Radical interval: Difference between revisions

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A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''2 ''e''3 … ''e''p }} is a fractional monzo, then it represents 2''e''2 3''e''3 … ''p''''e''''p'' just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 21/13 3-1/13 57/26. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)1/26. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.

A '''fractional monzo''' or '''fmonzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''2 ''e''3 … ''e''p }} is a fractional monzo, then it represents 2''e''2 3''e''3 … ''p''''e''''p'' just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 21/13 3-1/13 57/26. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)1/26. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.

Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.

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-A '''fractional monzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.
+A '''fractional monzo''' or '''fmonzo''' is like an ordinary [[monzo]] except that coefficients have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>. By taking a dot product with {{val| cents (2) cents (3) … cents (p) }} (or in layman's terms, multiplying each monzo entry by the cent value of the corresponding prime) the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (1/13)×1200.0 - (1/13)×cents (3) + (7/26)×cents (5) = 696.1648 cents.
 Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.