User:Squib/Unnamed music theory: Difference between revisions

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Although this is useful and allows for true 5-limit JI, it removes the important property of isomorphism, meaning that intervals available in some places may not be available in others. So instead, the sets are tuned to be equally spaced, keeping the major third approximately in tune while making the tuning equal again.

Then, since the tuning is equal, instead of adding new sets of pyth and retuning them to be equal, we equivalently divide the existing set. Thus, all pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1.

Then, since the tuning is equal, instead of adding new sets of pyth and retuning them to be equal, we equivalently divide the existing set. Thus, all pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1. This divides 3-limit JI analogously to how edos devide 2-limit JI.

This avoids the unnecessary clutter and ~~numbers~~ associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences. For example, without already knowing that 243/242 is the difference between 3/2 and a stack of two 11/9s, it's very difficult to tell that tempering it out spits the fifth in half. It's much more natural to simply say that the fifth is split in half, and then that hemififth can be easily used as 11/9 without having to factor 243 first.

This avoids the unnecessary clutter and math associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences. For example, without already knowing that 243/242 is the difference between 3/2 and a stack of two 11/9s, it's very difficult to tell that tempering it out spits the fifth in half. It's much more natural to simply say that the fifth is split in half, and then that hemififth can be easily used as 11/9 without having to factor 243 first.

section break?

With edos, simply specifying the number of divisions is enough, since there is only 1 way to arrange any number of copies of 2-limit JI isomorphically. But here, there are multiple ways. The number of tunings that use N copies of pyth is equal to the sum of the unique factors of N.

(I don't know how much more I can explain without the diagrams.)

==Derivation==

@@ Line 21: / Line 21: @@
 Although this is useful and allows for true 5-limit JI, it removes the important property of isomorphism, meaning that intervals available in some places may not be available in others. So instead, the sets are tuned to be equally spaced, keeping the major third approximately in tune while making the tuning equal again.
-Then, since the tuning is equal, instead of adding new sets of pyth and retuning them to be equal, we equivalently divide the existing set. Thus, all pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1.
+Then, since the tuning is equal, instead of adding new sets of pyth and retuning them to be equal, we equivalently divide the existing set. Thus, all pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1. This divides 3-limit JI analogously to how edos devide 2-limit JI.
-This avoids the unnecessary clutter and numbers associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences. For example, without already knowing that 243/242 is the difference between 3/2 and a stack of two 11/9s, it's very difficult to tell that tempering it out spits the fifth in half. It's much more natural to simply say that the fifth is split in half, and then that hemififth can be easily used as 11/9 without having to factor 243 first.
+This avoids the unnecessary clutter and math associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences. For example, without already knowing that 243/242 is the difference between 3/2 and a stack of two 11/9s, it's very difficult to tell that tempering it out spits the fifth in half. It's much more natural to simply say that the fifth is split in half, and then that hemififth can be easily used as 11/9 without having to factor 243 first.
+section break?
+With edos, simply specifying the number of divisions is enough, since there is only 1 way to arrange any number of copies of 2-limit JI isomorphically. But here, there are multiple ways. The number of tunings that use N copies of pyth is equal to the sum of the unique factors of N.
+(I don't know how much more I can explain without the diagrams.)
 ==Derivation==

User:Squib/Unnamed music theory: Difference between revisions

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