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Analog computer

An analog computer or analogue computer is a type of computation machine (computer) that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (analog signals) to model the problem being solved. In contrast, digital computers represent varying quantities symbolically and by discrete values of both time and amplitude (digital signals).

Analog computers can have a very wide range of complexity. Slide rules and nomograms are the simplest, while naval gunfire control computers and large hybrid digital/analog computers were among the most complicated. Complex mechanisms for process control and protective relays used analog computation to perform control and protective functions.

Analog computers were widely used in scientific and industrial applications even after the advent of digital computers, because at the time they were typically much faster, but they started to become obsolete as early as the 1950s and 1960s, although they remained in use in some specific applications, such as aircraft flight simulators, the flight computer in aircraft, and for teaching control systems in universities. Perhaps the most relatable example of analog computers are mechanical watches where the continuous and periodic rotation of interlinked gears drives the second, minute and hour needles in the clock. More complex applications, such as aircraft flight simulators and synthetic-aperture radar, remained the domain of analog computing (and hybrid computing) well into the 1980s, since digital computers were insufficient for the task.

Precursors
This is a list of examples of early computation devices considered precursors of the modern computers. Some of them may even have been dubbed 'computers' by the press, though they may fail to fit modern definitions.

The Antikythera mechanism, a type of device used to determine the positions of heavenly bodies known  as an orrery, was  described as an early mechanical analog computer by British physicist, information scientist, and historian of science Derek J. de Solla Price. It was discovered in 1901, in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to c. 150~100 BC, during the Hellenistic period. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later.

Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use. The planisphere was first described by Ptolemy in the 2nd century AD. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is often attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was effectively an analog computer capable of working out several different kinds of problems in spherical astronomy.

The sector, a calculating instrument used for solving problems in proportion, trigonometry, multiplication and division, and for various functions, such as squares and cube roots, was developed in the late 16th century and found application in gunnery, surveying and navigation.

The planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage.

The slide rule was invented around 1620–1630, shortly after the publication of the concept of the logarithm. It is a hand-operated analog computer for doing multiplication and division. As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions. Aviation is one of the few fields where slide rules are still in widespread use, particularly for solving time–distance problems in light aircraft.

In 1831–1835, mathematician and engineer Giovanni Plana devised a perpetual-calendar machine, which, through a system of pulleys and cylinders, could predict the perpetual calendar for every year from AD 0 (that is, 1 BC) to AD 4000, keeping track of leap years and varying day length.

The tide-predicting machine invented by Sir William Thomson in 1872 was of great utility to navigation in shallow waters. It used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location.

The differential analyser, a mechanical analog computer designed to solve differential equations by integration, used wheel-and-disc mechanisms to perform the integration. In 1876 James Thomson had already discussed the possible construction of such calculators, but he had been stymied by the limited output torque of the ball-and-disk integrators. Several systems followed, notably those of Spanish engineer Leonardo Torres Quevedo, who built various analog machines for solving real and complex roots of polynomials; and Michelson and Stratton, whose Harmonic Analyser performed Fourier analysis, but using an array of 80 springs rather than Kelvin integrators. This work led to the mathematical understanding of the Gibbs phenomenon of overshoot in Fourier representation near discontinuities. In a differential analyzer, the output of one integrator drove the input of the next integrator, or a graphing output. The torque amplifier was the advance that allowed these machines to work. Starting in the 1920s, Vannevar Bush and others developed mechanical differential analyzers.

Modern era
The Dumaresq was a mechanical calculating device invented around 1902 by Lieutenant John Dumaresq of the Royal Navy. It was an analog computer that related vital variables of the fire control problem to the movement of one's own ship and that of a target ship. It was often used with other devices, such as a Vickers range clock to generate range and deflection data so the gun sights of the ship could be continuously set. A number of versions of the Dumaresq were produced of increasing complexity as development proceeded.

By 1912, Arthur Pollen had developed an electrically driven mechanical analog computer for fire-control systems, based on the differential analyser. It was used by the Imperial Russian Navy in World War I.

Starting in 1929, AC network analyzers were constructed to solve calculation problems related to electrical power systems that were too large to solve with numerical methods at the time. These were essentially scale models of the electrical properties of the full-size system. Since network analyzers could handle problems too large for analytic methods or hand computation, they were also used to solve problems in nuclear physics and in the design of structures. More than 50 large network analyzers were built by the end of the 1950s.

World War II era gun directors, gun data computers, and bomb sights used mechanical analog computers. In 1942 Helmut Hölzer built a fully electronic analog computer at Peenemünde Army Research Center  as an embedded control system (mixing device) to calculate V-2 rocket trajectories from the accelerations and orientations (measured by gyroscopes) and to stabilize and guide the missile. Mechanical analog computers were very important in gun fire control in World War II, the Korean War and well past the Vietnam War; they were made in significant numbers.

In the period 1930–1945 in the Netherlands, Johan van Veen developed an analogue computer to calculate and predict tidal currents when the geometry of the channels are changed. Around 1950, this idea was developed into the Deltar, a hydraulic analogy computer supporting the closure of estuaries in the southwest of the Netherlands (the Delta Works).

The FERMIAC was an analog computer invented by physicist Enrico Fermi in 1947 to aid in his studies of neutron transport. Project Cyclone was an analog computer developed by Reeves in 1950 for the analysis and design of dynamic systems. Project Typhoon was an analog computer developed by RCA in 1952. It consisted of over 4,000 electron tubes and used 100 dials and 6,000 plug-in connectors to program. The MONIAC Computer was a hydraulic analogy of a national economy first unveiled in 1949.

Computer Engineering Associates was spun out of Caltech in 1950 to provide commercial services using the "Direct Analogy Electric Analog Computer" ("the largest and most impressive general-purpose analyzer facility for the solution of field problems") developed there by Gilbert D. McCann, Charles H. Wilts, and Bart Locanthi.

Educational analog computers illustrated the principles of analog calculation. The Heathkit EC-1, a $199 educational analog computer, was made by the Heath Company, US c. 1960. It was programmed using patch cords that connected nine operational amplifiers and other components. General Electric also marketed an "educational" analog computer kit of a simple design in the early 1960s consisting of two transistor tone generators and three potentiometers wired such that the frequency of the oscillator was nulled when the potentiometer dials were positioned by hand to satisfy an equation. The relative resistance of the potentiometer was then equivalent to the formula of the equation being solved. Multiplication or division could be performed, depending on which dials were inputs and which was the output. Accuracy and resolution was limited and a simple slide rule was more accurate. However, the unit did demonstrate the basic principle.

Analog computer designs were published in electronics magazines. One example is the PEAC (Practical Electronics analogue computer), published in Practical Electronics in the January 1968 edition. Another more modern hybrid computer design was published in Everyday Practical Electronics in 2002. An example described in the EPE hybrid computer was the flight of a VTOL aircraft such as the Harrier jump jet. The altitude and speed of the aircraft were calculated by the analog part of the computer and sent to a PC via a digital microprocessor and displayed on the PC screen.

In industrial process control, analog loop controllers were used to automatically regulate temperature, flow, pressure, or other process conditions. The technology of these controllers ranged from purely mechanical integrators, through vacuum-tube and solid-state devices, to emulation of analog controllers by microprocessors.

Electronic analog computers

The similarity between linear mechanical components, such as springs and dashpots (viscous-fluid dampers), and electrical components, such as capacitors, inductors, and resistors is striking in terms of mathematics. They can be modeled using equations of the same form.

However, the difference between these systems is what makes analog computing useful. Complex systems often are not amenable to pen-and-paper analysis, and require some form of testing or simulation. Complex mechanical systems, such as suspensions for racing cars, are expensive to fabricate and hard to modify. And taking precise mechanical measurements during high-speed tests adds further difficulty.

By contrast, it is very inexpensive to build an electrical equivalent of a complex mechanical system, to simulate its behavior. Engineers arrange a few operational amplifiers (op amps) and some passive linear components to form a circuit that follows the same equations as the mechanical system being simulated. All measurements can be taken directly with an oscilloscope. In the circuit, the (simulated) stiffness of the spring, for instance, can be changed by adjusting the parameters of an integrator. The electrical system is an analogy to the physical system, hence the name, but it is much less expensive than a mechanical prototype, much easier to modify, and generally safer.

The electronic circuit can also be made to run faster or slower than the physical system being simulated. Experienced users of electronic analog computers said that they offered a comparatively intimate control and understanding of the problem, relative to digital simulations. Electronic analog computers are especially well-suited to representing situations described by differential equations. Historically, they were often used when a system of differential equations proved very difficult to solve by traditional means. As a simple example, the dynamics of a spring-mass system can be described by the equation $$m \ddot y + d \dot y +cy = mg$$, with $$y$$ as the vertical position of a mass $$m$$, $$d$$ the damping coefficient, $$c$$ the spring constant and $$g$$ the gravity of Earth. For analog computing, the equation is programmed as $$\ddot y = - \tfrac{d}{m} \dot y - \tfrac{c}{m} y - g$$. The equivalent analog circuit consists of two integrators for the state variables $$-\dot y$$ (speed) and $$y$$ (position), one inverter, and three potentiometers.

Electronic analog computers have drawbacks: the value of the circuit's supply voltage limits the range over which the variables may vary (since the value of a variable is represented by a voltage on a particular wire). Therefore, each problem must be scaled so its parameters and dimensions can be represented using voltages that the circuit can supply —e.g., the expected magnitudes of the velocity and the position of a spring pendulum. Improperly scaled variables can have their values "clamped" by the limits of the supply voltage. Or if scaled too small, they can suffer from higher noise levels. Either problem can cause the circuit to produce an incorrect simulation of the physical system. (Modern digital simulations are much more robust to widely varying values of their variables, but are still not entirely immune to these concerns: floating-point digital calculations support a huge dynamic range, but can suffer from imprecision if tiny differences of huge values lead to numerical instability.)

The precision of the analog computer readout was limited chiefly by the precision of the readout equipment used, generally three or four significant figures. (Modern digital simulations are much better in this area. Digital arbitrary-precision arithmetic can provide any desired degree of precision.) However, in most cases the precision of an analog computer is absolutely sufficient given the uncertainty of the model characteristics and its technical parameters.

Many small computers dedicated to specific computations are still part of industrial regulation equipment, but from the 1950s to the 1970s, general-purpose analog computers were the only systems fast enough for real time simulation of dynamic systems, especially in the aircraft, military and aerospace field.

In the 1960s, the major manufacturer was Electronic Associates of Princeton, New Jersey, with its 231R Analog Computer (vacuum tubes, 20 integrators) and subsequently its EAI 8800 Analog Computer (solid state operational amplifiers, 64 integrators). Its challenger was Applied Dynamics of Ann Arbor, Michigan.

Although the basic technology for analog computers is usually operational amplifiers (also called "continuous current amplifiers" because they have no low frequency limitation), in the 1960s an attempt was made in the French ANALAC computer to use an alternative technology: medium frequency carrier and non dissipative reversible circuits.

In the 1970s, every large company and administration concerned with problems in dynamics had an analog computing center, such as:
 * In the US: NASA (Huntsville, Houston), Martin Marietta (Orlando), Lockheed, Westinghouse, Hughes Aircraft
 * In Europe: CEA (French Atomic Energy Commission), MATRA, Aérospatiale, BAC (British Aircraft Corporation).

Construction
An analog computing machine consists of several main components:

 * Signal sources: These are blocks that generate analog signals, such as voltage or current, to represent input data and operations.
 * Amplifiers: Amplifiers are used to boost analog signals and maintain their amplitudes throughout the system. They amplify weak input signals and compensate for signal losses during transmission.
 * Filters: Filters are used to modify the spectrum of signals by suppressing or amplifying specific frequencies. They allow the isolation or suppression of certain signal components depending on the computational requirements.
 * Modulators and demodulators: Modulators convert information into analog signals that can be transmitted through a communication channel, and demodulators perform the reverse transformation, recovering the original data from modulated signals.
 * Adders and multipliers: Adders and multipliers perform arithmetic operations on analog signals. They can be used for mathematical operations such as addition, multiplication, integration, and differentiation.
 * Storage and memory: Analog computing machines can use various forms of information storage, such as capacitors or inductors, to store intermediate results and memory.
 * Feedback and control: Feedback and control blocks are used to maintain the stability and accuracy of the analog computing machine. They may include regulation systems and error correction.
 * Patch panel: Analog computing machines also feature a patch panel or patch field. A patch panel is a physical structure on which connectors or contacts are placed to connect various components and modules within the system.

On the patch panel, various connections and routes can be set and switched to configure the machine and determine signal flows. This allows users to flexibly configure and reconfigure the analog computing system to perform specific tasks.

Patch panels are used to control data flows, connect and disconnect connections between various blocks of the system, including signal sources, amplifiers, filters, and other components. They provide convenience and flexibility in configuring and experimenting with analog computations.

Patch panels can be presented as a physical panel with connectors or, in more modern systems, as a software interface that allows virtual management of signal connections and routes.

 * Hardware interfaces: Interfaces provide means of interaction with the machine, for example, for parameter control or data transmission.
 * Output device: this device is designed to present the results of analog computations in a convenient form for the user or to transmit the obtained data to other systems.

Output devices in analog machines can vary depending on the specific goals of the system. For example, they could be graphical indicators, oscilloscopes, graphic recording devices, TV connection module, voltmeter, etc. These devices allow for the visualization of analog signals and the representation of the results of measurements or mathematical operations.

 * Power source and stabilizers.

These are just general blocks that can be found in a typical analog computing machine. The actual configuration and components may vary depending on the specific implementation and the intended use of the machine.

Analog–digital hybrids
Analog computing devices are fast; digital computing devices are more versatile and accurate. The idea behind an analog-digital hybrid is to combine the two processes for the best efficiency. An example of such hybrid elementary device is the hybrid multiplier, where one input is an analog signal, the other input is a digital signal and the output is analog. It acts as an analog potentiometer, upgradable digitally. This kind of hybrid technique is mainly used for fast dedicated real time computation when computing time is very critical, as signal processing for radars and generally for controllers in embedded systems.

In the early 1970s, analog computer manufacturers tried to tie together their analog computers with a digital computers to get the advantages of the two techniques. In such systems, the digital computer controlled the analog computer, providing initial set-up, initiating multiple analog runs, and automatically feeding and collecting data. The digital computer may also participate to the calculation itself using analog-to-digital and digital-to-analog converters.

The largest manufacturer of hybrid computers was Electronic Associates. Their hybrid computer model 8900 was made of a digital computer and one or more analog consoles. These systems were mainly dedicated to large projects such as the Apollo program and Space Shuttle at NASA, or Ariane in Europe, especially during the integration step where at the beginning everything is simulated, and progressively real components replace their simulated parts.

Only one company was known as offering general commercial computing services on its hybrid computers, CISI of France, in the 1970s.

The best reference in this field is the 100,000 simulation runs for each certification of the automatic landing systems of Airbus and Concorde aircraft.

After 1980, purely digital computers progressed more and more rapidly and were fast enough to compete with analog computers. One key to the speed of analog computers was their fully parallel computation, but this was also a limitation. The more equations required for a problem, the more analog components were needed, even when the problem wasn't time critical. "Programming" a problem meant interconnecting the analog operators; even with a removable wiring panel this was not very versatile.

Mechanical analog computers
While a wide variety of mechanisms have been developed throughout history, some stand out because of their theoretical importance, or because they were manufactured in significant quantities.

Most practical mechanical analog computers of any significant complexity used rotating shafts to carry variables from one mechanism to another. Cables and pulleys were used in a Fourier synthesizer, a tide-predicting machine, which summed the individual harmonic components. Another category, not nearly as well known, used rotating shafts only for input and output, with precision racks and pinions. The racks were connected to linkages that performed the computation. At least one U.S. Naval sonar fire control computer of the later 1950s, made by Librascope, was of this type, as was the principal computer in the Mk. 56 Gun Fire Control System.

Online, there is a remarkably clear illustrated reference (OP 1140) that describes the fire control computer mechanisms. For adding and subtracting, precision miter-gear differentials were in common use in some computers; the Ford Instrument Mark I Fire Control Computer contained about 160 of them.

Integration with respect to another variable was done by a rotating disc driven by one variable. Output came from a pick-off device (such as a wheel) positioned at a radius on the disc proportional to the second variable. (A carrier with a pair of steel balls supported by small rollers worked especially well. A roller, its axis parallel to the disc's surface, provided the output. It was held against the pair of balls by a spring.)

Arbitrary functions of one variable were provided by cams, with gearing to convert follower movement to shaft rotation.

Functions of two variables were provided by three-dimensional cams. In one good design, one of the variables rotated the cam. A hemispherical follower moved its carrier on a pivot axis parallel to that of the cam's rotating axis. Pivoting motion was the output. The second variable moved the follower along the axis of the cam. One practical application was ballistics in gunnery.

Coordinate conversion from polar to rectangular was done by a mechanical resolver (called a "component solver" in US Navy fire control computers). Two discs on a common axis positioned a sliding block with pin (stubby shaft) on it. One disc was a face cam, and a follower on the block in the face cam's groove set the radius. The other disc, closer to the pin, contained a straight slot in which the block moved. The input angle rotated the latter disc (the face cam disc, for an unchanging radius, rotated with the other (angle) disc; a differential and a few gears did this correction).

Referring to the mechanism's frame, the location of the pin corresponded to the tip of the vector represented by the angle and magnitude inputs. Mounted on that pin was a square block.

Rectilinear-coordinate outputs (both sine and cosine, typically) came from two slotted plates, each slot fitting on the block just mentioned. The plates moved in straight lines, the movement of one plate at right angles to that of the other. The slots were at right angles to the direction of movement. Each plate, by itself, was like a Scotch yoke, known to steam engine enthusiasts.

During World War II, a similar mechanism converted rectilinear to polar coordinates, but it was not particularly successful and was eliminated in a significant redesign (USN, Mk. 1 to Mk. 1A).

Multiplication was done by mechanisms based on the geometry of similar right triangles. Using the trigonometric terms for a right triangle, specifically opposite, adjacent, and hypotenuse, the adjacent side was fixed by construction. One variable changed the magnitude of the opposite side. In many cases, this variable changed sign; the hypotenuse could coincide with the adjacent side (a zero input), or move beyond the adjacent side, representing a sign change.

Typically, a pinion-operated rack moving parallel to the (trig.-defined) opposite side would position a slide with a slot coincident with the hypotenuse. A pivot on the rack let the slide's angle change freely. At the other end of the slide (the angle, in trig. terms), a block on a pin fixed to the frame defined the vertex between the hypotenuse and the adjacent side.

At any distance along the adjacent side, a line perpendicular to it intersects the hypotenuse at a particular point. The distance between that point and the adjacent side is some fraction that is the product of 1 the distance from the vertex, and 2 the magnitude of the opposite side.

The second input variable in this type of multiplier positions a slotted plate perpendicular to the adjacent side. That slot contains a block, and that block's position in its slot is determined by another block right next to it. The latter slides along the hypotenuse, so the two blocks are positioned at a distance from the (trig.) adjacent side by an amount proportional to the product.

To provide the product as an output, a third element, another slotted plate, also moves parallel to the (trig.) opposite side of the theoretical triangle. As usual, the slot is perpendicular to the direction of movement. A block in its slot, pivoted to the hypotenuse block positions it.

A special type of integrator, used at a point where only moderate accuracy was needed, was based on a steel ball, instead of a disc. It had two inputs, one to rotate the ball, and the other to define the angle of the ball's rotating axis. That axis was always in a plane that contained the axes of two movement pick-off rollers, quite similar to the mechanism of a rolling-ball computer mouse (in that mechanism, the pick-off rollers were roughly the same diameter as the ball). The pick-off roller axes were at right angles.

A pair of rollers "above" and "below" the pick-off plane were mounted in rotating holders that were geared together. That gearing was driven by the angle input, and established the rotating axis of the ball. The other input rotated the "bottom" roller to make the ball rotate.

Essentially, the whole mechanism, called a component integrator, was a variable-speed drive with one motion input and two outputs, as well as an angle input. The angle input varied the ratio (and direction) of coupling between the "motion" input and the outputs according to the sine and cosine of the input angle.

Although they did not accomplish any computation, electromechanical position servos were essential in mechanical analog computers of the "rotating-shaft" type for providing operating torque to the inputs of subsequent computing mechanisms, as well as driving output data-transmission devices such as large torque-transmitter synchros in naval computers.

Other readout mechanisms, not directly part of the computation, included internal odometer-like counters with interpolating drum dials for indicating internal variables, and mechanical multi-turn limit stops.

Considering that accurately controlled rotational speed in analog fire-control computers was a basic element of their accuracy, there was a motor with its average speed controlled by a balance wheel, hairspring, jeweled-bearing differential, a twin-lobe cam, and spring-loaded contacts (ship's AC power frequency was not necessarily accurate, nor dependable enough, when these computers were designed).

Electronic analog computers
Electronic analog computers typically have front panels with numerous jacks (single-contact sockets) that permit patch cords (flexible wires with plugs at both ends) to create the interconnections that define the problem setup. In addition, there are precision high-resolution potentiometers (variable resistors) for setting up (and, when needed, varying) scale factors. In addition, there is usually a zero-center analog pointer-type meter for modest-accuracy voltage measurement. Stable, accurate voltage sources provide known magnitudes.

Typical electronic analog computers contain anywhere from a few to a hundred or more operational amplifiers ("op amps"), named because they perform mathematical operations. Op amps are a particular type of feedback amplifier with very high gain and stable input (low and stable offset). They are always used with precision feedback components that, in operation, all but cancel out the currents arriving from input components. The majority of op amps in a representative setup are summing amplifiers, which add and subtract analog voltages, providing the result at their output jacks. As well, op amps with capacitor feedback are usually included in a setup; they integrate the sum of their inputs with respect to time.

Integrating with respect to another variable is the nearly exclusive province of mechanical analog integrators; it is almost never done in electronic analog computers. However, given that a problem solution does not change with time, time can serve as one of the variables.

Other computing elements include analog multipliers, nonlinear function generators, and analog comparators.

Electrical elements such as inductors and capacitors used in electrical analog computers had to be carefully manufactured to reduce non-ideal effects. For example, in the construction of AC power network analyzers, one motive for using higher frequencies for the calculator (instead of the actual power frequency) was that higher-quality inductors could be more easily made. Many general-purpose analog computers avoided the use of inductors entirely, re-casting the problem in a form that could be solved using only resistive and capacitive elements, since high-quality capacitors are relatively easy to make.

The use of electrical properties in analog computers means that calculations are normally performed in real time (or faster), at a speed determined mostly by the frequency response of the operational amplifiers and other computing elements. In the history of electronic analog computers, there were some special high-speed types.

Nonlinear functions and calculations can be constructed to a limited precision (three or four digits) by designing function generators—special circuits of various combinations of resistors and diodes to provide the nonlinearity. Typically, as the input voltage increases, progressively more diodes conduct.

When compensated for temperature, the forward voltage drop of a transistor's base-emitter junction can provide a usably accurate logarithmic or exponential function. Op amps scale the output voltage so that it is usable with the rest of the computer.

Any physical process that models some computation can be interpreted as an analog computer. Some examples, invented for the purpose of illustrating the concept of analog computation, include using a bundle of spaghetti as a model of sorting numbers; a board, a set of nails, and a rubber band as a model of finding the convex hull of a set of points; and strings tied together as a model of finding the shortest path in a network. These are all described in Dewdney (1984).

Components

Analog computers often have a complicated framework, but they have, at their core, a set of key components that perform the calculations. The operator manipulates these through the computer's framework.

Key hydraulic components might include pipes, valves and containers.

Key mechanical components might include rotating shafts for carrying data within the computer, miter gear differentials, disc/ball/roller integrators, cams (2-D and 3-D), mechanical resolvers and multipliers, and torque servos.

Key electrical/electronic components might include:
 * precision resistors and capacitors
 * operational amplifiers
 * multipliers
 * potentiometers
 * fixed-function generators

The core mathematical operations used in an electric analog computer are:
 * addition
 * integration with respect to time
 * inversion
 * multiplication
 * exponentiation
 * logarithm
 * division

In some analog computer designs, multiplication is much preferred to division. Division is carried out with a multiplier in the feedback path of an Operational Amplifier.

Differentiation with respect to time is not frequently used, and in practice is avoided by redefining the problem when possible. It corresponds in the frequency domain to a high-pass filter, which means that high-frequency noise is amplified; differentiation also risks instability.

Limitations
In general, analog computers are limited by non-ideal effects. An analog signal is composed of four basic components: DC and AC magnitudes, frequency, and phase. The real limits of range on these characteristics limit analog computers. Some of these limits include the operational amplifier offset, finite gain, and frequency response, noise floor, non-linearities, temperature coefficient, and parasitic effects within semiconductor devices. For commercially available electronic components, ranges of these aspects of input and output signals are always figures of merit.

Decline
In the 1950s to 1970s, digital computers based on first vacuum tubes, transistors, integrated circuits and then micro-processors became more economical and precise. This led digital computers to largely replace analog computers. Even so, some research in analog computation is still being done. A few universities still use analog computers to teach control system theory. The American company Comdyna manufactured small analog computers. At Indiana University Bloomington, Jonathan Mills has developed the Extended Analog Computer based on sampling voltages in a foam sheet. At the Harvard Robotics Laboratory, analog computation is a research topic. Lyric Semiconductor's error correction circuits use analog probabilistic signals. Slide rules are still used as flight computers in flight training.

Resurgence
With the development of very-large-scale integration (VLSI) technology, Yannis Tsividis' group at Columbia University has been revisiting analog/hybrid computers design in standard CMOS process. Two VLSI chips have been developed, an 80th-order analog computer (250 nm) by Glenn Cowan in 2005 and a 4th-order hybrid computer (65 nm) developed by Ning Guo in 2015, both targeting at energy-efficient ODE/PDE applications. Glenn's chip contains 16 macros, in which there are 25 analog computing blocks, namely integrators, multipliers, fanouts, few nonlinear blocks. Ning's chip contains one macro block, in which there are 26 computing blocks including integrators, multipliers, fanouts, ADCs, SRAMs and DACs. Arbitrary nonlinear function generation is made possible by the ADC+SRAM+DAC chain, where the SRAM block stores the nonlinear function data. The experiments from the related publications revealed that VLSI analog/hybrid computers demonstrated about 1–2 orders magnitude of advantage in both solution time and energy while achieving accuracy within 5%, which points to the promise of using analog/hybrid computing techniques in the area of energy-efficient approximate computing. In 2016, a team of researchers developed a compiler to solve differential equations using analog circuits.

Analog computers are also used in neuromorphic computing, and in 2021 a group of researchers have shown that a specific type of artificial neural network called a spiking neural network was able to work with analog neuromorphic computers.

Practical examples
These are examples of analog computers that have been constructed or practically used:

 * Boeing B-29 Superfortress Central Fire Control System
 * Deltar
 * E6B flight computer
 * Kerrison Predictor
 * Leonardo Torres y Quevedo's Analogue Calculating Machines based on "fusee sans fin"
 * Librascope, aircraft weight and balance computer
 * Mechanical computer
 * Mechanical watch
 * Mechanical integrators, for example, the planimeter
 * Nomogram
 * Norden bombsight
 * Rangekeeper, and related fire control computers
 * Scanimate
 * Torpedo Data Computer
 * Torquetum
 * Water integrator
 * MONIAC, economic modelling
 * Ishiguro Storm Surge Computer

Analog (audio) synthesizers can also be viewed as a form of analog computer, and their technology was originally based in part on electronic analog computer technology. The ARP 2600's Ring Modulator was actually a moderate-accuracy analog multiplier.

The Simulation Council (or Simulations Council) was an association of analog computer users in US. It is now known as The Society for Modeling and Simulation International. The Simulation Council newsletters from 1952 to 1963 are available online and show the concerns and technologies at the time, and the common use of analog computers for missilry.

title: Analog computer


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Audio

Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to:

Sound

 * Audio signal, an electrical representation of sound
 * Audio frequency, a frequency in the audio spectrum
 * Digital audio, representation of sound in a form processed and/or stored by computers or digital electronics
 * Audio, audible content (media) in audio production and publishing
 * Semantic audio, extraction of symbols or meaning from audio
 * Stereophonic audio, method of sound reproduction that creates an illusion of multi-directional audible perspective
 * Audio equipment

Entertainment

 * AUDIO (group), an American R&B band of 5 brothers formerly known as TNT Boyz and as B5
 * Audio (album), an album by the Blue Man Group
 * Audio (magazine), a magazine published from 1947 to 2000
 * Audio (musician), British drum and bass artist
 * "Audio" (song), a song by LSD

Computing

 * HTML audio, identified by the tag

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Audio


text:
Minute and second of arc

A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol ′, is a unit of angular measurement equal to $0.2909⁄1000$ of one degree. Since one degree is $mm⁄m$ of a turn, or complete rotation, one arcminute is $1⁄60$ of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth circumference is very near $\overline{6}$. A minute of arc is $\pi⁄10800$ of a radian.

A second of arc, arcsecond (arcsec), or arc second, denoted by the symbol ″, is $\pi·1000⁄10800$ of an arcminute, $9⁄600$ of a degree, $1⁄21600$ of a turn, and $1⁄60$ (about $1⁄360$) of a radian.

These units originated in Babylonian astronomy as sexagesimal (base 60) subdivisions of the degree; they are used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, navigation, land surveying, and marksmanship.

To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas) and microarcsecond (μas), for instance, are commonly used in astronomy. For a three-dimensional area such as on a sphere, square arcminutes or seconds may be used.

Symbols and abbreviations
The prime symbol ′ designates the arcminute, though a single quote ' (U+0027) is commonly used where only ASCII characters are permitted. One arcminute is thus written as 1′. It is also abbreviated as arcmin or amin.

Similarly, double prime ″ (U+2033) designates the arcsecond, though a double quote " (U+0022) is commonly used where only ASCII characters are permitted. One arcsecond is thus written as 1″. It is also abbreviated as arcsec or asec.

In celestial navigation, seconds of arc are rarely used in calculations, the preference usually being for degrees, minutes, and decimals of a minute, for example, written as 42° 25.32′ or 42° 25.322′. This notation has been carried over into marine GPS and aviation GPS receivers, which normally display latitude and longitude in the latter format by default.

Common examples
The average apparent diameter of the full Moon is about 31 arcminutes, or 0.52°.

One arcminute is the approximate distance two contours can be separated by, and still be distinguished by, a person with 20/20 vision.

One arcsecond is the approximate angle subtended by a U.S. dime coin (18 mm) at a distance of 4 km. An arcsecond is also the angle subtended by
 * an object of diameter $1⁄21,600$ at a distance of one astronomical unit,
 * an object of diameter $21,600 nmi$ at one light-year,
 * an object of diameter one astronomical unit ($\pi⁄10,800$) at a distance of one parsec, per the definition of the latter.

One milliarcsecond is about the size of a half dollar, seen from a distance equal to that between the Washington Monument and the Eiffel Tower.

One microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth.

One nanoarcsecond is about the size of a penny on Neptune's moon Triton as observed from Earth.

Also notable examples of size in arcseconds are:
 * Hubble Space Telescope has calculational resolution of 0.05 arcseconds and actual resolution of almost 0.1 arcseconds, which is close to the diffraction limit.
 * At crescent phase, Venus measures between 60.2 and 66 seconds of arc.

History
The concepts of degrees, minutes, and seconds—as they relate to the measure of both angles and time—derive from Babylonian astronomy and time-keeping. Influenced by the Sumerians, the ancient Babylonians divided the Sun's perceived motion across the sky over the course of one full day into 360 degrees. Each degree was subdivided into 60 minutes and each minute into 60 seconds. Thus, one Babylonian degree was equal to four minutes in modern terminology, one Babylonian minute to four modern seconds, and one Babylonian second to $1⁄60$ (approximately 0.067) of a modern second.

Astronomy

Since antiquity, the arcminute and arcsecond have been used in astronomy: in the ecliptic coordinate system as latitude (β) and longitude (λ); in the horizon system as altitude (Alt) and azimuth (Az); and in the equatorial coordinate system as declination (δ). All are measured in degrees, arcminutes, and arcseconds. The principal exception is right ascension (RA) in equatorial coordinates, which is measured in time units of hours, minutes, and seconds.

Contrary to what one might assume, minutes and seconds of arc do not directly relate to minutes and seconds of time, in either the rotational frame of the Earth around its own axis (day), or the Earth's rotational frame around the Sun (year). The Earth's rotational rate around its own axis is 15 minutes of arc per minute of time (360 degrees / 24 hours in day); the Earth's rotational rate around the Sun (not entirely constant) is roughly 24 minutes of time per minute of arc (from 24 hours in day), which tracks the annual progression of the Zodiac. Both of these factor in what astronomical objects you can see from surface telescopes (time of year) and when you can best see them (time of day), but neither are in unit correspondence. For simplicity, the explanations given assume a degree/day in the Earth's annual rotation around the Sun, which is off by roughly 1%. The same ratios hold for seconds, due to the consistent factor of 60 on both sides.

The arcsecond is also often used to describe small astronomical angles such as the angular diameters of planets (e.g. the angular diameter of Venus which varies between 10″ and 60″); the proper motion of stars; the separation of components of binary star systems; and parallax, the small change of position of a star or Solar System body as the Earth revolves about the Sun. These small angles may also be written in milliarcseconds (mas), or thousandths of an arcsecond. The unit of distance called the parsec, abbreviated from the parallax angle of one arc second, was developed for such parallax measurements. The distance from the Sun to a celestial object is the reciprocal of the angle, measured in arcseconds, of the object's apparent movement caused by parallax.

The European Space Agency's astrometric satellite Gaia, launched in 2013, can approximate star positions to 7 microarcseconds (μas).

Apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red giant with a diameter of 0.05″. Because of the effects of atmospheric blurring, ground-based telescopes will smear the image of a star to an angular diameter of about 0.5″; in poor conditions this increases to 1.5″ or even more. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1″. Techniques exist for improving seeing on the ground. Adaptive optics, for example, can produce images around 0.05″ on a 10 m class telescope.

Space telescopes are not affected by the Earth's atmosphere but are diffraction limited. For example, the Hubble Space Telescope can reach an angular size of stars down to about 0.1″.

Cartography
Minutes (′) and seconds (″) of arc are also used in cartography and navigation. At sea level one minute of arc along the equator equals exactly one geographical mile (not to be confused with international mile or statute mile) along the Earth's equator or approximately 1 nmi. A second of arc, one sixtieth of this amount, is roughly 30 m. The exact distance varies along meridian arcs or any other great circle arcs because the figure of the Earth is slightly oblate (bulges a third of a percent at the equator).

Positions are traditionally given using degrees, minutes, and seconds of arcs for latitude, the arc north or south of the equator, and for longitude, the arc east or west of the Prime Meridian. Any position on or above the Earth's reference ellipsoid can be precisely given with this method. However, when it is inconvenient to use base-60 for minutes and seconds, positions are frequently expressed as decimal fractional degrees to an equal amount of precision. Degrees given to three decimal places ($1⁄3,600$ of a degree) have about $1⁄1,296,000$ the precision of degrees-minutes-seconds ($\pi⁄648,000$ of a degree) and specify locations within about 120 m. For navigational purposes positions are given in degrees and decimal minutes, for instance The Needles lighthouse is at 50º 39.734’N 001º 35.500’W.

Property cadastral surveying
Related to cartography, property boundary surveying using the metes and bounds system and cadastral surveying relies on fractions of a degree to describe property lines' angles in reference to cardinal directions. A boundary "mete" is described with a beginning reference point, the cardinal direction North or South followed by an angle less than 90 degrees and a second cardinal direction, and a linear distance. The boundary runs the specified linear distance from the beginning point, the direction of the distance being determined by rotating the first cardinal direction the specified angle toward the second cardinal direction. For example, North 65° 39′ 18″ West 85.69 feet would describe a line running from the starting point 85.69 feet in a direction 65° 39′ 18″ (or 65.655°) away from north toward the west.

Firearms

The arcminute is commonly found in the firearms industry and literature, particularly concerning the precision of rifles, though the industry refers to it as minute of angle (MOA). It is especially popular as a unit of measurement with shooters familiar with the imperial measurement system because 1 MOA subtends a circle with a diameter of 1.047 inches (which is often rounded to just 1 inch) at 100 yards (1.047 in at 100 yd or 2.908 cm at 100 m), a traditional distance on American target ranges. The subtension is linear with the distance, for example, at 500 yards, 1 MOA subtends 5.235 inches, and at 1000 yards 1 MOA subtends 10.47 inches. Since many modern telescopic sights are adjustable in half ($1⁄206,264.8$), quarter ($1⁄360$) or eighth ($17.453 mrad$) MOA increments, also known as clicks, zeroing and adjustments are made by counting 2, 4 and 8 clicks per MOA respectively.

For example, if the point of impact is 3 inches high and 1.5 inches left of the point of aim at 100 yards (which for instance could be measured by using a spotting scope with a calibrated reticle, or a target delineated for such purposes), the scope needs to be adjusted 3 MOA down, and 1.5 MOA right. Such adjustments are trivial when the scope's adjustment dials have a MOA scale printed on them, and even figuring the right number of clicks is relatively easy on scopes that click in fractions of MOA. This makes zeroing and adjustments much easier:
 * To adjust a $1⁄60$ MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 × 2 = 6 clicks down and 1.5 x 2 = 3 clicks right
 * To adjust a $290.888 μrad$ MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 4 = 12 clicks down and 1.5 × 4 = 6 clicks right
 * To adjust a $1⁄60$ MOA scope 3 MOA down and 1.5 MOA right, the scope needs to be adjusted 3 x 8 = 24 clicks down and 1.5 × 8 = 12 clicks right

Another common system of measurement in firearm scopes is the milliradian (mrad). Zeroing an mrad based scope is easy for users familiar with base ten systems. The most common adjustment value in mrad based scopes is $1⁄3600$ mrad (which approximates $4.848 μrad$ MOA).
 * To adjust a $1⁄3600000$ mrad scope 0.9 mrad down and 0.4 mrad right, the scope needs to be adjusted 9 clicks down and 4 clicks right (which equals approximately 3 and 1.5 MOA respectively).

One thing to be aware of is that some MOA scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA on the scope knobs corresponds to exactly 1 inch of impact adjustment on a target at 100 yards, rather than the mathematically correct 1.047 inches. This is commonly known as the Shooter's MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards, this error compounds significantly on longer range shots that may require adjustment upwards of 20–30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss.

The physical group size equivalent to m minutes of arc can be calculated as follows: group size = tan($4.848 nrad$) × distance. In the example previously given, for 1 minute of arc, and substituting 3,600 inches for 100 yards, 3,600 tan($0$) ≈ 1.047 inches. In metric units 1 MOA at 100 metres ≈ 2.908 centimetres.

Sometimes, a precision-oriented firearm's performance will be measured in MOA. This simply means that under ideal conditions (i.e. no wind, high-grade ammo, clean barrel, and a stable mounting platform such as a vise or a benchrest used to eliminate shooter error), the gun is capable of producing a group of shots whose center points (center-to-center) fit into a circle, the average diameter of circles in several groups can be subtended by that amount of arc. For example, a 1 MOA rifle should be capable, under ideal conditions, of repeatably shooting 1-inch groups at 100 yards. Most higher-end rifles are warrantied by their manufacturer to shoot under a given MOA threshold (typically 1 MOA or better) with specific ammunition and no error on the shooter's part. For example, Remington's M24 Sniper Weapon System is required to shoot 0.8 MOA or better, or be rejected from sale by quality control.

Rifle manufacturers and gun magazines often refer to this capability as sub-MOA, meaning a gun consistently shooting groups under 1 MOA. This means that a single group of 3 to 5 shots at 100 yards, or the average of several groups, will measure less than 1 MOA between the two furthest shots in the group, i.e. all shots fall within 1 MOA. If larger samples are taken (i.e., more shots per group) then group size typically increases, however this will ultimately average out. If a rifle was truly a 1 MOA rifle, it would be just as likely that two consecutive shots land exactly on top of each other as that they land 1 MOA apart. For 5-shot groups, based on 95% confidence, a rifle that normally shoots 1 MOA can be expected to shoot groups between 0.58 MOA and 1.47 MOA, although the majority of these groups will be under 1 MOA. What this means in practice is if a rifle that shoots 1-inch groups on average at 100 yards shoots a group measuring 0.7 inches followed by a group that is 1.3 inches, this is not statistically abnormal.

The metric system counterpart of the MOA is the milliradian (mrad or 'mil'), being equal to $4.848 prad$ of the target range, laid out on a circle that has the observer as centre and the target range as radius. The number of milliradians on a full such circle therefore always is equal to 2 × $\pi$ × 1000, regardless the target range. Therefore, 1 MOA ≈ 0.2909 mrad. This means that an object which spans 1 mrad on the reticle is at a range that is in metres equal to the object's size in millimetres (e.g. an object of 100 mm subtending 1 mrad is 100 metres away). So there is no conversion factor required, contrary to the MOA system. A reticle with markings (hashes or dots) spaced with a one mrad apart (or a fraction of a mrad) are collectively called a mrad reticle. If the markings are round they are called mil-dots.

In the table below conversions from mrad to metric values are exact (e.g. 0.1 mrad equals exactly 10 mm at 100 metres), while conversions of minutes of arc to both metric and imperial values are approximate.

 * 1′ at 100 yards is about 1.047 inches
 * 1′ ≈ 0.291 mrad (or 29.1 mm at 100 m, approximately 30 mm at 100 m)
 * 1 mrad ≈ 3.44′, so $725.27 km$ mrad ≈ $45,866,916 km$′
 * 0.1 mrad equals exactly 1 cm at 100 m, or exactly 0.36 inches at 100 yards

Human vision
In humans, 20/20 vision is the ability to resolve a spatial pattern separated by a visual angle of one minute of arc, from a distance of twenty feet. A 20/20 letter subtends 5 minutes of arc total.

Materials
The deviation from parallelism between two surfaces, for instance in optical engineering, is usually measured in arcminutes or arcseconds. In addition, arcseconds are sometimes used in rocking curve (ω-scan) x ray diffraction measurements of high-quality epitaxial thin films.

Manufacturing
Some measurement devices make use of arcminutes and arcseconds to measure angles when the object being measured is too small for direct visual inspection. For instance, a toolmaker's optical comparator will often include an option to measure in "minutes and seconds".

title: Minute and second of arc


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Alberto Giacometti

Alberto Giacometti (,,  ; 10 October 1901 – 11 January 1966) was a Swiss sculptor, painter, draftsman and printmaker. Beginning in 1922, he lived and worked mainly in Paris but regularly visited his hometown Borgonovo to see his family and work on his art.

Giacometti was one of the most important sculptors of the 20th century. His work was particularly influenced by artistic styles such as Cubism and Surrealism. Philosophical questions about the human condition, as well as existential and phenomenological debates played a significant role in his work. Around 1935 he gave up on his Surrealist influences to pursue a more deepened analysis of figurative compositions. Giacometti wrote texts for periodicals and exhibition catalogues and recorded his thoughts and memories in notebooks and diaries. His critical nature led to self-doubt about his own work and his self-perceived inability to do justice to his own artistic vision. His insecurities nevertheless remained a powerful motivating artistic force throughout his entire life.

Between 1938 and 1944 Giacometti's sculptures had a maximum height of seven centimeters (2.75 inches). Their small size reflected the actual distance between the artist's position and his model. In this context he self-critically stated: "But wanting to create from memory what I had seen, to my terror the sculptures became smaller and smaller". After World War II, Giacometti created his most famous sculptures: his extremely tall and slender figurines. These sculptures were subject to his individual viewing experience—between an imaginary yet real, a tangible yet inaccessible space.

In Giacometti's whole body of work, his painting constitutes only a small part. After 1957, however, his figurative paintings were equally as present as his sculptures. The almost monochrome paintings of his late work do not refer to any other artistic styles of modernity.

Early life
Giacometti was born in Borgonovo, Switzerland, the eldest of four children of Giovanni Giacometti, a well-known post-Impressionist painter, and Annetta Giacometti-Stampa. He was a descendant of Protestant refugees escaping the inquisition. Coming from an artistic background, he was interested in art from an early age and was encouraged by his father and godfather. Alberto attended the Geneva School of Fine Arts. His brothers Diego (1902–1985) and Bruno (1907–2012) would go on to become artists and architects as well. Additionally, his cousin Zaccaria Giacometti, later professor of constitutional law and chancellor of the University of Zurich, grew up together with them, having been orphaned at the age of 12 in 1905.

Career

In 1922, he moved to Paris to study under the sculptor Antoine Bourdelle, an associate of Rodin. It was there that Giacometti experimented with Cubism and Surrealism and came to be regarded as one of the leading Surrealist sculptors. Among his associates were Miró, Max Ernst, Picasso, Bror Hjorth, and Balthus.

Between 1936 and 1940, Giacometti concentrated his sculpting on the human head, focusing on the sitter's gaze. He preferred models he was close to—his sister and the artist Isabel Rawsthorne (then known as Isabel Delmer). This was followed by a phase in which his statues of Isabel became stretched out; her limbs elongated. Obsessed with creating his sculptures exactly as he envisioned through his unique view of reality, he often carved until they were as thin as nails and reduced to the size of a pack of cigarettes, much to his consternation. A friend of his once said that if Giacometti decided to sculpt you, "he would make your head look like the blade of a knife". During World War II, Giacometti took refuge in Switzerland. There, in 1946, he met Annette Arm, a secretary for the Red Cross. They married in 1949.

After his marriage his tiny sculptures became larger, but the larger they grew, the thinner they became. For the remainder of Giacometti's life, Annette was his main female model. His paintings underwent a parallel procedure. The figures appear isolated and severely attenuated, as the result of continuous reworking.

He frequently revisited his subjects: one of his favourite models was his younger brother Diego, with whom he shared his studio in Paris.

Later years
In 1958 Giacometti was asked to create a monumental sculpture for the Chase Manhattan Bank building in New York, which was beginning construction. Although he had for many years "harbored an ambition to create work for a public square", he "had never set foot in New York, and knew nothing about life in a rapidly evolving metropolis. Nor had he ever laid eyes on an actual skyscraper", according to his biographer James Lord. Giacometti's work on the project resulted in the four figures of standing women—his largest sculptures—entitled Grande femme debout I through IV (1960). The commission was never completed, however, because Giacometti was unsatisfied by the relationship between the sculpture and the site, and abandoned the project. In 1962, Giacometti was awarded the grand prize for sculpture at the Venice Biennale, and the award brought with it worldwide fame. Even when he had achieved popularity and his work was in demand, he still reworked models, often destroying them or setting them aside to be returned to years later. The prints produced by Giacometti are often overlooked but the catalogue raisonné, Giacometti – The Complete Graphics and 15 Drawings by Herbert Lust (Tudor 1970), comments on their impact and gives details of the number of copies of each print. Some of his most important images were in editions of only 30 and many were described as rare in 1970.

In his later years Giacometti's works were shown in a number of large exhibitions throughout Europe. Riding a wave of international popularity, and despite his declining health, he traveled to the United States in 1965 for an exhibition of his works at the Museum of Modern Art in New York. As his last work he prepared the text for the book Paris sans fin, a sequence of 150 lithographs containing memories of all the places where he had lived.

Death
Giacometti died in 1966 of heart disease (pericarditis) and chronic obstructive pulmonary disease at the Kantonsspital in Chur, Switzerland. His body was returned to his birthplace in Borgonovo, where he was interred close to his parents.

With no children, Annette Giacometti became the sole holder of his property rights. She worked to collect a full listing of authenticated works by her late husband, gathering documentation on the location and manufacture of his works and working to fight the rising number of counterfeited works. When she died in 1993, the Fondation Giacometti was set up by the French state.

In May 2007 the executor of his widow's estate, former French foreign minister Roland Dumas, was convicted of illegally selling Giacometti's works to a top auctioneer, Jacques Tajan, who was also convicted. Both were ordered to pay €850,000 to the Alberto and Annette Giacometti Foundation.

Artistic analysis

Regarding Giacometti's sculptural technique and according to the Metropolitan Museum of Art: "The rough, eroded, heavily worked surfaces of Three Men Walking (II), 1949, typify his technique. Reduced, as they are, to their very core, these figures evoke lone trees in winter that have lost their foliage. Within this style, Giacometti would rarely deviate from the three themes that preoccupied him—the walking man; the standing, nude woman; and the bust—or all three, combined in various groupings."

In a letter to Pierre Matisse, Giacometti wrote: "Figures were never a compact mass but like a transparent construction". In the letter, Giacometti writes about how he looked back at the realist, classical busts of his youth with nostalgia, and tells the story of the existential crisis which precipitated the style he became known for.

"[I rediscovered] the wish to make compositions with figures. For this I had to make (quickly I thought; in passing), one or two studies from nature, just enough to understand the construction of a head, of a whole figure, and in 1935 I took a model. This study should take, I thought, two weeks and then I could realize my compositions...I worked with the model all day from 1935 to 1940...Nothing was as I imagined. A head, became for me an object completely unknown and without dimensions."

Since Giacometti achieved exquisite realism with facility when he was executing busts in his early adolescence, Giacometti's difficulty in re-approaching the figure as an adult is generally understood as a sign of existential struggle for meaning, rather than as a technical deficit.

Giacometti was a key player in the Surrealist art movement, but his work resists easy categorization. Some describe it as formalist, others argue it is expressionist or otherwise having to do with what Deleuze calls "blocs of sensation" (as in Deleuze's analysis of Francis Bacon). Even after his excommunication from the Surrealist group, while the intention of his sculpting was usually imitation, the end products were an expression of his emotional response to the subject. He attempted to create renditions of his models the way he saw them, and the way he thought they ought to be seen. He once said that he was sculpting not the human figure but "the shadow that is cast".

Scholar William Barrett in Irrational Man: A Study in Existential Philosophy (1962), argues that the attenuated forms of Giacometti's figures reflect the view of 20th century modernism and existentialism that modern life is increasingly empty and devoid of meaning. "All the sculptures of today, like those of the past, will end one day in pieces...So it is important to fashion one's work carefully in its smallest recess and charge every particle of matter with life."

A 2011–2012 exhibition at the Pinacothèque de Paris focused on showing how Giacometti was inspired by Etruscan art.

Walking Man and other human figures
Giacometti is best known for the bronze sculptures of tall, thin human figures, made in the years 1945 to 1960. Giacometti was influenced by the impressions he took from the people hurrying in the big city. People in motion he saw as "a succession of moments of stillness".

The emaciated figures are often interpreted as an expression of the existential fear, insignificance and loneliness of mankind. The mood of fear in the period of the 1940s and the Cold War is reflected in this figure. It feels sad, lonely and difficult to relate to.

Exhibitions
Giacometti's work has been the subject of numerous solo exhibitions including the High Museum of Art, Atlanta (1970); Centre Pompidou, Paris (2007–2008); Pushkin Museum, Moscow "The Studio of Alberto Giacometti: Collection of the Fondation Alberto et Annette Giacometti" (2008); Kunsthal Rotterdam (2008); Fondation Beyeler, Basel (2009); Buenos Aires (2012); Kunsthalle Hamburg (2013); Pera Museum, Istanbul (2015); Tate Modern, London (2017); Vancouver Art Gallery, "Alberto Giacometti: A Line Through Time" (2019); National Gallery of Ireland, Dublin (2022).

The National Portrait Gallery, London's first solo exhibition of Giacometti's work, Pure Presence opened to five star reviews on 13 October 2015 (to 10 January 2016, in honour of the fiftieth anniversary of the artist's death). From April 2019, the Prado Museum in Madrid, has been highlighting Giacometti in an exhibition.

Public collections
Giacometti's work is displayed in numerous public collections, including:

 * Albright-Knox Art Gallery, Buffalo
 * Art Institute of Chicago
 * Baltimore Museum of Art, Baltimore, Maryland
 * Bechtler Museum of Modern Art, Charlotte, North Carolina
 * Berggruen Museum, Berlin
 * Botero Museum, Bogotá, Colombia
 * Bündner Kunstmuseum Chur, Switzerland
 * Carnegie Museum of Art, Pittsburgh
 * Detroit Institute of Arts
 * Fondation Beyeler, Basel
 * Hirshhorn Museum and Sculpture Garden, Washington D.C.
 * Holstebro, Denmark
 * J. Paul Getty Museum, Los Angeles, California
 * Johnson Museum of Art, Cornell University
 * Kunsthaus Zürich
 * Kunstmuseum Basel
 * Leeum, Samsung Museum of Art, South Korea
 * Los Angeles County Museum of Art
 * Louisiana Museum of Modern Art, Denmark
 * Minneapolis Institute of Art
 * Museum of Modern Art, New York
 * Museum of Fine Arts, Boston
 * National Gallery of Art, Washington D.C.
 * National Gallery of Canada, Ottawa
 * North Carolina Museum of Art, Raleigh, North Carolina
 * Sainsbury Centre for Visual Arts, University of East Anglia
 * Scottsdale Museum of Contemporary Art, Scottsdale, Arizona
 * Solomon R. Guggenheim Museum, New York
 * Tate, London
 * Tehran Museum of Contemporary Art, Iran
 * University of Michigan Museum of Art
 * Wadsworth Atheneum, Hartford
 * Walker Art Center, Minneapolis
 * Vancouver Art Gallery
 * Yale University Art Gallery, New Haven

Art foundations
The Fondation Alberto et Annette Giacometti, having received a bequest from Alberto Giacometti's widow Annette, holds a collection of circa 5,000 works, frequently displayed around the world through exhibitions and long-term loans. A public interest institution, the Foundation was created in 2003 and aims at promoting, disseminating, preserving and protecting Alberto Giacometti's work.

The Alberto-Giacometti-Stiftung established in Zürich in 1965, holds a smaller collection of works acquired from the collection of the Pittsburgh industrialist G. David Thompson.

Notable sales
According to record Giacometti has sold the two most expensive sculptures in history.

In November 2000 a Giacometti bronze, Grande Femme Debout I, sold for $14.3 million. Grande Femme Debout II was bought by the Gagosian Gallery for $27.4 million at Christie's auction in New York City on 6 May 2008.

L'Homme qui marche I, a life-sized bronze sculpture of a man, became one of the most expensive works of art, and at the time was the most expensive sculpture ever sold at auction. It was in February 2010, when it sold for £65 million (US$104.3 million) at Sotheby's, London. Grande tête mince, a large bronze bust, sold for $53.3 million just three months later.

L'Homme au doigt (Pointing Man) sold for $126 million (£81,314,455.32), or $141.3 million with fees, in Christie's May 2015, "Looking Forward to the Past" sale in New York City. The work had been in the same private collection for 45 years. As of now it is the most expensive sculpture sold at auction.

After being showcased on the BBC programme Fake or Fortune, a plaster sculpture, titled Gazing Head, sold in 2019 for half a million pounds.

In April 2021, Giacometti's small-scale bronze sculpture, Nu debout II (1953), was sold from a Japanese private collection and went for £1.5 million ($2 million), against an estimate of £800,000 ($1.1 million).

Other legacy
Giacometti created the monument on the grave of Gerda Taro at Père Lachaise Cemetery.

According to a lecture by Michael Peppiatt at Cambridge University on 8 July 2010, Giacometti, who had a friendship with author/playwright Samuel Beckett, created a tree for the set of a 1961 Paris production of Waiting for Godot.

Giacometti and his sculpture L'Homme qui marche I appear on the former 100 Swiss franc banknote.

In 2001 he was included in the Painting the Century: 101 Portrait Masterpieces 1900–2000 exhibition held at the National Portrait Gallery, London.

Giacometti's sculptural style has featured in advertisements for various financial institutions, starting in 1987 with the Shoes ad for Royal Bank of Scotland directed by Gerry Anderson.

The 2017 movie Final Portrait retells the story of his friendship with the biographer James Lord. Giacometti is played by Geoffrey Rush.

General sources

 * Jacques Dupin (1962). Alberto Giacometti, Paris, Maeght
 * Reinhold Hohl (1971). Alberto Giacometti, Stuttgart: Gerd Hatje
 * Die Sammlung der Alberto Giacometti-Stiftung (1990), Zürich, Zürcher Kunstgesellschaft
 * Alberto Giacometti (1991–92). ''Sculptures – peintures – dessins. Paris, Musée d'art moderne de la Ville de Paris,.
 * Jean Soldini (1993). Alberto Giacometti. Le colossal, la mère, le sacré, Lausanne, L'Age d'Homme
 * David Sylvester (1996) Looking at Giacometti, Henry Holt & Co.
 * Alberto Giacometti 1901–1966. Kunsthalle Wien, 1996
 * James Lord (1997). Giacometti: A Biography, Farrar, Straus and Giroux
 * Alberto Giacometti. Kunsthaus Zürich, 2001. New York: The Museum of Modern Art, 2001–2002.
 * Yves Bonnefoy (2006). Alberto Giacometti: A Biography of His Work, New edition, Flammarion

title: Alberto Giacometti


text:
Anthem

An anthem is a musical composition of celebration, usually used as a symbol for a distinct group, particularly the national anthems of countries. Originally, and in music theory and religious contexts, it also refers more particularly to short sacred choral work (still frequently seen in Sacred Harp and other types of shape note singing) and still more particularly to a specific form of liturgical music. In this sense, its use began c. 1550 in English-speaking churches; it uses English language words, in contrast to the originally Roman Catholic 'motet' which sets a Latin text.

Etymology
Anthem is derived from the Greek ἀντίφωνα (antíphōna) via Old English antefn. Both words originally referred to antiphons, a call-and-response style of the singing. The adjectival form is "anthemic".

History
Anthems were originally a form of liturgical music. In the Church of England, the rubric appoints them to follow the third collect at morning and evening prayer. Several anthems are included in the British coronation service. The words are selected from Holy Scripture or in some cases from the Liturgy and the music is generally more elaborate and varied than that of psalm or hymn tunes. Being written for a trained choir rather than the congregation, the Anglican anthem is analogous to the motet of the Catholic and Lutheran Churches but represents an essentially English musical form. Anthems may be described as "verse", "full", or "full with verse", depending on whether they are intended for soloists, the full choir, or both. Another way of describing an anthem is that it is a piece of music written specifically to fit a certain accompanying text, and it is often difficult to make any other text fit that same melodic arrangement. It also often changes melody and/or meter, frequently multiple times within a single song, and is sung straight through from start to finish, without repeating the melody for following verses like a normal song (although certain sections may be repeated when marked). An example of an anthem with multiple meter shifts, fuguing, and repeated sections is "Claremont", or "Vital Spark of Heav'nly Flame". Another well known example is William Billing's "Easter Anthem", also known as "The Lord Is Risen Indeed!" after the opening lines. This anthem is still one of the more popular songs in the Sacred Harp tune book.

The anthem developed as a replacement for the Catholic "votive antiphon" commonly sung as an appendix to the main office to the Blessed Virgin Mary or other saints.

Notable composers of liturgical anthems: historic context
During the Elizabethan period, notable anthems were composed by Thomas Tallis, William Byrd, Tye, and Farrant but they were not mentioned in the Book of Common Prayer until 1662 when the famous rubric "In quires and places where they sing here followeth the Anthem" first appears. Early anthems tended to be simple and homophonic in texture, so that the words could be clearly heard. During the 17th century, notable anthems were composed by Orlando Gibbons, Henry Purcell, and John Blow, with the verse anthem becoming the dominant musical form of the Restoration. In the 18th century, famed anthems were composed by Croft, Boyce, James Kent, James Nares, Benjamin Cooke, and Samuel Arnold. In the 19th century, Samuel Sebastian Wesley wrote anthems influenced by contemporary oratorio which stretch to several movements and last twenty minutes or longer. Later in the century, Charles Villiers Stanford used symphonic techniques to produce a more concise and unified structure. Many anthems have been written since then, generally by specialists in organ music rather than composers, and often in a conservative style. Major composers have usually written anthems in response to commissions and for special occasions: for instance Edward Elgar's 1912 "Great is the Lord" and 1914 "Give unto the Lord" (both with orchestral accompaniment); Benjamin Britten's 1943 "Rejoice in the Lamb" (a modern example of a multi-movement anthem, today heard mainly as a concert piece); and, on a much smaller scale, Ralph Vaughan Williams's 1952 "O Taste and See" written for the coronation of Queen Elizabeth II. With the relaxation of the rule, in England at least, that anthems should only be in English, the repertoire has been greatly enhanced by the addition of many works from the Latin repertoire.

Types
The word "anthem" is commonly used to describe any celebratory song or composition for a distinct group, as in national anthems. Further, some songs are artistically styled as anthems, whether or not they are used as such, including Marilyn Manson's "Irresponsible Hate Anthem", Silverchair's "Anthem for the Year 2000", and Toto's "Child's Anthem".

National anthem
A national anthem (also state anthem, national hymn, national song, etc.) is generally a patriotic musical composition that evokes and eulogizes the history, traditions, and struggles of a country's people, recognized either by that state's government as the official national song, or by convention through use by the people. The majority of national anthems are marches or hymns in style. The countries of Latin America, Central Asia, and Europe tend towards more ornate and operatic pieces, while those in the Middle East, Oceania, Africa, and the Caribbean use a simpler fanfare. Some countries that are devolved into multiple constituent states have their own official musical compositions for them (such as with the United Kingdom, Russian Federation, and the former Soviet Union); their constituencies' songs are sometimes referred to as national anthems even though they are not sovereign states.

Flag anthem
A flag anthem is generally a patriotic musical composition that extols and praises a flag, typically one of a country, in which case it is sometimes called a national flag anthem. It is often either sung or performed during or immediately before the raising or lowering of a flag during a ceremony. Most countries use their respective national anthems or some other patriotic song for this purpose. However, some countries, particularly in South America, use a separate flag anthem for such purposes. Not all countries have flag anthems. Some used them in the past but no longer do so, such as Iran, China, and South Africa. Flag anthems can be officially codified in law, or unofficially recognized by custom and convention. In some countries, the flag anthem may be just another song, and in others, it may be an official symbol of the state akin to a second national anthem, such as in Taiwan.

Sports anthem
Many pop songs are used as sports anthems, notably including Queen's "We Are the Champions" and "We Will Rock You", and some sporting events have their own anthems, most notably including UEFA Champions League.

Shared anthems
Although anthems are used to distinguish states and territories, there are instances of shared anthems. "Nkosi Sikelel' iAfrika" became a pan-African liberation anthem and was later adopted as the national anthem of five countries in Africa including Zambia, Tanzania, Namibia and Zimbabwe after independence. Zimbabwe and Namibia have since adopted new national anthems. Since 1997, the South African national anthem has been a hybrid song combining new English lyrics with extracts of "Nkosi Sikelel' iAfrika" and the former state anthem "Die Stem van Suid-Afrika".

For North and South Korea, the folk song Arirang is considered a shared anthem for both countries. For example, it was played when the two Koreas marched together during the 2018 Winter Olympics.

"Hymn to Liberty" is the longest national anthem in the world by length of text. In 1865, the first three stanzas and later the first two officially became the national anthem of Greece and later also that of the Republic of Cyprus.

"Forged from the Love of Liberty" was composed as the national anthem for the short-lived West Indies Federation (1958–1962) and was adopted by Trinidad and Tobago when it became independent in 1962.

"Esta É a Nossa Pátria Bem Amada" is the national anthem of Guinea-Bissau and was also the national anthem of Cape Verde until 1996.

"Oben am jungen Rhein", the national anthem of Liechtenstein, is set to the tune of "God Save the King/Queen". Other anthems that have used the same melody include "Heil dir im Siegerkranz" (Germany), "Kongesangen" (Norway), "My Country, 'Tis of Thee" (United States), "Rufst du, mein Vaterland" (Switzerland), "E Ola Ke Alii Ke Akua" (Hawaii), and "The Prayer of Russians".

The Estonian anthem "Mu isamaa, mu õnn ja rõõm" is set to a melody composed in 1848 by Fredrik (Friedrich) Pacius which is also that of the national anthem of Finland: "Maamme" ("Vårt Land" in Swedish). It is also considered to be the ethnic anthem for the Livonian people with lyrics "Min izāmō, min sindimō" ("My Fatherland, my native land").

"Hey, Slavs" is dedicated to Slavic peoples. Its first lyrics were written in 1834 under the title "Hey, Slovaks" ("Hej, Slováci") by Samuel Tomášik and it has since served as the ethnic anthem of the Pan-Slavic movement, the organizational anthem of the Sokol physical education and political movement, the national anthem of Yugoslavia and the transitional anthem of the State Union of Serbia and Montenegro. The song is also considered to be the second, unofficial anthem of the Slovaks. Its melody is based on Mazurek Dąbrowskiego, which has also been the anthem of Poland since 1926, but the Yugoslav variation is much slower and more accentuated.

Between 1991 and 1994 "Deșteaptă-te, române!" was the national anthem of both Romania (which adopted it in 1990) and Moldova, but in the case of the latter it was replaced by the current Moldovan national anthem, "Limba noastră". Between 1975 and 1977, the national anthem of Romania "E scris pe tricolor Unire" shared the same melody as the national anthem of Albania "Himni i Flamurit", which is the melody of a Romanian patriotic song "Pe-al nostru steag e scris Unire".

The modern national anthem of Germany, "Das Lied der Deutschen", uses the same tune as the 19th- and early 20th-century Austro-Hungarian imperial anthem "Gott erhalte Franz den Kaiser".

The "Hymn of the Soviet Union", was used until its dissolution in 1991, and was given new words and adopted by the Russian Federation in 2000 to replace an instrumental national anthem that had been introduced in 1990.

"Bro Gozh ma Zadoù", the regional anthem of Brittany and, "Bro Goth Agan Tasow", the Cornish regional anthem, are sung to the same tune as that of the Welsh regional anthem "Hen Wlad Fy Nhadau", with similar words.

For parts of states
Some countries, such as the former Soviet Union, Spain, and the United Kingdom, among others, are held to be unions of several "nations" by various definitions. Each of the different "nations" may have their own anthem and these songs may or may not be officially recognized; these compositions are typically referred to as regional anthems though may be known by other names as well (e.g. "state songs" in the United States).

Austria
In Austria, the situation is similar to that in Germany. The regional anthem of Upper Austria, the "Hoamatgsang" ("Chant of the Homeland"), is notable as the only (official) German-language anthem written – and sung – entirely in dialect.

Belgium
In Belgium, Wallonia uses "Le Chant des Wallons" and Flanders uses "De Vlaamse Leeuw".

Brazil
Most of the Brazilian states have official anthems. Minas Gerais uses an adapted version of the traditional Italian song "Vieni sul mar" as its unofficial anthem. During the Vargas Era (1937–1945) all regional symbols including anthems were banned, but they were legalized again by the Eurico Gaspar Dutra government.

Canada
The Canadian province of Newfoundland and Labrador, having been the independent Dominion of Newfoundland before 1949, also has its own regional anthem from its days as a dominion and colony of the UK, the "Ode to Newfoundland". It was the only Canadian province with its own anthem until 2010, when Prince Edward Island adopted the 1908 song "The Island Hymn" as its provincial anthem.

Czechoslovakia
Czechoslovakia had a national anthem composed of two parts, the Czech anthem followed by one verse of the Slovak one. After the dissolution of Czechoslovakia, the Czech Republic adopted its own regional anthem as its national one, whereas Slovakia did so with slightly changed lyrics and an additional stanza.

Germany
In Germany, many of the Länder (states) have their own anthems, some of which predate the unification of Germany in 1871. A prominent example is the Hymn of Bavaria, which also has the status of an official anthem (and thus enjoys legal protection). There are also several unofficial regional anthems, like the "Badnerlied" and the "Niedersachsenlied".

India
Some of the states and union territories of India have officially adopted their own state anthem for use during state government functions.

Malaysia
All the individual states of Malaysia have their own anthems.

Mexico
In Mexico, after the national anthem was established in 1854, most of the states of the federation adopted their own regional anthems, which often emphasize heroes, virtues or particular landscapes. In particular, the regional anthem of Zacatecas, the "Marcha de Zacatecas", is one of the more well-known of Mexico's various regional anthems.

Serbia and Montenegro
In 2004 and 2005 respectively, the Montenegrin and Serbian regions of Serbia and Montenegro adopted their own regional anthems. When the two regions both became independent sovereign states in mid-2006, their regional anthems became their national anthems.

Soviet Union
Fourteen of the fifteen constituent states of the Soviet Union had their own official song which was used at events connected to that region, and also written and sung in that region's own language. The Russian Soviet Federative Socialist Republic used the Soviet Union's national anthem as its regional anthem ("The Internationale" from 1917 to 1944 and the "National Anthem of the Soviet Union" from 1944 to 1990) until 1990, the last of the Soviet constituent states to do so. After the Soviet Union disbanded in the early 1990s, some of its former constituent states, now sovereign nations in their own right, retained the melodies of their old Soviet-era regional anthems until replacing them or, in some cases, still use them today.

Unlike most national anthems, few of which were composed by renowned composers, the Soviet Union's various regional anthems were composed by some of the best Soviet composers, including world-renowned Gustav Ernesaks (Estonia), Aram Khachaturian (Armenia), Otar Taktakishvili (Georgia), and Uzeyir Hajibeyov (Azerbaijan).

The lyrics present great similarities, all having mentions to Vladimir Lenin (and most, in their initial versions, to Joseph Stalin, the Armenian and Uzbek anthems being exceptions), to the guiding role of the Communist Party of the Soviet Union, and to the brotherhood of the Soviet peoples, including a specific reference to the friendship of the Russian people (the Estonian, Georgian and Karelo-Finnish anthems were apparently an exception to this last rule).

Some of the Soviet regional anthems' melodies can be sung in the Soviet Union anthem lyrics (Ukrainian and Belarus are the most fitted in this case).

Most of these regional anthems were replaced with new national ones during or after the dissolution of the Soviet Union; Belarus, Kazakhstan (until 2006), Tajikistan, Turkmenistan (until 1997), and Uzbekistan kept the melodies, but with different lyrics. Russia itself had abandoned the Soviet hymn, replacing it with a tune by Glinka. However, with Vladimir Putin coming to power, the old Soviet tune was restored, with new lyrics written to it.

Like the hammer and sickle and red star, the public performance of the anthems of the Soviet Union's various regional anthems the national anthem of the Soviet Union itself are considered as occupation symbols as well as symbols of totalitarianism and state terror by several countries formerly either members of or occupied by the Soviet Union. Accordingly, Latvia, Lithuania, Hungary, and Ukraine  have banned those anthems amongst other things deemed to be symbols of fascism, socialism, communism, and the Soviet Union and its republics. In Poland, dissemination of items which are “media of fascist, communist, or other totalitarian symbolism” was criminalized in 1997. However, in 2011 the Constitutional Tribunal found this sanction to be unconstitutional. In contrast to this treatment of the symbolism, promotion of fascist, communist and other totalitarian ideology remains illegal. Those laws do not apply to the anthems of Russia, Belarus, Uzbekistan, Kazakhstan, and Tajikistan which used the melody with different lyrics.

Spain
In Spain, the situation is similar to that in Austria and Germany. Unlike the national anthem, most of the anthems of the autonomous communities have words. All are official. Three prominent examples are "Els Segadors" of Catalonia, "Eusko Abendaren Ereserkia" of the Basque Country, and "Os Pinos" of Galicia, all written and sung in the local languages.

United Kingdom
The United Kingdom's national anthem is "God Save the King" but its constituent countries and Crown Dependencies also have their own equivalent songs which have varying degrees of official recognition. England, Scotland, Wales, and Northern Ireland each have anthems which are played at occasions such as sports matches and official events.

 * England - "God Save the King" is usually presumed to be, and often played as, the English regional anthem; but "Jerusalem", "I Vow To Thee, My Country" and "Land of Hope and Glory" are also sung. "Jerusalem" is used as England's anthem at the Commonwealth Games.
 * Scotland variously uses "Flower of Scotland", "Auld Lang Syne", and "Scotland the Brave" as its unofficial national anthems. "Flower of Scotland" is used as Scotland's anthem at the Commonwealth Games and international football and rugby matches.
 * Wales has sung "Hen Wlad Fy Nhadau" since 1856 when it was written by father and son Evan and James James. The music and a Breton translation, "Bro Gozh ma Zadoù", were adopted by Brittany as its anthem; and there is also a Cornish version, "Bro Goth agan Tasow", sung alongside "Trelawney" as an unofficial Cornish anthem. In Wales, "Hen Wlad fy Nhadau" is sometimes accompanied by the hymn, "Guide Me, O thou Great Redeemer" (also referred to as "Bread of Heaven" from repeated words in its first verse), especially at rugby matches.
 * Northern Ireland currently uses "God Save the King" as its anthem at international football matches and uses "Danny Boy/Londonderry Air" at the Commonwealth Games.

The Isle of Man, a Crown dependency, uses "God Save the King" as a Royal anthem, but also has its own local anthem, "O Land of Our Birth" (Manx: "O Halloo Nyn Ghooie").

United States
Although the United States has "The Star-Spangled Banner" as its official national anthem, all except two of its constituent states and territories also have their own regional anthem (referred to by most US states as a "state song"), along with Washington, DC. The two exceptions are New Jersey, which has never had an official state song, and Maryland, which rescinded "Maryland, My Maryland" in 2021 due to its racist language and has yet to adopt a replacement.

The state songs are selected by each state legislature, and/or state governor, as a symbol (or emblem) of that particular US state.

Some US states have more than one official state song, and may refer to some of their official songs by other names; for example, Arkansas officially has two state songs, plus a state anthem, and a state historical song. Tennessee has the most state songs, with 12 official state songs and an official bicentennial rap.

Arizona has a song that was written specifically as a state anthem in 1915, as well as the 1981 country hit "Arizona", which it adopted as the alternate state anthem in 1982.

Two individuals, Stephen Foster, and John Denver, have written or co-written two state songs. Foster's two state songs, "Old Folks at Home" (better known as "Swanee Ribber" or "Suwannee River"), adopted by Florida, and "My Old Kentucky Home" are among the best-known songs in the US On March 12, 2007, the Colorado Senate passed a resolution to make Denver's trademark 1972 hit "Rocky Mountain High" one of the state's two official state songs, sharing duties with its predecessor, "Where the Columbines Grow". On March 7, 2014, the West Virginia Legislature approved a resolution to make Denver's "Take Me Home, Country Roads" one of four official state songs of West Virginia. Governor Earl Ray Tomblin signed the resolution into law on March 8, 2014. Additionally, Woody Guthrie wrote or co-wrote two state folk songs – Roll On, Columbia, Roll On and Oklahoma Hills – but they have separate status from the official state songs of Washington and Oklahoma, respectively. Other well-known state songs include "Yankee Doodle", "You Are My Sunshine", "Rocky Top", and "Home on the Range"; a number of others are popular standards, including "Oklahoma" (from the Rodgers and Hammerstein musical), Hoagy Carmichael's "Georgia on My Mind", "Tennessee Waltz", "Missouri Waltz", and "On the Banks of the Wabash, Far Away". Many of the others are much less well-known, especially outside the state.

New Jersey has no official state song, while Virginia's previous state song, "Carry Me Back to Old Virginny", adopted in 1940, was later rescinded in 1997 due to its racist language by the Virginia General Assembly. In 2015, "Our Great Virginia" was made the new state song of Virginia.

Iowa ("The Song of Iowa") uses the tune from the song "O Tannenbaum" as the melody to its official state song.

Yugoslavia
In Yugoslavia, each of the country's constituent states (except for Bosnia and Herzegovina) had the right to have its own anthem, but only the Croatian one actually did so initially, later joined by the Slovene one on the brink of the breakup of Yugoslavia. Before 1989, Macedonia did not officially use a regional anthem, even though one was proclaimed during the World War II by the Anti-fascist Assembly for the National Liberation of Macedonia (ASNOM).

International organizations
Larger entities also sometimes have anthems, in some cases known as 'international anthems'. Lullaby is the official anthem of UNICEF composed by Steve Barakatt. "The Internationale" is the organizational anthem of various socialist movements. Before March 1944, it was also the anthem of the Soviet Union and the Comintern. ASEAN Way is the official anthem of ASEAN. The tune of the "Ode to Joy" from Beethoven's Symphony No. 9 is the official anthem of the European Union and of the Council of Europe. Let's All Unite and Celebrate is the official anthem of the African Union ("Let Us All Unite and Celebrate Together").

The Olympic Movement also has its own organizational anthem. Esperanto speakers at meetings often use the song "La Espero" as their linguistic anthem. The first South Asian Anthem by poet-diplomat Abhay K may inspire SAARC to come up with an official SAARC Anthem.

"Ireland's Call" was commissioned as the sporting anthem of both the Ireland national rugby union team and the Ireland national rugby league team, which are composed of players from both jurisdictions on the island of Ireland, in response to dissatisfaction among Northern Ireland unionists with the use of the Irish national anthem. "Ireland's Call" has since been used by some other all-island bodies.

An international anthem also unifies a group of organizations sharing the same appellation such as the International Anthem of the Royal Golf Clubs composed by Steve Barakatt. Same applies to the European Broadcasting Union: the prelude of Te Deum in D Major by Marc-Antoine Charpentier is played before each official Eurovision and Euroradio broadcast. The prelude's first bars are heavily associated with the Eurovision Song Contest.

Global anthem
Various artists have created "Earth anthems" for the entire planet, typically extolling the ideas of planetary consciousness. Though UNESCO have praised the idea of a global anthem, the United Nations has never adopted an official song.

title: Anthem