Radical interval: Difference between revisions

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Fractional monzos can be used to notate any number that can be expressed as a root, so they can be used to express the degrees of [[Equal-step tuning|equal tunings]]. For example, 12edo's fifth can be expressed as [7/12⟩, and the Bohlen-Pierce supermajor third may be expressed as [0 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 [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]], 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, fractional monzos 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⟩.

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, fractional monzos 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⟩.

== Use in projection matrices ==

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 Fractional monzos can be used to notate any number that can be expressed as a root, so they can be used to express the degrees of [[Equal-step tuning|equal tunings]]. For example, 12edo's fifth can be expressed as [7/12⟩, and the Bohlen-Pierce supermajor third may be expressed as [0 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 [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]], 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, fractional monzos 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⟩.
+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, fractional monzos 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⟩.
 == Use in projection matrices ==