1.
Harry Partch
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Harry Partch was an American composer, music theorist, and creator of musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century composers in the West to work systematically with microtonal scales. He built custom-made instruments in these tunings on which to play his compositions, to play his music, Partch built a large number of unique instruments, with such names as the Chromelodeon, the Quadrangularis Reversum, and the Zymo-Xyl. Partch described his music as corporeal, and distinguished it from abstract music, ancient Greek theatre and Japanese Noh and kabuki heavily influenced his music theatre. Encouraged by his mother, Partch learned several instruments at a young age, by fourteen, he was composing, and in particular took to setting dramatic situations. He dropped out of the University of Southern Californias School of Music in 1922 over dissatisfaction with the quality of his teachers, in 1930, he burned all his previous compositions in a rejection of the European concert tradition. Partch frequently moved around the US, early in his career, he was a transient worker, and sometimes a hobo, later he depended on grants, university appointments, and record sales to support himself. In 1970, supporters created the Harry Partch Foundation to administer Partchs music, Partch was born on June 24,1901, in Oakland, California. His parents were Virgil Franklin Partch and Jennie, the Presbyterian couple were missionaries, and served in China from 1888 to 1893, and again from 1895 to 1900, when they fled the Boxer Rebellion. Partch moved with his family to Arizona for his mothers health and his father worked for the Immigration Service there, and they settled in the small town of Benson. It was still the Wild West there in the twentieth century. Nearby, there were native Yaqui people, whose music he heard and his mother sang to him in Mandarin Chinese, and he heard and sang songs in Spanish and the Yaqui language. His mother encouraged her children to music, and he learned the mandolin, violin, piano, reed organ. His mother taught him to read music, the family moved to Albuquerque, New Mexico, in 1913, where Partch seriously studied the piano. He had work playing keyboards for silent films while he was in high school, by 14, he was composing for the piano. He early found an interest in writing music for dramatic situations, and often cited the lost composition Death, Partch graduated from high school in 1919. The family moved to Los Angeles in 1919 following the death of Partchs father, there, his mother was killed in a trolley accident in 1920. He enrolled in the University of Southern Californias School of Music in 1920 and he moved to San Francisco and studied books on music in the libraries there and continued to compose
2.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
3.
Yuri Landman
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Yuri Landman started as a comic book artist and made his debut in the comics field in 1997 with Je Mag Alles Met Me Doen. In the follow-up, released in 1998, Het Verdiende Loon, for the second title he received the 1998 Breda Prize, an award for rising new comic artists in the Netherlands. Since then he has published no other comic books, together with Cees van Appeldoorn, he formed the lo-fi band Zoppo playing bass and prepared guitar in 1997. After 2 albums and several 7” singles, Landman left the band in 2000, Landman then formed the noise band Avec Aisance with drummer/producer Valentijn Höllander and released a CD, Vivre dans l’aisance in 2004. After quitting Avec-A in 2006, he focused mainly on instrument building, Landman is musically untrained and cannot play chords. While with Avec-A, Landman began creating and building several experimental string instruments, including electric zithers, electric Cymbalum, in the period between 2000-2005, Landman created 9 prototype instruments. In 2006 he changed his focus and stopped to perform. The Moodswinger was the first instrument Landman made for the band Liars, after the Moodswinger, he started making more instruments for other bands as well. From November 2006 to January 2007 Landman finished 2 copies of The Moonlander, the Springtime is an experimental electric guitar with seven strings and three outputs. The first prototype of this instrument, created in 2008, was made for guitar player Laura-Mary Carter of Blood Red Shoes, afterwards he also made copies for Lou Barlow and dEUS Mauro Pawlowski. For John Schmersal of Enon he built the Twister guitar, a version of the Springtime. In 2009 he finished instruments for The Dodos, Liam Finn, HEALTH, Micachu, for The Dodos and Finn he created electric 24 string drum guitars called the Tafelberg and for Andrews an electric 17 string harp guitar called the Burner guitar. He also started to again after a Perpignan Festival hosted by Vincent Moon. Meanwhile, he continued to build instruments for such as These Are Powers, Women. For his own musical career he develops a 25-meter electric long-string instrument, the Moodswinger led to a research on harmonics theory. He published a clarifying 3rd Bridge diagram related to this subject in 2012, besides his output as an inventor and noise musician, he starts to focus on music education, and participatory art. At first Landman started giving lectures at venues, festivals. He published an extensive 8 chapter guide on how to prepare a guitar and his lectures and presentations with his instruments lead to a request in 2009 for a practical building workshop
4.
Music theory
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Music theory is the study of the practices and possibilities of music. The term is used in three ways in music, though all three are interrelated. The first is what is otherwise called rudiments, currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, Theory in this sense is treated as the necessary preliminary to the study of harmony, counterpoint, and form. The second is the study of writings about music from ancient times onwards, Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. However, this medieval discipline became the basis for tuning systems in later centuries, Music theory as a practical discipline encompasses the methods and concepts composers and other musicians use in creating music. The development, preservation, and transmission of music theory in this sense may be found in oral and written music-making traditions, musical instruments, and other artifacts. In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia and ancient China, have also considered music theory in more formal ways such as written treatises, in modern academia, music theory is a subfield of musicology, the wider study of musical cultures and history. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a person who researches, teaches, or writes articles about music theory is a music theorist. University study, typically to the M. A. or Ph. D level, is required to teach as a music theorist in a US or Canadian university. Methods of analysis include mathematics, graphic analysis, and especially analysis enabled by Western music notation, comparative, descriptive, statistical, and other methods are also used. See for instance Paleolithic flutes, Gǔdí, and Anasazi flute, several surviving Sumerian and Akkadian clay tablets include musical information of a theoretical nature, mainly lists of intervals and tunings. The scholar Sam Mirelman reports that the earliest of these dates from before 1500 BCE. Further, All the Mesopotamian texts are united by the use of a terminology for music that, much of Chinese music history and theory remains unclear. The earliest texts about Chinese music theory are inscribed on the stone and they include more than 2800 words describing theories and practices of music pitches of the time. The bells produce two intertwined pentatonic scales three tones apart with additional pitches completing the chromatic scale, Chinese theory starts from numbers, the main musical numbers being twelve, five and eight. Twelve refers to the number of pitches on which the scales can be constructed, the Lüshi chunqiu from about 239 BCE recalls the legend of Ling Lun. On order of the Yellow Emperor, Ling Lun collected twelve bamboo lengths with thick, blowing on one of these like a pipe, he found its sound agreeable and named it huangzhong, the Yellow Bell
5.
Moodswinger
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The Moodswinger is a twelve-string electric zither with an additional third bridge designed by Yuri Landman. The rod which functions as the third bridge divides the strings into two sections to cause an overtone multiphonic sound, one of the copies of the instrument is part of the collection of the Musical Instrument Museum in Phoenix, Arizona. In March 2006 Landman was contacted by the noise band Liars to make an instrument for them, after 6 months he finished two copies of The Moodswinger, an electric 12-string 3rd-bridge overtone koto, one for guitarist/drummer Aaron Hemphill and one for himself. Although it closely resembles a guitar, it is actually a zither. The pickup and electronics are built into the neck instead of in the body like usual electric guitars, in 2008 the Moodswinger II was released as a serial product. Jessie Stein of The Luyas owns a copy, in 2009 Landman created a derivative version of the instrument called the Home Swinger, for workshops at festivals, where participants built their own copy within four hours. In 2010 the Musical Instrument Museum in Phoenix included a Moodswinger as well as a Home Swinger in their collection as two of the pieces of the Dutch section of musical instrument inventions, the 3rd bridge divides the strings into two segments with different pitches. Depending on where the string is played, a bell-like harmonic second tone is created, the string resonates more or less when the back side is struck, depending on the position of the 3rd bridge along the string. This can be explained by resonance and microtonality. At harmonic nodal positions, the string more than at other positions. For instance, dividing the string 1/3 + 2/3 creates a clear overtone, the color dotted scale indicates the simple-number ratios up to the 7 limit On these positions just intoned harmonic dyads occur. The Moodswinger is focused on a playing technique. A mathematical scale is added to specify 23 harmonic positions on the strings, because the instrument has 12 strings, tuned in a circle of fourths, it is always possible to play every note of the equal tempered scale. However some positions have a + or - indication, because the equal tempered scale is not a perfect well-tempered scale, the tuning of this instrument is a circle of fourths, E-A-D-G-C-F-A#-D#-G#-C#-F#-B, arranged in 3 clusters of 4 strings each to make the field of strings better readable. Because of this tuning all five neighbouring strings form a harmonic pentatonic scale and this allows a very easy fingerpicking technique without picking false notes, if the right key is chosen. On the Home Swinger the same color system occurs, the sound of a 3rd-bridged string is a combination of 3 tones. A soft-sounding attack tone of the string part hit at the body side, the diagram below shows the tone combinations of the overtone and the low tone of the counterpart. The attack tone is in most positions exact the same note as the overtone, exceptions are 3/4, 3/5, 3/7 and 5/7
6.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
7.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
8.
Lattice (music)
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In musical tuning, a lattice is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern, each point on the lattice corresponds to a ratio. The lattice can be two-, three-, or n-dimensional, with each corresponding to a different prime-number partial or chroma. The points in a lattice represent pitch classes, and the connectors in a lattice represent the intervals between them, repeatedly adding the same vector moves you further in the same direction. Lattices in just intonation are theoretically infinite, however, lattices are sometimes also used to notate limited subsets that are particularly interesting. Examples of musical lattices include the Tonnetz of Euler and Hugo Riemann, musical intervals in just intonation are related to those in equal tuning by Adriaan Fokkers Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv Wilson, the limit is the highest prime number used in the ratios that define the intervals used by a tuning. Here is a template he used to generate what he called a “Euler“ lattice after where he drew his inspiration, each prime harmonic has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure. Another feature of the template is worth pointing out, harmonically generated intervals will always appear above the fundamental and the subharmonics below. With a 9-limit system the direction will be both above and to the right, leaving the other quadrant for more complex ratios with the opposite with the subharmonic and this makes it quite easy to understand what is being represented. Erv Wilson would commonly use 10-squares-to-the-inch graph paper and that way, he had room to notate both ratios and often the scale degree, which explains why he didnt use a template where all the numbers where divided by 2. The scale degree always followed a period or dot to separate it from the ratios, numerous examples appear throughout the Wilson Archives Tonality diamond Johnston, Ben. Rational Structure in Music, Maximum Clarity and Other Writings on Music, edited by Bob Gilmore
9.
Musical tuning
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In music, there are two common meanings for tuning, Tuning practice, the act of tuning an instrument or voice. Tuning systems, the systems of pitches used to tune an instrument. 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. Out of tune refers to a pitch/tone that is too high or too low in relation to a given reference pitch. While an instrument might be in relative to its own range of notes. Some instruments become out of tune with damage or time and must be readjusted or repaired, different methods of sound production require different methods of adjustment, Tuning to a pitch with ones voice is called matching pitch and is the most basic skill learned in ear training. Turning pegs to increase or decrease the tension on strings so as to control the pitch, instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs, while others such as the violin can be tuned manually. Modifying the length or width of the tube of an instrument, brass instrument, pipe, bell. The sounds of instruments such as cymbals are inharmonic—they have irregular overtones not conforming to the harmonic series. 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. Symphony orchestras and concert bands tend to tune to an A or a B♭, respectively, interference beats are used to objectively measure the accuracy of tuning. As the two 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 the highest string of a cello at the middle 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 fifth between the fundamentals of the two strings. In music, the open string refers to the fundamental note of the unstopped. The strings of a guitar are tuned to fourths, as are the strings of the bass guitar. Violin, viola, and cello strings are tuned to fifths, however, non-standard tunings exist to change the sound of the instrument or create other playing options
10.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations
11.
Octave
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In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are written using eight notes. For example, the C major scale is typically written C D E F G A B C, two notes separated by an octave have the same letter name and are of the same pitch class. Three commonly cited examples of melodies featuring the perfect octave as their opening interval are Singin in the Rain, Somewhere Over the Rainbow, the interval between the first and second harmonics of the harmonic series is an octave. The octave has occasionally referred to as a diapason. To emphasize that it is one of the intervals, the octave is designated P8. The octave above or below a note is sometimes abbreviated 8a or 8va, 8va bassa. For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, the ratio of frequencies of two notes an octave apart is therefore 2,1. Further octaves of a note occur at 2n times the frequency of that note, such as 2,4,8,16, etc. and the reciprocal of that series. For example,55 Hz and 440 Hz are one and two away from 110 Hz because they are 1⁄2 and 4 times the frequency, respectively. After the unison, the octave is the simplest interval in music, the human ear tends to hear both notes as being essentially the same, due to closely related harmonics. Notes separated by a ring together, adding a pleasing sound to music. For this reason, notes an octave apart are given the note name in the Western system of music notation—the name of a note an octave above A is also A. The conceptualization of pitch as having two dimensions, pitch height and pitch class, inherently include octave circularity, thus all C♯s, or all 1s, in any octave are part of the same pitch class. Octave equivalency is a part of most advanced cultures, but is far from universal in primitive. The languages in which the oldest extant written documents on tuning are written, leon Crickmore recently proposed that The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week
12.
Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion, a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the forms a gradation. Consonance is associated with sweetness, pleasantness and acceptability and dissonance with harshness, unpleasantness, as Hindemith stressed, The two concepts have never been completely explained, and for a thousand years the definitions have varied. The opposition can be made in different contexts, In acoustics or psychophysiology, in modern times, it usually is based on the perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds. In music, even if the opposition often is founded on the preceding, objective distinction, it often is subjective, conventional, cultural. A major second would be considered dissonant if it occurred in a J. S, Bach prelude from the 1700s, however, the same interval may sound consonant in the context of a Claude Debussy piece from the early 1900s or an atonal contemporary piece. For this reason, consonance and dissonance have been considered particularly in the case of Western polyphonic music, most historical definitions of consonance and dissonance since about the 16th century have stressed their pleasant/unpleasant, or agreeable/disagreeable character. In addition, the oppositions pleasant/unpleasant or agreeable/disagreeable evidence a confusion between the concepts of dissonance and of noise and these include, Frequency ratios, with ratios of lower simple numbers being more consonant than those that are higher. Many of these definitions do not require exact integer tunings, only approximation, coincidence of partials, with consonance being a greater coincidence of partials. By this definition, consonance is dependent not only on the width of the interval between two notes, but also on the spectral distribution and thus sound quality of the notes. Thus, a note and the note one octave higher are highly consonant because the partials of the note are also partials of the lower note. Although Helmholtzs work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings, fusion, perception of unity or tonal fusion between two notes. A stable tone combination is a consonance, consonances are points of arrival, rest, an unstable tone combination is a dissonance, its tension demands an onward motion to a stable chord. Thus dissonant chords are active, traditionally they have been considered harsh and have expressed pain, grief, in Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance. Both consonance and dissonance are words applied to harmony, chords, and intervals and, by extension, to melody, tonality, and even rhythm and metre. Nevertheless, the ideas of dissonance, consonance, and resolution exist in some form in all musical traditions that have a concept of melody, harmony. Dissonance being the complement of consonance it may be defined, as above, as non-coincidence of partials, lack of fusion or pattern matching, or as complexity. For many musicians and composers, the ideas of dissonance
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