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

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If your temperament's dimensionality <math>d</math> is 6 or less (within the 13-limit), you can take advantage of this table I've prepared, and use this simplified method:

# 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.

# Find the correct cell in Figure 6b below using your temperament's <math>d</math> and <math>g</math>, which stands for '''grade'''. Grade is like rank or nullity, but it is generic; if you are taking the dual of a multimap, you would use rank as the grade, and if you are taking the dual of a multicomma, you would use nullity as the grade. 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 <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.

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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.

An important observation to make about multicommas and multimaps is that — for a given temperament — they always have the same count of terms. This may surprise you, since the rank and nullity for a temperament are often different, and the length of the multimap comes from the rank while the length of the multicomma comes from the nullity. But there’s a simple explanation for this. In either case, the length is not directly equal to the rank or nullity, but to the dimensionality choose the rank or nullity. And there’s a pattern to combinations that can be visualized in the symmetry of rows of Pascal’s triangle: <math>{d \choose n}</math> is always equal to <math>{d \choose {d - n}}</math>, or in other words, <math>{d \choose n}</math> is always equal to <math>{d \choose r}</math>. Here are some examples:

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 If your temperament's dimensionality <span><math>d</math></span> is 6 or less (within the 13-limit), you can take advantage of this table I've prepared, and use this simplified method:
-# 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.
+# Find the correct cell in Figure 6b below using your temperament's <span><math>d</math></span> and <span><math>g</math></span>, which stands for '''grade'''. Grade is like rank or nullity, but it is generic; if you are taking the dual of a multimap, you would use rank as the grade, and if you are taking the dual of a multicomma, you would use nullity as the grade. 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 <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.
@@ Line 1,362: / Line 1,362: @@
 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.
 An important observation to make about multicommas and multimaps is that — for a given temperament — they always have the same count of terms. This may surprise you, since the rank and nullity for a temperament are often different, and the length of the multimap comes from the rank while the length of the multicomma comes from the nullity. But there’s a simple explanation for this. In either case, the length is not directly equal to the rank or nullity, but to the dimensionality choose the rank or nullity. And there’s a pattern to combinations that can be visualized in the symmetry of rows of Pascal’s triangle: <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose {d - n}}</math></span>, or in other words, <span><math>{d \choose n}</math></span> is always equal to <span><math>{d \choose r}</math></span>. Here are some examples: