produce musical tones. Resonators suspended underneath the bars amplify their sound and this instrument is a type of idiophone, but with a more resonant and lower-pitched tessitura than the xylophone. The marimba was developed in Central America by African slaves, and descended from its ancestral balafon instrument, Marimba is now the national instrument of Guatemala and Costa Rica. Modern uses of the marimba include solo performances, woodwind and brass ensembles, marimba concertos, jazz ensembles, marching band, drum and bugle corps, contemporary composers have used the unique sound of the marimba more and more in recent years. A player of the marimba can be called a marimbist or a marimba player, xylophones are widely used in music of west and central Africa. The name marimba stems from Bantu marimba or malimba, xylophone, the instrument itself is present, but is called balafon or heri in Mali and/or Guinea, while its known as gyil among the Akan peoples in and around Ghana. The word marimba and derivative words is used widely in East, other sources credit the creation of the marimba and the kalimba to Queen Marimba of the Wakambi people, who live south of Lake Victoria. In the Swahili language imba means song, Marimba, is said to be the mother of song and the creator of all the instruments, including the marimba. Mama means mother in Kiswahili, so it makes sense that the word mother would be combined with the word for song. The karimba is also said to have created by Queen Marimba. In much of East & Central Africa the karimba is seen as a version of the marimba. Diatonic xylophones were introduced to Central America in the 16th or 17th century, the first historical record of Mayan musicians using gourd resonator marimbas in Guatemala was made in 1680, by the historian Domingo Juarros. It became more widespread during the 18th and 19th centuries, as Mayan, in 1821, the marimba was proclaimed the national instrument of Guatemala in its independence proclamation. The gourd resonators were later replaced by harmonic wooden boxes, variants with slats made of steel, glass or bamboo instead of wood appeared during the 19th century. In 1892, Corazón de Jesús Borras Moreno, a musician from Chiapas, expanded marimba to include the chromatic scale by adding another row of sound bars, the name marimba was later applied to the orchestra instrument inspired by the Latin American model. In the United States, companies like Deagan and Leedy company adapted the Latino American instruments for use in western music, metal tubes were used as resonators, fine-tuned by rotating metal discs at the bottom, lowest note tubes were U-shaped. The marimbas were first used for music and dance, such as Vaudeville theater. Clair Omar Musser was a proponent of marimba in the United States at the time
19.
Vibraphone
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The vibraphone is a musical instrument in the struck idiophone subfamily of the percussion family. The vibraphone resembles the xylophone, marimba, and glockenspiel, each bar is paired with a resonator tube that has a motor-driven butterfly valve at its upper end. The valves are mounted on a shaft, which produces a tremolo or vibrato effect while spinning. The vibraphone also has a sustain pedal similar to that on a piano, with the pedal up, the bars are all damped and produce a shortened sound. With the pedal down, they sound for several seconds, the vibraphone is commonly used in jazz music, in which it often plays a featured role and was a defining element of the sound of mid-20th-century Tiki lounge exotica, as popularized by Arthur Lyman. It is the second most popular solo keyboard percussion instrument in music, after the marimba. It is an instrument in the modern percussion section for orchestras. The first musical instrument called vibraphone was marketed by the Leedy Manufacturing Company in the United States in 1921, however, this instrument differed in significant details from the instrument now called the vibraphone. The Leedy vibraphone achieved a degree of popularity after it was used in the novelty recordings of Aloha Oe and this popularity led J. C. Deagan, Inc. in 1927 to ask its Chief Tuner, Henry Schluter, to develop a similar instrument. Schluters design was more popular than the Leedy design, and has become the template for all instruments now called vibraphone, however, when Deagan began marketing Schluters instrument in 1928, they called it the vibraharp. The name derived from similar aluminum bars that were mounted vertically, since Deagan trademarked the name, others were obliged to use the earlier vibraphone for their instruments incorporating the newer design. The name confusion continues, even to the present, but over time vibraphone became significantly more popular than vibraharp, by 1974, the Directory of the D. C. Federation of Musicians listed 39 vibraphone players and 3 vibraharp players, the initial purpose of the vibraphone was to add to the large arsenal of percussion sounds used by vaudeville orchestras for novelty effects. This use was quickly overwhelmed in the 1930s by its development as a jazz instrument, the use of the vibraphone in jazz was pioneered by Paul Barbarin, the drummer with Luis Russells band. Bergerault, of Ligueil, France also began manufacturing vibraphones in the 1930s, Deagan struck endorsement deals with many of the leading players, including Lionel Hampton and Milt Jackson. The Deagan company went out of business in the 1980s, its trademark, Yamaha continues to make percussion instruments based on Deagan designs. In 1948, the Musser Mallet Company was founded by Clair Omar Musser, the Musser company continues to manufacture vibraphones as part of the Ludwig Drum Company. The standard modern instrument has a range of three octaves, from the F below middle C, larger three-and-a-half or four octave models from the C below middle C are also becoming more common
20.
Timpani
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Timpani, or kettledrums, are musical instruments in the percussion family. A type of drum, they consist of a skin called a head stretched over a large bowl traditionally made of copper and they are played by striking the head with a specialized drum stick called a timpani stick or timpani mallet. Timpani evolved from military drums to become a staple of the orchestra by the last third of the 18th century. Today, they are used in many types of ensembles, including concert bands, marching bands, orchestras. Timpani is an Italian plural, the singular of which is timpano and they are also often incorrectly termed timpanis. A musician who plays the timpani is a timpanist, alternative spellings with y in place of either or both is—tympani, tympany, or timpany—are occasionally encountered in older English texts. Although the word timpani has been adopted in the English language. The German word for timpani is Pauken, the French and Spanish is timbales, the Ashanti pair of talking drums are known as atumpan. The tympanum is defined in the Etymologiae of St. Isidore of Seville, est enim pars media symphoniae in similitudinem cribri. Tympanum autem dictum quod medium est, unde, et margaritum medium tympanum dicitur, et ipsum ut symphonia ad virgulam percutitur. The tympanum is skin or hide stretched over a wooden vessel which extends out. It is said by the symphonias to resemble a sieve, but has also likened to half a pearl. It is struck with a wand, beating time for the symphonia, the reference comparing the tympanum to half a pearl is borrowed from Pliny the Elder. The basic timpani drum consists of a drumhead stretched across the opening of a bowl made of copper or, in less expensive models, fiberglass. In the Sachs–Hornbostel classification, the timpani are thus considered membranophones, the heads tension can be adjusted by loosening or tightening the rods. Most timpani have six to eight tension rods, the shape of the bowl contributes to the quality of the drum. For example, hemispheric bowls produce brighter tones while parabolic bowls produce darker tones, another factor that affects the timbre of the drum is the quality of the bowls surface. Copper bowls may have a smooth, machined surface or a surface with many small dents hammered into it
21.
Standing bell
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Singing bowls are a type of bell, specifically classified as a standing bell. Singing bowls are used worldwide for meditation, music, relaxation, Singing bowls were historically made throughout Asia, especially Nepal, China and Japan. They are closely related to decorative bells made along the Silk Road from the Near East to Western Asia, today they are made in Nepal, India, Japan, China and Korea. In some Buddhist practices, singing bowls are used as a signal to begin, some practitioners use the singing bowl to accompany the wooden fish during chanting, striking it when a particular phrase is chanted. In Japan and Vietnam, singing bowls are used during chanting and may also mark the passage of time or signal a change in activity. In Japan, singing bowls are used in funeral rites. Every Japanese temple holds a singing bowl, Singing bowls are found on altars and in meditation rooms worldwide. There arent any traditional texts about singing bowls so far as we know, all known references to them are strictly modern. However, a few pieces of art dating from several centuries ago depict singing bowls in detail, including Tibetan paintings, some Tibetan rinpoches and monks use singing bowls in monasteries and meditation centers today. Singing bowls from at least the 15th century are found in private collections, bronze bells from Asia have been discovered as early as the 8th–10th century BC and singing bowls are thought to go back in the Himalayas to the 10th-12th century AD. Singing bowls are played by striking the rim of the bowl with a padded mallet and they can also be played by the friction of rubbing a wood, plastic, or leather wrapped mallet around the rim of the bowl to emphasize the harmonic overtones and a continuous singing sound. Both antique and new bowls are used as an aid to meditation. They are also used in yoga, music therapy, sound healing, religious services, performance, antique singing bowls produce harmonic overtones creating an effect that is unique to the instrument. The subtle yet complex multiple harmonic frequencies are a special quality caused by variations in the shape of the hand made singing bowls and they may display abstract decorations like lines, rings and circles engraved into the surface. Decoration may appear outside the rim, inside the bottom, around the top of the rim, Singing bowls are still manufactured today in the traditional way as well as with modern manufacturing techniques. New bowls may be plain or decorated and they sometimes feature religious iconography and spiritual motifs and symbols, such as the Tibetan mantra Om mani padme hum, images of Buddhas, and Ashtamangala. New singing bowls are made in two processes, hand hammering is the traditional method of creating singing bowls which is still used to make new bowls. The modern method is by sand casting and then machine lathing, machine lathing can be done only with brass, so machine lathed singing bowls are made with modern techniques and modern brass alloy
22.
Overtone
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An overtone is any frequency greater than the fundamental frequency of a sound. Using the model of Fourier analysis, the fundamental and the overtones together are called partials, harmonics, or more precisely, harmonic partials, are partials whose frequencies are integer multiples of the fundamental. These overlapping terms are used when discussing the acoustic behavior of musical instruments. The model of Fourier analysis provides for the inclusion of inharmonic partials, when a resonant system such as a blown pipe or plucked string is excited, a number of overtones may be produced along with the fundamental tone. In simple cases, such as for most musical instruments, the frequencies of these tones are the same as the harmonics. Examples of exceptions include the drum, – a timpani whose first overtone is about 1.6 times its fundamental resonance frequency, gongs and cymbals. The human vocal tract is able to produce highly variable amplitudes of the overtones, called formants, most oscillators, from a guitar string to a flute, will naturally vibrate at a series of distinct frequencies known as normal modes. The lowest normal mode frequency is known as the fundamental frequency, often, when an oscillator is excited by, for example, plucking a guitar string, it will oscillate at several of its modal frequencies at the same time. So when a note is played, this gives the sensation of hearing other frequencies above the lowest frequency, timbre is the quality that gives the listener the ability to distinguish between the sound of different instruments. The timbre of an instrument is determined by which overtones it emphasizes and that is to say, the relative volumes of these overtones to each other determines the specific flavor or color of sound of that family of instruments. The intensity of each of these overtones is rarely constant for the duration of a note, over time, different overtones may decay at different rates, causing the relative intensity of each overtone to rise or fall independent of the overall volume of the sound. A carefully trained ear can hear these changes even in a single note and this is why the timbre of a note may be perceived differently when played staccato or legato. A driven non-linear oscillator, such as the folds, a blown wind instrument, or a bowed violin string will oscillate in a periodic. This generates the impression of sound at integer multiple frequencies of the known as harmonics, or more precisely. Thus, in music, overtones are often called harmonics, depending upon how the string is plucked or bowed, different overtones can be emphasized. However, some overtones in some instruments may not be of an integer multiplication of the fundamental frequency. High quality instruments are built in such a manner that their individual notes do not create disharmonious overtones. This is illustrated by the following, Consider a guitar string and this dead length actually varies from string to string, being more pronounced with thicker and/or stiffer strings
23.
Synthesizer
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A synthesizer is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones. Synthesizers may either imitate instruments like piano, Hammond organ, flute, vocals, natural sounds like ocean waves, etc. or generate new electronic timbres. Synthesizers without built-in controllers are called sound modules, and are controlled via USB, MIDI or CV/gate using a controller device. Synthesizers use various methods to generate electronic signals, synthesizers were first used in pop music in the 1960s. In the 1970s, synths were used in disco, especially in the late 1970s, in the 1980s, the invention of the relatively inexpensive, mass market Yamaha DX7 synth made synthesizers widely available. 1980s pop and dance music often made use of synthesizers. In the 2010s, synthesizers are used in genres of pop, rock. Contemporary classical music composers from the 20th and 21st century write compositions for synthesizer, the beginnings of the synthesizer are difficult to trace, as it is difficult to draw a distinction between synthesizers and some early electric or electronic musical instruments. One of the earliest electric musical instruments, the telegraph, was invented in 1876 by American electrical engineer Elisha Gray. He accidentally discovered the sound generation from a self-vibrating electromechanical circuit and this musical telegraph used steel reeds with oscillations created by electromagnets transmitted over a telegraph line. Gray also built a simple loudspeaker device into later models, consisting of a diaphragm in a magnetic field. This instrument was a remote electromechanical musical instrument that used telegraphy, though it lacked an arbitrary sound-synthesis function, some have erroneously called it the first synthesizer. In 1897, Thaddeus Cahill invented the Teleharmonium, which used dynamos, and was capable of additive synthesis like the Hammond organ, however, Cahills business was unsuccessful for various reasons, and similar but more compact instruments were subsequently developed, such as electronic and tonewheel organs. In 1906, American engineer, Lee De Forest ushered in the electronics age and he invented the first amplifying vacuum tube, called the Audion tube. This led to new entertainment technologies, including radio and sound films, ondes Martenot and Trautonium were continuously developed for several decades, finally developing qualities similar to later synthesizers. In the 1920s, Arseny Avraamov developed various systems of graphic sonic art, in 1938, USSR engineer Yevgeny Murzin designed a compositional tool called ANS, one of the earliest real-time additive synthesizers using optoelectronics. The earliest polyphonic synthesizers were developed in Germany and the United States, during the three years that Hammond manufactured this model,1,069 units were shipped, but production was discontinued at the start of World War II. Both instruments were the forerunners of the electronic organs and polyphonic synthesizers
24.
Normal mode
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A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies and these fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge or molecule, has a set of modes and their natural frequencies that depend on its structure, materials. The most general motion of a system is a superposition of its normal modes, the modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other, when relating to music, normal modes of vibrating instruments are called harmonics or overtones. The concept of normal modes also finds application in optics, quantum mechanics, most dynamical system can be excited under several modes. Each mode is characterized by one or several frequencies, according the modal variable field, for example, a vibrating rope in the 2D space is defined by a single-frequency, but a vibrating rope in the 3D space is defined by two frequencies -2D axial displacement-. For a given amplitude on the variable, each mode will store a specific amount of energy. From all the modes of a system, the normal or dominant mode of a system, will be the mode storing the minimum amount of energy. Or equivalently, for a given stored amount of energy, will be the mode imposing the maximum amplitude of the modal variable, a mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of waves in the vibration. For example, if a beam with both ends pinned displayed a mode shape of half of a sine wave it would be vibrating in mode 1. If it had a full sine wave it would be vibrating in mode 2, in a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have a radial coordinate and an angular coordinate, if one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because half of the disk is considered due to the antisymmetric nature of a disks vibration in the angular direction. Thus, measuring 180° along the direction you would encounter a half wave. So the mode number of the system is 2-1 or 1-2, depending on which coordinate is considered the first, in linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies and different mode shapes, in a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero
25.
Cone
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
26.
Cylinder
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
27.
Wave
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In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium. Frequency refers to the addition of time, wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations, around almost fixed locations, there are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed, restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors, when the molecules collide, they also bounce away from each other. This keeps the molecules from continuing to travel in the direction of the wave, the second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves, in addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. While mechanical waves can be transverse and longitudinal, all electromagnetic waves are transverse in free space. A single, all-encompassing definition for the wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value, however, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a results in a blurred line. The term wave is often understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium and it may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity and this difference in origin introduces certain wave characteristics particular to the properties of the medium involved
28.
Vibrations
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Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem, the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable, for example, the motion of a fork, the reed in a woodwind instrument or harmonica. In many cases, however, vibration is undesirable, wasting energy, for example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the parts, uneven friction. Careful designs usually minimize unwanted vibrations, the studies of sound and vibration are closely related. Sound, or pressure waves, are generated by vibrating structures, hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a system is set in motion with an initial input. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork, the mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, the periodic input can be a harmonic or a non-harmonic disturbance. Damped vibration, When the energy of a system is gradually dissipated by friction and other resistances. The vibrations gradually reduce or change in frequency or intensity or cease, Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT is attached to the table of a shaker, Vibration testing is performed to examine the response of a device under test to a defined vibration environment. The measured response may be life, resonant frequencies or squeak. Squeak and rattle testing is performed with a type of quiet shaker that produces very low sound levels while under operation. For relatively low frequency forcing, servohydraulic shakers are used, for higher frequencies, electrodynamic shakers are used. Generally, one or more input or control points located on the DUT-side of a fixture is kept at a specified acceleration, other response points experience maximum vibration level or minimum vibration level
29.
Stretched tuning
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In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2,1 ratio, are tuned slightly farther apart. For a stretched tuning the octave is greater than a factor of 2, for example, the piano features both stretched harmonics and, to accommodate those, stretched fundamentals. The fundamental note and its harmonics sound together, and the relationships among them strongly affect the perceived tone or timbre of the instrument. Each note on the keyboard has its own separate vibrating element whose tension and/or length, in electric pianos, the motion of the vibrating element is sensed by an electromagnetic pickup and amplified electronically. In tuning, the relationship between two notes is determined by evaluating their common harmonics, for example, we say two notes are an octave apart when the fundamental frequency of the upper note exactly matches the second harmonic of the lower note. On instruments strung with wire, however, neither of these assumptions is valid. Inharmonicity refers to the difference between the theoretical and actual frequencies of the harmonics or overtones of a vibrating tine or string, the theoretical frequency of the second harmonic is twice the fundamental frequency, and of the third harmonic is three times the fundamental frequency, and so on. But on metal strings, tines, and reeds, the frequencies of those harmonics are slightly higher. A digital emulation of these instruments must recreate this inharmonicity if it is to sound convincing, when a stretched wire string is excited into motion by plucking or striking, a complex wave travels outward to the ends of the string. As it travels outward, this initial impulse forces the wire out of its resting position all along its length, after the impulse has passed, each part of the wire immediately begins to return toward its resting position, which means vibration has been induced. Meanwhile, the impulse is reflected at both ends of the string and travels back toward the center. On the way, it interacts with the vibrations it induced on the initial pass. When the reflected impulses encounter each other, their interaction again cancels some components, as a rule, the amplitude of its vibration is less for higher harmonics than for lower, meaning that higher harmonics are softer—though the details of this differ from instrument to instrument. The exact combination of different harmonics and their amplitudes is a factor affecting the timbre or tone quality of a particular musical tone. In the theoretical string, however, the force acting to return a part of the string to its rest position is the tension between its ends. If you try bending a short piece of wire or guitar string slightly with your fingers. In a vibrating string, that adds to the effect of string tension in returning a given part of the string toward its rest position. The result is a frequency of vibration higher than the theoretical frequency, and because the wires resistance to bending increases as its length decreases, its effect is greater in higher harmonics than in lower
30.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
31.
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
32.
Geometric progression
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For example, the sequence 2,6,18,54. is a geometric progression with common ratio 3. Similarly 10,5,2.5,1.25. is a sequence with common ratio 1/2. Examples of a sequence are powers rk of a fixed number r, such as 2k. The general form of a sequence is a, a r, a r 2, a r 3, a r 4, … where r ≠0 is the common ratio. The n-th term of a sequence with initial value a. Such a geometric sequence also follows the relation a n = r a n −1 for every integer n ≥1. Generally, to whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative. For instance 1, −3,9, −27,81, the behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is, Positive, the terms will all be the sign as the initial term. Negative, the terms will alternate between positive and negative, greater than 1, there will be exponential growth towards positive or negative infinity. 1, the progression is a constant sequence, between −1 and 1 but not zero, there will be exponential decay towards zero. −1, the progression is an alternating sequence Less than −1, for the absolute values there is exponential growth towards infinity, due to the alternating sign. Geometric sequences show exponential growth or exponential decay, as opposed to the growth of an arithmetic progression such as 4,15,26,37,48. This result was taken by T. R. Malthus as the foundation of his Principle of Population. A geometric series is the sum of the numbers in a geometric progression, for example,2 +10 +50 +250 =2 +2 ×5 +2 ×52 +2 ×53. The formula works for any real numbers a and r. For example, −2 π +4 π2 −8 π3 = −2 π +2 +3 = −2 π1 − = −2 π1 +2 π ≈ −214.855
33.
Perfect fifth
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In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3,2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five notes in a diatonic scale. The perfect fifth spans seven semitones, while the diminished fifth spans six, for example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. Play The perfect fifth may be derived from the series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and it occurs above the root of all major and minor chords and their extensions. Until the late 19th century, it was referred to by one of its Greek names. Its inversion is the perfect fourth, the octave of the fifth is the twelfth. The term perfect identifies the perfect fifth as belonging to the group of perfect intervals, so called because of their simple pitch relationships and their high degree of consonance. However, when using correct enharmonic spelling, the fifth in Pythagorean tuning or meantone temperament is actually not a perfect fifth. The perfect unison has a pitch ratio 1,1, the perfect octave 2,1, the perfect fourth 4,3, within this definition, other intervals may also be called perfect, for example a perfect third or a perfect major sixth. In terms of semitones, these are equivalent to the tritone, the justly tuned pitch ratio of a perfect fifth is 3,2, meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a violin is tuned, if adjacent strings are adjusted to the ratio of 3,2, the result is a smooth and consonant sound. Keyboard instruments such as the piano normally use a version of the perfect fifth. In 12-tone equal temperament, the frequencies of the perfect fifth are in the ratio 7 or approximately 1.498307. An equally tempered fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents. Kepler explored musical tuning in terms of ratios, and defined a lower imperfect fifth as a 40,27 pitch ratio. His lower perfect fifth ratio of 1.4815 is much more imperfect than the equal temperament tuning of 1.498, the perfect fifth is a basic element in the construction of major and minor triads, and their extensions
34.
Perfect fourth
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In classical music from Western culture, a fourth is a musical interval encompassing four staff positions, and the perfect fourth is a fourth spanning five semitones. For example, the interval from C to the next F is a perfect fourth, as the note F lies five semitones above C. Diminished and augmented fourths span the same number of staff positions, the perfect fourth may be derived from the harmonic series as the interval between the third and fourth harmonics. The term perfect identifies this interval as belonging to the group of perfect intervals, so called because they are neither major nor minor, up until the late 19th century, the perfect fourth was often called by its Greek name, diatessaron. Its most common occurrence is between the fifth and upper root of all major and minor triads and their extensions. A perfect fourth in just intonation corresponds to a ratio of 4,3, or about 498 cents, while in equal temperament a perfect fourth is equal to five semitones. A helpful way to recognize a fourth is to hum the starting of the Bridal Chorus from Wagners Lohengrin. Other examples are the first two notes of the Christmas carol Hark, the Herald Angels Sing or El Cóndor Pasa, and, for a descending perfect fourth, the second and third notes of O Come All Ye Faithful. The perfect fourth is a perfect interval like the unison, octave, and perfect fifth, in common practice harmony, however, it is considered a stylistic dissonance in certain contexts, namely in two-voice textures and whenever it appears above the bass. Conventionally, adjacent strings of the bass and of the bass guitar are a perfect fourth apart when unstopped, as are all pairs. Sets of tom-tom drums are also tuned in perfect fourths. The 4,3 just perfect fourth arises in the C major scale between G and C, play The use of perfect fourths and fifths to sound in parallel with and to thicken the melodic line was prevalent in music prior to the European polyphonic music of the Middle Ages. In the 13th century, the fourth and fifth together were the concordantiae mediae after the unison and octave, in the 15th century the fourth came to be regarded as dissonant on its own, and was first classed as a dissonance by Johannes Tinctoris in his Terminorum musicae diffinitorium. In practice, however, it continued to be used as a consonance when supported by the interval of a third or fifth in a lower voice. Modern acoustic theory supports the medieval interpretation insofar as the intervals of unison, octave, the octave has the ratio of 2,1, for example the interval between a at A440 and a at 880 Hz, giving the ratio 880,440, or 2,1. The fifth has a ratio of 3,2, and its complement has the ratio of 3,4, ancient and medieval music theorists appear to have been familiar with these ratios, see for example their experiments on the Monochord. In early western polyphony, these simpler intervals were generally preferred, however, in its development between the 12th and 16th centuries, In the earliest stages, these simple intervals occur so frequently that they appear to be the favourite sound of composers. Later, the more complex intervals move gradually from the margins to the centre of musical interest
35.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician
36.
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
37.
Marin Mersenne
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Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the father of acoustics. Mersenne, an ordained priest, had contacts in the scientific world and has been called the center of the world of science. Marin Mersenne was born of peasant parents near Oizé, Maine and he was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars, and, after studying theology, between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of LAnnonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval and he corresponded with Giovanni Doni, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a defender of Galileo, assisting him in translations of some of his mechanical works. For four years, Mersenne devoted himself entirely to philosophic and theological writing and it is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus and he taught theology and philosophy at Nevers and Paris. He was not afraid to cause disputes among his friends in order to compare their views. In 1635 Mersenne met with Tommaso Campanella, but concluded that he could teach nothing in the sciences but still he has a good memory, Mersenne asked if René Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640,1641 and 1645, in 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendis position. Among his correspondents were Descartes, Galilei, Roberval, Pascal, Beeckman and he died September 1 through complications arising from a lung abscess. Some history scientists suggest he died for having drunk a huge quantity of water, along with Descartes. It was written as a commentary on the Book of Genesis, at first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism and he mentions Martin Del Rios Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi, a popular amongst Renaissance neo-platonists. Whilst allowing for an interpretation of the Cabala, he wholeheartedly condemned its magical application—particularly to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target, Fludd responded with Sophia cum moria certamen, wherein Fludd admits his involvement with the Rosicrucians
38.
Western world
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The Western world or the West is a term usually referring to different nations, depending on the context, most often including at least part of Europe. There are many accepted definitions about what they all have in common, the Western world is also known as the Occident. The concept of the Western part of the earth has its roots in Greco-Roman civilization in Europe, before the Cold War era, the traditional Western viewpoint identified Western Civilization with the Western Christian countries and culture. Its political usage was changed by the antagonism during the Cold War in the mid-to-late 20th Century. The term originally had a literal geographic meaning, Western culture was influenced by many older great civilizations of the ancient Near East, such as Phoenicia, Minoan Crete, Sumer, Babylonia, and also Ancient Egypt. It originated in the Mediterranean basin and its vicinity, Greece, over time, their associated empires grew first to the east and west to include the rest of Mediterranean and Black Sea coastal areas, conquering and absorbing. Later, they expanded to the north of the Mediterranean Sea to include Western, Central, numerous times, this expansion was accompanied by Christian missionaries, who attempted to proselytize Christianity. There is debate among some as to whether Latin America is in a category of its own, specifically, Western culture may imply, a Biblical Christian cultural influence in spiritual thinking, customs and either ethic or moral traditions, around the Post-Classical Era and after. European cultural influences concerning artistic, musical, folkloric, ethic and oral traditions, the concept of Western culture is generally linked to the classical definition of the Western world. In this definition, Western culture is the set of literary, scientific, political, artistic, much of this set of traditions and knowledge is collected in the Western canon. The term has come to apply to countries whose history is marked by European immigration or settlement, such as the Americas, and Oceania. The geopolitical divisions in Europe that created a concept of East and West originated in the Roman Empire, Roman Catholic Western and Central Europe, as such, maintained a distinct identity particularly as it began to redevelop during the Renaissance. Even following the Protestant Reformation, Protestant Europe continued to see itself as more tied to Roman Catholic Europe than other parts of the civilized world. Use of the term West as a cultural and geopolitical term developed over the course of the Age of Exploration as Europe spread its culture to other parts of the world. Additionally, closer contacts between the West and Asia and other parts of the world in recent times have continued to cloud the use, herodotus considered the Persian Wars of the early 5th century BC a conflict of Europa versus Asia. The terms West and East were not used by any Greek author to describe that conflict, the Great Schism and the Fourth Crusade confirmed this deviation. The Renaissance in the West emerged partly from currents within the Roman Empire, Ancient Rome was a civilization that grew from a city-state founded on the Italian Peninsula about the 9th century BC to a massive empire straddling the Mediterranean Sea. In its 12-century existence, Roman civilization shifted from a monarchy, to a republic, nonetheless, despite its great legacy, a number of factors led to the eventual decline of the Roman Empire
39.
Semitone
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A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, for example, C is adjacent to C♯, the interval between them is a semitone. In music theory, a distinction is made between a diatonic semitone, or minor second and a semitone or augmented unison. In twelve-tone equal temperament all semitones are equal in size, in other tuning systems, semitone refers to a family of intervals that may vary both in size and name. In quarter-comma meantone, seven of them are diatonic, and 117.1 cents wide, while the five are chromatic. 12-tone scales tuned in just intonation typically define three or four kinds of semitones, for instance, Asymmetric five-limit tuning yields chromatic semitones with ratios 25,24 and 135,128, and diatonic semitones with ratios 16,15 and 27,25. The condition of having semitones is called hemitonia, that of having no semitones is anhemitonia, a musical scale or chord containing semitones is called hemitonic, one without semitones is anhemitonic. The minor second occurs in the scale, between the third and fourth degree, and between the seventh and eighth degree. It is also called the diatonic semitone because it occurs between steps in the diatonic scale, the minor second is abbreviated m2. Its inversion is the major seventh, listen to a minor second in equal temperament. Here, middle C is followed by D♭, which is a tone 100 cents sharper than C, melodically, this interval is very frequently used, and is of particular importance in cadences. In the perfect and deceptive cadences it appears as a resolution of the leading-tone to the tonic, in the plagal cadence, it appears as the falling of the subdominant to the mediant. It also occurs in many forms of the cadence, wherever the tonic falls to the leading-tone. Harmonically, the interval usually occurs as some form of dissonance or a tone that is not part of the functional harmony. It may also appear in inversions of a seventh chord. In unusual situations, the second can add a great deal of character to the music. For instance, Frédéric Chopins Étude Op.25, No.5 opens with a melody accompanied by a line that plays fleeting minor seconds and these are used to humorous and whimsical effect, which contrasts with its more lyrical middle section. This eccentric dissonance has earned the piece its nickname, the wrong note étude and this kind of usage of the minor second appears in many other works of the Romantic period, such as Modest Mussorgskys Ballet of the Unhatched Chicks
40.
Paul Hindemith
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Paul Hindemith was a prolific German composer, violist, violinist, teacher and conductor. Notable compositions include his song cycle Das Marienleben, Der Schwanendreher for Viola and Orchestra, Hindemiths most popular work, both on record and in the concert hall, is likely the Symphonic Metamorphosis of Themes by Carl Maria von Weber, written in 1943. Born in Hanau, near Frankfurt am Main, Hindemith was taught the violin as a child and he entered Frankfurts Hoch’sche Konservatorium, where he studied violin with Adolf Rebner, as well as conducting and composition with Arnold Mendelssohn and Bernhard Sekles. At first he supported himself by playing in bands and musical-comedy groups. He became deputy leader of the Frankfurt Opera Orchestra in 1914 and he played second violin in the Rebner String Quartet from 1914. Hindemith was conscripted into the German army in September,1917, there he was assigned to play bass drum in the regiment band, and also formed a string quartet. In May 1918 he was deployed to the front in Flanders, after the armistice he returned to Frankfurt and the Rebner Quartet. In 1921 he founded the Amar Quartet, playing viola, in 1922, some of his pieces were played in the International Society for Contemporary Music festival at Salzburg, which first brought him to the attention of an international audience. The following year, he began to work as an organizer of the Donaueschingen Festival, in 1927 he was appointed Professor at the Berliner Hochschule für Musik in Berlin. Hindemith wrote the music for Hans Richters 1928 avant-garde film Ghosts Before Breakfast, although the score was subsequently lost, in 1929 he played the solo part in the premiere of William Waltons Viola Concerto, after Lionel Tertis, for whom it was written, turned it down. Hindemith did not stay in Turkey as long as many other émigrés, nevertheless, he greatly influenced the developments of Turkish musical life, the Ankara State Conservatory owes much to his efforts. In fact, Hindemith was regarded as a master by young Turkish musicians and he was appreciated. Towards the end of the 1930s, he made tours in America as a viola and viola damore soloist. Hindemiths relationship to the Nazis is a complicated one, the controversy around his work continued throughout the thirties, with the composer falling in and out of favor with the Nazi hierarchy, he finally emigrated to Switzerland in 1938. This development seems to have been supported by the Nazi regime, it may have got him out of the way, yet at the same time he propagated a German view of musical history. In 1940, Hindemith emigrated to the United States, at the same time that he was codifying his musical language, his teaching and compositions began to be affected by his theories, according to critics like Ernest Ansermet. During this time he gave the Charles Eliot Norton Lectures at Harvard. Hindemith had a friendship with Erich Katz, whose own compositions were influenced by him
41.
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
42.
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
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Combination tone
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A combination tone is a psychoacoustic phenomenon of an additional tone or tones that are artificially perceived when two real tones are sounded at the same time. Their discovery is credited to the violinist Giuseppe Tartini and so are also called Tartini tones, combination tones are heard when two pure tones, differing in frequency by about 50 cycles per second or more, sound together at sufficient intensity. Combination tones can also be produced electronically by combining two signals in a circuit that has nonlinear distortion, such as a subject to clipping or a ring modulator. One way a difference tone can be heard is when two tones with fairly complete sets of harmonics make a just fifth and this can be explained as an example of the missing fundamental phenomenon. If f is the fundamental frequency, then 2 f would be the frequency of the lower tone. Since a fifth corresponds to a ratio of 2,3. When both tones are sounded, there are components with frequencies of 2 f,3 f,4 f,6 f,8 f,9 f, the missing fundamental is heard because so many of these components refer to it. The specific phenomenon that Tartini discovered was physical, sum and difference tones are thought to be caused sometimes by the non-linearity of the inner ear. This causes intermodulation distortion of the frequencies which enter the ear. They are combined linearly, generating relatively faint components with frequencies equal to the sums and differences of whole multiples of the original frequencies. Any components which are heard are usually lower, with the most commonly heard frequency being just the tone, f 2 − f 1. However, experiments show evidence that even when using headphones providing a single pure tone to each ear separately, since the peculiar, non-linear physics of the ear doesnt come into play in this case, it is thought that this must be a separate, neural phenomenon. Heinz Bohlen proposed what is now known as the Bohlen–Pierce scale on the basis of combination tones, a resultant tone is produced when any two loud and sustained musical sounds are heard at the same time. In pipe organs, this is done by having two pipes, one pipe of the note being played, and another related, typically at its fifth. The result is a pitch at a common subharmonic of the pitches played and this effect is useful especially in the lowest ranks of the pipe organ where cost or space could prohibit having a rank of such low pitch. For example, a 64 pipe would be costly and take up at least 32 of space for each pipe, using a resultant tone for such low pitches would eliminate the cost and space factor, but would not sound as full as a true 64 pipe. This effect is most often used in the lowest octave of the organ only and it can vary from highly effective to disappointing depending on several factors, primarily the skill of the organ voicer, and the acoustics of the room the instrument is installed in. Uses combination tones for pitch recognition, if certain intervals are played in double-stop, the program can display its Tartini-tone
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Tritone
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In music theory, the tritone is strictly defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B, according to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale, a tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave, for instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half, in classical music, the tritone is a harmonic and melodic dissonance and is important in the study of musical harmony. The tritone can be used to avoid traditional tonality, Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality. Contrarily, the found in the dominant seventh chord helps establish the tonality of a composition. These contrasting uses exhibit the flexibility, ubiquity, and distinctness of the tritone in music, the condition of having tritones is called tritonia, that of having no tritones is atritonia. A musical scale or chord containing tritones is called tritonic, one without tritones is atritonic, since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each starting from a different pitch and spanning six semitones. According to a complex but widely used naming convention, six of them are classified as augmented fourths, under that convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions. The augmented fourth and diminished fifth are defined as the produced by widening the perfect fourth. They both span six semitones, and they are the inverse of each other, meaning that their sum is equal to one perfect octave. In twelve-tone equal temperament, the most commonly used tuning system, in most other tuning systems, they are not equivalent, and neither is exactly equal to half an octave. Any augmented fourth can be decomposed into three whole tones, for instance, the interval F–B is an augmented fourth and can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. It is not possible to decompose a diminished fifth into three adjacent whole tones, the reason is that a whole tone is a major second, and according to a rule explained elsewhere, the composition of three seconds is always a fourth. To obtain a fifth, it is necessary to add another second, for instance, using the notes of the C major scale, the diminished fifth B–F can be decomposed into the four adjacent intervals B–C, C–D, D–E, and E–F. Using the notes of a scale, B–F may be also decomposed into the four adjacent intervals B–C♯, C♯–D♯, D♯–E♯. Notice that the diminished second is formed by two enharmonically equivalent notes
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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
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