Temperament addition: Difference between revisions

Line 101:

===Diagrammatic explanation===

(~~WIP~~)

====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</math> + <math>n</math> = <math>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:

{| class="wikitable"

|+

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|       

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;"|      

|-

|style="background-color: #ccffcc;"|      

|}

This represents a <math>d=4</math> temperament. These diagrams are grade-agnostic, which is to say that they are agnostic as to which side counts the <math>r</math> and which side counts the <math>n</math>. Initially, though, it is probably simpler to explain things by arbitrary assigning each side of the line or the other of the two grades. So let's just say that the count of rows above the line is <math>r</math> and the count below the line is <math>n</math>, so we can then say that this diagram represents a rank-1, nullity-3 temperament.

Actually, this diagram represents more than just a single temperament. It represents a relationship between a pair of temperaments (which, of course, must have the same [[dimensions]]). Green coloration indicates linearly dependent basis vectors between this pair of temperaments, and red coloration indicates linearly ''in''dependent basis vectors between the same pair of temperaments.

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 rank-1 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; A plus A simply equals A again, and A minus A 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 of the rank, rank-1 or otherwise.

From this information, we can see that the comma bases of any randomly selected pair of ''different'' 7-limit ETs are going to share 2 vectors, or in other words, their linear dependence basis 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.

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 shared 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 linear-independence (in the integer count sense) on either side of duality is always equal. There's no notion of a max or min here, as there is with grade or the linear-dependence (again, in the integer count sense); the linear-independence 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).

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 linear dependence, but ''two'' such possibilities. For example, this diagram equally represents both <math>d=4, r=1, n=3, l=1</math> as well as <math>d=4, r=3, n=1, l=1</math> (where <math>l</math> is the linear-independence). Which is another way of saying we could vertically mirror it without changing it.

With the black bar always either in the top half or exactly in the center, we can see that the emanating red band will always either hit the top edge of the square grid first, or they will hit both the top and bottom edges of it simultaneously. So this is how these diagrams visually convey the fact that the linear-independence <math>l</math> between two temperaments will always be less than or equal to their min-grade <math>m</math>: because a situation where <math>m>l</math> would visually look like the red band spilling past the edges of the square grid.

We could also say that two temperaments are linearly dependent on each other when their linear-independence is less than their ''max''-grade.

Perhaps more importantly, we can also see from these diagrams that any pair of <math>m=1</math> temperaments will be addable. Because if they are <math>m=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=1</math>, and that's the definition of addability.

====A simple 5-limit example====

Let's back-pedal to 5-limit for a simple illustrative example.

{| class="wikitable"

|+

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;|      

|}

This diagram shows us that any two 5-limit ETs (rank-1) 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 [[projective tuning space]] diagram from [[The 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 (nullity-1) 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 the 5-limit, 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 5-limit temperaments that are nontrivial (<math>0 < n < d</math> and <math>0 < r < d</math>) will be addable.

====Completing the suite of 7-limit examples====

Okay, back to 7-limit. We've already looked at the <math>m=1</math> possibility (which, for any <math>d</math>, there will only ever be one of). So let's start looking at the possibilities where <math>m=2</math>, which in the case of <math>d=4</math> leaves us only one pair of values for <math>r</math> and <math>n</math>: both being 2.

{| class="wikitable"

|+

|style="background-color: #ccffcc;|      

|-

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;|      

|}

But even with <math>d</math>, <math>r</math>, and <math>n</math> fixed, we still have more than one possibility for <math>l</math>. The above diagram shows <math>l=1</math>. The below diagram shows <math>l=2</math>.

{| class="wikitable"

|+

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ffcccc;|      

|}

In the former possibility, where <math>l=1</math> (and therefore the temperaments are addable), we have a pair of different 7-limit rank-2 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=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 line, 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.

====11-limit diagrams finally reveal important relationships====

So how about we go to the 11-limit, i.e. <math>d=5</math>. As usual, starting with <math>m=1</math>:

{| class="wikitable"

|+

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;|      

|-

|style="background-color: #ccffcc;|      

|-

|style="background-color: #ccffcc;|      

|}

Just as with the <math>l=1</math> diagrams given for <math>d=3</math> and <math>d=5</math>, we can see these are addable temperaments.

Now let's look at <math>d=5</math> but with <math>m=2</math>. This presents two possibilities. First, <math>l=1</math>:

{| class="wikitable"

|+

|style="background-color: #ccffcc;|      

|-

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;|      

|-

|style="background-color: #ccffcc;|      

|}

And second, <math>l=2</math>:

{| class="wikitable"

|+

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ffcccc;|      

|-

|style="background-color: #ccffcc;|      

|}

Here's where things really get interesting. Because in both of these cases, the pairs of temperaments represented are linearly dependent on each other (i.e. either their mappings are linearly dependent, their comma bases are linearly dependent, or both). And so far, every possibility where temperaments have been linearly dependent, they have also been <math>l=1</math>, and therefore addable. But if you look at the second case here, we are <math>l=2</math>, but since <math>d=5</math>, the temperaments still manage to be linearly dependent. So this is the first example of a linearly dependent temperament pairing which is not addable.

====Back to the 3-limit, for a surprisingly tricky example====

Beyond the 11-limit, these diagrams get cumbersome to prepare, and cease to reveal further insights. But if we step back down to the 3-limit, a place simpler than anywhere we've looked so far, we actually find another surprisingly tricky example, which is hopefully still illuminating.

So the 3-limit, i.e. <math>d=2</math> presents another case — like the <math>d=5</math>, <math>m=2</math>, <math>l=2</math> case shared most recently above — where the properties "linearly dependent" and "addable" do not agree. But while in the other case, we had a temperament pair that was linearly dependent yet not addable, in this <math>d=2</math> (and therefore <math>m=1</math>, <math>l=1</math>) case, it is the other way around: addable yet linearly independent!

{| class="wikitable"

|+

|style="background-color: #ffcccc; border-bottom: 3px solid black;"|   

|-

|style="background-color: #ffcccc;"|    

|}

Basically, in the case of <math>d=2</math>, the max-grade (in non-trivial cases, i.e. not JI or the unison temperament) is 1, so any two different ETs or commas you pick are going to be linearly independent (because the only way they could be linearly dependent would be to be the same temperament). And yet we know we can still entry-wise add them to new vectors that are [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Decomposability|decomposable]], because they're already vectors (decomposing means to express a [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#From_vectors_to_multivectors|multivector]] in the form of a list of monovectors, so decomposing a multivector that's already a monovector like this is tantamount to merely putting array braces around it.)

====Conclusion====

This explanation has hopefully helped get a grip on what addability AKA linear-independence-1 is like. But it still hasn't quite explained why linear-independence-1 is one and the same thing as addability. We will look at this in another section soon.

===Geometric explanation===

@@ Line 101: / Line 101: @@
 ===Diagrammatic explanation===
-(WIP)
+====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</math> + <math>n</math> = <math>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:
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|-
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|style="background-color: #ccffcc;"|
+|}
+This represents a <math>d=4</math> temperament. These diagrams are grade-agnostic, which is to say that they are agnostic as to which side counts the <math>r</math> and which side counts the <math>n</math>. Initially, though, it is probably simpler to explain things by arbitrary assigning each side of the line or the other of the two grades. So let's just say that the count of rows above the line is <math>r</math> and the count below the line is <math>n</math>, so we can then say that this diagram represents a rank-1, nullity-3 temperament.
+Actually, this diagram represents more than just a single temperament. It represents a relationship between a pair of temperaments (which, of course, must have the same [[dimensions]]). Green coloration indicates linearly dependent basis vectors between this pair of temperaments, and red coloration indicates linearly ''in''dependent basis vectors between the same pair of temperaments.
+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 rank-1 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; A plus A simply equals A again, and A minus A 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 of the rank, rank-1 or otherwise.
+From this information, we can see that the comma bases of any randomly selected pair of ''different'' 7-limit ETs are going to share 2 vectors, or in other words, their linear dependence basis 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.
+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 shared 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 linear-independence (in the integer count sense) on either side of duality is always equal. There's no notion of a max or min here, as there is with grade or the linear-dependence (again, in the integer count sense); the linear-independence 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).
+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 linear dependence, but ''two'' such possibilities. For example, this diagram equally represents both <math>d=4, r=1, n=3, l=1</math> as well as <math>d=4, r=3, n=1, l=1</math> (where <math>l</math> is the linear-independence). Which is another way of saying we could vertically mirror it without changing it.
+With the black bar always either in the top half or exactly in the center, we can see that the emanating red band will always either hit the top edge of the square grid first, or they will hit both the top and bottom edges of it simultaneously. So this is how these diagrams visually convey the fact that the linear-independence <math>l</math> between two temperaments will always be less than or equal to their min-grade <math>m</math>: because a situation where <math>m>l</math> would visually look like the red band spilling past the edges of the square grid.
+We could also say that two temperaments are linearly dependent on each other when their linear-independence is less than their ''max''-grade.
+Perhaps more importantly, we can also see from these diagrams that any pair of <math>m=1</math> temperaments will be addable. Because if they are <math>m=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=1</math>, and that's the definition of addability.
+====A simple 5-limit example====
+Let's back-pedal to 5-limit for a simple illustrative example.
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|}
+This diagram shows us that any two 5-limit ETs (rank-1) 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 [[projective tuning space]] diagram from [[The 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 (nullity-1) 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 the 5-limit, 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 5-limit temperaments that are nontrivial (<math>0 < n < d</math> and <math>0 < r < d</math>) will be addable.
+====Completing the suite of 7-limit examples====
+Okay, back to 7-limit. We've already looked at the <math>m=1</math> possibility (which, for any <math>d</math>, there will only ever be one of). So let's start looking at the possibilities where <math>m=2</math>, which in the case of <math>d=4</math> leaves us only one pair of values for <math>r</math> and <math>n</math>: both being 2.
+{| class="wikitable"
+|+
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|-
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|}
+But even with <math>d</math>, <math>r</math>, and <math>n</math> fixed, we still have more than one possibility for <math>l</math>. The above diagram shows <math>l=1</math>. The below diagram shows <math>l=2</math>.
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|}
+In the former possibility, where <math>l=1</math> (and therefore the temperaments are addable), we have a pair of different 7-limit rank-2 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=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 line, 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.
+====11-limit diagrams finally reveal important relationships====
+So how about we go to the 11-limit, i.e. <math>d=5</math>. As usual, starting with <math>m=1</math>:
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|}
+Just as with the <math>l=1</math> diagrams given for <math>d=3</math> and <math>d=5</math>, we can see these are addable temperaments.
+Now let's look at <math>d=5</math> but with <math>m=2</math>. This presents two possibilities. First, <math>l=1</math>:
+{| class="wikitable"
+|+
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|-
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|}
+And second, <math>l=2</math>:
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|style="background-color: #ffcccc;|
+|-
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|style="background-color: #ccffcc;|
+|}
+Here's where things really get interesting. Because in both of these cases, the pairs of temperaments represented are linearly dependent on each other (i.e. either their mappings are linearly dependent, their comma bases are linearly dependent, or both). And so far, every possibility where temperaments have been linearly dependent, they have also been <math>l=1</math>, and therefore addable. But if you look at the second case here, we are <math>l=2</math>, but since <math>d=5</math>, the temperaments still manage to be linearly dependent. So this is the first example of a linearly dependent temperament pairing which is not addable.
+====Back to the 3-limit, for a surprisingly tricky example====
+Beyond the 11-limit, these diagrams get cumbersome to prepare, and cease to reveal further insights. But if we step back down to the 3-limit, a place simpler than anywhere we've looked so far, we actually find another surprisingly tricky example, which is hopefully still illuminating.
+So the 3-limit, i.e. <math>d=2</math> presents another case — like the <math>d=5</math>, <math>m=2</math>, <math>l=2</math> case shared most recently above — where the properties "linearly dependent" and "addable" do not agree. But while in the other case, we had a temperament pair that was linearly dependent yet not addable, in this <math>d=2</math> (and therefore <math>m=1</math>, <math>l=1</math>) case, it is the other way around: addable yet linearly independent!
+{| class="wikitable"
+|+
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|style="background-color: #ffcccc; border-bottom: 3px solid black;"|
+|-
+|style="background-color: #ffcccc;"|
+|style="background-color: #ffcccc;"|
+|}
+Basically, in the case of <math>d=2</math>, the max-grade (in non-trivial cases, i.e. not JI or the unison temperament) is 1, so any two different ETs or commas you pick are going to be linearly independent (because the only way they could be linearly dependent would be to be the same temperament). And yet we know we can still entry-wise add them to new vectors that are [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Decomposability|decomposable]], because they're already vectors (decomposing means to express a [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#From_vectors_to_multivectors|multivector]] in the form of a list of monovectors, so decomposing a multivector that's already a monovector like this is tantamount to merely putting array braces around it.)
+====Conclusion====
+This explanation has hopefully helped get a grip on what addability AKA linear-independence-1 is like. But it still hasn't quite explained why linear-independence-1 is one and the same thing as addability. We will look at this in another section soon.
 ===Geometric explanation===