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Chord

'''Chord''' is a term with multiple meanings across various fields. In music, a chord is a harmonic set of three or more pitches sounded simultaneously. In mathematics, particularly geometry, a chord is a straight line segment whose endpoints both lie on a given curve. In biology, a chord refers to the notochord, a flexible rod-like structure present in the embryos of all chordates. In physics and engineering, it denotes a principal horizontal member in a truss or the straight line connecting the leading and trailing edges of an airfoil. In programming and computing, Chord is a protocol for distributed hash tables in peer-to-peer networks (see Distributed systems subsection). The term also applies to notable individuals and fictional characters (see People section).

Music

Definition

In music theory, a chord is the simultaneous sounding of three or more pitches, creating a harmonic simultaneity that forms the vertical dimension of musical texture.^[1] The simplest and most common chord is the triad, comprising a root pitch, a third above it (either major or minor), and a fifth above the root, which together establish basic tonal relationships.^[2] The historical origins of chords trace back to ancient Greek theory, where Pythagoras (c. 570–495 BCE) identified consonant intervals through mathematical ratios—such as the octave (2:1) and perfect fifth (3:2)—via experiments with the monochord, linking music to cosmic order.^[3] This foundation evolved during the Renaissance with the rise of polyphony, as composers like Josquin des Prez layered independent voices to produce vertical harmonies, formalized by theorists such as Gioseffo Zarlino in his 1558 Le Istitutioni harmoniche, which emphasized proportional consonance in multi-voice compositions.^[3] By the 18th century, Jean-Philippe Rameau advanced modern harmonic principles in Traité de l'harmonie (1722), treating chords as fundamental units derived from natural overtones and assigning them functional roles within tonal keys.^[3] Chords function harmonically through consonance and dissonance: consonance arises from stable, pleasing interval combinations that evoke resolution, while dissonance introduces tension through clashing intervals that demand progression toward consonance.^[4] The major triad, with its major third and perfect fifth, represents a prototypical consonant harmony, conveying brightness and stability, whereas the minor triad, featuring a minor third, provides a consonant yet more subdued affective quality.^[5] The English term "chord" evolved from Middle English "cord," a spelling influenced by Latin chorda (gut string) and ultimately from Greek khordē (string), but semantically derived from "accord" to signify the agreement of sounding notes.^[6]

Types and construction

Musical chords are primarily constructed by stacking intervals in thirds, beginning with the simplest form, the triad, which consists of three notes: a root, a third above the root, and a fifth above the root.^[7] The most common triads are major, minor, diminished, and augmented, each defined by the quality of their intervals measured in semitones from the root. A major triad is formed by a root, a major third (4 semitones above the root), and a perfect fifth (7 semitones above the root), resulting in the interval structure 0-4-7; for example, a C major triad comprises the notes C-E-G.^[8] In contrast, a minor triad uses a root, a minor third (3 semitones), and a perfect fifth (7 semitones), yielding 0-3-7, as in C-E♭-G.^[8] The diminished triad features a root, minor third (3 semitones), and diminished fifth (6 semitones), creating 0-3-6, such as C-E♭-G♭, which produces a tense, unstable sound.^[8] An augmented triad consists of a root, major third (4 semitones), and augmented fifth (8 semitones), forming 0-4-8, like C-E-G♯, often used for dramatic effect due to its symmetrical structure.^[8] Seventh chords extend the triad by adding a seventh interval above the root, typically another third stacked on the fifth, resulting in four notes.^[9] The dominant seventh, common in functional harmony, builds on a major triad with a minor seventh (10 semitones), giving 0-4-7-10, as in C-E-G-B♭.^[8] A major seventh chord adds a major seventh (11 semitones) to a major triad, producing 0-4-7-11, exemplified by C-E-G-B.^[8] The minor seventh chord stacks a minor seventh (10 semitones) on a minor triad, yielding 0-3-7-10, such as C-E♭-G-B♭.^[8] Extended chords build further by adding notes beyond the seventh, continuing the pattern of thirds while often omitting certain chord tones for playability.^[10] Ninth chords incorporate a ninth (2 semitones above the octave, or 14 semitones from the root) above a seventh chord, compatible with major, minor, dominant, and diminished types; for instance, a dominant ninth might be C-E-G-B♭-D.^[10] Eleventh chords add an eleventh (5 semitones above the octave, or 17 semitones total), typically on minor or diminished seventh chords to avoid clashing with the major third, as in C-E♭-G-B♭-D-F.^[10] Thirteenth chords extend to a thirteenth (7 semitones above the octave, or 21 semitones), often on dominant or major seventh chords, with omissions like the fifth or ninth for voicing, such as C-E-G-B♭-D-F-A.^[10] Inversions rearrange the chord notes so that a note other than the root is in the bass, altering the voicing without changing the chord's identity; voicings refer to the specific spacing and order of notes for instruments like piano or guitar.^[11] For triads, root position places the root in the bass (e.g., C-E-G), first inversion puts the third in the bass (E-G-C), and second inversion the fifth (G-C-E).^[11] Seventh chords have an additional third inversion with the seventh in the bass (e.g., for Cmaj7: B-C-E-G).^[11] Spread voicings, common on piano and guitar, distribute notes across octaves for fuller sound, such as voicing a C major seventh as C-G-B-E to emphasize openness.^[11] Non-triadic chords deviate from third-based construction, encompassing clusters and polychords, particularly in 20th-century atonal music.^[12] Tone clusters stack adjacent notes (seconds) for dense dissonance, as in secundal harmony.^[12] Polychords superimpose two or more triads, often separated by register, to create complex sonorities.^[12] Arnold Schoenberg employed such structures in his free atonal works, like Pierrot Lunaire (1912), to maximize dissonance and avoid tonal resolution, influencing expressionist composition.^[12]

Mathematics

Geometry

In geometry, a chord is defined as a straight line segment connecting two points on a curve, with the term most commonly applied to such segments on a circle. This figure contrasts with the arc, which is the curved path between the same points along the circle's boundary.^[13] Within a circle of radius r, several key properties govern chords. The perpendicular line drawn from the circle's center to a chord bisects that chord, dividing it into two equal segments; this follows from the symmetry of the circle and the equal radii forming isosceles triangles. Additionally, the length of a chord subtending a central angle \theta (in radians) is given by the formula $2r \sin(\theta/2), derived from trigonometric considerations in the isosceles triangle formed by the two radii and the chord. Angles inscribed in the circle that subtend the same chord are equal, a consequence of the inscribed angle theorem.^[13]^[14]^[15] Important theorems involving chords include the intersecting chords theorem, which states that if two chords intersect inside a circle, the products of the lengths of their respective segments are equal: for chords AB and CD intersecting at P, AP \cdot PB = CP \cdot PD. This is a direct application of similar triangles formed by the intersecting lines. Relatedly, the power of a point theorem asserts that for a point inside the circle, the product of the segment lengths of any chord passing through that point is constant, equal to r^2 - d^2 where d is the distance from the center to the point; this value remains invariant regardless of the chord's direction.^[13]^[16] Chords find applications in more complex curves such as cycloids and ellipses. In a cycloid—the path traced by a point on a circle rolling along a straight line—chords connect points on the generating circle to analyze parametric properties, such as the equality of certain segment lengths to chord measures in the circle. For ellipses, an analogous intersecting chords theorem holds, where the rectangles formed by pairs of intersecting chords are proportional to the squares of parallel diameters, extending circle properties to these conic sections. Historically, Archimedes employed chords as sides of inscribed and circumscribed regular polygons (up to 96 sides) in his Measurement of a Circle to approximate the value of \pi, bounding it between $223/71 and [22/7](/page/22/7).^[17]^[18]^[19] Diagrams illustrating chords typically depict a circle with a chord AB, the center O, and the perpendicular from O to the chord's midpoint M, highlighting the bisection and forming right triangles OMA and OMB. Another common visualization shows two intersecting chords within the circle, labeling segment lengths to demonstrate the theorem, or inscribed angles subtending the chord to show equality. Such figures emphasize the chord's role in bridging straight-line geometry with curved boundaries.^[13]

Advanced concepts

In graph theory, chordal graphs are undirected graphs in which every cycle of length four or greater contains a chord connecting two non-adjacent vertices in the cycle.^[20] This property ensures that chordal graphs are perfect graphs, meaning their chromatic number equals the size of the maximum clique.^[21] A key characterization is the existence of a perfect elimination ordering (PEO), an ordering of vertices such that, for each vertex, its later neighbors form a clique.^[21] Chordal graphs admit efficient algorithms for optimization problems, including minimum coloring and maximum clique detection, running in linear time relative to the number of edges.^[22] In semidefinite programming, chordal structure exploits sparsity by decomposing the problem into smaller dense subproblems along a clique tree, enabling scalable solutions for large-scale applications like control theory and machine learning.^[21] A fundamental trigonometric identity expresses the length of a chord subtending a central angle \theta in the unit circle as $2 \sin(\theta/2).^[23] This formula arises from considering the isosceles triangle formed by two radii and the chord, halving it to apply the half-angle sine relation.^[24] It facilitates computations in circular geometries and appears in Fourier analysis for evaluating distances between points on the unit circle, such as phase differences in periodic signals where the chord length quantifies angular separation in frequency domain representations.^[25] In complex analysis, chords play a role through the chordal metric on the Riemann sphere, which extends the complex plane by adding a point at infinity and equips it with a metric invariant under Möbius transformations.^[26] For points z, w \in \mathbb{C} \cup \{\infty\}, the chordal distance is defined as d(z, w) = \frac{|z - w|}{\sqrt{(1 + |z|^2)(1 + |w|^2)}}, corresponding to the Euclidean chord length between stereographic projections on the unit sphere.^[27] This metric is crucial in conformal mapping, as it preserves angles and provides a complete metric space for studying holomorphic functions on Riemann surfaces, facilitating proofs of compactness and uniformization theorems.^[26] Chord diagrams are combinatorial objects used in knot theory to represent links and compute Vassiliev invariants, consisting of a circle with non-intersecting chords connecting pairs of points that model double points in knot projections.^[28] Each diagram encodes the Gauss code of a knot or link via chord endpoints on the circle, with intersections of chords (but not the circle) forbidden except at endpoints.^[29] Seminal work by Vassiliev introduced these diagrams to generate finite-type invariants through singularity resolution, where relations like 4T and 1T simplify diagrams while preserving invariant values.^[28] Applications include deriving weight systems from Lie algebras, enabling classification of low-degree knot invariants beyond classical polynomials.^[29] In computational geometry, algorithms for computing chord lengths support polygon approximation of curves, where a smooth curve is discretized into a polygon by selecting vertices such that the error, measured by maximum deviation from chords to the curve, is minimized.^[30] A dynamic programming approach solves the uniform-cost variant in O(n^2) time for n sample points, optimizing vertex selection to balance chord lengths against approximation error.^[30] For non-uniform costs, such as minimizing total chord length subject to error bounds, approximation algorithms achieve constant-factor guarantees, with applications in computer-aided design and geographic information systems for efficient curve representation.^[30]

Science and technology

Biology

The notochord is a flexible, rod-like embryonic midline structure defining the phylum Chordata, composed of large vacuolated notochordal cells surrounded by an extracellular matrix sheath rich in collagen, laminin, and proteoglycans.^[31] This structure provides both mechanical support as a hydrostatic skeleton and signaling functions during early development.^[31] In embryogenesis, the notochord plays a critical role in inducing neural tube formation by secreting signaling molecules such as Sonic hedgehog (Shh), which patterns the ventral neural tube and floor plate.^[31] It also offers axial elongation and support to the developing embryo, acting as a primitive skeleton before the vertebral column forms.^[32] Evolutionarily, the notochord is a synapomorphy of chordates, present in all members including invertebrate forms like lancelets (cephalochordates) where it persists into adulthood, and tunicates (urochordates) where it appears transiently in larvae.^[33] In vertebrates, it serves as the precursor to the vertebral column, with surrounding sclerotome cells differentiating into vertebrae that envelop and replace much of the notochord.^[33] In human embryos, the notochord forms around weeks 3-4 and is transient, largely degenerating by birth as its cells condense to form the nucleus pulposus—the gel-like core of intervertebral discs that provides shock absorption in the spine.^[32] Malformations of the notochord, often due to disruptions in planar cell polarity pathways like VANGL1 and VANGL2 mutations, can impair neural tube closure and contribute to spinal defects such as spina bifida.^[34]

Physics and engineering

In aerodynamics, the chord length refers to the straight-line distance between the leading and trailing edges of an airfoil, serving as a fundamental dimension for analyzing lift, drag, and overall wing performance.^[35] This measurement defines the chord line, which provides a reference for the airfoil's angle of attack and camber. For aircraft with tapered or swept wings, the mean aerodynamic chord (MAC) represents the average chord length across the wing, weighted by the local lift distribution, and is crucial for determining the center of gravity position to ensure longitudinal stability.^[36] Typically, the center of gravity is positioned at 15-25% of the MAC to optimize controllability and prevent instability during flight.^[37] In structural engineering, chords form the principal longitudinal members of truss systems, such as those used in bridges, where the top chord primarily resists compressive forces and the bottom chord handles tensile loads under applied weights.^[38] In a Warren truss, these parallel chords are connected by equilateral triangular diagonals that alternate between tension and compression, enabling efficient load distribution without vertical members and making it suitable for spans up to 100 meters.^[39] This design, patented in 1848 by James Warren, exemplifies how chords contribute to the truss's rigidity and economy of material in civil infrastructure.^[40] Astronomy employs the chord concept in stellar occultations to measure planetary and asteroid diameters precisely. During an occultation, when a celestial body passes in front of a star, observers on Earth record the timing of the light's disappearance and reappearance, defining a chord as the straight-line path across the body's silhouette.^[41] Multiple such chords from different observation sites allow reconstruction of the body's shape and size; for instance, a 2022 multi-chord observation of the near-Earth asteroid (3200) Phaethon yielded projected dimensions of 6.12 ± 0.07 km × 4.14 ± 0.07 km by fitting the chord lengths to an ellipsoidal model.^[42] This method provides sub-kilometer accuracy, far superior to radar or thermal imaging for small bodies.^[43] In physics, the chord approximation simplifies the analysis of pendulum motion beyond the small-angle limit. For larger amplitudes, replacing the arc of swing with the straight chord connecting the endpoints adjusts the effective length, yielding a more accurate period estimate via T \approx 2\pi \sqrt{\frac{l}{g}} \left(1 + \frac{\theta_0^2}{16}\right), where l is the chord length, g is gravity, and \theta_0 is the initial angle.^[44] This approach, derived from geometric considerations, reduces error compared to the simple harmonic approximation for angles up to 45 degrees.^[45] Engineering applications of chords extend to historical and modern contexts, such as in wind turbine blades where chord length varies along the span to optimize aerodynamic efficiency. In contemporary designs, blades taper from a wider chord near the root (up to 3-4 meters) to a narrower tip, enhancing lift at low wind speeds while minimizing drag at rated conditions.^[46]

Programming and computing

Input devices

Chorded input devices enable users to enter characters, commands, or actions by pressing multiple keys or buttons simultaneously, analogous to striking a chord on a piano keyboard to produce a harmonious sound. These devices prioritize compactness and efficiency over sequential keying, allowing for reduced hardware size while supporting complex inputs through predefined combinations.^[47] The history of chorded keyboards traces back to the late 19th century, with early applications in telegraphy and stenography for rapid transcription. Devices like the stenotype machine, developed around 1877 for court reporting, used a chorded layout where operators pressed up to 10 keys at once to phonetically encode syllables or words, achieving speeds over 200 words per minute after extensive training.^[48] Modern chorded keyboards evolved from these foundations, incorporating ergonomic designs for computing. The BAT keyboard, introduced by Infogrip around 1986, features a one-handed setup with seven keys on a base for resting the hand, enabling users to generate alphanumeric characters and macros through key combinations; it supports efficient typing with practice and is particularly suited for accessibility needs.^[49] Similarly, the Twiddler, first prototyped in 1992 for wearable computing, employs a 3x4 button grid operated by one hand, where letters are entered via single or dual-key chords and frequent words via multi-character chords, allowing expert users to reach 60-65 words per minute.^[50] Mouse chording extends the chord principle to graphical user interfaces, where holding multiple mouse buttons simultaneously triggers specific functions. In the 1968 oN-Line System (NLS) developed by Douglas Engelbart, a five-button chorded keyset paired with a three-button mouse allowed users to perform editing commands by chording while manipulating the cursor, demonstrating early integration of simultaneous inputs for productivity.^[51] In contemporary computer-aided design (CAD) software, such as AutoCAD, chording techniques like holding the middle mouse button with wheel scrolling enable panning and zooming, while combining buttons with keyboard modifiers (e.g., Shift + right-click) accesses context-sensitive menus for object manipulation. In gaming, mouse chording facilitates combo actions, such as pressing left and right buttons together to execute special attacks in first-person shooters, enhancing responsiveness without interrupting movement controls.^[52] Chorded input devices offer advantages in compactness and one-handed operation, making them ideal for portable or space-constrained environments, such as wearable computers or assistive technologies for users with mobility impairments. For instance, the BAT and Twiddler reduce key count to 7-12, minimizing physical strain and enabling input while the other hand performs tasks like mouse control, with potential speeds rivaling traditional keyboards after training.^[47] However, they present disadvantages including a learning curve requiring substantial practice due to memorization of chord mappings—ranging from several hours for basic use to longer for proficiency—and higher initial error rates for novice users, as there are no visual cues like QWERTY labels. In accessibility applications, chorded keyboards like on-screen variants or the BAT aid individuals with physical disabilities by allowing customizable chord assignments for reduced fatigue, though adoption depends on user-specific training.^[47] Contemporary implementations adapt chorded principles to emerging interfaces, particularly in virtual reality (VR) and mobile systems. In VR, hand controllers like those for Oculus or HTC Vive support chorded text entry by mapping finger grips or button combinations to keyboard layouts, enabling one-handed typing without additional hardware, as demonstrated in mappings for consumer VR setups. Wearable solutions, such as the chording glove, embed keys on fingers for gesture-based input in immersive environments, detecting simultaneous presses via sensors to input text or commands. On mobile devices, multi-touch gesture systems emulate chording through simultaneous taps or swipes on screens, facilitating quick shortcuts in apps while maintaining portability.^[53]

Distributed systems

Chord is a distributed hash table (DHT) protocol designed for scalable key-value storage and lookup in peer-to-peer (P2P) networks, enabling efficient mapping of keys to nodes without centralized control.^[54] Introduced in 2001 by Ion Stoica, Robert Morris, David Karger, M. Frans Kaashoek, and Hari Balakrishnan, it organizes nodes in a logical ring structure using consistent hashing, where both nodes and keys are assigned m-bit identifiers from a hash function, forming a circular identifier space of size $2^m.^[54] This approach ensures that keys are stored on the successor node responsible for the interval they fall into, supporting applications requiring decentralized data distribution.^[54] Each node in Chord maintains a finger table containing O(\log N) pointers (where N is the number of nodes), each pointing to the successor node at a distance of $2^i (for i = 0 to m-1) along the identifier circle from the current node.^[54] Routing proceeds via a binary search-like mechanism: to find a key, a node forwards the query to the finger that most closely precedes the key's identifier, halving the remaining distance on average and achieving a lookup path length of O(\log N) hops with high probability.^[54] This structure keeps per-node state and message overhead logarithmic in network size, facilitating scalability.^[54] Chord's algorithms for node management include join and leave procedures that maintain the ring's consistency. When a node joins, it initializes its successor and finger table by querying an existing node, then notifies affected nodes to update their tables, requiring O(\log^2 N) messages; leaves are handled by updating predecessors and successors.^[54] Stabilization runs periodically on each node to verify and correct successor pointers in the face of failures or concurrent changes, ensuring the ring remains accurate with O(\log N) messages per stabilization round.^[54] These mechanisms provide fault tolerance, as the system operates correctly even if nodes fail, by relying on redundant paths in the finger tables.^[54] The protocol supports applications such as distributed indexes for keyword search (similar to early systems like Gnutella or Napster), cooperative file mirroring for load-balanced content distribution, and time-shared storage for intermittently connected devices. As of 2025, Chord continues to influence research, with implementations in languages like Rust for high-performance data replication and applications in distributed real-time rendering.^[54]^[55]^[56] Chord has influenced subsequent P2P systems, including variants of other DHTs like Kademlia used in file-sharing networks such as BitTorrent for decentralized peer discovery and content retrieval. To address load imbalances from skewed key distributions, Chord employs virtual nodes, where each physical node hosts multiple virtual identifiers (e.g., O(\log N) per node), improving key uniformity—reducing the maximum load to 1.1 times the average in simulations with 1,000 nodes and 100,000 keys using 20 virtual nodes.^[54] Chord's advantages include its decentralization, eliminating single points of failure, and fault tolerance to node churn, with lookups succeeding despite up to a constant fraction of failures.^[54] Compared to prefix-based protocols like Pastry and Tapestry, Chord's consistent hashing approach is simpler in implementation and more robust to concurrent joins and failures, though it shares the O(\log N) lookup cost.^[54]

People

Notable individuals

Chord Overstreet (born February 17, 1989) is an American actor and musician best known for portraying Sam Evans, a quarterback-turned-glee club member, on the Fox musical comedy-drama series Glee from 2009 to 2015.^[57] Born in Nashville, Tennessee, Overstreet grew up in a family immersed in country music; he is the son of Paul Overstreet, a prominent country singer-songwriter known for hits like "Daddy's Come Around," and Julie Miller, a makeup artist.^[58] His siblings include actor and musician Nash Overstreet, a member of the pop-rock band Hot Chelle Rae, in whose music video for "Tonight Tonight" Chord made a cameo appearance. Overstreet's early exposure to music shaped his career, leading him to pursue both acting and songwriting. After Glee, he transitioned into a solo music career, signing with Safehouse Records and Island Records; his debut single, "Homeland," was released on August 26, 2016.^[59] He followed with the 2017 single "Hold On," which achieved double platinum certification and marked his first major success as an independent artist. Overstreet has released several EPs, including Hold On - EP (2017), Stone Man - EP (2018), and Tree House Tapes - EP (2019), blending pop, folk, and country influences reflective of his upbringing. In recent years, he has continued releasing singles, including "To Break A Good Man" (November 2024), "Barely Hanging On" (July 2025), and "Why Do I Drink?" (September 2025).^[60]^[61] Through his role on Glee, Overstreet contributed to the show's broader cultural influence in mainstreaming musical performances and inspiring renewed interest in musical theater among younger audiences.^[62] He has described the experience as an intensive "bootcamp" that honed his skills in live singing and acting, preparing him for subsequent projects like the Apple TV+ series Acapulco (2021–2025).^[63]^[64] The surname Chord is rare and primarily of English origin, possibly derived from Old French "corde" meaning string or rope, or as an Anglicized form of Gaelic "ceard" referring to a craftsman or tinker.^[65] Historical records show limited notable figures bearing the surname, with no major documented individuals in science, invention, or the arts beyond contemporary uses like Overstreet's first name, which evokes musical terminology.^[66]

Fictional characters

In the Marvel Comics universe, Andrew Chord is a former U.S. Army sergeant who served during the Siancong War and became entangled with the mystical Dragon's Breadth cult.^[67] Married to cult member Miyami, he fathered twins Janelle and Aaron Chord, later raising Dwayne Taylor (Night Thrasher) as a guardian after being mystically compelled to kill Taylor's parents to fulfill a cult pact.^[68] As a skilled combatant, marksman, and inventor, Chord mentored the New Warriors team, designing equipment like Night Thrasher's suit, though his actions were often influenced by the cult's schemes.^[69] He eventually reconciled with his family before his death.^[67] Janelle Chord, also known as Silhouette, is Andrew's daughter and a mutant descendant of the Dragon's Breadth cult, granting her Darkforce manipulation powers including teleportation, phasing, flight, and energy constructs.^[67] First appearing as a vigilante alongside her brother Aaron (Midnight's Fire) and Night Thrasher, she joined the New Warriors and participated in missions against threats like the Folding Circle.^[69] Paralyzed after a botched operation, she adapted her martial arts expertise for upper-body combat and later aligned with the Secret Avengers during the Superhuman Registration Act.^[67] Her abilities and family ties often symbolize hidden mystical forces and interdimensional connections in superhero narratives.^[70] Aaron Chord, known as Midnight's Fire, shares the family legacy as Andrew's son and Janelle's twin, possessing similar superhuman strength, speed, and Darkforce abilities.^[67] A vigilante and New Warriors member, he frequently collaborated with his sister against cult-related villains, embodying themes of fraternal unity amid supernatural tension.^[69] In film, Chord appears as the primary antagonist in the 2014 techno-thriller Open Windows, portrayed by Neil Maskell. Operating as a cyberterrorist under the alias Chord (sometimes specified as Simon Chord), he manipulates protagonist Nick Chambers through online deception, escalating into a sadistic game of control and surveillance.^[71] This role highlights digital intrusion and psychological tension, drawing loosely from musical metaphors of discord.^[72] Fictional characters named Chord remain scarce outside niche superhero comics and indie thrillers, primarily serving disambiguation in media with no major cultural icons comparable to those in music or other fields. Their portrayals often evoke unity or conflict through familial or manipulative dynamics, rooted in post-1980s genre storytelling.^[69]

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[25]
Ptolemy's Table of Chords: Trigonometry in the Second Century
With this theorem, Ptolemy produced three corollaries from which more chord lengths could be calculated: the chord of the difference of two arcs, the chord of ...
[26]
[PDF] EE 261 - The Fourier Transform and its Applications
1 Bracewell, for example, starts right off with the Fourier transform and picks up a little on Fourier series later. Page 8. 2. Chapter 1 Fourier Series ... use.
[27]
[PDF] A concise course in complex analysis and Riemann surfaces ...
Preface v. Chapter 1. From i to z: the basics of complex analysis. 1. 1. The field of complex numbers. 1. 2. Differentiability and conformality.
[28]
[PDF] Extended Complex Plane and Riemann Sphere
The chordal metric is a metric that measures chordal distance between two points on the sphere, which is an established result (Ahlfors, 1979), we show clear ...
[29]
[PDF] Introduction to Vassiliev Knot Invariants First draft. Comments ...
Part 1. Fundamentals. Chapter 1. Knots and their relatives. 15. §1.1. Definitions and examples. 15. §1.2. Isotopy. 16. §1.3. Plane knot diagrams.
[30]
[PDF] Introduction to Vassiliev Knot Invariants
They explained the relation between the Jones polynomial and finite type invariants2 and emphasized the role of the algebra of chord diagrams. M. Kontsevich ...
[31]
Computational-geometric methods for polygonal approximations of ...
In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the ...Missing: chordal | Show results with:chordal
[32]
The notochord: structure and functions - PMC - PubMed Central
The notochord is an embryonic midline structure common to all members of the phylum Chordata, providing both mechanical and signaling cues to the developing ...
[33]
Update on the Notochord Including its Embryology, Molecular ...
Apr 4, 2017 · One of the notochord's most important roles in embryonic development is its patterning of the neural tube (Figure 3). The neural tube arises ...
[34]
Evolution of the notochord - PMC - PubMed Central
Oct 5, 2015 · A notochord is characteristic of developing chordates (which comprise amphioxus, tunicates and vertebrates), and, more arguably, is also found in some other ...Missing: significance | Show results with:significance
[35]
Spina Bifida: A Review of the Genetics, Pathophysiology and ...
Jun 6, 2022 · Therefore, notochord cells lacking the expression of VANGL1 and VANG2 have random cilia distribution, leading to turbulent nodal flow and ...
[36]
Geometry Definitions
The straight line drawn from the leading to trailing edges of the airfoil is called the chord line.
[37]
Mean Aerodynamic Chord (MAC) | SKYbrary Aviation Safety
Mean Aerodynamic Chord (MAC) is the average chord length of a tapered, swept wing. It's a 2D representation of the wing, and its position is often important.
[38]
[PDF] aircraft cg and
The Aerodynamic Center is where lift and drag act. The Mean Aerodynamic Chord (MAC) is the mean center of lift. The 25-26% point of MAC is best for competition ...
[39]
Warren Truss: What is it? And How to Calculate it?
What Is the Warren Truss? · Top chord · Bottom chord · Diagonals (sometimes called strut or tie depending on compression or tensions) ...What Is the Warren Truss? · Static System · Warren Truss Analysis
[40]
The Warren Truss - Structure Magazine
But they are merely trusses with parallel chords and diagonals, or rather, oblique members, with only one series of obliques, and without verticals, except to ...
[41]
Truss Series: Warren Truss - Garrett's Bridges
All you have to do is lay down your top and bottom chords, and glue on the truss members directly on top of the top and bottom chords. The example bridge that I ...
[42]
A multi-chord stellar occultation by the large trans-Neptunian object ...
Stellar occultations are one of the most accurate ground-based methods to directly determine the size and shape of Solar System objects, down to kilometric ...<|separator|>
[43]
Multi-chord observation of stellar occultation by the near-Earth ...
Dec 20, 2022 · This is the first time that this many multiple chord observations have been made for such a small asteroid (it has a diameter of 5–6 km).
[44]
Asteroid sizes determined with thermophysical model and stellar ...
Occultations resulting in only one or two positive chords provide limited constraints on size. Multichord (with at least three chords) stellar occultation ...
[45]
3.5: Predicting the Period of a Pendulum - Mathematics LibreTexts
Jul 17, 2022 · To make a more accurate approximation, replace the arc with the chord (a straight but nonvertical line). What is the resulting approximation for ...
[46]
[PDF] 3.5 Pendulum period - MIT
Feb 10, 2009 · A more accurate approximation replaces the arc with the chord (a straight but non-vertical line). What is the resulting approximation for ...
[47]
Effects of blade airfoil chord length and rotor diameter on ...
Feb 15, 2023 · In this study, the effects of airfoil chord length and rotor diameter (circumference) on the aerodynamic characteristics of SB-VAWTs were explored by numerical ...
[48]
Calculating blade chord length and twist angle - NREL Forum
Feb 11, 2016 · I am designing a 3-MW with a rotor diameter of 115.7 m and nominal speed of 12 m/s. My question is about calculating c (chord length) when designing the blade.
[49]
The Aqueducts and Water Supply of Ancient Rome - PubMed Central
The construction of the aqueduct involved cutting a tunnel through a hill of solid rock by excavating from both sides simultaneously. The tunnel itself did not ...Missing: chords | Show results with:chords
[50]
[PDF] CASE STUDY 2: CHORD KEYBOARDS - Bill Buxton
Jul 29, 2013 · But there are some disadvantages, which I believe led to their abandonment at PARC/Xerox: ... chording keyboard, n keys gives us access to n ...
[51]
History of Writing Machines - Stenograph, L.L.C.
The Ireland Stenotype machine was the first practical machine with a totally depressible keyboard. The ability to write words phonetically and numbers at a ...Missing: chorded | Show results with:chorded
[52]
BAT chording keyboard - CHM Revolution - Computer History Museum
Instead of having a separate key for each character, this commercial “chording” keyset expects the user to memorize combinations of keys and press them ...<|separator|>
[53]
[PDF] Twiddler Typing: One-Handed Chording Text Entry for Mobile Phones
Each letter of the alphabet can be typed on the Twiddler by pressing one or two keys concur- rently. The Twiddler also has the feature of multi–character chords ...
[54]
Douglas Engelbart and 'The Mother of All Demos' - Brown CS
Engelbart's research team developed the mouse to navigate and manipulate the system, and a one-handed chording keyboard so the user could still perform commands ...
[55]
Assigning AutoCAD commands to mouse buttons - Autodesk
Jun 5, 2025 · To assign AutoCAD commands to a standard three-button mouse, see About Customizing Mouse Buttons. Other mouse buttons are controlled by the mouse driver.
[56]
Design and Implementation of a Chorded On-Screen Keyboard for ...
Aug 7, 2025 · The purposes of this study were to design an alternative on-screen keyboard for people with physical impairments and to evaluate the ...
[57]
Design and Implementation of a Chorded-Keyboard Mapping for ...
This paper presents a novel controller-centric text input system that allows for chording key selection using existing consumer VR hardware.
[58]
Chord: A scalable peer-to-peer lookup service for internet applications
This paper presents Chord, a distributed lookup protocol that addresses this problem. Chord provides support for just one operation: given a key, it maps ...Missing: original | Show results with:original
[59]
Chord Overstreet - IMDb
Chord Overstreet was born in Nashville, Tennessee, to Julie (Miller), a make-up artist, and Paul Overstreet, a country musician and songwriter.
[60]
Chord Overstreet - Biography - IMDb
Chord Overstreet was born in Nashville, Tennessee, to Julie (Miller), a make-up artist, and Paul Overstreet, a country musician and songwriter.
[61]
Tonight Tonight (Hot Chelle Rae song) - Wikipedia
Nash's brother Chord Overstreet also appears in the video. Critical reception. edit. "Tonight Tonight" was received well by critics, with Carolyn Giannini ...Track listing · Personnel · Charts · Certifications
[62]
Chord Overstreet - Fame Studios
Sep 20, 2024 · Chord began a career as a solo musical artist. On August 26, 2016, he released his debut single, “Homeland”, through Safehouse and Island Records.
[63]
Chord Overstreet Lyrics, Songs, and Albums - Genius
Chord Overstreet 100 · Stone Man - EP · Hold On - EP · Tree House Tapes - EP.Missing: career | Show results with:career
[64]
Chord Overstreet Interview on New Video & 'Glee' Impact | Billboard
Jun 26, 2017 · Chord Overstreet Talks Vibey Video for 'Hold On (Remix)' & How 'Glee' Impacted His Solo Career. After releasing the pool party-based video for ...Missing: theater | Show results with:theater
[65]
Chord Overstreet Says Glee Was Like 'Bootcamp' for His Budding ...
Jun 22, 2017 · “It was a bootcamp in the best way. I learned more than I could have ever learned from acting or singing classes,” he says of performing as part ...
[66]
Chord History, Family Crest & Coats of Arms - HouseOfNames
Early Origins and Etymology of Chord. The surname Chord was first found in Somerset at Chard, a borough, market-town, and parish, and the head of a union, ...
[67]
Chord Surname Origin, Meaning & Last Name History - Forebears
It is borne by around 1 in 26,026,950 people. The surname occurs predominantly in The Americas, where 83 percent of Chord live; 82 percent live in North America ...
[68]
https://www.marvel.com/characters/night-thrasher-dwayne-taylor
[69]
Night Thrasher (Dwayne Taylor) Powers, Enemies, History - Marvel
Dwayne is born to parents Daryl and Melody Ann Taylor, who sadly perish at the hands of Andrew Chord, who is mystically compelled by the Dragon's Breadth cult ...
[70]
New Warriors Members, Enemies, Powers | Marvel
Mentored by his legal guardians-retired mercenary Andrew Chord and enigmatic housekeeper Tai. Dwayne fought crime as Night Thrasher, forging a vigilante ...
[71]
Folding Circle Members, Enemies, Powers - Marvel.com
Several decades after Miyami's birth, an American military unit known as the "Half-Fulls" (comprised of Private Diego Casseas, Sgt. Andrew Chord, Lt. Mark ...
[72]
Open Windows | Cast and Crew - Rotten Tomatoes
A fan (Elijah Wood) of a sexy actress (Sasha Grey) becomes a pawn (Neil Maskell) in another man's plan to follow her every move.Missing: Simon | Show results with:Simon
[73]
OPEN WINDOWS (2014) - CULTURE CRYPT
Mar 19, 2014 · A mysterious man identified as Simon Chord contacts Nick through the computer. Claiming that Jill cancelled her dinner arrangement with Nick, ...