Douglas Blumeyer's RTT How-To: Difference between revisions

Line 386:

Cents and Hertz values can readily be converted between one form and the other, so it’s the second difference which is more important. It’s their size. If we do convert 12-ET’s generator to cents so we can compare it with meantone’s generator at 12-ET, we can see that 12-ET’s generator is 100¢ (log₂1.059 × 1200¢ = 100¢) while meantone’s generator at 12-ET is 500¢ (5/12 × 1200¢ = 500¢). What is the explanation for this difference?

Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. ~~Is anything clicking yet?~~

Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.

Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.

@@ Line 386: / Line 386: @@
 Cents and Hertz values can readily be converted between one form and the other, so it’s the second difference which is more important. It’s their size. If we do convert 12-ET’s generator to cents so we can compare it with meantone’s generator at 12-ET, we can see that 12-ET’s generator is 100¢ (log₂1.059 × 1200¢ = 100¢) while meantone’s generator at 12-ET is 500¢ (5/12 × 1200¢ = 500¢). What is the explanation for this difference?
-Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. Is anything clicking yet?
+Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.
 Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.