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

Line 409:

For another rank-1 example, we could call 7-ET “meantone|dicot”, because it is the intersection between meantone and dicot temperaments. It’s not merely at that intersection, it is the intersection.

We can conclude that there’s no “blackwood|compton” temperament, because those two lines are parallel. In other words, it’s impossible to temper out the blackwood comma and compton comma simultaneously. How could it ever be the case that 12 fifths take you back where you started yet also 5 fifths take you back where you started?

We can conclude that there’s no “blackwood|compton” temperament, because those two lines are parallel. In other words, it’s impossible to temper out the blackwood comma and compton comma simultaneously. How could it ever be the case that 12 fifths take you back where you started yet also 5 fifths take you back where you started?<ref>As you can confirm using the matrix tools you'll learn soon, technically speaking you ''can'' temper them both out at the same time... but it'll only be by using 0-EDO, i.e. a system with only a single pitch. For more information see [[trivial temperaments]].</ref>

Similarly, we can express rank-2 temperaments in terms of rank-1 temperaments. Have you ever heard the expression “two points make a line”? Well, if we choose two ETs from PTS, then there is one and only one line that runs through both of them. So, by choosing those ETs, we can be understood to be describing the rank-2 temperament along that line, or in other words, the one and only temperament whose comma both of those ETs temper out.

Line 424:

Don’t worry: we’re not going 4D just yet. We’ve still got plenty we can cover using only the 5-limit. But we may put away PTS for a couple sections. It’s matrix time.

== matrices ==

@@ Line 409: / Line 409: @@
 For another rank-1 example, we could call 7-ET “meantone|dicot”, because it is the intersection between meantone and dicot temperaments. It’s not merely at that intersection, it is the intersection.
-We can conclude that there’s no “blackwood|compton” temperament, because those two lines are parallel. In other words, it’s impossible to temper out the blackwood comma and compton comma simultaneously. How could it ever be the case that 12 fifths take you back where you started yet also 5 fifths take you back where you started?
+We can conclude that there’s no “blackwood|compton” temperament, because those two lines are parallel. In other words, it’s impossible to temper out the blackwood comma and compton comma simultaneously. How could it ever be the case that 12 fifths take you back where you started yet also 5 fifths take you back where you started?<ref>As you can confirm using the matrix tools you'll learn soon, technically speaking you ''can'' temper them both out at the same time... but it'll only be by using 0-EDO, i.e. a system with only a single pitch. For more information see [[trivial temperaments]].</ref>
 Similarly, we can express rank-2 temperaments in terms of rank-1 temperaments. Have you ever heard the expression “two points make a line”? Well, if we choose two ETs from PTS, then there is one and only one line that runs through both of them. So, by choosing those ETs, we can be understood to be describing the rank-2 temperament along that line, or in other words, the one and only temperament whose comma both of those ETs temper out.
@@ Line 424: / Line 424: @@
 Don’t worry: we’re not going 4D just yet. We’ve still got plenty we can cover using only the 5-limit. But we may put away PTS for a couple sections. It’s matrix time.
 == matrices ==