Kite's thoughts on pergens: Difference between revisions

Line 312:

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.

=Applications=

Line 1,413:

! | tuning's rank

! | notation

! | notation's rank

! | notation's rank<br>if no enharmonics

! | # of enharmonics needed

! | # of enharmonics<br>needed

! | enharmonics

|-

@@ Line 312: / Line 312: @@
 Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).
-The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.
+The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible. Using 7 instead of 5 in the pergen is very common for rank-3. See [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]] for more examples.
 =Applications=
@@ Line 1,413: / Line 1,413: @@
 ! | tuning's rank
 ! | notation
-! | notation's rank
+! | notation's rank<br>if no enharmonics
-! | # of enharmonics needed
+! | # of enharmonics<br>needed
 ! | enharmonics
 |-