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

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</math>

And I’m just going to ~~flip~~ the order of those two:

And I’m just going to switch the order of those two:

<math>

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# Convert each of those combinations to a square <math>r×r</math> matrix by slicing a column for each prime out of the mapping and putting them together.

# Take each matrix's determinant.

# ~~Flip~~ the sign of every result if the first result is negative.

# Switch the sign of every result if the first non-zero result is negative.

# Extract the GCD from these results.

# Set the results inside <math>r</math> brackets

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|}

And so our results are <math>-2</math>, <math>3</math>, <math>1</math>, <math>-11</math>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all, or mean there's anything different about the temperament). And so we ~~flip~~ the signs<ref>If it helps you, you could think of this sign-~~flipping~~ step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <math>2</math>, <math>-3</math>, <math>-1</math>, <math>11</math>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multicovector|rank=3|2 -3 -1 11}}.

And so our results are <math>-2</math>, <math>3</math>, <math>1</math>, <math>-11</math>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all, or mean there's anything different about the temperament). And so we switch the signs<ref>If it helps you, you could think of this sign-switching step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <math>2</math>, <math>-3</math>, <math>-1</math>, <math>11</math>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multicovector|rank=3|2 -3 -1 11}}.

As for getting from the multimap back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multimap without the mapping anyway.

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# Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.

# We can calculate a multicomma from a comma basis matrix much in the same way we can calculate a multimap from a mapping matrix

# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and ~~negating~~ some of them.

# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and switching the signs of some of them.

[[File:Algebra notation.png|300px|thumb|right|'''Figure 6a.''' RTT bracket notation comparison.]]

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# Find the correct cell in Table 6a 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 -, ~~negate~~ the sign.

# Match up the terms of your multimap with these symbols. If the symbol is +, do nothing. If the symbol is -, switch the sign (positive to negative, or negative to positive).

# Reverse the order.

# Reverse the order of the terms.

# Set the result in the proper count of brackets.

{| class="wikitable"

|+ '''Table 6a.''' Complement sign ~~flipping~~ sequences by rank and dimensionality

|+ '''Table 6a.''' Complement sign-switching sequences by rank and dimensionality

! colspan="2" rowspan="2" |

! colspan="7" |<math>d</math>

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

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

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

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

# Now we set the result in the proper count of brackets: {{vector|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 Table 6a. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like 1 -1 1 -1 1 and 1 -1 1 -1 1; other times not, like 1 -1 1 -1 1 -1 1 1 -1 1 and 1 -1 1 1 -1 1 -1 1 -1 1. (Well, and sometimes they’re reverses of each other, but then ~~flipped~~ signs so that the first time is always 1.)

Note the Pascal’s triangle shape to the numbers in Table 6a. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like 1 -1 1 -1 1 and 1 -1 1 -1 1; other times not, like 1 -1 1 -1 1 -1 1 1 -1 1 and 1 -1 1 1 -1 1 -1 1 -1 1. (Well, and sometimes they’re reverses of each other, but then switched signs so that the first time is always 1.)

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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If you need to do this process for a higher dimensionality than 6, then you'll need to understand how I found the symbols for each cell of Figure 6a. Here's how:

If you review the seven steps in the process for taking the complement, you may notice that a lot of it is busywork that will never change from one multimap to another. It all amounts to a specific sequence of 1’s and -1’s corresponding to a given rank and dimensionality. In consideration of this, I have gone ahead and prepared a table with the sequences you need to multiply the terms of your multimap by before ~~flipping~~ them:

If you review the seven steps in the process for taking the complement, you may notice that a lot of it is busywork that will never change from one multimap to another. It all amounts to a specific sequence of 1’s and -1’s corresponding to a given rank and dimensionality. In consideration of this, I have gone ahead and prepared a table with the sequences you need to multiply the terms of your multimap by before reversing them:

# 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>.

@@ Line 902: / Line 902: @@
 </math>
-And I’m just going to flip the order of those two:
+And I’m just going to switch the order of those two:
 <math>
@@ Line 1,137: / Line 1,137: @@
 # Convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together.
 # Take each matrix's determinant.
-# Flip the sign of every result if the first result is negative.
+# Switch the sign of every result if the first non-zero result is negative.
 # Extract the GCD from these results.
 # Set the results inside <span><math>r</math></span> brackets
@@ Line 1,239: / Line 1,239: @@
 |}
-And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all, or mean there's anything different about the temperament). And so we flip the signs<ref>If it helps you, you could think of this sign-flipping step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multicovector|rank=3|2 -3 -1 11}}.
+And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all, or mean there's anything different about the temperament). And so we switch the signs<ref>If it helps you, you could think of this sign-switching step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multicovector|rank=3|2 -3 -1 11}}.
 As for getting from the multimap back to the mapping, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never have the multimap without the mapping anyway.
@@ Line 1,250: / Line 1,250: @@
 # Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.
 # We can calculate a multicomma from a comma basis matrix much in the same way we can calculate a multimap from a mapping matrix
-# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and negating some of them.
+# We can convert between multimaps and multicommas using an operation called “taking the '''complement'''”<ref>Elsewhere on the wiki you may find the complement operation called "taking [[the dual]]", or even the dual of a multimap being called simply "the dual". In these materials, I am using the dual to refer to the general case, while the specific case of the dual of a multimap is a multicomma and the operation to get from one of these to its dual is called taking the complement (whereas to get to the dual of a mapping, which is a comma basis, the operation is called taking the null-space).</ref><ref>You may also sometimes see "Hodge dual" used where you'd expect to see the complement operation. The Hodge star operation, or Hodge dual operation, is not another name for the complement operation. It is a linear algebra operation which works as a limited substitute for the exterior algebra operation. The limitation is that it only works when the rank is 2. This is because when rank is 2, bicovectors can be represented as skew-symmetric matrices (see: https://en.wikipedia.org/wiki/Bivector#Matrices), which gives you access to some extra linear algebra utilities such as Hodge star.</ref>, which basically involves reversing the order of terms and switching the signs of some of them.
 [[File:Algebra notation.png|300px|thumb|right|'''Figure 6a.''' RTT bracket notation comparison.]]
@@ Line 1,263: / Line 1,263: @@
 # Find the correct cell in Table 6a 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 -, negate the sign.
+# Match up the terms of your multimap with these symbols. If the symbol is +, do nothing. If the symbol is -, switch the sign (positive to negative, or negative to positive).
-# Reverse the order.
+# Reverse the order of the terms.
 # Set the result in the proper count of brackets.
 {| class="wikitable"
-|+ '''Table 6a.''' Complement sign flipping sequences by rank and dimensionality
+|+ '''Table 6a.''' Complement sign-switching sequences by rank and dimensionality
 ! colspan="2" rowspan="2" |
 ! colspan="7" |<span><math>d</math></span>
@@ Line 1,349: / Line 1,349: @@
 # We have d=3, r=2, so the correct cell contains the symbols + - +.
-# Matching these symbols up with the terms of our multimap, we don't negate 1, we do negate 4 to -4, and we don't negate the second 4.
+# Matching these symbols up with the terms of our multimap, we don't switch the sign of 1, we do switch the sign of 4 to -4, and we don't switch the sign of the second 4.
 # Now we reverse 1 -4 4 to 4 -4 1.
 # Now we set the result in the proper count of brackets: {{vector|4 -4 1}}
@@ Line 1,357: / Line 1,357: @@
 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 Table 6a. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like 1 -1 1 -1 1 and 1 -1 1 -1 1; other times not, like 1 -1 1 -1 1 -1 1 1 -1 1 and 1 -1 1 1 -1 1 -1 1 -1 1. (Well, and sometimes they’re reverses of each other, but then flipped signs so that the first time is always 1.)
+Note the Pascal’s triangle shape to the numbers in Table 6a. Also note that the mirrored results within each dimensionality are reverses of each other. Sometimes that means they’re identical, like 1 -1 1 -1 1 and 1 -1 1 -1 1; other times not, like 1 -1 1 -1 1 -1 1 1 -1 1 and 1 -1 1 1 -1 1 -1 1 -1 1. (Well, and sometimes they’re reverses of each other, but then switched signs so that the first time is always 1.)
 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 1,400: / Line 1,400: @@
 If you need to do this process for a higher dimensionality than 6, then you'll need to understand how I found the symbols for each cell of Figure 6a. Here's how:
-If you review the seven steps in the process for taking the complement, you may notice that a lot of it is busywork that will never change from one multimap to another. It all amounts to a specific sequence of 1’s and -1’s corresponding to a given rank and dimensionality. In consideration of this, I have gone ahead and prepared a table with the sequences you need to multiply the terms of your multimap by before flipping them:
+If you review the seven steps in the process for taking the complement, you may notice that a lot of it is busywork that will never change from one multimap to another. It all amounts to a specific sequence of 1’s and -1’s corresponding to a given rank and dimensionality. In consideration of this, I have gone ahead and prepared a table with the sequences you need to multiply the terms of your multimap by before reversing them:
 # 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>.