Temperament addition: Difference between revisions

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For <math>g_{\text{min}}>1</math> temperaments, temperament arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_.5Bmath.5D.5Cmin.28g.29.3D1.5B.2Fmath.5D|beyond <math>g_{\text{min}}=1</math> section]] later.

= Applications =

[[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 temperaments each defined by 2 vectors may be added as long as the other vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 vectors is able to be added by virtue of all other vectors being the same.]]

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 81/80 × 250/243, and dicot tempers out 81/80 × 243/250.

Similar reasoning is possible for the mapping-rows of mappings ₋ the analogs of the commas of comma bases ₋ but are less intuitive to describe.

Ultimately, this effect is the primary application of temperament arithmetic. With temperament arithmetic, you're essentially never really able to do anything meaningful beyond entry-wise adding a pair of (mono)vectors. What changes from situation to situation is how many other vectors there are alongside, in the vector sets representing the temperament, whether there are 0 other vectors or 2 or 5. As we'll learn later, any other vectors beyond the first ones are always required to be the same between all the summed or differenced temperaments.

== Fokker groups ==

According to some sources, these properties are discussed in terms of "Fokker groups" on this page: [[Fokker block]].

=A note on variance=

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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]]). 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.

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.

=Versus meet and join=

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(Examples WIP)

~~= Applications =~~

The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments. According to some sources, these properties are discussed in terms of "Fokker groups" on this page: [[Fokker block]].

= References =

@@ Line 128: / Line 128: @@
 For <math>g_{\text{min}}>1</math> temperaments, temperament arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_.5Bmath.5D.5Cmin.28g.29.3D1.5B.2Fmath.5D|beyond <math>g_{\text{min}}=1</math> section]] later.
+= Applications =
+[[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 temperaments each defined by 2 vectors may be added as long as the other vectors match. In the third row we see a continued development of this idea, where a pair of temperaments each defined by 3 vectors is able to be added by virtue of all other vectors being the same.]]
+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 81/80 × 250/243, and dicot tempers out 81/80 × 243/250.
+Similar reasoning is possible for the mapping-rows of mappings ₋ the analogs of the commas of comma bases ₋ but are less intuitive to describe.
+Ultimately, this effect is the primary application of temperament arithmetic. With temperament arithmetic, you're essentially never really able to do anything meaningful beyond entry-wise adding a pair of (mono)vectors. What changes from situation to situation is how many other vectors there are alongside, in the vector sets representing the temperament, whether there are 0 other vectors or 2 or 5. As we'll learn later, any other vectors beyond the first ones are always required to be the same between all the summed or differenced temperaments.
+== Fokker groups ==
+According to some sources, these properties are discussed in terms of "Fokker groups" on this page: [[Fokker block]].
 =A note on variance=
@@ Line 163: / Line 179: @@
 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 <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.
+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.
 =Versus meet and join=
@@ Line 738: / Line 754: @@
 (Examples WIP)
-= Applications =
-The temperament that results from summing or diffing two temperaments, as stated above, has similar properties to the original two temperaments. According to some sources, these properties are discussed in terms of "Fokker groups" on this page: [[Fokker block]].
 = References =