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

Line 471:

Now how in the world could that matrix represent the same temperament as {{val|19 30 44}}? Well, they’re two different ways of describing it. {{val|19 30 44}}, as we know, tells us how many generator steps it takes to reach each prime approximation. This matrix, it turns out, is an equivalent way of stating the same information. This matrix is a minimal representation of the null-space of that mapping, or in other words, of all the commas it tempers out.

This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{monzo|-4 4 ₋1}} + {{monzo|10 1 ₋5}} = {{monzo|6 5 ₋6}}, which is the same thing as <span><math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></span></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!

This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{monzo|-4 4 -1}} + {{monzo|10 1 -5}} = {{monzo|6 5 -6}}, which is the same thing as <span><math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></span></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!

Try one. How about the hanson comma, {{monzo|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{monzo|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this is technically called “[https://en.wikipedia.org/wiki/Gaussian_elimination Gaussian elimination]”. Feel free to work through any other examples yourself.

@@ Line 471: / Line 471: @@
 Now how in the world could that matrix represent the same temperament as {{val|19 30 44}}? Well, they’re two different ways of describing it. {{val|19 30 44}}, as we know, tells us how many generator steps it takes to reach each prime approximation. This matrix, it turns out, is an equivalent way of stating the same information. This matrix is a minimal representation of the null-space of that mapping, or in other words, of all the commas it tempers out.
-This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{monzo|-4 4 ₋1}} + {{monzo|10 1 ₋5}} = {{monzo|6 5 ₋6}}, which is the same thing as <span><math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></span></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!
+This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{monzo|-4 4 -1}} + {{monzo|10 1 -5}} = {{monzo|6 5 -6}}, which is the same thing as <span><math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></span></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!
 Try one. How about the hanson comma, {{monzo|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{monzo|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this is technically called “[https://en.wikipedia.org/wiki/Gaussian_elimination Gaussian elimination]”. Feel free to work through any other examples yourself.