Temperament addition: Difference between revisions
Cmloegcmluin (talk | contribs) →Negation: don't use normal uppercase letter for an abstraction that's not a matrix |
Cmloegcmluin (talk | contribs) m update templates and formatting |
||
| Line 213: | Line 213: | ||
==Getting to the side of duality with <math>g_{\text{min}}=1</math>== | ==Getting to the side of duality with <math>g_{\text{min}}=1</math>== | ||
We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{ | We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{rket|{{map|1 0 -4}} {{map|0 1 4}}}} and the porcupine mapping {{rket|{{map|1 2 3}} {{map|0 3 5}}}} each consist of two vectors. So these representations require additional labor to compute. But their duals are easy! If we simply find a comma basis for each of these mappings, we get [{{vector|4 -4 1}}] and [{{vector|1 -5 3}}]. In this form, the temperaments can be entry-wise added, to [{{vector|5 -9 4}}] as we saw earlier. And if in the end we're still after a mapping, since we started with mappings, we can take the dual of this comma basis, to find the mapping {{rket|{{map|1 1 1}} {{map|0 4 9}}}}. | ||
==Negation== | ==Negation== | ||
| Line 257: | Line 257: | ||
<span style="color: #3C8031;">Linear dependence</span> has been defined for the matrices and multivectors that represent temperaments, but it can also be defined for temperaments themselves. The conditions of temperament addition motivate a definition of <span style="color: #3C8031;">linear dependence</span> for temperaments whereby temperaments are considered <span style="color: #3C8031;">linearly dependent</span> if ''either of their mappings or their comma bases are <span style="color: #3C8031;">linearly dependent</span>''<ref>or — equivalently, in EA — either their multimaps or their multicommas are <span style="color: #3C8031;">linearly dependent</span></ref>. | <span style="color: #3C8031;">Linear dependence</span> has been defined for the matrices and multivectors that represent temperaments, but it can also be defined for temperaments themselves. The conditions of temperament addition motivate a definition of <span style="color: #3C8031;">linear dependence</span> for temperaments whereby temperaments are considered <span style="color: #3C8031;">linearly dependent</span> if ''either of their mappings or their comma bases are <span style="color: #3C8031;">linearly dependent</span>''<ref>or — equivalently, in EA — either their multimaps or their multicommas are <span style="color: #3C8031;">linearly dependent</span></ref>. | ||
For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ | For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{rket|{{map|5 8 12}}}} and {{rket|{{map|7 11 16}}}} may at first seem to be <span style="color: #B6321C;">linearly independent</span>, because the basis vectors visible in their mappings are clearly <span style="color: #B6321C;">linearly independent</span> (when comparing two vectors, the only way they could be <span style="color: #3C8031;">linearly dependent</span> is if they are multiples of each other, as discussed [[Linear dependence#Linear dependence between individual vectors|here]]). And indeed their ''mappings'' are <span style="color: #B6321C;">linearly independent</span>. But these two ''temperaments'' are <span style="color: #3C8031;">linearly ''de''pendent</span>, because if we consider their corresponding comma bases, we will find that they <span style="color: #3C8031;">share</span> the basis vector of the meantone comma {{vector|4 -4 1}}. | ||
To make this point visually, we could say that two temperaments are <span style="color: #3C8031;">linearly dependent</span> if they intersect in one or the other of tone space and tuning space. So you have to check both views.<ref>You may be wondering — what about two temperaments which are parallel in tone or tuning space, e.g. compton and blackwood in tuning space? Their comma bases are each <math>n=1</math>, and they merge to give a <math>n=2</math> [[comma basis]], which corresponds to a <math>r=1</math> mapping, which means it should appear as an ET point on the PTS diagram. But how could that be? Well, here's their comma-merge: | To make this point visually, we could say that two temperaments are <span style="color: #3C8031;">linearly dependent</span> if they intersect in one or the other of tone space and tuning space. So you have to check both views.<ref>You may be wondering — what about two temperaments which are parallel in tone or tuning space, e.g. compton and blackwood in tuning space? Their comma bases are each <math>n=1</math>, and they merge to give a <math>n=2</math> [[comma basis]], which corresponds to a <math>r=1</math> mapping, which means it should appear as an ET point on the PTS diagram. But how could that be? Well, here's their comma-merge: [{{vector|1 0 0}} {{vector|0 1 0}}], and so that corresponding mapping is {{rket|{{map|0 0 1}}}}. So it's some degenerate ET. I suppose we could say it's the point at infinity away from the center of the diagram.</ref> | ||
===3. <span style="color: #B6321C;">Linear independence</span> between temperaments=== | ===3. <span style="color: #B6321C;">Linear independence</span> between temperaments=== | ||
| Line 339: | Line 339: | ||
===Example=== | ===Example=== | ||
For our example, let’s look at septimal meantone plus flattone. The canonical forms of these temperaments are {{ | For our example, let’s look at septimal meantone plus flattone. The canonical forms of these temperaments are {{rket|{{map|1 0 -4 -13}} {{map|0 1 4 10}}}} and {{rket|{{map|1 0 -4 17}} {{map|0 1 4 -9}}}}. | ||
'''0. Counterexample.''' Before we try following the detailed instructions just described above, let's do the counterexample, to illustrate why we have to follow them at all. Simple entry-wise addition of these two mapping matrices gives {{ | '''0. Counterexample.''' Before we try following the detailed instructions just described above, let's do the counterexample, to illustrate why we have to follow them at all. Simple entry-wise addition of these two mapping matrices gives {{rket|{{map|2 0 -8 4}} {{map|0 2 8 1}}}}, which is not the correct answer: | ||
| Line 368: | Line 368: | ||
And it's wrong not only because it is clearly enfactored (at least one factor of 2, that is visible in the first vector). The full explanation of why this is the wrong answer is beyond the scope of this example (the nature of correctness here is discussed in the section [[Temperament addition#Addition on non-addable temperaments]]). However, if we now follow through with the instructions described above, we can find the correct answer. | And it's wrong not only because it is clearly enfactored (at least one factor of 2, that is visible in the first vector). The full explanation of why this is the wrong answer is beyond the scope of this example (the nature of correctness here is discussed in the section [[Temperament addition#Addition on non-addable temperaments]]). However, if we now follow through with the instructions described above, we can find the correct answer. | ||
'''1. Find the linear-dependence basis.''' We know where to start: first find the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> and put each of these two mappings into a form that includes it explicitly. In this case, their <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> consists of a single vector: <span style="color: #3C8031;">{{ | '''1. Find the linear-dependence basis.''' We know where to start: first find the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> and put each of these two mappings into a form that includes it explicitly. In this case, their <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> consists of a single vector: <span style="color: #3C8031;">{{rket|{{map|19 30 44 53}}}}</span>. | ||
'''2. Reproduce the original temperament.''' The original matrices had two vectors, so as our second step, we pad out these matrices by drawing from vectors from the original matrices, starting from their first vectors, so now we have [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|1 0 -4 -13}}⟩ and [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|1 0 -4 17}}⟩. We could choose any vectors from the original matrices, as long as they are <span style="color: #B6321C;">linearly independent</span> from the ones we already have; if one is not, skip it and move on. In this case the first vectors are both fine, though. | '''2. Reproduce the original temperament.''' The original matrices had two vectors, so as our second step, we pad out these matrices by drawing from vectors from the original matrices, starting from their first vectors, so now we have [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|1 0 -4 -13}}⟩ and [<span style="color: #3C8031;">{{map|19 30 44 53}}</span> {{map|1 0 -4 17}}⟩. We could choose any vectors from the original matrices, as long as they are <span style="color: #B6321C;">linearly independent</span> from the ones we already have; if one is not, skip it and move on. In this case the first vectors are both fine, though. | ||
| Line 506: | Line 506: | ||
=== Initial example: canonical form === | === Initial example: canonical form === | ||
Clearly, two non-addable temperaments may still be entry-wise added. For example, the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> for the canonical comma bases for septimal meantone | Clearly, two non-addable temperaments may still be entry-wise added. For example, the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> for the canonical comma bases for septimal meantone [{{vector|4 -4 1 0}} {{vector|13 -10 0 1}}] and septimal blackwood [{{vector|-8 5 0 0}} {{vector|-6 2 0 1}}] is empty, meaning their <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> <math>=2</math>, and therefore they aren't addable. Yet we can still do entry-wise addition as if they were: | ||
| Line 596: | Line 596: | ||
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 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! | ||
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 | 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. | ||
=== Third example: reordering of canonical form === | === Third example: reordering of canonical form === | ||
| Line 640: | Line 640: | ||
And the canonical form of | And the canonical form of [{{vector|-2 -2 1 1}} {{vector|5 -5 0 1}}] is [{{vector|-7 3 1 0}} {{vector|5 -5 0 1}}], so that's yet another possible temperament resulting from adding these non-addable temperaments. | ||
=== Fourth example: other side of duality === | === Fourth example: other side of duality === | ||
We can even experience this without changing basis. Let's just compare the results we get from the canonical form of these two temperaments, on either side of duality. The first example we worked through happened to be their canonical comma bases. So now let's look at their canonical mappings. Septimal meantone's is {{ | We can even experience this without changing basis. Let's just compare the results we get from the canonical form of these two temperaments, on either side of duality. The first example we worked through happened to be their canonical comma bases. So now let's look at their canonical mappings. Septimal meantone's is {{rket|{{map|1 0 -4 -13}} {{map|0 1 4 10}}}} and septimal blackwood's is {{rket|{{map|5 8 0 14}} {{map|0 0 1 0}}}}. So what temperament do we get by summing these? | ||
| Line 676: | Line 676: | ||
In order to compare this result directly with our other three results, let's take the dual of this {{ | In order to compare this result directly with our other three results, let's take the dual of this {{rket|{{map|6 8 -4 1}} {{map|0 1 5 10}}}}, which is [{{vector|22 -15 3 0}} {{vector|41 -30 2 2}}] (in canonical form), so we can see that's yet a fourth possible result<ref> | ||
It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{ | It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{rket|{{map|97 152 220 259}} {{map|-30 -47 -68 -80}}}} + {{rket|{{map|-95 -152 -212 -266}} {{map|30 48 67 84}}}}.</ref> | ||
=== Summary === | === Summary === | ||
| Line 731: | Line 731: | ||
What we're experiencing here is the effect first discussed in the early section [[Temperament addition#The temperaments are addable]]: since entry-wise addition of matrices is a operation defined on matrices, not bases, we get different results for different bases. | What we're experiencing here is the effect first discussed in the early section [[Temperament addition#The temperaments are addable]]: since entry-wise addition of matrices is a operation defined on matrices, not bases, we get different results for different bases. | ||
This in stark contrast to the situation when you have addable temperaments; once you get them into the form with the explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> and only the single <span style="color: #B6321C;">linearly independent basis vector</span>, you will get the same resultant temperament regardless of which side of duality you add them on — the duals stay in sync, we could say — and regardless of which basis we choose.<ref>Note that different bases ''are'' possible for addable temperaments, e.g. the simplest addable forms for 5-limit meantone and porcupine are [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|-2 -3 -4}}</span>⟩ + [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|1 2 3}}</span>⟩ = {{ | This in stark contrast to the situation when you have addable temperaments; once you get them into the form with the explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> and only the single <span style="color: #B6321C;">linearly independent basis vector</span>, you will get the same resultant temperament regardless of which side of duality you add them on — the duals stay in sync, we could say — and regardless of which basis we choose.<ref>Note that different bases ''are'' possible for addable temperaments, e.g. the simplest addable forms for 5-limit meantone and porcupine are [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|-2 -3 -4}}</span>⟩ + [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|1 2 3}}</span>⟩ = {{rket|{{map|14 22 32}} {{map|-1 -1 -1}}}} which canonicalizes to {{rket|{{map|1 1 1}} {{map|0 4 9}}}}. But [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|-9 -14 -20}}</span>⟩ + [<span style="color: #3C8031;">{{map|7 11 16}}</span> <span style="color: #B6321C;">{{map|1 2 3}}</span>⟩ also works (in the meantone mapping, we've added one copy of the first vector to the second), giving {{rket|{{map|14 22 32}} {{map|-8 -12 -17}}}} which also canonicalizes to {{rket|{{map|1 1 1}} {{map|0 4 9}}}}; in fact, as long as the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> is explicit and neither matrix is enfactored, the entry-wise addition will work out fine.</ref> | ||
And so we can see that despite immediate appearances, while it seems like we can simply do entry-wise addition on temperaments with more than one <span style="color: #B6321C;">basis vector not in common</span>, this does not give us reliable results per temperament. | And so we can see that despite immediate appearances, while it seems like we can simply do entry-wise addition on temperaments with more than one <span style="color: #B6321C;">basis vector not in common</span>, this does not give us reliable results per temperament. | ||
| Line 1,264: | Line 1,264: | ||
| colspan="1" rowspan="5" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | | colspan="1" rowspan="5" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
[{{vector|<math>a</math> <math>b</math> <math>c</math>}}] | |||
! rowspan="5" | | ! rowspan="5" | | ||
| style="background-color: #BED5BA;"|<math>a</math> | | style="background-color: #BED5BA;"|<math>a</math> | ||
| Line 1,476: | Line 1,476: | ||
| colspan="1" rowspan="7" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | | colspan="1" rowspan="7" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
[{{vector|<math>a</math> <math>b</math> <math>c</math> <math>d</math> <math>e</math>}} | |||
{{vector|<math>f</math> <math>g</math> <math>h</math> <math>i</math> <math>j</math>}}] | {{vector|<math>f</math> <math>g</math> <math>h</math> <math>i</math> <math>j</math>}}] | ||
| style="background-color: #BED5BA;"|<math>r_1</math> | | style="background-color: #BED5BA;"|<math>r_1</math> | ||
| Line 1,954: | Line 1,954: | ||
<math>\text{nullity}(\text{union}(C_1, C_2)) = \text{nullity}(\text{ | <math>\text{nullity}(\text{union}(C_1, C_2)) = \text{nullity}(\text{nullspace}(\text{intersection}(M_1, M_2)))</math> | ||
| Line 1,960: | Line 1,960: | ||
<math>\text{rank}(M_1) + \text{rank}(M_2) - \text{rank}(\text{intersection}(M_1, M_2)) - r = \text{nullity}(\text{ | <math>\text{rank}(M_1) + \text{rank}(M_2) - \text{rank}(\text{intersection}(M_1, M_2)) - r = \text{nullity}(\text{nullspace}(\text{intersection}(M_1, M_2))) - n</math> | ||
| Line 1,966: | Line 1,966: | ||
<math>\text{nullity}(\text{ | <math>\text{nullity}(\text{nullspace}(\text{intersection}(M_1, M_2))) + \text{rank}(\text{intersection}(M_1, M_2)) = d</math> | ||
Now solve Equation D for <math>\text{nullity}(\text{ | Now solve Equation D for <math>\text{nullity}(\text{nullspace}(\text{intersection}(M_1, M_2)))</math>, and substitute that result into Equation B: | ||