Temperament addition: Difference between revisions

Line 135:

The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments.

Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't ~~temper out~~ the meantone comma itself, nor the porcupine comma itself, but instead ~~tempers out~~ whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that ~~tempers out~~ neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot ~~tempers out~~ 80/81 × 250/243, and dicot ~~tempers out~~ 80/81 × 243/250.

Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish.

Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; for more information on this, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Scale trees]]. So if you understand the effects on individual maps, then you can apply those to changes of maps within a more complex temperament.

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And — at first glance — the result may even seem to be what we were looking for: a temperament which ~~tempers out~~

And — at first glance — the result may even seem to be what we were looking for: a temperament which makes

# neither the meantone comma {{vector|4 -4 1 0}} nor the Pythagorean limma {{vector|-8 5 0 0}}, but does ~~temper out~~ the just diatonic semitone {{vector|-4 1 1 0}}; and

# neither the meantone comma {{vector|4 -4 1 0}} nor the Pythagorean limma {{vector|-8 5 0 0}} vanish, but does make the just diatonic semitone {{vector|-4 1 1 0}} vanish; and

# neither Harrison's comma {{vector|13 -10 0 1}} nor Archytas' comma {{vector|-6 2 0 1}}, but does ~~temper out~~ the laruru negative second {{vector|7 -8 0 2}}.

# neither Harrison's comma {{vector|13 -10 0 1}} nor Archytas' comma {{vector|-6 2 0 1}} vanish, but does make the laruru negative second {{vector|7 -8 0 2}} vanish.

But while these two monovector additions have worked out individually, the full result cannot truly be said to be the "temperament sum" of septimal meantone and blackwood. And here follows a demonstration why it cannot.

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=== Second example: alternate form ===

Let's try summing two completely different comma bases for these temperaments and see what we get. So septimal meantone can also be represented by the comma basis consisting of the diesis {{vector|1 2 -3 1}} and the hemimean comma {{vector|-6 0 5 -2}} (which is another way of saying that septimal meantone also ~~tempers out~~ those commas). And septimal blackwood can also be represented by the septimal third-tone {{vector|2 -3 0 1}} and the cloudy comma {{vector|-14 0 0 5}}. So here's those two bases' entry-wise sum:

Let's try summing two completely different comma bases for these temperaments and see what we get. So septimal meantone can also be represented by the comma basis consisting of the diesis {{vector|1 2 -3 1}} and the hemimean comma {{vector|-6 0 5 -2}} (which is another way of saying that septimal meantone also makes those commas vanish). And septimal blackwood can also be represented by the septimal third-tone {{vector|2 -3 0 1}} and the cloudy comma {{vector|-14 0 0 5}}. So here's those two bases' entry-wise sum:

Line 594:

This works out for the individual monovectors too, that is, it now ~~tempers out~~ none of the input commas anymore, but instead their sums. But what we're looking at here ''is not a comma basis for the same temperament'' as we got the first time!

This works out for the individual monovectors too, that is, it now makes none of the input commas vanish anymore, but instead their sums. But what we're looking at here ''is not a comma basis for the same temperament'' as we got the first time!

We can confirm this by putting both results into [[canonical form]]. That's exactly what canonical form is for: confirming whether or not two matrices are representations of the same temperament! The first result happens to already be in canonical form, so that's [{{vector|-4 1 1 0}} {{vector|7 -8 0 2}}]. This second result [{{vector|3 -1 -3 2}} {{vector|-20 0 5 3}}] doesn't match that, but we can't be sure whether we don't have a match until we put it into canonical form. So its canonical form is [{{vector|-49 3 19 0}} {{vector|-23 1 8 1}}], which doesn't match, and so these are decidedly not the same temperament.

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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.

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 makes to vanish is this shared vector. The diagram also tells us that any two 5-limit temperaments that make only a single comma vanish 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 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.

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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 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 make to vanish, 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 vector away from the black bar, temperament addition 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.

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===Geometric explanation===

We've presented a diagrammatic illustration of the behavior of linear-independence <math>l_{\text{ind}}</math> with respect to temperament dimensions. But some of the results might have seemed surprising. For instance, when looking at the diagram for <math>d=4, g_{\text{min}}=1, g_{\text{max}}=3</math>, it might have seemed intuitive enough that the the red band could not extend beyond the square grid, but then again, why shouldn't it be possible to have, say, two 7-limit ETs which ~~temper out~~ only a single comma in common? Perhaps it doesn't seem clear that this is impossible, and that they must ~~temper out~~ two commas in common (and of course the infinitude of combinations of these two commas). If this is as unclear to you as it was to the author when exploring this topic, then this explanatory section is for you! Here, we will use geometrical representations of temperaments to hone our intuitions about the possible combinations of dimensions and linear-independence <math>l_{\text{ind}}</math> of temperaments.

We've presented a diagrammatic illustration of the behavior of linear-independence <math>l_{\text{ind}}</math> with respect to temperament dimensions. But some of the results might have seemed surprising. For instance, when looking at the diagram for <math>d=4, g_{\text{min}}=1, g_{\text{max}}=3</math>, it might have seemed intuitive enough that the the red band could not extend beyond the square grid, but then again, why shouldn't it be possible to have, say, two 7-limit ETs which make only a single comma in common vanish? Perhaps it doesn't seem clear that this is impossible, and that they must make two commas in common vanish (and of course the infinitude of combinations of these two commas). If this is as unclear to you as it was to the author when exploring this topic, then this explanatory section is for you! Here, we will use geometrical representations of temperaments to hone our intuitions about the possible combinations of dimensions and linear-independence <math>l_{\text{ind}}</math> of temperaments.

In this approach, we’re actually not going to focus directly on the linear-independence <math>l_{\text{ind}}</math> of temperaments. Instead, we're going to look at the linear-''de''pendence <math>l_{\text{dep}}</math> of matrices representing temperaments such as mappings and comma bases, and then compute the linear-independence <math>l_{\text{ind}}</math> from it and the grade <math>g</math>. As we’ve established, the linear-dependence <math>l_{\text{dep}}</math> differs from one side of duality to the other, so we’ll only be looking at one side of duality at a time.

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Think of it this way: geometric points are zero-dimensional, simply representing a position in space, whereas linear algebra vectors are one-dimensional, representing both a magnitude and direction; the way vectors manage to encode this extra dimension without providing any additional information is by being understood to describe this position in space ''relative to an origin''. Well, so we'll now switch our interpretation of these objects to the geometric one, where the vector's entries are nothing more than a coordinate for a point in space. And the "projection" involved in projective vector space essentially positions us at this discarded origin, looking out from it upon every individual point, which accomplishes the same feat, in a visual way.

Perhaps an example may help clarify this setup. Suppose we've got an (x,y,z) space, and two coordinates (5,8,12) and (7,11,16). You should recognize these as the simple maps for 5-ET and 7-ET, usually written as {{map|5 8 12}} and {{map|7 11 16}}, respectively. Ask for the equation of the plane defined by the three points (5,8,12), (7,11,16), and the origin (0,0,0) and you'll get -4x + 4y -1z = 0, which clearly shows us the entries of the meantone comma. That's because meantone temperament can be defined by these two maps. 5-limit JI is a 3D space, and meantone temperament, as a rank-2 temperament, would be a 2D plane. But we don't normally need to think of the map corresponding to the origin, where everything is ~~tempered out~~, including meantone. So we can just assume it, and think of a 2D plane as being defined by only 2 points, which in a view projected (from the origin) will look like just the line connecting (5,8,12) and (7,11,16).

Perhaps an example may help clarify this setup. Suppose we've got an (x,y,z) space, and two coordinates (5,8,12) and (7,11,16). You should recognize these as the simple maps for 5-ET and 7-ET, usually written as {{map|5 8 12}} and {{map|7 11 16}}, respectively. Ask for the equation of the plane defined by the three points (5,8,12), (7,11,16), and the origin (0,0,0) and you'll get -4x + 4y -1z = 0, which clearly shows us the entries of the meantone comma. That's because meantone temperament can be defined by these two maps. 5-limit JI is a 3D space, and meantone temperament, as a rank-2 temperament, would be a 2D plane. But we don't normally need to think of the map corresponding to the origin, where everything is made to vanish, including meantone. So we can just assume it, and think of a 2D plane as being defined by only 2 points, which in a view projected (from the origin) will look like just the line connecting (5,8,12) and (7,11,16).

So, we've set the stage for our projective vector spaces. We will now be looking at representations of temperaments as counts of vector sets, and then using this scheme to convert them to primitive geometric forms. We'll place two of each form into the space, representing the two temperaments whose addability is being checked. Then we will observe their possible ''intersections'' depending on how they're oriented in space, and it's these intersections that represent their linear-dependence. When the dimension of the intersection is then converted back to a vector set count, then we have their linear-dependence <math>l_{\text{dep}}</math>, for this side of duality, anyway (remember, unlike the linear-independence <math>l_{\text{ind}}</math>, this value isn't necessarily the same on both sides of duality). We can finally subtract the linear-dependence from the grade (vector count) to get the linear-indepedence, in order to determine if the two temperaments are addable.

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Let's look at temperaments represented by matrices with 1 vector first (<math>g=1</math>). Yet again, we find ourselves with two separate points, but now we find them in a space that's not a line, not a plane, but a volume. This doesn't change <math>l_{\text{ind}}</math><math> = g = 1</math>, so we're not even going to show it, or any further cases of <math>g=1</math>. These are all addable.

And when <math>g=3</math>, because this is paired with <math>g=1</math> from the min and max values, we should expect to get the same answer as with <math>g=1</math>. And indeed, it will check out that way. Because two <math>g=3</math> temperaments will be planes in this volume, and the intersection of two planes is a line. Which means that <math>l_{\text{dep}}</math><math> = 2</math>. And so <math>l_{\text{ind}}</math><math> = g</math> <math> - </math> <math>l_{\text{dep}}</math> <math>= 3 -</math> <math>2</math> <math>= 1</math>. And here's where our geometric approach begins to pay off! This was the example given at the beginning that might seem unintuitive when relying only on the diagrammatic approach. But here we can see clearly that there would be no way for two planes in a volume to intersect only at a point, which proves the fact that two 7-limit ETs could never only ~~temper out~~ a single comma in common.

And when <math>g=3</math>, because this is paired with <math>g=1</math> from the min and max values, we should expect to get the same answer as with <math>g=1</math>. And indeed, it will check out that way. Because two <math>g=3</math> temperaments will be planes in this volume, and the intersection of two planes is a line. Which means that <math>l_{\text{dep}}</math><math> = 2</math>. And so <math>l_{\text{ind}}</math><math> = g</math> <math> - </math> <math>l_{\text{dep}}</math> <math>= 3 -</math> <math>2</math> <math>= 1</math>. And here's where our geometric approach begins to pay off! This was the example given at the beginning that might seem unintuitive when relying only on the diagrammatic approach. But here we can see clearly that there would be no way for two planes in a volume to intersect only at a point, which proves the fact that two 7-limit ETs could never only make a single comma in common vanish.

[[File:D4 g3 dep2.png|200px]]

@@ Line 135: / Line 135: @@
 The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments.
-Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't temper out the meantone comma itself, nor the porcupine comma itself, but instead tempers out whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that tempers out neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot tempers out 80/81 × 250/243, and dicot tempers out 80/81 × 243/250.
+Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish.
 Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; for more information on this, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Scale trees]]. So if you understand the effects on individual maps, then you can apply those to changes of maps within a more complex temperament.
@@ Line 546: / Line 546: @@
-And — at first glance — the result may even seem to be what we were looking for: a temperament which tempers out
+And — at first glance — the result may even seem to be what we were looking for: a temperament which makes
-# neither the meantone comma {{vector|4 -4 1 0}} nor the Pythagorean limma {{vector|-8 5 0 0}}, but does temper out the just diatonic semitone {{vector|-4 1 1 0}}; and
+# neither the meantone comma {{vector|4 -4 1 0}} nor the Pythagorean limma {{vector|-8 5 0 0}} vanish, but does make the just diatonic semitone {{vector|-4 1 1 0}} vanish; and
-# neither Harrison's comma {{vector|13 -10 0 1}} nor Archytas' comma {{vector|-6 2 0 1}}, but does temper out the laruru negative second {{vector|7 -8 0 2}}.
+# neither Harrison's comma {{vector|13 -10 0 1}} nor Archytas' comma {{vector|-6 2 0 1}} vanish, but does make the laruru negative second {{vector|7 -8 0 2}} vanish.
 But while these two monovector additions have worked out individually, the full result cannot truly be said to be the "temperament sum" of septimal meantone and blackwood. And here follows a demonstration why it cannot.
@@ Line 554: / Line 554: @@
 === Second example: alternate form ===
-Let's try summing two completely different comma bases for these temperaments and see what we get. So septimal meantone can also be represented by the comma basis consisting of the diesis {{vector|1 2 -3 1}} and the hemimean comma {{vector|-6 0 5 -2}} (which is another way of saying that septimal meantone also tempers out those commas). And septimal blackwood can also be represented by the septimal third-tone {{vector|2 -3 0 1}} and the cloudy comma {{vector|-14 0 0 5}}. So here's those two bases' entry-wise sum:
+Let's try summing two completely different comma bases for these temperaments and see what we get. So septimal meantone can also be represented by the comma basis consisting of the diesis {{vector|1 2 -3 1}} and the hemimean comma {{vector|-6 0 5 -2}} (which is another way of saying that septimal meantone also makes those commas vanish). And septimal blackwood can also be represented by the septimal third-tone {{vector|2 -3 0 1}} and the cloudy comma {{vector|-14 0 0 5}}. So here's those two bases' entry-wise sum:
@@ Line 594: / Line 594: @@
-This works out for the individual monovectors too, that is, it now tempers out none of the input commas anymore, but instead their sums. But what we're looking at here ''is not a comma basis for the same temperament'' as we got the first time!
+This works out for the individual monovectors too, that is, it now makes none of the input commas vanish anymore, but instead their sums. But what we're looking at here ''is not a comma basis for the same temperament'' as we got the first time!
 We can confirm this by putting both results into [[canonical form]]. That's exactly what canonical form is for: confirming whether or not two matrices are representations of the same temperament! The first result happens to already be in canonical form, so that's [{{vector|-4 1 1 0}} {{vector|7 -8 0 2}}]. This second result [{{vector|3 -1 -3 2}} {{vector|-20 0 5 3}}] doesn't match that, but we can't be sure whether we don't have a match until we put it into canonical form. So its canonical form is [{{vector|-49 3 19 0}} {{vector|-23 1 8 1}}], which doesn't match, and so these are decidedly not the same temperament.
@@ Line 820: / Line 820: @@
 |}
-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.
+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 makes to vanish is <span style="color: #3C8031;">this shared vector</span>. The diagram also tells us that any two 5-limit temperaments that make only a single comma vanish 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 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.
@@ Line 888: / Line 888: @@
 |}
-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 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 make to vanish, 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 vector away from the black bar, temperament addition 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.
@@ Line 1,048: / Line 1,048: @@
 ===Geometric explanation===
-We've presented a diagrammatic illustration of the behavior of <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> with respect to temperament dimensions. But some of the results might have seemed surprising. For instance, when looking at the diagram for <math>d=4, g_{\text{min}}=1, g_{\text{max}}=3</math>, it might have seemed intuitive enough that the the <span style="color: #3C8031;">red band</span> could not extend beyond the square grid, but then again, why shouldn't it be possible to have, say, two 7-limit ETs which temper out only a single comma in common? Perhaps it doesn't seem clear that this is impossible, and that they must temper out two commas in common (and of course the infinitude of combinations of these two commas). If this is as unclear to you as it was to the author when exploring this topic, then this explanatory section is for you! Here, we will use geometrical representations of temperaments to hone our intuitions about the possible combinations of dimensions and <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> of temperaments.
+We've presented a diagrammatic illustration of the behavior of <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> with respect to temperament dimensions. But some of the results might have seemed surprising. For instance, when looking at the diagram for <math>d=4, g_{\text{min}}=1, g_{\text{max}}=3</math>, it might have seemed intuitive enough that the the <span style="color: #3C8031;">red band</span> could not extend beyond the square grid, but then again, why shouldn't it be possible to have, say, two 7-limit ETs which make only a single comma in common vanish? Perhaps it doesn't seem clear that this is impossible, and that they must make two commas in common vanish (and of course the infinitude of combinations of these two commas). If this is as unclear to you as it was to the author when exploring this topic, then this explanatory section is for you! Here, we will use geometrical representations of temperaments to hone our intuitions about the possible combinations of dimensions and <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> of temperaments.
 In this approach, we’re actually not going to focus directly on the <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> of temperaments. Instead, we're going to look at the <span style="color: #3C8031;">linear-''de''pendence <math>l_{\text{dep}}</math></span> of matrices representing temperaments such as mappings and comma bases, and then compute the <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span> from it and the grade <math>g</math>. As we’ve established, the <span style="color: #3C8031;">linear-dependence <math>l_{\text{dep}}</math></span> differs from one side of duality to the other, so we’ll only be looking at one side of duality at a time.
@@ Line 1,096: / Line 1,096: @@
 Think of it this way: geometric points are zero-dimensional, simply representing a position in space, whereas linear algebra vectors are one-dimensional, representing both a magnitude and direction; the way vectors manage to encode this extra dimension without providing any additional information is by being understood to describe this position in space ''relative to an origin''. Well, so we'll now switch our interpretation of these objects to the geometric one, where the vector's entries are nothing more than a coordinate for a point in space. And the "projection" involved in projective vector space essentially positions us at this discarded origin, looking out from it upon every individual point, which accomplishes the same feat, in a visual way.
-Perhaps an example may help clarify this setup. Suppose we've got an (x,y,z) space, and two coordinates (5,8,12) and (7,11,16). You should recognize these as the simple maps for 5-ET and 7-ET, usually written as {{map|5 8 12}} and {{map|7 11 16}}, respectively. Ask for the equation of the plane defined by the three points (5,8,12), (7,11,16), and the origin (0,0,0) and you'll get -4x + 4y -1z = 0, which clearly shows us the entries of the meantone comma. That's because meantone temperament can be defined by these two maps. 5-limit JI is a 3D space, and meantone temperament, as a rank-2 temperament, would be a 2D plane. But we don't normally need to think of the map corresponding to the origin, where everything is tempered out, including meantone. So we can just assume it, and think of a 2D plane as being defined by only 2 points, which in a view projected (from the origin) will look like just the line connecting (5,8,12) and (7,11,16).
+Perhaps an example may help clarify this setup. Suppose we've got an (x,y,z) space, and two coordinates (5,8,12) and (7,11,16). You should recognize these as the simple maps for 5-ET and 7-ET, usually written as {{map|5 8 12}} and {{map|7 11 16}}, respectively. Ask for the equation of the plane defined by the three points (5,8,12), (7,11,16), and the origin (0,0,0) and you'll get -4x + 4y -1z = 0, which clearly shows us the entries of the meantone comma. That's because meantone temperament can be defined by these two maps. 5-limit JI is a 3D space, and meantone temperament, as a rank-2 temperament, would be a 2D plane. But we don't normally need to think of the map corresponding to the origin, where everything is made to vanish, including meantone. So we can just assume it, and think of a 2D plane as being defined by only 2 points, which in a view projected (from the origin) will look like just the line connecting (5,8,12) and (7,11,16).
 So, we've set the stage for our projective vector spaces. We will now be looking at representations of temperaments as counts of vector sets, and then using this scheme to convert them to primitive geometric forms. We'll place two of each form into the space, representing the two temperaments whose addability is being checked. Then we will observe their possible <span style="color: #3C8031;">''intersections''</span> depending on how they're oriented in space, and it's these <span style="color: #3C8031;">intersections that represent their linear-dependence</span>. When the dimension of the <span style="color: #3C8031;">intersection</span> is then converted back to a vector set count, then we have their <span style="color: #3C8031;">linear-dependence <math>l_{\text{dep}}</math></span>, for this side of duality, anyway (remember, unlike the <span style="color: #B6321C;">linear-independence <math>l_{\text{ind}}</math></span>, this value isn't necessarily the same on both sides of duality). We can finally subtract the <span style="color: #3C8031;">linear-dependence</span> from the grade (vector count) to get the <span style="color: #B6321C;">linear-indepedence</span>, in order to determine if the two temperaments are addable.
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 Let's look at temperaments represented by matrices with 1 vector first (<math>g=1</math>). Yet again, we find ourselves with two separate points, but now we find them in a space that's not a line, not a plane, but a volume. This doesn't change <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span><math> = g = 1</math>, so we're not even going to show it, or any further cases of <math>g=1</math>. These are all addable.
-And when <math>g=3</math>, because this is paired with <math>g=1</math> from the min and max values, we should expect to get the same answer as with <math>g=1</math>. And indeed, it will check out that way. Because two <math>g=3</math> temperaments will be planes in this volume, and the intersection of two planes is a line. Which means that <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span><math> = 2</math>. And so <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span><math> = g</math> <math> - </math> <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span> <math>= 3 -</math> <span style="color: #3C8031;"><math>2</math></span> <math>= 1</math>. And here's where our geometric approach begins to pay off! This was the example given at the beginning that might seem unintuitive when relying only on the diagrammatic approach. But here we can see clearly that there would be no way for two planes in a volume to intersect only at a point, which proves the fact that two 7-limit ETs could never only temper out a single comma in common.
+And when <math>g=3</math>, because this is paired with <math>g=1</math> from the min and max values, we should expect to get the same answer as with <math>g=1</math>. And indeed, it will check out that way. Because two <math>g=3</math> temperaments will be planes in this volume, and the intersection of two planes is a line. Which means that <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span><math> = 2</math>. And so <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span><math> = g</math> <math> - </math> <span style="color: #3C8031;"><math>l_{\text{dep}}</math></span> <math>= 3 -</math> <span style="color: #3C8031;"><math>2</math></span> <math>= 1</math>. And here's where our geometric approach begins to pay off! This was the example given at the beginning that might seem unintuitive when relying only on the diagrammatic approach. But here we can see clearly that there would be no way for two planes in a volume to intersect only at a point, which proves the fact that two 7-limit ETs could never only make a single comma in common vanish.
 [[File:D4 g3 dep2.png|200px]]