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Tensegrity Lego. it’s crazy So the string between the purple piece is supporting the weight of the upper part, while the corner strings keeps it from flipping. The single supporting string creates an unstable equilibrium, which requires the two side strings to hold things steady.

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Nyckelharpa has existed in one form or another format least 600 years, and is generally considered an instrument of Swedes. Given how Christianity came later to Sweden, its earliest iterations can easily be associated with Norse gods and secular life in the Nordic lands. There were multiple versions of stringed instrument, as it has evolved over time, to today where it could be mistaken for an instrumental marriage between a bowed instrument (violin, viola, cello, bass) and a keyboard such as a piano, harpsichord or perhaps an accordion. It’s unclear exactly what earliest versions of Nyckelharpa looked like. There are just three surviving instruments from 16th Century, but there are earlier depictions of an instrument that resembles the boxy, oblong contraption. Church paintings in Denmark, Sweden and Italy, dating to early 1400s, and later (about 1590) in a church in Hildesheim, Germany, seem to show the instrument. An even older example in ecclesiastical art dates from about 1350 on a carved relief on a gate into Kallunge Church in Gotland, Sweden, which shows what might be two Nyckelharpa players in what we are to assume is a performance of pre-Reformation sacred music (academics stop short of declaring these to be definitively Nyckelharpa). The physical instrument evolved in various iterations over time, as evidenced by addition of strings, now in three rows: the three melody strings, a drone C, and 12 resonance (sympathetic) strings, which were more modern addition (16th Century). The keys, played with the left hand while the right hand bows the strings, serve as frets when a key is depressed. This changes the pitch of the string, similar to what the violinists’ hands do on a violin. The nyckelharpa belongs to the same instrument family as French vielle and the English hurdy gurdy. Nyckelharpa has wooden keys that slide under the strings and have tangents set perpendicularly to the keys that reach up and stop (shorten) melody string. It’s sort of like moveable frets that move to meet the string, rather than pressing the string against a fingerboard or frets. A short bow is used. The oldest “evidence” of nyckelharpa use is a relief on one of the gates to Källunge church on Gotland from about 1350 AD, depicting two nyckelharpa players. There are three surviving examples of medieval nyckelharpa: one found in the town of Mora in Dalarna, Sweden, one found in Vefsen, Norway (it hangs in the Musik Museum in Stockholm); and one found in Esse, Finland. Moraharpa is dated 1526, and hangs in the Zorn Museum in Mora. They have one row of keys (although some keys stop two strings, making them a mix of melody and drone strings) and two drone strings. Moraharpa is shaped like a lute, while the Vefsenharpa and Esseharpa are shaped more like modern nyckelharpa. Recent evidence suggests that Moraharpa was probably built in early 1600’s, using wood that was already about 100 years old. It was likely made as a copy of the picture in the German book (picture forthcoming). So it appears that lute-shaped tradition is broken, while the long/narrow tradition is unbroken. A few people still play this version of the instrument today, including Anders Stake (Anders Norrude) in group “Hedningarna”. Perhaps under the influence of the viola d’amore which was popular in Uppsala and Stockholm in late 1600’s and early 1700’s, resonance strings were added. Sympathetic strings were brought from the near east to England circa 1600, and were thought to be a fashionable innovation. On the nyckelharpa, the resonance strings are set lower in the bridge, so the bow does not touch them. Oldest surviving example of enkelharpa is from 1777. Mixturharpa is similar to enkelharpa, except that 2nd string also has tangents (on the same key at the tangent for the 1st string) that stop the string and make the notes. Not many people today play this version of nyckelharpa, opting for older or younger versions. © American Nyckelharpa Association #archaeohistories

Archaeo - Histories

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String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

Mathelirium

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