Radical interval: Difference between revisions

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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 [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 follow 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, slendric can be described as equating [3 0 0 -1⟩ and [-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 in the 2.4ed5 subgroup.

== Fmonzos in projection matrices ==

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 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 [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 follow 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, slendric can be described as equating [3 0 0 -1⟩ and [-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 in the 2.4ed5 subgroup.
 == Fmonzos in projection matrices ==