User:Squib/Unnamed music theory: Difference between revisions

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==Theory==

The simplest tuning is Pythagorean, which is 2.3 JI. Although this is generally thought of as notes created by stacking 2/1 and 3/1, here 3/2 and 4/3 are combined instead. This is much more natural.

~~All pitches in a tuning are~~ created ~~by multiplying together some~~ (~~perhaps fractional~~) ~~number~~ of ~~2/1 and 3~~/1. ~~This avoids the unnecessary clutter and numbers associated with RTT by specifying which notes~~ are ~~available in a tuning~~ instead of ~~specifying equivalences~~.

To use prime 5, which cannot be created from fifths and fourths (I'll get to meantone later), multiple sets of Pythagorean are used, offset by 5/4. However, this is difficult to use practically since most keyboards are not three-dimensional. So instead, sets of pyth are added interlaced with the original, such that the layout is still two-dimensional.

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.

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.

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.

==Derivation==

@@ Line 15: / Line 15: @@
 ==Theory==
+The simplest tuning is Pythagorean, which is 2.3 JI. Although this is generally thought of as notes created by stacking 2/1 and 3/1, here 3/2 and 4/3 are combined instead. This is much more natural.
-All pitches in a tuning are created by multiplying together some (perhaps fractional) number of 2/1 and 3/1. This avoids the unnecessary clutter and numbers associated with RTT by specifying which notes are available in a tuning instead of specifying equivalences.
+To use prime 5, which cannot be created from fifths and fourths (I'll get to meantone later), multiple sets of Pythagorean are used, offset by 5/4. However, this is difficult to use practically since most keyboards are not three-dimensional. So instead, sets of pyth are added interlaced with the original, such that the layout is still two-dimensional.
-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.
+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.
+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.
 ==Derivation==

User:Squib/Unnamed music theory: Difference between revisions

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