User:Squib/Unnamed music theory: Difference between revisions

(4 intermediate revisions by the same user not shown)

Line 16:

==Theory==

~~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~~.

This theory organizes pitches as points on a 2D lattice defined by primes 2 and 3, then divides that lattice into a fixed number of pitch classes ("colors") which can be used to approximate other primes. The result of this is a clearly specified tuning and layout that uses colors to intuitively show harmonic relationships and progression.

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==

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.

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.

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 divides 3-limit JI analogously to how edos devide 2-limit JI.

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.

~~==Derivation==~~

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.)

===Overview===

Line 73:

Line 87:

[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]

==Random==

In two dimensions, the number of tunings with N colors is the sum of the unique factors of N. So there are 6+3+2+1=12 6-color tunings in two dimensions. The amount of *three*-dimensional tunings with N colors is the sum of the factors of N times the sum of *their* factors. So the number of 3D 6-color tunings is 6(6+3+2+1)+3(3+1)+2(2+1)+1(1) = 91.

@@ Line 16: / Line 16: @@
 ==Theory==
-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.
+This theory organizes pitches as points on a 2D lattice defined by primes 2 and 3, then divides that lattice into a fixed number of pitch classes ("colors") which can be used to approximate other primes. The result of this is a clearly specified tuning and layout that uses colors to intuitively show harmonic relationships and progression.
-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==
+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.
+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.
+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 divides 3-limit JI analogously to how edos devide 2-limit JI.
+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.
-==Derivation==
+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.)
 ===Overview===
@@ Line 73: / Line 87: @@
 [[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]
+==Random==
+In two dimensions, the number of tunings with N colors is the sum of the unique factors of N. So there are 6+3+2+1=12 6-color tunings in two dimensions. The amount of *three*-dimensional tunings with N colors is the sum of the factors of N times the sum of *their* factors. So the number of 3D 6-color tunings is 6(6+3+2+1)+3(3+1)+2(2+1)+1(1) = 91.

User:Squib/Unnamed music theory: Difference between revisions

Squib (talk | contribs)