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| 2^1/2 × 3^-1/13 }}). 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'''. 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''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 2^1/13 3^-1/13 5^7/26. 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, 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| log₂(2) log₂(3) … log₂(''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⋅log₂(3) + (7/26)⋅1200⋅log₂(5) {{=}} 696.1648 cents }}.

Vectors in [[monzos and interval space|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 ''n''-th root of a positive rational number which corresponds to it.

== Tunings in terms of radical intervals ==

Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo| 7/12 }} or 2^7/12, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo| 0 3/13 }} or 3^3/13.

What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by {{monzo| 7/12 -3/13 }}. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like [[slendric]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo| 3 0 0 -1 }} and {{monzo| -1/3 1/3 }}.

=== Radical subgroups ===

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 ==

 

[[Category:Regular temperament theory]]

[[Category:Math]]

[[Category:Tuning]]