User:2^67-1/TempClean sandbox/Pythagorean tuning: Difference between revisions
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The '''3-limit''' consists of [[interval]]s that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as '''Pythagorean tuning''', and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. A 3-limit interval is also known as a Pythagorean interval. | The '''3-limit''' consists of [[interval]]s that are either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as <math>2^a \cdot 3^b</math>, where ''a'' and ''b'' can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are [[3/2]], [[4/3]], [[9/8]]. Confining intervals to the 3-limit is known as '''Pythagorean tuning''', and the Pythagorean tuning used in Europe during the Middle Ages is the seed out of which grew the common-practice tradition of Western music, as well as genres derived from it. A 3-limit interval is also known as a Pythagorean interval. | ||
Pythagorean tuning forms the basis of most systems of diatonic interval categories. | |||
== EDO approximation == | == EDO approximation == | ||
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== Approaches == | == Approaches == | ||
There are | There are many possible approaches to Pythagorean tuning, and each approach is associated with a different Pythagorean equave. The two most widely-used are [[octave]]-based and [[tritave]]-based Pythagorean. | ||
[[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53. | [[Octave]]-based Pythagorean tuning is essentially how it is used in the common-practice. This gives MOS sizes of 2, 3, 5 ([[2L 3s]] pentic), 7 ([[5L 2s]] diatonic), 12 ([[5L 7s]] chromatic), 17 ([[12L 5s]] enharmonic), 29, 41, and 53. | ||
[[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8, 11, 19, | [[Tritave]]-based Pythagorean tuning is an approach described in [https://arxiv.org/abs/1709.00375 this paper] by M. Schmidmeier. This gives MOS sizes of 2, 3, 5, 8 (3L 5s), 11 (8L 3s), 19 (8L 11s), 27 (19L 8s), 46, and 65. The 11-note scale can be regarded as the diatonic-like scale of tritave-equivalent Pythagorean, and the 19-note scale can be regarded as its respective chromatic-like scale. | ||
== Table of intervals == | == Table of intervals == | ||