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Lattice (music)
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 (i.e., a pitch, or an interval with respect to some other point on the lattice). The lattice can be two-, three-, or ''n''-dimensional, with each dimension corresponding to a different prime-number partial ." When listed in a spreadsheet a lattice may be referred to as a tuning table. The points in a lattice represent pitch classes (or pitches if octaves are represented), and the connectors in a lattice represent the intervals between them. The connecting lines in a lattice display intervals as vectors, so that a line of the same length and angle always has the same intervalic relationship between the points it connects, no matter where it occurs in the lattice. Repeatedly adding the same vector (repeatedly stacking the same interval) moves you further in the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tonnetz
In musical tuning and harmony, the (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. History through 1900 The ''Tonnetz'' originally appeared in Leonhard Euler's 1739 . Euler's ''Tonnetz'', pictured at left, shows 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, , discusses the relationships between keys, presenting them in a network analogous to Euler's ''Tonnetz'', but showing keys rather than notes. The ''Tonnetz'' itself was rediscovered in 1858 by Ernst Naumann in his ., and was disseminated in an 1866 treatise of Arthur von Oettingen. Oettingen and the influential musicologist Hugo Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fokker Periodicity Blocks
Fokker periodicity blocks are a concept in tuning theory used to mathematically relate musical intervals in just intonation to those in equal tuning. They are named after Adriaan Daniël Fokker. These are included as the primary subset of what Erv Wilson refers to as constant structures, where "each interval occurs always subtended by the same number of steps". The basic idea of Fokker's periodicity blocks is to represent just ratios as points on a lattice, and to find vectors in the lattice which represent very small intervals, known as commas. Treating pitches separated by a comma as equivalent "folds" the lattice, effectively reducing its dimension by one; mathematically, this corresponds to finding the quotient group of the original lattice by the sublattice generated by the commas. For an ''n''-dimensional lattice, identifying ''n'' linearly independent commas reduces the dimension of the lattice to zero, meaning that the number of pitches in the lattice is finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unison
Unison (stylised as UNISON) is a Great Britain, British trade union. Along with Unite the Union, Unite, Unison is one of the two largest trade unions in the United Kingdom, with over 1.2 million members who work predominantly in public services, including local government, education, health and outsourcing, outsourced services. The union was formed in 1993 when three public sector trade unions, the National Association of Local Government Officers, National and Local Government Officers Association (NALGO), the National Union of Public Employees (NUPE) and the Confederation of Health Service Employees (COHSE) merged. UNISON's current general secretary is Christina McAnea, who replaced Dave Prentis in 2021. Members and organisation Members of UNISON are typically from industries within the public sector and generally cover both full-time and part-time support and administrative staff. The majority of people joining UNISON are workers within sectors such as local government, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Fourth
A fourth is a interval (music), musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished fourth, Diminished and Tritone, augmented fourths span the same number of staff positions, but consist of a different number of semitones (four and six, respectively). The perfect fourth may be derived from the Harmonic series (music), harmonic series as the interval between the third and fourth harmonics. The term ''perfect'' identifies this interval as belonging to the group of perfect intervals, so called because they are neither major nor minor. A perfect fourth in just intonation corresponds to a pitch ratio of 4:3, or about 498 cent (music), cents (), while in equal temperam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tritone
In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. Narrowly defined, each of these whole tones must be a step in the scale (music), scale, so by this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. More broadly, a tritone is also commonly defined as any interval with a width of three whole tones (spanning six semitones in the chromatic scale), regardless of scale degrees. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B (from F to the B above it, also called augment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Major Seventh
In music from Western culture, a seventh is a interval (music), musical interval encompassing seven staff positions (see Interval (music)#Number, Interval number for more details), and the major seventh is one of two commonly occurring sevenths. It is qualified as ''major'' because it is the larger of the two. The major seventh spans eleven semitones, its smaller counterpart being the minor seventh, spanning ten semitones. For example, the interval from C to B is a major seventh, as the note B lies eleven semitones above C, and there are seven staff positions from C to B. Diminished seventh, Diminished and Augmented seventh, augmented sevenths span the same number of staff positions, but consist of a different number of semitones (nine and twelve). The easiest way to locate and identify the major seventh is from the octave rather than the unison, and it is suggested that one sings the octave first.Keith Wyatt, Carl Schroeder, Joe Elliott (2005). ''Ear Training for the Contempora ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Major Third
In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two Whole step, whole steps. Along with the minor third, the major third is one of two commonly occurring thirds. It is described as ''major'' because it is the larger interval of the two: The major third spans four semitones, whereas the minor third only spans three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished third, Diminished and augmented thirds are shown on the musical staff the same number of lines and spaces apart, but contain a different number of semitones in pitch (two and five). Harmonic and non-harmonic thirds The major third may be derived from the harmonic series (music), harmonic series as the interval be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Major Sixth
In music theory, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as ''major'' because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names (C, D, E, F, G, A) and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths (such as C to A and C to A) span the same number of note letter names and staff positions, but consist of a different number of semitones (seven and ten, respectively). A commonly cited example of a melody featuring the major sixth as its opening is " My Bonnie Lies Over the Ocean".Blake Neely, ''Piano For Dummies'', se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Of Fifths
In music theory, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the sequence is: C, G, D, A, E, B, F/G, C/D, G/A, D/E, A/B, F, and C. This order places the most closely related key signatures adjacent to one another. Twelve-tone equal temperament tuning divides each octave into twelve equivalent semitones, and the circle of fifths leads to a C seven octaves above the starting point. If the fifths are tuned with an exact frequency ratio of 3:2 (the system of tuning known as just intonation), this is not the case (the circle does not "close"). Definition The circle of fifths organizes pitches in a sequence of perfect fifths, generally shown as a circle with the pitches (and their corresponding keys) in clockwise order. It can be viewed in a counterclockwise direction as a circle of fourths. Harmonic progres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal to if is exactly a prime number, and equal to otherwise. That is, the number of prime numbers less than , plus half if equals a prime. Growth rate Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately \frac where is the natural logarithm, in the sense that \lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is \lim_\frac=1 where is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octave Equivalence
In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The interval between the first and second harmonics of the harmonic series is an octave. In Western music notation, notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class. To emphasize that it is one of the perfect intervals (including unison, perfect fourth, and perfect fifth), the octave is designated P8. Other interval qualities are also possible, though rare. The octave above or below an indicated note is sometimes abbreviated ''8a'' or ''8va'' (), ''8va bassa'' (, sometimes also ''8vb''), or simply ''8'' for the octave in the direction indicated by placing this mark above or below the staff. Ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
