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

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# Find the correct cell in Figure 6b below using your temperament's <math>d</math> and <math>r</math> (rank). This cell should contain the same number of symbols as there are terms of your multimap.

# Match up the terms of your multimap with these symbols. If the symbol is +, do nothing. If the symbol is -, change the sign (positive to negative, or negative to positive; you could think of it like multiplying by either +1 or -1).

# Match up the terms of your multimap with these symbols. If the symbol is <math>+</math>, do nothing. If the symbol is <math>-</math>, change the sign (positive to negative, or negative to positive; you could think of it like multiplying by either +1 or -1).

# Reverse the order of the terms.

# Set the result in the proper count of brackets.

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So in this case:

# We have d=3, r=2, so the correct cell contains the symbols +-+.

# We have d=3, r=2, so the correct cell contains the symbols <math>+-+</math>.

# Matching these symbols up with the terms of our multimap, we don't change the sign of 1, we do change the sign of 4 to -4, and we don't change the sign of the second 4.

# Now we reverse 1 -4 4 to 4 -4 1.

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What’s the proper count of brackets though? Well, the total count of brackets on the multicomma and multimap for a temperament must always sum to the dimensionality of the system from which you tempered. It’s the same thing as <math>d - n = r</math>, just phrased as <math>r + n = d</math>, and where <math>r</math> should be the bracket count for the multimap and <math>n</math> should be the bracket count for the multicomma. So with 5-limit meantone, with dimensionality 3, there should be 3 total pairs of brackets. If 2 are on the multimap, then only 1 are on the multicomma.

Note the Pascal’s triangle shape to the numbers in Figure 6b. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like +-+-+ and +-+-+; other times not, like +-++-+-+-+ and +-+-+-++-+.

Note the Pascal’s triangle shape to the numbers in Figure 6b. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like <math>+-+-+</math> and <math>+-+-+</math>; other times not, like <math>+-++-+-+-+</math> and <math>+-+-+-++-+</math>.

If you’re instead converting a multicomma to a multimap, then you can think of it a couple different ways. Either use <math>n</math> as <math>r</math> when looking up in this table, and then reverse the result, or find <math>r</math> by subtracting <math>n</math> from <math>d</math> and then look it up.

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# Take the rank, halved, rounded up. In our case, <math>\lceil \frac{r}{2} \rceil = \lceil \frac{2}{2} \rceil = \lceil 1 \rceil = 1</math>. Save that result for later. Let’s call it <math>x</math>.

# Find the lexicographic combinations of <math>r</math> primes again: (2,3), (2,5), (3,5). Except this time we don’t want the primes themselves, but their indices in the list of primes. So: <math>(1,2)</math>, <math>(1,3)</math>, <math>(2,3)</math>.

# Find the lexicographic combinations of <math>r</math> primes again: <math>(2,3)</math>, <math>(2,5)</math>, <math>(3,5)</math>. Except this time we don’t want the primes themselves, but their indices in the list of primes. So: <math>(1,2)</math>, <math>(1,3)</math>, <math>(2,3)</math>.

# Take the sums of these sets of indices, and to each sum, also add <math>x</math>. So <math>1+2+x</math>, <math>1+3+x</math>, <math>2+3+x</math> = <math>1+2+1</math>, <math>1+3+1</math>, <math>2+3+1</math> = <math>4</math>, <math>5</math>, <math>6</math>.

# Even terms become +'s and odd terms become -'s.

# Even terms become <math>+</math>'s and odd terms become <math>-</math>'s.

Yes, it's a lot of busywork. I could probably write a program to do it faster than I took explaining it. Maybe I will one day.

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 # Find the correct cell in Figure 6b below using your temperament's <span><math>d</math></span> and <span><math>r</math></span> (rank). This cell should contain the same number of symbols as there are terms of your multimap.
-# Match up the terms of your multimap with these symbols. If the symbol is +, do nothing. If the symbol is -, change the sign (positive to negative, or negative to positive; you could think of it like multiplying by either +1 or -1).
+# Match up the terms of your multimap with these symbols. If the symbol is <span><math>+</math></span>, do nothing. If the symbol is <span><math>-</math></span>, change the sign (positive to negative, or negative to positive; you could think of it like multiplying by either +1 or -1).
 # Reverse the order of the terms.
 # Set the result in the proper count of brackets.
@@ Line 1,342: / Line 1,342: @@
 So in this case:
-# We have d=3, r=2, so the correct cell contains the symbols +-+.
+# We have d=3, r=2, so the correct cell contains the symbols <span><math>+-+</math></span>.
 # Matching these symbols up with the terms of our multimap, we don't change the sign of 1, we do change the sign of 4 to -4, and we don't change the sign of the second 4.
 # Now we reverse 1 -4 4 to 4 -4 1.
@@ Line 1,351: / Line 1,351: @@
 What’s the proper count of brackets though? Well, the total count of brackets on the multicomma and multimap for a temperament must always sum to the dimensionality of the system from which you tempered. It’s the same thing as <span><math>d - n = r</math></span>, just phrased as <span><math>r + n = d</math></span>, and where <span><math>r</math></span> should be the bracket count for the multimap and <span><math>n</math></span> should be the bracket count for the multicomma. So with 5-limit meantone, with dimensionality 3, there should be 3 total pairs of brackets. If 2 are on the multimap, then only 1 are on the multicomma.
-Note the Pascal’s triangle shape to the numbers in Figure 6b. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like +-+-+ and +-+-+; other times not, like +-++-+-+-+ and +-+-+-++-+.
+Note the Pascal’s triangle shape to the numbers in Figure 6b. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like <span><math>+-+-+</math></span> and <span><math>+-+-+</math></span>; other times not, like <span><math>+-++-+-+-+</math></span> and <span><math>+-+-+-++-+</math></span>.
 If you’re instead converting a multicomma to a multimap, then you can think of it a couple different ways. Either use <span><math>n</math></span> as <span><math>r</math></span> when looking up in this table, and then reverse the result, or find <span><math>r</math></span> by subtracting <span><math>n</math></span> from <span><math>d</math></span> and then look it up.
@@ Line 1,395: / Line 1,395: @@
 # Take the rank, halved, rounded up. In our case, <span><math>\lceil \frac{r}{2} \rceil = \lceil \frac{2}{2} \rceil = \lceil 1 \rceil = 1</math></span>. Save that result for later. Let’s call it <span><math>x</math></span>.
-# Find the lexicographic combinations of <span><math>r</math></span> primes again: (2,3), (2,5), (3,5). Except this time we don’t want the primes themselves, but their indices in the list of primes. So: <span><math>(1,2)</math></span>, <span><math>(1,3)</math></span>, <span><math>(2,3)</math></span>.
+# Find the lexicographic combinations of <span><math>r</math></span> primes again: <span><math>(2,3)</math></span>, <span><math>(2,5)</math></span>, <span><math>(3,5)</math></span>. Except this time we don’t want the primes themselves, but their indices in the list of primes. So: <span><math>(1,2)</math></span>, <span><math>(1,3)</math></span>, <span><math>(2,3)</math></span>.
 # Take the sums of these sets of indices, and to each sum, also add <span><math>x</math></span>. So <span><math>1+2+x</math></span>, <span><math>1+3+x</math></span>, <span><math>2+3+x</math></span> = <span><math>1+2+1</math></span>, <span><math>1+3+1</math></span>, <span><math>2+3+1</math></span> = <span><math>4</math></span>, <span><math>5</math></span>, <span><math>6</math></span>.
-# Even terms become +'s and odd terms become -'s.
+# Even terms become <span><math>+</math></span>'s and odd terms become <span><math>-</math></span>'s.
 Yes, it's a lot of busywork. I could probably write a program to do it faster than I took explaining it. Maybe I will one day.