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The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.
An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. [3]
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
Music education is a field of practice in which educators are trained for careers as elementary or secondary music teachers, school or music conservatory ensemble directors. Music education is also a research area in which scholars do original research on ways of teaching and learning music.
Music without subjects/themes, or without recognizable, repeating, and developing subjects/themes, is called athematic. Examples include the pre-twelve-tone or early atonal works of Arnold Schoenberg, Anton Webern, Alban Berg, and Alois Hába. Schoenberg once said that, "intoxicated by the enthusiasm of having freed music from the shackles of ...
Educational music, is a genre of music in which songs, lyrics, or other musical elements are used as a method of teaching and/or learning. It has been shown in research to promote learning. Additionally, music study in general has been shown to improve academic performance of students.
Basic Concepts in Music Education is a landmark work published in the USA 1958 as the Fifty-Seventh Yearbook of the National Society for the Study of Education. In 1954, the Music Educators National Conference ( MENC ) had formed its Commission on Basic Concepts in an attempt to seek a more soundly-based philosophical foundation.
Throughout the history of music education, many music educators have adopted and implemented technology in the classroom. Alice Keith and D.C. Boyle were said to be the first music educators in the United States to use the radio for teaching music. Keith wrote Listening in on the Masters, which was a broadcast music appreciation course. [44]