With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as musical keyboards.
Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world. Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The 19 equal temperament , first proposed and used by Guillaume Costeley in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal semitone equal temperament at the cost of a flatter fifth.
The overall effect is one of greater consonance. Twenty-four equal temperament , with twenty-four equally spaced tones, is widespread in the pedagogy and notation of Arabic music. However, in theory and practice, the intonation of Arabic music conforms to rational ratios , as opposed to the irrational ratios of equally tempered systems. While any analog to the equally tempered quarter tone is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, or neutral second , frequently occur. These neutral seconds, however, vary slightly in their ratios dependent on maqam , as well as geography.
Indeed, Arabic music historian Habib Hassan Touma has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture. Musical set theory uses the language of mathematical set theory in an elementary way to organize musical objects and describe their relationships.
To analyze the structure of a piece of typically atonal music using musical set theory, one usually starts with a set of tones, which could form motives or chords.
By applying simple operations such as transposition and inversion , one can discover deep structures in the music. Operations such as transposition and inversion are called isometries because they preserve the intervals between tones in a set. Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music.
For example, the pitch classes in an equally tempered octave form an abelian group with 12 elements. It is possible to describe just intonation in terms of a free abelian group. Transformational theory is a branch of music theory developed by David Lewin. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.
Theorists have also proposed musical applications of more sophisticated algebraic concepts. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on a Grassmannian. So the chromatic scale can be thought of as a torsor for the group. From Wikipedia, the free encyclopedia. Relationships between music and mathematics. Main article: Musical form. Main articles: Musical tuning and Musical temperament. Main article: Set theory music.
Main article: Abstract algebra. Music portal Mathematics portal. The Music of the Arabs. Portland, OR: Amadeus Press. Music theory. List of music theorists. These designs also enhance the art of calligraphy. As well as being stunning to look at, Celtic knot designs have a rich cultural heritage and can convey a variety of meanings. Whether you decide to wear one or simply create your own design, you will be helping to carry on a lengthy and noble tradition.
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Various Celtic Knot Interpretations Most scholars believe that Celtic knots were developed for religious and secular purposes. Celtic Patterns Knotwork Interlace This pattern is said to represent the interconnection of life and our place within the universe. Popular Celtic Knot Designs For individuals of Welsh, Irish and Scottish heritage, Celtic knot designs can be seen as a source of great cultural pride.
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