Temperament addition: Difference between revisions

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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.

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]]). ~~Essentially,~~ it~~'s the same as saying that all the~~ vectors representing the ~~temperaments being~~ summed or differenced must match except for one vector in each. ~~But~~ we can ~~at least say here~~ that ~~any~~ set of <math>g_{\text{min}}=1</math> temperaments are addable<ref>or they are all the same temperament, in which case they <span style="color: #3C8031;">share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</span></ref>, ~~fortunately~~, so we don't need to worry about it in that case.

[[File:Addability.png|300px|thumb|left|In the first row, we see the sum of two vectors. In the second row, we see how a pair of temperaments each defined by 2 basis vectors may be added as long as the other basis vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 basis vectors is able to be added by virtue of all other basis vectors being the same.]]

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, and so a detailed discussion of it will be deferred to a later section (here: [[Temperament arithmetic#Addability]]). But let us at least say here what it essentially means. The basis vectors representing the summed or differenced temperaments ''must all match, except for one non-matching vector in each''. Said another way, any number of matching vectors is allowed in the basis alongside, but ultimately we're only ever able to perform arithmetic on (mono)vectors — the single non-matching vectors from each temperament.

We can gain some intuition about this addability condition by thinking about these non-matching vectors — the ones that are changing — as if they were themselves a basis for a temperament, and then recalling what we know about bases: that when a basis consists of two or more vectors, then an infinitude of other bases for the same subspace exist (such as how there are multiple forms for a rank-2 temperament mapping); whereas when a basis consists of only a single vector, then there is only one possible basis. Finally, we must recognize that entry-wise matrix arithmetic is an operation defined on matrices, not bases; and so entry-wise matrix arithmetic can give different results when performed on different bases for the same subspace. The only way for temperament arithmetic to work reliably, therefore, is to only perform it on matrices where the basis for what is changing has only a single possible representation, and that is only the case when only one basis vector is changing.

Any set of <math>g_{\text{min}}=1</math> temperaments are addable<ref>or they are all the same temperament, in which case they <span style="color: #3C8031;">share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</span></ref>, because the side of duality where <math>g=1</math> will satisfy this condition, so we don't need to worry in detail about it in that case. Or in other words, <math>g_{\text{min}}=1</math> temperaments can be represented by monovectors, and we have no problem doing entry-wise arithmetic on those.

=Versus meet and join=

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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.
-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]]). Essentially, it's the same as saying that all the vectors representing the temperaments being summed or differenced must match except for one vector in each. But we can at least say here that any set of <math>g_{\text{min}}=1</math> temperaments are addable<ref>or they are all the same temperament, in which case they <span style="color: #3C8031;">share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</span></ref>, fortunately, so we don't need to worry about it in that case.
+[[File:Addability.png|300px|thumb|left|In the first row, we see the sum of two vectors. In the second row, we see how a pair of temperaments each defined by 2 basis vectors may be added as long as the other basis vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 basis vectors is able to be added by virtue of all other basis vectors being the same.]]
+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, and so a detailed discussion of it will be deferred to a later section (here: [[Temperament arithmetic#Addability]]). But let us at least say here what it essentially means. The basis vectors representing the summed or differenced temperaments ''must all match, except for one non-matching vector in each''. Said another way, any number of matching vectors is allowed in the basis alongside, but ultimately we're only ever able to perform arithmetic on (mono)vectors — the single non-matching vectors from each temperament.
+We can gain some intuition about this addability condition by thinking about these non-matching vectors — the ones that are changing — as if they were themselves a basis for a temperament, and then recalling what we know about bases: that when a basis consists of two or more vectors, then an infinitude of other bases for the same subspace exist (such as how there are multiple forms for a rank-2 temperament mapping); whereas when a basis consists of only a single vector, then there is only one possible basis. Finally, we must recognize that entry-wise matrix arithmetic is an operation defined on matrices, not bases; and so entry-wise matrix arithmetic can give different results when performed on different bases for the same subspace. The only way for temperament arithmetic to work reliably, therefore, is to only perform it on matrices where the basis for what is changing has only a single possible representation, and that is only the case when only one basis vector is changing.
+Any set of <math>g_{\text{min}}=1</math> temperaments are addable<ref>or they are all the same temperament, in which case they <span style="color: #3C8031;">share all the same basis vectors and could perhaps be said to be ''completely'' linearly dependent.</span></ref>, because the side of duality where <math>g=1</math> will satisfy this condition, so we don't need to worry in detail about it in that case. Or in other words, <math>g_{\text{min}}=1</math> temperaments can be represented by monovectors, and we have no problem doing entry-wise arithmetic on those.
 =Versus meet and join=