Radical interval: Difference between revisions
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A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. For the sake of clarity, monzos representing radical intervals are called '''fractional monzos | A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals. | ||
=== Fractional monzos === | |||
For the sake of clarity, monzos representing radical intervals are called '''fractional monzos'''. Mathematically, fractional monzos behave the same as ordinary [[monzo]]s, except that elements 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>, which may also be written as 1\26ed312500/9. | |||
By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}. | By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}. | ||
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A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup. | A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup. | ||
== | == Fractional monzos in projection matrices == | ||
{{Main| Projection matrices }} | {{Main| Projection matrices }} | ||