In music , there are two common meanings for tuning :
58-743: 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
116-750: A , {\displaystyle g:b\to a,} that is, f g = 1 b {\displaystyle fg=1_{b}} and g f = 1 a . {\displaystyle gf=1_{a}.} Two categories C and D are isomorphic if there exist functors F : C → D {\displaystyle F:C\to D} and G : D → C {\displaystyle G:D\to C} which are mutually inverse to each other, that is, F G = 1 D {\displaystyle FG=1_{D}} (the identity functor on D ) and G F = 1 C {\displaystyle GF=1_{C}} (the identity functor on C ). In
174-473: A , b ) ↦ ( 3 a + 4 b ) mod 6. {\displaystyle (a,b)\mapsto (3a+4b)\mod 6.} For example, ( 1 , 1 ) + ( 1 , 0 ) = ( 0 , 1 ) , {\displaystyle (1,1)+(1,0)=(0,1),} which translates in the other system as 1 + 3 = 4. {\displaystyle 1+3=4.} Even though these two groups "look" different in that
232-448: 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
290-449: A concrete category (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the category of topological spaces or categories of algebraic objects (like the category of groups , the category of rings , and the category of modules ), an isomorphism must be bijective on the underlying sets . In algebraic categories (specifically, categories of varieties in
348-428: 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
406-427: A slide rule with a logarithmic scale. Consider the group ( Z 6 , + ) , {\displaystyle (\mathbb {Z} _{6},+),} the integers from 0 to 5 with addition modulo 6. Also consider the group ( Z 2 × Z 3 , + ) , {\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),}
464-549: 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
522-811: A language that may be used to unify the approach to these different aspects of the basic idea. Let R + {\displaystyle \mathbb {R} ^{+}} be the multiplicative group of positive real numbers , and let R {\displaystyle \mathbb {R} } be the additive group of real numbers. The logarithm function log : R + → R {\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} } satisfies log ( x y ) = log x + log y {\displaystyle \log(xy)=\log x+\log y} for all x , y ∈ R + , {\displaystyle x,y\in \mathbb {R} ^{+},} so it
580-447: 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
638-643: 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
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#1732793934849696-409: A rational number as a set (equivalence class). The universal property of the rational numbers is essentially that they form a field that contains the integers and does not contain any proper subfield. It results that given two fields with these properties, there is a unique field isomorphism between them. This allows identifying these two fields, since every property of one of them can be transferred to
754-633: 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
812-494: Is while another is and no one isomorphism is intrinsically better than any other. On this view and in this sense, these two sets are not equal because one cannot consider them identical : one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism. Also, integers and even numbers are isomorphic as ordered sets and abelian groups (for addition), but cannot be considered equal sets, since one
870-487: Is reflexive , irreflexive , symmetric , antisymmetric , asymmetric , transitive , total , trichotomous , a partial order , total order , well-order , strict weak order , total preorder (weak order), an equivalence relation , or a relation with any other special properties, if and only if R is. For example, R is an ordering ≤ and S an ordering ⊑ , {\displaystyle \scriptstyle \sqsubseteq ,} then an isomorphism from X to Y
928-445: 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
986-508: Is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a universal property ), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every prime number p , all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. The term isomorphism
1044-504: Is a group homomorphism . The exponential function exp : R → R + {\displaystyle \exp :\mathbb {R} \to \mathbb {R} ^{+}} satisfies exp ( x + y ) = ( exp x ) ( exp y ) {\displaystyle \exp(x+y)=(\exp x)(\exp y)} for all x , y ∈ R , {\displaystyle x,y\in \mathbb {R} ,} so it too
1102-425: Is a proper subset of the other. On the other hand, when sets (or other mathematical objects ) are defined only by their properties, without considering the nature of their elements, one often considers them to be equal. This is generally the case with solutions of universal properties . For example, the rational numbers are usually defined as equivalence classes of pairs of integers, although nobody thinks of
1160-486: Is a bijection preserving addition, scalar multiplication, and inner product. In early theories of logical atomism , the formal relationship between facts and true propositions was theorized by Bertrand Russell and Ludwig Wittgenstein to be isomorphic. An example of this line of thinking can be found in Russell's Introduction to Mathematical Philosophy . In cybernetics , the good regulator or Conant–Ashby theorem
1218-520: Is a bijective function f : X → Y {\displaystyle f:X\to Y} such that f ( u ) ⊑ f ( v ) if and only if u ≤ v . {\displaystyle f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.} Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism . If X = Y , {\displaystyle X=Y,} then this
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#17327939348491276-432: Is a homomorphism that has an inverse that is also a homomorphism, log {\displaystyle \log } is an isomorphism of groups. The log {\displaystyle \log } function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms , or using
1334-454: Is a homomorphism. The identities log exp x = x {\displaystyle \log \exp x=x} and exp log y = y {\displaystyle \exp \log y=y} show that log {\displaystyle \log } and exp {\displaystyle \exp } are inverses of each other. Since log {\displaystyle \log }
1392-437: Is a relation-preserving automorphism . In algebra , isomorphisms are defined for all algebraic structures . Some are more specifically studied; for example: Just as the automorphisms of an algebraic structure form a group , the isomorphisms between two algebras sharing a common structure form a heap . Letting a particular isomorphism identify the two structures turns this heap into a group. In mathematical analysis ,
1450-439: Is a structure-preserving mapping (a morphism ) between two structures of the same type that can be reversed by an inverse mapping . Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'. The interest in isomorphisms lies in the fact that two isomorphic objects have
1508-521: Is an isomorphism, the relation that two mathematical objects are isomorphic is an equivalence relation . An equivalence class given by isomorphisms is commonly called an isomorphism class . Examples of isomorphism classes are plentiful in mathematics. However, there are circumstances in which the isomorphism class of an object conceals vital information about it. Although there are cases where isomorphic objects can be considered equal, one must distinguish equality and isomorphism . Equality
1566-420: 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
1624-402: 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
1682-431: Is mainly used for algebraic structures . In this case, mappings are called homomorphisms , and a homomorphism is an isomorphism if and only if it is bijective . In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example: Category theory , which can be viewed as a formalization of the concept of mapping between structures, provides
1740-712: 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 . Isomorphic In mathematics , an isomorphism
1798-421: Is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system. In category theory , given a category C , an isomorphism is a morphism f : a → b {\displaystyle f:a\to b} that has an inverse morphism g : b →
Tonnetz - Misplaced Pages Continue
1856-440: 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
1914-720: Is when two objects are the same, and therefore everything that is true about one object is true about the other. On the other hand, isomorphisms are related to some structure, and two isomorphic objects share only the properties that are related to this structure. For example, the sets A = { x ∈ Z ∣ x 2 < 2 } and B = { − 1 , 0 , 1 } {\displaystyle A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}} are equal ; they are merely different representations—the first an intensional one (in set builder notation ), and
1972-658: The Chinese remainder theorem . If one object consists of a set X with a binary relation R and the other object consists of a set Y with a binary relation S then an isomorphism from X to Y is a bijective function f : X → Y {\displaystyle f:X\to Y} such that: S ( f ( u ) , f ( v ) ) if and only if R ( u , v ) {\displaystyle \operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)} S
2030-690: The Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations. In graph theory , an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the "edge structure" in the sense that there is an edge from vertex u to vertex v in G if and only if there is an edge from f ( u ) {\displaystyle f(u)} to f ( v ) {\displaystyle f(v)} in H . See graph isomorphism . In mathematical analysis, an isomorphism between two Hilbert spaces
2088-525: 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
2146-418: 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
2204-579: 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
2262-470: 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
2320-450: 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
2378-433: 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
Tonnetz - Misplaced Pages Continue
2436-496: 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
2494-427: 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
2552-582: 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
2610-451: 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
2668-599: 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
2726-417: 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
2784-865: The ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the x -coordinate is modulo 2 and addition in the y -coordinate is modulo 3. These structures are isomorphic under addition, under the following scheme: ( 0 , 0 ) ↦ 0 ( 1 , 1 ) ↦ 1 ( 0 , 2 ) ↦ 2 ( 1 , 0 ) ↦ 3 ( 0 , 1 ) ↦ 4 ( 1 , 2 ) ↦ 5 {\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}} or in general (
2842-431: 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
2900-451: 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
2958-414: The same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures
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#17327939348493016-456: The second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets { A , B , C } {\displaystyle \{A,B,C\}} and { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} are not equal since they do not have the same elements. They are isomorphic as sets, but there are many choices (in fact 6) of an isomorphism between them: one isomorphism
3074-399: The sense of universal algebra ), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces). Since a composition of isomorphisms is an isomorphism, since the identity is an isomorphism and since the inverse of an isomorphism
3132-496: The sets contain different elements, they are indeed isomorphic : their structures are exactly the same. More generally, the direct product of two cyclic groups Z m {\displaystyle \mathbb {Z} _{m}} and Z n {\displaystyle \mathbb {Z} _{n}} is isomorphic to ( Z m n , + ) {\displaystyle (\mathbb {Z} _{mn},+)} if and only if m and n are coprime , per
3190-410: 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
3248-445: 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
3306-585: 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
3364-414: 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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