Temperament addition: Difference between revisions

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[[File:Visualization of temperament arithmetic.png|500px|right|thumb|A visualization of temperament arithmetic on projective tuning space.]]

This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping rows.

This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping-rows.

These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15=20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5=10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.

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====How to read the diagrams====

The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that form a basis for preimage intervals (this basis is typically called "the [[generator]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> ~~rows~~ corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).

The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that form a basis for preimage intervals (this basis is typically called "the [[generator]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).

So consider this first example of such a diagram:

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So, in this case, the two ET maps are linearly independent. This should be unsurprising; because ET maps are constituted by only a single vector (they're <math>r=1</math> by definition), if they ''were'' linearly dependent, then they'd necessarily be the ''same'' exact ET! Temperament arithmetic on two of the same ET is never interesting; <math>T_1</math> plus <math>T_1</math> simply equals <math>T_1</math> again, and <math>T_1</math> minus <math>T_1</math> is undefined. That said, if we ''were'' to represent temperament arithmetic between two of the same temperament on such a diagram as this, then every cell would be green. And this is true regardless whether <math>r=1</math> or otherwise.

From this information, we can see that the comma bases of any randomly selected pair of ''different'' <math>d=4</math> ETs are going to share 2 vectors, or in other words, their <math>L_{\text{dep}}</math> will have two basis vectors. In terms of the diagram, we're saying that they'll always have two green-colored ~~rows~~ under the black bar.

From this information, we can see that the comma bases of any randomly selected pair of ''different'' <math>d=4</math> ETs are going to share 2 vectors, or in other words, their <math>L_{\text{dep}}</math> will have two basis vectors. In terms of the diagram, we're saying that they'll always have two green-colored vectors under the black bar.

These diagrams are a good way to understand which temperament relationships are possible and which aren't, where by a "relationship" here we mean a particular combination of their matching dimensions and their linear-independence integer count. A good way to use these diagrams for this purpose is to imagine the red coloration emanating away from the black bar in both directions simultaneously, one pair of ~~rows~~ at a time. Doing it like this captures the fact, as previously stated, that the <math>l_{\text{ind}}</math> on either side of duality is always equal. There's no notion of a max or min here, as there is with <math>g</math> or <math>l_{\text{dep}}</math>; the <math>l_{\text{ind}}</math> on either side is always the same, so we can capture it with a single number, which counts the red ~~rows~~ on just one half (that is, half of the total count of red ~~rows~~, or half of the width of the red band in the middle of the grid).

These diagrams are a good way to understand which temperament relationships are possible and which aren't, where by a "relationship" here we mean a particular combination of their matching dimensions and their linear-independence integer count. A good way to use these diagrams for this purpose is to imagine the red coloration emanating away from the black bar in both directions simultaneously, one pair of vectors at a time. Doing it like this captures the fact, as previously stated, that the <math>l_{\text{ind}}</math> on either side of duality is always equal. There's no notion of a max or min here, as there is with <math>g</math> or <math>l_{\text{dep}}</math>; the <math>l_{\text{ind}}</math> on either side is always the same, so we can capture it with a single number, which counts the red vectors on just one half (that is, half of the total count of red vectors, or half of the width of the red band in the middle of the grid).

There's no need to look at diagrams like this where the black bar is below the center. This is because, even though for convenience we're currently treating the top half as <math>r</math> and the bottom half as <math>n</math>, these diagrams are ultimately grade-agnostic. So we could say that each one essentially represents not just one possibility for the relationship between two temperaments' dimensions and <math>l_{\text{ind}}</math>, but ''two'' such possibilities. Again, this diagram equally represents both <math>d=4, r=1, n=3, </math><math>l_{\text{ind}}=1</math> as well as <math>d=4, r=3, n=1, </math><math>l_{\text{ind}}=1</math>. Which is another way of saying we could vertically mirror it without changing it.

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We could also say that two temperaments are linearly dependent on each other when <math>l_{\text{ind}}</math><math><g_{\text{max}}</math>, that is, their linear-independence is less than their ''max''-grade.

Perhaps more importantly, we can also see from these diagrams that any pair of <math>g_{\text{min}}=1</math> temperaments will be addable. Because if they are <math>g_{\text{min}}=1</math>, then the furthest the red band can extend from the black bar is 1 ~~row~~, and 1 mirrored set of red ~~rows~~ means <math>l_{\text{ind}}=1</math>, and that's the definition of addability.

Perhaps more importantly, we can also see from these diagrams that any pair of <math>g_{\text{min}}=1</math> temperaments will be addable. Because if they are <math>g_{\text{min}}=1</math>, then the furthest the red band can extend from the black bar is 1 vector, and 1 mirrored set of red vectors means <math>l_{\text{ind}}=1</math>, and that's the definition of addability.

====A simple <math>d=3</math> example====

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This diagram shows us that any two <math>d=3</math>, <math>g_{\text{min}}=1</math> temperaments (like 5-limit ETs) will be linearly dependent, i.e. their comma bases will share one vector. You may already know this intuitively if you are familiar with the 5-limit [[projective tuning space]] diagram from the [[Paul_Erlich#Papers|Middle Path]] paper, which shows how we can draw a line through any two ETs and that line will represent a temperament, and the single comma that temperament tempers out is this shared vector. The diagram also tells us that any two 5-limit temperaments that temper out only a single comma will also be linearly dependent, for the opposite reason: their ''mappings'' will always share one vector.

And we can see that there are no other diagrams of interest for <math>d=3</math>, because there's no sense in looking at diagrams with no red band, but we can't extend the red band any further than 1 ~~row~~ on each side without going over the edge, and we can't lower the black bar any further without going below the center. So we're done. And our conclusion is that any pair of different <math>d=3</math> temperaments that are nontrivial (<math>0 < n < d=3</math> and <math>0 < r < d=3</math>) will be addable.

And we can see that there are no other diagrams of interest for <math>d=3</math>, because there's no sense in looking at diagrams with no red band, but we can't extend the red band any further than 1 vector on each side without going over the edge, and we can't lower the black bar any further without going below the center. So we're done. And our conclusion is that any pair of different <math>d=3</math> temperaments that are nontrivial (<math>0 < n < d=3</math> and <math>0 < r < d=3</math>) will be addable.

====Completing the suite of <math>d=4</math> examples====

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In the former possibility, where <math>l_{\text{ind}}=1</math> (and therefore the temperaments are addable), we have a pair of different <math>d=4</math>, <math>r=2</math> temperaments where we can find a single comma that both temperaments temper out, and — equivalently — we can find one ET that supports both temperaments.

In the latter possibility, where <math>l_{\text{ind}}=2</math>, neither side of duality shares any vectors in common. And so we've encountered our first example that is not addable. In other words, if the red band ever extends more than 1 ~~row~~ away from the black bar, temperament arithmetic is not possible. So <math>d=4</math> is the first time we had enough room (half of <math>d</math>) to support that condition.

In the latter possibility, where <math>l_{\text{ind}}=2</math>, neither side of duality shares any vectors in common. And so we've encountered our first example that is not addable. In other words, if the red band ever extends more than 1 vector away from the black bar, temperament arithmetic is not possible. So <math>d=4</math> is the first time we had enough room (half of <math>d</math>) to support that condition.

We have now exhausted the possibility space for <math>d=4</math>. We can't extend either the red band or the black bar any further.

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=== Sintel's proof of the linear-independence conjecture===

If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (~~i'll use~~ A and B as their rowspace equivalently):

If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (using A and B as their rowspace equivalently):

dim(A + B) - m = dim(ker(A) + ker(B)) - (n-m)

@@ Line 141: / Line 141: @@
 [[File:Visualization of temperament arithmetic.png|500px|right|thumb|A visualization of temperament arithmetic on projective tuning space.]]
-This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping rows.
+This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping-rows.
 These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15=20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5=10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.
@@ Line 231: / Line 231: @@
 ====How to read the diagrams====
-The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that form a basis for preimage intervals (this basis is typically called "the [[generator]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> rows corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).
+The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that form a basis for preimage intervals (this basis is typically called "the [[generator]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).
 So consider this first example of such a diagram:
@@ Line 270: / Line 270: @@
 So, in this case, the two ET maps are <span style="color: #B6321C;">linearly independent</span>. This should be unsurprising; because ET maps are constituted by only a single vector (they're <math>r=1</math> by definition), if they ''were'' <span style="color: #3C8031;">linearly dependent</span>, then they'd necessarily be the ''same'' exact ET! Temperament arithmetic on two of the same ET is never interesting; <math>T_1</math> plus <math>T_1</math> simply equals <math>T_1</math> again, and <math>T_1</math> minus <math>T_1</math> is undefined. That said, if we ''were'' to represent temperament arithmetic between two of the same temperament on such a diagram as this, then every cell would be green. And this is true regardless whether <math>r=1</math> or otherwise.
-From this information, we can see that the comma bases of any randomly selected pair of ''different'' <math>d=4</math> ETs are going to <span style="color: #3C8031;">share 2 vectors</span>, or in other words, their <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> will have two basis vectors. In terms of the diagram, we're saying that they'll always have two <span style="color: #3C8031;">green-colored rows</span> under the black bar.
+From this information, we can see that the comma bases of any randomly selected pair of ''different'' <math>d=4</math> ETs are going to <span style="color: #3C8031;">share 2 vectors</span>, or in other words, their <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> will have two basis vectors. In terms of the diagram, we're saying that they'll always have two <span style="color: #3C8031;">green-colored vectors</span> under the black bar.
-These diagrams are a good way to understand which temperament relationships are possible and which aren't, where by a "relationship" here we mean a particular combination of their matching dimensions and their linear-independence integer count. A good way to use these diagrams for this purpose is to imagine the <span style="color: #B6321C;">red coloration</span> emanating away from the black bar in both directions simultaneously, one pair of rows at a time. Doing it like this captures the fact, as previously stated, that the <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> on either side of duality is always equal. There's no notion of a max or min here, as there is with <math>g</math> or <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span>; the <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> on either side is always the same, so we can capture it with a single number, which counts the <span style="color: #B6321C;">red rows</span> on just one half (that is, half of the total count of <span style="color: #B6321C;">red rows</span>, or half of the width of the <span style="color: #B6321C;">red band</span> in the middle of the grid).
+These diagrams are a good way to understand which temperament relationships are possible and which aren't, where by a "relationship" here we mean a particular combination of their matching dimensions and their linear-independence integer count. A good way to use these diagrams for this purpose is to imagine the <span style="color: #B6321C;">red coloration</span> emanating away from the black bar in both directions simultaneously, one pair of vectors at a time. Doing it like this captures the fact, as previously stated, that the <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> on either side of duality is always equal. There's no notion of a max or min here, as there is with <math>g</math> or <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span>; the <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> on either side is always the same, so we can capture it with a single number, which counts the <span style="color: #B6321C;">red vectors</span> on just one half (that is, half of the total count of <span style="color: #B6321C;">red vectors</span>, or half of the width of the <span style="color: #B6321C;">red band</span> in the middle of the grid).
 There's no need to look at diagrams like this where the black bar is below the center. This is because, even though for convenience we're currently treating the top half as <math>r</math> and the bottom half as <math>n</math>, these diagrams are ultimately grade-agnostic. So we could say that each one essentially represents not just one possibility for the relationship between two temperaments' dimensions and <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span>, but ''two'' such possibilities. Again, this diagram equally represents both <math>d=4, r=1, n=3, </math><span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span> as well as <math>d=4, r=3, n=1, </math><span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span>. Which is another way of saying we could vertically mirror it without changing it.
@@ Line 280: / Line 280: @@
 We could also say that two temperaments are <span style="color: #3C8031;">linearly dependent</span> on each other when <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span><math><g_{\text{max}}</math>, that is, their <span style="color: #B6321C;">linear-independence</span> is less than their ''max''-grade.
-Perhaps more importantly, we can also see from these diagrams that any pair of <math>g_{\text{min}}=1</math> temperaments will be addable. Because if they are <math>g_{\text{min}}=1</math>, then the furthest the <span style="color: #B6321C;">red band</span> can extend from the black bar is 1 row, and 1 mirrored set of <span style="color: #B6321C;">red rows</span> means <span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span>, and that's the definition of addability.
+Perhaps more importantly, we can also see from these diagrams that any pair of <math>g_{\text{min}}=1</math> temperaments will be addable. Because if they are <math>g_{\text{min}}=1</math>, then the furthest the <span style="color: #B6321C;">red band</span> can extend from the black bar is 1 vector, and 1 mirrored set of <span style="color: #B6321C;">red vectors</span> means <span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span>, and that's the definition of addability.
 ====A simple <math>d=3</math> example====
@@ Line 308: / Line 308: @@
 This diagram shows us that any two <math>d=3</math>, <math>g_{\text{min}}=1</math> temperaments (like 5-limit ETs) will be <span style="color: #3C8031;">linearly dependent</span>, i.e. their comma bases will <span style="color: #3C8031;">share</span> one vector. You may already know this intuitively if you are familiar with the 5-limit [[projective tuning space]] diagram from the [[Paul_Erlich#Papers|Middle Path]] paper, which shows how we can draw a line through any two ETs and that line will represent a temperament, and the single comma that temperament tempers out is <span style="color: #3C8031;">this shared vector</span>. The diagram also tells us that any two 5-limit temperaments that temper out only a single comma will also be <span style="color: #3C8031;">linearly dependent</span>, for the opposite reason: their ''mappings'' will always <span style="color: #3C8031;">share</span> one vector.
-And we can see that there are no other diagrams of interest for <math>d=3</math>, because there's no sense in looking at diagrams with no <span style="color: #B6321C;">red band</span>, but we can't extend the <span style="color: #B6321C;">red band</span> any further than 1 row on each side without going over the edge, and we can't lower the black bar any further without going below the center. So we're done. And our conclusion is that any pair of different <math>d=3</math> temperaments that are nontrivial (<math>0 < n < d=3</math> and <math>0 < r < d=3</math>) will be addable.
+And we can see that there are no other diagrams of interest for <math>d=3</math>, because there's no sense in looking at diagrams with no <span style="color: #B6321C;">red band</span>, but we can't extend the <span style="color: #B6321C;">red band</span> any further than 1 vector on each side without going over the edge, and we can't lower the black bar any further without going below the center. So we're done. And our conclusion is that any pair of different <math>d=3</math> temperaments that are nontrivial (<math>0 < n < d=3</math> and <math>0 < r < d=3</math>) will be addable.
 ====Completing the suite of <math>d=4</math> examples====
@@ Line 376: / Line 376: @@
 In the former possibility, where <span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span> (and therefore the temperaments are addable), we have a pair of different <math>d=4</math>, <math>r=2</math> temperaments where we can find a single comma that both temperaments temper out, and — equivalently — we can find one ET that supports both temperaments.
-In the latter possibility, where <span style="color: #B6321C;"><math>l_{\text{ind}}=2</math></span>, neither side of duality <span style="color: #3C8031;">shares</span> any vectors in common. And so we've encountered our first example that is not addable. In other words, if the <span style="color: #B6321C;">red band</span> ever extends more than 1 row away from the black bar, temperament arithmetic is not possible. So <math>d=4</math> is the first time we had enough room (half of <math>d</math>) to support that condition.
+In the latter possibility, where <span style="color: #B6321C;"><math>l_{\text{ind}}=2</math></span>, neither side of duality <span style="color: #3C8031;">shares</span> any vectors in common. And so we've encountered our first example that is not addable. In other words, if the <span style="color: #B6321C;">red band</span> ever extends more than 1 vector away from the black bar, temperament arithmetic is not possible. So <math>d=4</math> is the first time we had enough room (half of <math>d</math>) to support that condition.
 We have now exhausted the possibility space for <math>d=4</math>. We can't extend either the <span style="color: #B6321C;">red band</span> or the black bar any further.
@@ Line 546: / Line 546: @@
 === Sintel's proof of the linear-independence conjecture===
-If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (i'll use A and B as their rowspace equivalently):
+If A and B are mappings from Z^n to Z^m, with n > m, A, B full rank (using A and B as their rowspace equivalently):
 dim(A + B) - m = dim(ker(A) + ker(B)) - (n-m)