Temperament addition: Difference between revisions
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Cmloegcmluin (talk | contribs) →Diagrammatic explanation: improve diagrams |
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Temperament arithmetic is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament arithmetic will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments. | Temperament arithmetic is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament arithmetic will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments. | ||
Matching the dimensions is only the first of two conditions on the possibility of temperament arithmetic. The second condition is that the temperaments must all be '''addable'''. This condition is trickier, though, and so a detailed discussion of it will be deferred to a later section (here: [[Temperament arithmetic#Addability]]). But we can at least say here that any set of min | Matching the dimensions is only the first of two conditions on the possibility of temperament arithmetic. The second condition is that the temperaments must all be '''addable'''. This condition is trickier, though, and so a detailed discussion of it will be deferred to a later section (here: [[Temperament arithmetic#Addability]]). But we can at least say here that any set of <math>\min(g)=1</math> temperaments are addable<ref>or they are all the same temperament, in which case they share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</ref>, fortunately, so we don't need to worry about it in that case. | ||
==Versus meet and join== | ==Versus meet and join== | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | | rowspan="4" |<math>d = 4</math> | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | | style="border-bottom: 3px solid black;"|<math>\min(g) = 1</math> | ||
|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; border-bottom: 3px solid black;" | ↑ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;| | | rowspan="3" |<math>\max(g) = 3</math> | ||
|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;" | ↓ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ccffcc;"| | |style="background-color: #ccffcc;"| | ||
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|style="background-color: #ccffcc;"| | |style="background-color: #ccffcc;"| | ||
|style="background-color: #ccffcc;"| | |style="background-color: #ccffcc;"| | ||
| rowspan="2" | | |||
|- | |- | ||
|style="background-color: #ccffcc;"| | |style="background-color: #ccffcc;"| | ||
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|} | |} | ||
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>. | 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>. So we are showing them as <math>\min(g)</math> and <math>\max(g)</math> instead. We could say there's a variation on the rank-nullity theorem whereby <math>\min(g) + \max(g) = d</math>, just as <math>r + n = d</math>. So we can then say that this diagram represents either a <math>r=1</math>, <math>n=3</math> temperament, or perhaps a <math>n=1</math>, <math>r=3</math> temperament. | ||
But actually, this diagram represents more than just a single temperament. It represents a relationship between a pair of temperaments (which have the same [[dimensions]], non-grade-agnostically, i.e. not a pairing of a <math>r=1</math>, <math>n=3</math> temperament with a <math>r=3</math>, <math>n=1</math> temperament). 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 <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; 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 whether <math>r=1</math> or otherwise. | 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; 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 whether <math>r=1</math> or otherwise. | ||
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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). | 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. | 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. Again, 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>. 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 between two temperaments will always be less than or equal to their min-grade: because a situation where <math>\min(g)>l</math> would visually look like the red band spilling past the edges of the square grid. | 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 between two temperaments will always be less than or equal to their min-grade: because a situation where <math>\min(g)>l</math> would visually look like the red band spilling past the edges of the square grid. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | | rowspan="3" |<math>d = 3</math> | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | |style="border-bottom: 3px solid black;"|<math>\min(g) = 1</math> | ||
|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;" | ↑ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;| | | rowspan="2" |<math>\max(g) = 2</math> | ||
|style="background-color: #ffcccc;| | | style="background-color: #ffcccc;" | ↓ | ||
|style="background-color: #ffcccc;| | | style="background-color: #ffcccc;" | ↓ | ||
| style="background-color: #ffcccc;" | ↓ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|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 diagram shows us that any two <math>d=3</math>, <math> | This diagram shows us that any two <math>d=3</math>, <math>\min(g)=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 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 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. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| rowspan="4" |<math>d = 4</math> | |||
| rowspan="2" style="border-bottom: 3px solid black;" |<math>\min(g) = 2</math> | |||
|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: #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; border-bottom: 3px solid black;"| ↑ | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | |style="background-color: #ffcccc; border-bottom: 3px solid black;"| ↑ | ||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;| | | rowspan="2" |<math>\max(g) = 2</math> | ||
|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;" | ↓ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
| | |||
|} | |} | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc;| | | rowspan="4" |<math>d = 4</math> | ||
|style="background-color: #ffcccc;| | | rowspan="2" style="border-bottom: 3px solid black;" |<math>\min(g) = 2</math> | ||
|style="background-color: #ffcccc;| | | style="background-color: #ffcccc;" | ↑ | ||
|style="background-color: #ffcccc;| | | style="background-color: #ffcccc;" | ↑ | ||
| style="background-color: #ffcccc;" | ↑ | |||
| style="background-color: #ffcccc;" | ↑ | |||
| rowspan="2" |<math>l = 2</math> | |||
|- | |- | ||
|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; 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;| | | rowspan="2" |<math>\max(g) = 2</math> | ||
|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;" | ↓ | |||
| rowspan="2" |<math>l = 2</math> | |||
|- | |- | ||
|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;| ↓ | ||
|} | |} | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | | rowspan="5" |<math>d = 5</math> | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | |style="border-bottom: 3px solid black;" |<math>\min(g) = 1</math> | ||
|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; border-bottom: 3px solid black;" | ↑ | ||
| style="background-color: #ffcccc; border-bottom: 3px solid black;" | ↑ | |||
| style="background-color: #ffcccc; border-bottom: 3px solid black;" | ↑ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;| | | rowspan="4" |<math>\max(g) = 4</math> | ||
|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;" | ↓ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
| rowspan="3" | | |||
|- | |- | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| rowspan="5" |<math>d = 5</math> | |||
| rowspan="2" style="border-bottom: 3px solid black;" |<math>\min(g) = 2</math> | |||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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|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; 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;"| ↑ | ||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;| | | rowspan="3" |<math>\max(g) = 3</math> | ||
|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;" | ↓ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
| | |||
|- | |- | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
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|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
| | |||
|} | |} | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc;| | | rowspan="5" |<math>d = 5</math> | ||
|style="background-color: #ffcccc;| | | rowspan="2" style="border-bottom: 3px solid black;" |<math>\min(g) = 2</math> | ||
|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;" | ↑ | |||
| rowspan="2" |<math>l = 2</math> | |||
|- | |- | ||
|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; 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;| | | rowspan="3" |<math>\max(g) = 3</math> | ||
|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;" | ↓ | |||
| rowspan="2" |<math>l = 2</math> | |||
|- | |- | ||
|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;| | ||
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|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
|style="background-color: #ccffcc;| | |style="background-color: #ccffcc;| | ||
| | |||
|} | |} | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | | rowspan="2" |<math>d = 2</math> | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| | |style="border-bottom: 3px solid black;" |<math>\min(g) = 1</math> | ||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| ↑ | |||
|style="background-color: #ffcccc; border-bottom: 3px solid black;"| ↑ | |||
|<math>l = 1</math> | |||
|- | |- | ||
|style="background-color: #ffcccc;"| | |<math>\max(g) = 1</math> | ||
|style="background-color: #ffcccc;"| | |style="background-color: #ffcccc;"| ↓ | ||
|style="background-color: #ffcccc;"| ↓ | |||
|<math>l = 1</math> | |||
|} | |} | ||