In music , there are two common meanings for tuning :
77-721: In musical tuning and harmony , the Tonnetz (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music. The Tonnetz originally appeared in Leonhard Euler 's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae . Euler's Tonnetz , pictured at left, shows
154-448: A B♭ , respectively, provided by the principal oboist or clarinetist , who tune to the keyboard if part of the performance. When only strings are used, then the principal string (violinist) typically has sounded the tuning pitch, but some orchestras have used an electronic tone machine for tuning. Tuning can also be done through a prior recording; this method uses simultaneous audio. Interference beats are used to objectively measure
231-580: A Riemannian manifold , as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when L = Z 2 {\displaystyle \mathbb {Z} ^{2}} : R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , which can also be described as the Cartesian plane under the identifications ( x , y ) ~ ( x + 1, y ) ~ ( x , y + 1) . This particular flat torus (and any uniformly scaled version of it)
308-449: A closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If
385-510: A fiber bundle over S (the Hopf bundle ). The surface described above, given the relative topology from R 3 {\displaystyle \mathbb {R} ^{3}} , is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R 3 {\displaystyle \mathbb {R} ^{3}} from
462-428: A guitar are normally tuned to fourths (excepting the G and B strings in standard tuning, which are tuned to a third), as are the strings of the bass guitar and double bass . Violin , viola , and cello strings are tuned to fifths . However, non-standard tunings (called scordatura ) exist to change the sound of the instrument or create other playing options. To tune an instrument, often only one reference pitch
539-593: A maximal torus ; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G . Toroidal groups are examples of protori , which (like tori) are compact connected abelian groups, which are not required to be manifolds . Automorphisms of T are easily constructed from automorphisms of the lattice Z n {\displaystyle \mathbb {Z} ^{n}} , which are classified by invertible integral matrices of size n with an integral inverse; these are just
616-752: A product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem , giving: A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r , V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . {\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}} These formulas are
693-409: A torus ( pl. : tori or toruses ) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut . If the axis of revolution does not touch
770-521: A 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical. These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads . A flat torus
847-549: A few differing tones. As the number of tones is increased, conflicts arise in how each tone combines with every other. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses. It is impossible to tune the twelve-note chromatic scale so that all intervals are pure. For instance, three pure major thirds stack up to 125 / 64 , which at 1 159 cents
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#1732869714829924-492: A flat torus into 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals , yielding
1001-447: A pitch (i.e., using pitch classes ). Under equal temperament, the never-ending series of ascending fifths mentioned earlier becomes a cycle. Neo-Riemannian theorists typically assume enharmonic equivalence (in other words, A♭ = G♯), and so the two-dimensional plane of the 19th-century Tonnetz cycles in on itself in two different directions, and is mathematically isomorphic to a torus . Neo-Riemannian theorists have also used
1078-643: A pitch/tone that is either too high ( sharp ) or too low ( flat ) in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered 'in tune' if it does not match the chosen reference pitch. Some instruments become 'out of tune' with temperature, humidity, damage, or simply time, and must be readjusted or repaired. Different methods of sound production require different methods of adjustment: The sounds of some instruments, notably unpitched percussion instrument such as cymbals , are of indeterminate pitch , and have irregular overtones not conforming to
1155-429: A rectangle together, choosing the other two sides instead will cause the same reversal of orientation. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian ). The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points . Every conformal structure on the 2-torus can be represented as such
1232-406: A regular torus. For example, in the following map: If R and P in the above flat torus parametrization form a unit vector ( R , P ) = (cos( η ), sin( η )) then u , v , and 0 < η < π /2 parameterize the unit 3-sphere as Hopf coordinates . In particular, for certain very specific choices of a square flat torus in the 3-sphere S , where η = π /4 above, the torus will partition
1309-454: A so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. In
1386-544: A sphere — by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, M* may be constructed by glueing together two congruent geodesic triangles in
1463-622: A tone to the E ♭ so as to have the most accented note of the main theme sound on an open string. In Mahler's Symphony No. 4 , the solo violin is tuned one whole step high to produce a harsh sound evoking Death as the Fiddler. In Bartók's Contrasts , the violin is tuned G ♯ -D-A-E ♭ to facilitate the playing of tritones on open strings. American folk violinists of the Appalachians and Ozarks often employ alternate tunings for dance songs and ballads. The most commonly used tuning
1540-456: A torus is a closed surface defined as the product of two circles : S × S . This can be viewed as lying in C and is a subset of the 3-sphere S of radius √2. This topological torus is also often called the Clifford torus . In fact, S is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S as
1617-405: A torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle ). Torus
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#17328697148291694-425: A torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of
1771-426: A torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R , the distance from the center of the coordinate system, and θ and φ , angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in
1848-563: A torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric . It can not be analytically embedded ( smooth of class C , 2 ≤ k ≤ ∞ ) into Euclidean 3-space. Mapping it into 3 -space requires one to stretch it, in which case it looks like
1925-463: A two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points . In fact, the conformal type of the torus is determined by the cross-ratio of the four points. The torus has a generalization to higher dimensions, the n-dimensional torus , often called the n -torus or hypertorus for short. (This is the more typical meaning of
2002-459: Is R n {\displaystyle \mathbb {R} ^{n}} modulo the action of the integer lattice Z n {\displaystyle \mathbb {Z} ^{n}} (with the action being taken as vector addition). Equivalently, the n -torus is obtained from the n -dimensional hypercube by gluing the opposite faces together. An n -torus in this sense is an example of an n- dimensional compact manifold . It
2079-445: Is A-E-A-E. Likewise banjo players in this tradition use many tunings to play melody in different keys. A common alternative banjo tuning for playing in D is A-D-A-D-E. Many Folk guitar players also used different tunings from standard, such as D-A-D-G-A-D, which is very popular for Irish music. A musical instrument that has had its pitch deliberately lowered during tuning is said to be down-tuned or tuned down . Common examples include
2156-1005: Is a Latin word for "a round, swelling, elevation, protuberance". A torus of revolution in 3-space can be parametrized as: x ( θ , φ ) = ( R + r cos θ ) cos φ y ( θ , φ ) = ( R + r cos θ ) sin φ z ( θ , φ ) = r sin θ {\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\\end{aligned}}} using angular coordinates θ , φ ∈ [ 0 , 2 π ) , {\displaystyle \theta ,\varphi \in [0,2\pi ),} representing rotation around
2233-458: Is a member of the Lie group SO(4). It is known that there exists no C (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of
2310-408: Is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings , non-inflatable lifebuoys , ring doughnuts , and bagels . In topology , a ring torus is homeomorphic to the Cartesian product of two circles : S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , and the latter is taken to be
2387-411: Is a torus with the metric inherited from its representation as the quotient , R 2 {\displaystyle \mathbb {R} ^{2}} / L , where L is a discrete subgroup of R 2 {\displaystyle \mathbb {R} ^{2}} isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} . This gives the quotient the structure of
Tonnetz - Misplaced Pages Continue
2464-457: Is also an example of a compact abelian Lie group . This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups . This is due in part to the fact that in any compact Lie group G one can always find
2541-420: Is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships. Neo-Riemannian music theorists David Lewin and Brian Hyer revived the Tonnetz to further explore properties of pitch structures. Modern music theorists generally construct the Tonnetz in equal temperament and without distinction between octave transpositions of
2618-402: Is given. This reference is used to tune one string, to which the other strings are tuned in the desired intervals. On a guitar, often the lowest string is tuned to an E. From this, each successive string can be tuned by fingering the fifth fret of an already tuned string and comparing it with the next higher string played open. This works with the exception of the G string, which must be stopped at
2695-447: Is known as the "square" flat torus. This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into
2772-691: Is nearly a quarter tone away from the octave (1200 cents). So there is no way to have both the octave and the major third in just intonation for all the intervals in the same twelve-tone system. Similar issues arise with the fifth 3 / 2 , and the minor third 6 / 5 , or any other choice of harmonic-series based pure intervals. Many different compromise methods are used to deal with this, each with its own characteristics, and advantages and disadvantages. The main ones are: Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments . Torus In geometry ,
2849-545: Is the n -fold product of the circle, the n -torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, T n = ( S 1 ) n {\displaystyle \mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}} . The configuration space of unordered , not necessarily distinct points is accordingly the orbifold T n / S n {\displaystyle \mathbb {T} ^{n}/\mathbb {S} _{n}} , which
2926-440: Is the choice of number and spacing of frequency values used. Due to the psychoacoustic interaction of tones and timbres , various tone combinations sound more or less "natural" in combination with various timbres. For example, using harmonic timbres: More complex musical effects can be created through other relationships. The creation of a tuning system is complicated because musicians want to make music with more than just
3003-487: Is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip , the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle , with a twist ; equivalently, as a triangular prism whose top and bottom faces are connected with
3080-400: Is the standard 2-torus, T 2 {\displaystyle \mathbb {T} ^{2}} . And similar to the 2-torus, the n -torus, T n {\displaystyle \mathbb {T} ^{n}} can be described as a quotient of R n {\displaystyle \mathbb {R} ^{n}} under integral shifts in any coordinate. That is, the n -torus
3157-450: The Euler characteristic of the n -torus is 0 for all n . The cohomology ring H ( T n {\displaystyle \mathbb {T} ^{n}} , Z ) can be identified with the exterior algebra over the Z - module Z n {\displaystyle \mathbb {Z} ^{n}} whose generators are the duals of the n nontrivial cycles. As the n -torus
Tonnetz - Misplaced Pages Continue
3234-525: The Tonnetz appeared in the work of many late-19th century German music theorists. Oettingen and Riemann both conceived of the relationships in the chart being defined through just intonation , which uses pure intervals. One can extend out one of the horizontal rows of the Tonnetz indefinitely, to form a never-ending sequence of perfect fifths: F-C-G-D-A-E-B-F♯-C♯-G♯-D♯-A♯-E♯-B♯-F𝄪-C𝄪-G𝄪- (etc.) Starting with F, after 12 perfect fifths, one reaches E♯. Perfect fifths in just intonation are slightly larger than
3311-418: The Tonnetz to 19th-century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C♯-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing
3388-579: The Tonnetz to visualize non-tonal triadic relationships. For example, the diagonal going up and to the left from C in the diagram at the beginning of the article forms a division of the octave in three major thirds : C-A♭-E-C (the E is actually an F♭, and the final C a D♭♭). Richard Cohn argues that while a sequence of triads built on these three pitches (C major, A♭ major, and E major) cannot be adequately described using traditional concepts of functional harmony, this cycle has smooth voice leading and other important group properties which can be easily observed on
3465-470: The Tonnetz . The harmonic table note layout is a note layout that is topologically equivalent to the Tonnetz , and is used on several music instruments that allow playing major and minor chords with a single finger. The Tonnetz can be overlayed on the Wicki–Hayden note layout , where the major second can be found half way the major third. The Tonnetz is the dual graph of Schoenberg 's chart of
3542-450: The harmonic series . See § Tuning of unpitched percussion instruments . Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A tuning fork or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano is used (as its pitch cannot be adjusted for each performance). Symphony orchestras and concert bands usually tune to an A 440 or
3619-472: The hyperbolic plane along their (identical) boundaries, where each triangle has angles of π/2, π/3, and 0. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the Gauss-Bonnet theorem shows that the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space M* has area equal to π/3. The other two cusps occur at
3696-433: The snare drum . Tuning pitched percussion follows the same patterns as tuning any other instrument, but tuning unpitched percussion does not produce a specific pitch . For this reason and others, the traditional terms tuned percussion and untuned percussion are avoided in recent organology . A tuning system is the system used to define which tones , or pitches , to use when playing music . In other words, it
3773-454: The square root gives a quartic equation , ( x 2 + y 2 + z 2 + R 2 − r 2 ) 2 = 4 R 2 ( x 2 + y 2 ) . {\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).} The three classes of standard tori correspond to
3850-433: The " moduli space " of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space M may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle = π and the other has total angle = 2π/3. M may be turned into a compact space M* — topologically equivalent to
3927-406: The 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary . One example is the torus T defined by Other tori in S having this partitioning property include the square tori of the form Q ⋅ T , where Q is a rotation of 4-dimensional space R 4 {\displaystyle \mathbb {R} ^{4}} , or in other words Q
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#17328697148294004-496: The C and the E vertices. The dominant triad of A minor, E major (E-G♯-B) is diagonally across the E vertex, and shares no other vertices. One important point is that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared vertices, the more shared pitches the chord will have. This provides a visualization of the principle of parsimonious voice-leading, in which motions between chords are considered smoother when fewer pitches change. This principle
4081-427: The accuracy of tuning. As the two pitches approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected. For other intervals, this is dependent on the tuning system being used. Harmonics may be used to facilitate tuning of strings that are not themselves tuned to the unison. For example, lightly touching
4158-400: The circle, the surface has a ring shape and is called a torus of revolution , also known as a ring torus . If the axis of revolution is tangent to the circle, the surface is a horn torus . If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus ). If the axis of revolution passes through the center of
4235-437: The circle, the surface is a degenerate torus, a double-covered sphere . If the revolved curve is not a circle, the surface is called a toroid , as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings , inner tubes and ringette rings . A torus should not be confused with a solid torus , which is formed by rotating a disk , rather than a circle, around an axis. A solid torus
4312-579: The compromised fifths used in equal temperament tuning systems more common in the present. This means that when one stacks 12 fifths starting from F, the E♯ we arrive at will not be seven octaves above the F we started with. Oettingen and Riemann's Tonnetz thus extended on infinitely in every direction without actually repeating any pitches. In the twentieth century, composer-theorists such as Ben Johnston and James Tenney continued to developed theories and applications involving just-intoned Tonnetze . The appeal of
4389-471: The definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space , but another way to do this is the Cartesian product of the embedding of S 1 {\displaystyle S^{1}} in the plane with itself. This produces a geometric object called the Clifford torus , a surface in 4-space . In the field of topology ,
4466-451: The electric guitar and electric bass in contemporary heavy metal music , whereby one or more strings are often tuned lower than concert pitch . This is not to be confused with electronically changing the fundamental frequency , which is referred to as pitch shifting . Many percussion instruments are tuned by the player, including pitched percussion instruments such as timpani and tabla , and unpitched percussion instruments such as
4543-599: The fourth fret to sound B against the open B string above. Alternatively, each string can be tuned to its own reference tone. Note that while the guitar and other modern stringed instruments with fixed frets are tuned in equal temperament , string instruments without frets, such as those of the violin family, are not. The violin, viola, and cello are tuned to beatless just perfect fifths and ensembles such as string quartets and orchestras tend to play in fifths based Pythagorean tuning or to compensate and play in equal temperament, such as when playing with other instruments such as
4620-417: The highest string of a cello at the middle (at a node ) while bowing produces the same pitch as doing the same a third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the perfect fifth between the fundamentals of the two strings. In music , the term open string refers to the fundamental note of the unstopped, full string. The strings of
4697-406: The integral matrices with determinant ±1. Making them act on R n {\displaystyle \mathbb {R} ^{n}} in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n -torus is a free abelian group of rank n . The k -th homology group of an n -torus is a free abelian group of rank n choose k . It follows that
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#17328697148294774-430: The north pole of S . The torus can also be described as a quotient of the Cartesian plane under the identifications or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA B . The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: Intuitively speaking, this means that
4851-431: The piano. For example, the cello, which is tuned down from A220 , has three more strings (four total) and the just perfect fifth is about two cents off from the equal tempered perfect fifth, making its lowest string, C−, about six cents more flat than the equal tempered C. This table lists open strings on some common string instruments and their standard tunings from low to high unless otherwise noted. Violin scordatura
4928-451: The regions , and of course vice versa . Research into music cognition has demonstrated that the human brain uses a "chart of the regions" to process tonal relationships. Musical tuning Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is usually based on a fixed reference, such as A = 440 Hz . The term " out of tune " refers to
5005-431: The same as for a cylinder of length 2π R and radius r , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. Expressing the surface area and the volume by the distance p of an outermost point on
5082-487: The spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r , and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topologically ,
5159-410: The strings of the solo viola are raised one half-step, ostensibly to give the instrument a brighter tone so the solo violin does not overshadow it. Scordatura for the violin was also used in the 19th and 20th centuries in works by Niccolò Paganini , Robert Schumann , Camille Saint-Saëns , Gustav Mahler , and Béla Bartók . In Saint-Saëns' " Danse Macabre ", the high string of the violin is lower half
5236-449: The study of Riemann surfaces , one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature . In the case of a torus, the constant curvature must be zero. Then one defines
5313-1057: The surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q / 2 and r = p − q / 2 ), yields A = 4 π 2 ( p + q 2 ) ( p − q 2 ) = π 2 ( p + q ) ( p − q ) , V = 2 π 2 ( p + q 2 ) ( p − q 2 ) 2 = 1 4 π 2 ( p + q ) ( p − q ) 2 . {\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}} As
5390-439: The term " n -torus", the other referring to n holes or of genus n . ) Just as the ordinary torus is topologically the product space of two circles, the n -dimensional torus is topologically equivalent to the product of n circles. That is: The standard 1-torus is just the circle: T 1 = S 1 {\displaystyle \mathbb {T} ^{1}=\mathbb {S} ^{1}} . The torus discussed above
5467-426: The three possible aspect ratios between R and r : When R ≥ r , the interior ( x 2 + y 2 − R ) 2 + z 2 < r 2 {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}} of this torus is diffeomorphic (and, hence, homeomorphic) to
5544-478: The torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem , which was proven in the 1950s, an isometric C embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C (continuously differentiable) isometric embedding of
5621-534: The torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z {\displaystyle z} - axis is ( x 2 + y 2 − R ) 2 + z 2 = r 2 . {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.} Algebraically eliminating
5698-445: The triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a major third above F). Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst , discusses the relationships between keys, presenting them in a network analogous to Euler's Tonnetz , but showing keys rather than notes. The Tonnetz itself
5775-420: The tube and rotation around the torus' axis of revolution, respectively, where the major radius R {\displaystyle R} is the distance from the center of the tube to the center of the torus and the minor radius r {\displaystyle r} is the radius of the tube. The ratio R / r {\displaystyle R/r} is called the aspect ratio of
5852-574: Was employed in the 17th and 18th centuries by Italian and German composers, namely, Biagio Marini , Antonio Vivaldi , Heinrich Ignaz Franz Biber (who in the Rosary Sonatas prescribes a great variety of scordaturas, including crossing the middle strings), Johann Pachelbel and Johann Sebastian Bach , whose Fifth Suite For Unaccompanied Cello calls for the lowering of the A string to G. In Mozart 's Sinfonia Concertante in E-flat major (K. 364), all
5929-414: Was rediscovered in 1858 by Ernst Naumann in his Harmoniesystem in dualer Entwickelung ., and was disseminated in an 1866 treatise of Arthur von Oettingen . Oettingen and the influential musicologist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann ) explored the capacity of the space to chart harmonic modulation between chords and motion between keys. Similar understandings of
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