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

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=== Temperament merging ===

We’ve seen how 12-ET is found at the intersection of the meantone and augmented temperament lines, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the "intersection" of meantone and augmented. Using the pipe operator “|” to mean ~~“comma~~-~~merge”~~, then, we could call 12-ET “meantone|augmented”, read "meantone comma-merge augmented", or "meantone or augmented" for short. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.

We’ve seen how 12-ET is found at the intersection of the meantone and augmented temperament lines, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the "intersection" of meantone and augmented. Using the pipe operator “|” to mean “[[comma-merging|comma-merge]]”, then, we could call 12-ET “meantone|augmented”, read "meantone comma-merge augmented", or "meantone or augmented" for short. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.

For another rank-1 example, we could call 7-ET “meantone|dicot”, because it is the comma-merge of meantone and dicot temperaments.

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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.

For example, we could choose 7-ET and 12-ET. Looking at either 12-ET or 7-ET, we can see that many, many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there’s only one line which runs through both 7-ET and 12-ET, and that’s the meantone line. So of all the commas that 7-ET tempers out, and all the commas that 12-ET tempers out, there’s only a single one which they have in common, and that’s the meantone comma. Therefore we could give meantone temperament another name, and that’s “7&12”; in this case we use the ampersand operator, not the pipe. We can call this operator "map-merge", so we can read that "7 map-merge 12", or "7 and 12" for short.<ref>Elsewhere, you may see this described as "cross-breeding", where an ET map is a "breed".</ref>

For example, we could choose 7-ET and 12-ET. Looking at either 12-ET or 7-ET, we can see that many, many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there’s only one line which runs through both 7-ET and 12-ET, and that’s the meantone line. So of all the commas that 7-ET tempers out, and all the commas that 12-ET tempers out, there’s only a single one which they have in common, and that’s the meantone comma. Therefore we could give meantone temperament another name, and that’s “7&12”; in this case we use the ampersand operator, not the pipe. We can call this operator "[[map-merging|map-merge]]", so we can read that "7 map-merge 12", or "7 and 12" for short.<ref>Elsewhere, you may see this described as "cross-breeding", where an ET map is a "breed".</ref>

When specifying a rank-1 temperament in terms of two rank-2 temperaments, an obvious constraint is that the two rank-2 temperaments cannot be parallel. When specifying a rank-2 temperament in terms of two rank-1 temperaments, it seems like things should be more open-ended. Indeed, however, there is a special additional constraint on either method, and they’re related to each other. Let’s look at rank-2 as the map-merge of rank-1 first.

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 === Temperament merging ===
-We’ve seen how 12-ET is found at the intersection of the meantone and augmented temperament lines, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the "intersection" of meantone and augmented. Using the pipe operator “|” to mean “comma-merge”, then, we could call 12-ET “meantone|augmented”, read "meantone comma-merge augmented", or "meantone or augmented" for short. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.
+We’ve seen how 12-ET is found at the intersection of the meantone and augmented temperament lines, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the "intersection" of meantone and augmented. Using the pipe operator “|” to mean “[[comma-merging|comma-merge]]”, then, we could call 12-ET “meantone|augmented”, read "meantone comma-merge augmented", or "meantone or augmented" for short. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.
 For another rank-1 example, we could call 7-ET “meantone|dicot”, because it is the comma-merge of meantone and dicot temperaments.
@@ Line 434: / Line 434: @@
 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.
-For example, we could choose 7-ET and 12-ET. Looking at either 12-ET or 7-ET, we can see that many, many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there’s only one line which runs through both 7-ET and 12-ET, and that’s the meantone line. So of all the commas that 7-ET tempers out, and all the commas that 12-ET tempers out, there’s only a single one which they have in common, and that’s the meantone comma. Therefore we could give meantone temperament another name, and that’s “7&12”; in this case we use the ampersand operator, not the pipe. We can call this operator "map-merge", so we can read that "7 map-merge 12", or "7 and 12" for short.<ref>Elsewhere, you may see this described as "cross-breeding", where an ET map is a "breed".</ref>
+For example, we could choose 7-ET and 12-ET. Looking at either 12-ET or 7-ET, we can see that many, many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there’s only one line which runs through both 7-ET and 12-ET, and that’s the meantone line. So of all the commas that 7-ET tempers out, and all the commas that 12-ET tempers out, there’s only a single one which they have in common, and that’s the meantone comma. Therefore we could give meantone temperament another name, and that’s “7&12”; in this case we use the ampersand operator, not the pipe. We can call this operator "[[map-merging|map-merge]]", so we can read that "7 map-merge 12", or "7 and 12" for short.<ref>Elsewhere, you may see this described as "cross-breeding", where an ET map is a "breed".</ref>
 When specifying a rank-1 temperament in terms of two rank-2 temperaments, an obvious constraint is that the two rank-2 temperaments cannot be parallel. When specifying a rank-2 temperament in terms of two rank-1 temperaments, it seems like things should be more open-ended. Indeed, however, there is a special additional constraint on either method, and they’re related to each other. Let’s look at rank-2 as the map-merge of rank-1 first.