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

Line 319:

Let’s look at the other perfectly horizontal line on this diagram. It’s found about a quarter of the way down the diagram, and runs through the 10-ET and 20-ET we looked at earlier. This one’s called “[[blackwood]]”. Here, we can see that all of its ETs are all multiples of 5. In fact, [[5edo|5-ET]] itself is on this line, though we can only see a sliver of its giant numeral off the left edge of the diagram. Again, all of its maps have locked ratios between their mappings of prime 2 and prime 3: {{val|5 8}}, {{val|10 16}}, {{val|15 24}}, {{val|20 32}}, {{val|40 64}}, {{val|80 128}}, etc. You get the idea.

So what do these two temperaments have in common such that their lines are parallel? Well, they’re defined by commas, so why don’t we compare their commas. The compton comma is {{monzo|-19 12 0}}, and the blackwood comma is {{monzo|8 -5 0}}<ref>Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref>. What sticks out about these two commas is that they both have a 5-term of 0 ~~(this means that we can drop it if we want, and simply write them {{monzo|-19 12}} and {{monzo|8 -5}})~~. This means that when we ask the question “how many steps does this comma map to in a given ET”, the ET’s mapping of 5 is irrelevant. Whether we check it in {{val|40 63 93}} or {{val|40 63 94}}, the result is going to be the same. So if {{val|40 63 93}} tempers out blackwood, then {{val|40 63 94}} also tempers out blackwood. And if {{val|24 38 56}} tempers out compton, then {{val|24 38 55}} tempers out compton. And so on.

So what do these two temperaments have in common such that their lines are parallel? Well, they’re defined by commas, so why don’t we compare their commas. The compton comma is {{monzo|-19 12 0}}, and the blackwood comma is {{monzo|8 -5 0}}<ref>Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref>. What sticks out about these two commas is that they both have a 5-term of 0. This means that when we ask the question “how many steps does this comma map to in a given ET”, the ET’s mapping of 5 is irrelevant. Whether we check it in {{val|40 63 93}} or {{val|40 63 94}}, the result is going to be the same. So if {{val|40 63 93}} tempers out blackwood, then {{val|40 63 94}} also tempers out blackwood. And if {{val|24 38 56}} tempers out compton, then {{val|24 38 55}} tempers out compton. And so on.

Similar temperaments can be found which include only 2 of the 3 primes at once. Take “[[augmented]]”, for instance, running from bottom-left to top-right. This temperament is aligned with the 3-axis. This tells us several equivalent things: that relative changes to the mapping of 3 are irrelevant for augmented temperament, that the augmented comma has no 3’s in its prime factorization, and the ratios of the mappings of 2 and 5 are the same for any ET along this line. Indeed we find that the augmented comma is {{monzo|7 0 -3}}, or [[128/125]], which has no 3’s. And if we sample a few maps along this line, we find {{val|12 19 28}}, {{val|9 14 21}}, {{val|15 24 35}}, {{val|21 33 48}}, {{val|27 43 63}}, etc., for which there is no pattern to the 3-term, but the 2- and 5-terms for each are in a 3:7 ratio.

@@ Line 319: / Line 319: @@
 Let’s look at the other perfectly horizontal line on this diagram. It’s found about a quarter of the way down the diagram, and runs through the 10-ET and 20-ET we looked at earlier. This one’s called “[[blackwood]]”. Here, we can see that all of its ETs are all multiples of 5. In fact, [[5edo|5-ET]] itself is on this line, though we can only see a sliver of its giant numeral off the left edge of the diagram. Again, all of its maps have locked ratios between their mappings of prime 2 and prime 3: {{val|5 8}}, {{val|10 16}}, {{val|15 24}}, {{val|20 32}}, {{val|40 64}}, {{val|80 128}}, etc. You get the idea.
-So what do these two temperaments have in common such that their lines are parallel? Well, they’re defined by commas, so why don’t we compare their commas. The compton comma is {{monzo|-19 12 0}}, and the blackwood comma is {{monzo|8 -5 0}}<ref>Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref>. What sticks out about these two commas is that they both have a 5-term of 0 (this means that we can drop it if we want, and simply write them {{monzo|-19 12}} and  {{monzo|8 -5}}). This means that when we ask the question “how many steps does this comma map to in a given ET”, the ET’s mapping of 5 is irrelevant. Whether we check it in {{val|40 63 93}} or {{val|40 63 94}}, the result is going to be the same. So if {{val|40 63 93}} tempers out blackwood, then {{val|40 63 94}} also tempers out blackwood. And if {{val|24 38 56}} tempers out compton, then {{val|24 38 55}} tempers out compton. And so on.
+So what do these two temperaments have in common such that their lines are parallel? Well, they’re defined by commas, so why don’t we compare their commas. The compton comma is {{monzo|-19 12 0}}, and the blackwood comma is {{monzo|8 -5 0}}<ref>Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref>. What sticks out about these two commas is that they both have a 5-term of 0. This means that when we ask the question “how many steps does this comma map to in a given ET”, the ET’s mapping of 5 is irrelevant. Whether we check it in {{val|40 63 93}} or {{val|40 63 94}}, the result is going to be the same. So if {{val|40 63 93}} tempers out blackwood, then {{val|40 63 94}} also tempers out blackwood. And if {{val|24 38 56}} tempers out compton, then {{val|24 38 55}} tempers out compton. And so on.
 Similar temperaments can be found which include only 2 of the 3 primes at once. Take “[[augmented]]”, for instance, running from bottom-left to top-right. This temperament is aligned with the 3-axis. This tells us several equivalent things: that relative changes to the mapping of 3 are irrelevant for augmented temperament, that the augmented comma has no 3’s in its prime factorization, and the ratios of the mappings of 2 and 5 are the same for any ET along this line. Indeed we find that the augmented comma is {{monzo|7 0 -3}}, or [[128/125]], which has no 3’s. And if we sample a few maps along this line, we find {{val|12 19 28}}, {{val|9 14 21}}, {{val|15 24 35}}, {{val|21 33 48}}, {{val|27 43 63}}, etc., for which there is no pattern to the 3-term, but the 2- and 5-terms for each are in a 3:7 ratio.