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

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The immediate conclusion is that 12-EDO is not equipped to approximate the meantone comma directly as a melodic or harmonic interval, and this shouldn’t be surprising because 81/80 is only around 20¢, while the (smallest) step in 12-EDO is five times that.

But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1}} = {{monzo|-4 4}}, and then moving down one 5ave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one 5ave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].

But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1 0}} = {{monzo|-4 4 0}}, and then moving down one 5ave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one 5ave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].

From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''supports''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT.

But we’re still only talking about JI and EDOs. If you’re familiar with meantone as a historical temperament, you may be aware already that it is neither JI nor an EDO. Well, we’ve got a ways to go yet before we get there. One thing we can easily begin to do now, though, is this: refer to EDOs instead as ETs, or equal temperaments. The two terms are [[EDO_vs_ET|roughly synonymous]], but have different implications and connotations, and since we’re learning about temperament theory here, it would be best to use the local terminology. 12-ET it is, then.

=== approximating JI ===

@@ Line 69: / Line 69: @@
 The immediate conclusion is that 12-EDO is not equipped to approximate the meantone comma directly as a melodic or harmonic interval, and this shouldn’t be surprising because 81/80 is only around 20¢, while the (smallest) step in 12-EDO is five times that.
-But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1}} = {{monzo|-4 4}}, and then moving down one 5ave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one 5ave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].
+But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1 0}} = {{monzo|-4 4 0}}, and then moving down one 5ave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one 5ave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].
 From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''supports''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT.
 But we’re still only talking about JI and EDOs. If you’re familiar with meantone as a historical temperament, you may be aware already that it is neither JI nor an EDO. Well, we’ve got a ways to go yet before we get there. One thing we can easily begin to do now, though, is this: refer to EDOs instead as ETs, or equal temperaments. The two terms are [[EDO_vs_ET|roughly synonymous]], but have different implications and connotations, and since we’re learning about temperament theory here, it would be best to use the local terminology. 12-ET it is, then.
 === approximating JI ===