Temperament addition: Difference between revisions
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=Visualizing temperament arithmetic= | =Visualizing temperament arithmetic= | ||
[[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[merge]]s | [[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[temperament merging|merge]]s both A and B. A+B and A-B give the sum and difference, respectively.]] | ||
==Versus the wedge product== | ==Versus the wedge product== | ||
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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. | 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 | =Versus temperament merging= | ||
Like [[ | Like [[tempermaent merging]], temperament arithmetic takes temperaments as inputs and finds a new temperament sharing properties of the inputs. And they both can be understood as, in some sense, ''adding'' these input temperaments together. | ||
But there is a big difference between temperament arithmetic and | But there is a big difference between temperament arithmetic and merging. Temperament arithmetic is done using ''entry-wise'' addition (or subtraction), whereas merging is done using ''concatenation''. So the temperament sum of mappings with two rows each is a new mapping that still has exactly two rows, while the other hand, the merging of mappings with two rows each is a new mapping that has a total of four rows<ref>At least, this mapping would have a total of four rows before it is reduced. After reduction, it may end up with only three (or two if you map-merged a temperament with itself for some reason).</ref>. | ||
==The linear dependence connection== | ==The linear dependence connection== | ||
Another connection between temperament arithmetic and | Another connection between temperament arithmetic and merging is that they ''may'' involve checks for linear dependence. | ||
Temperament arithmetic, as stated earlier, always requires addability, which is a more complex property involving linear dependence. | Temperament arithmetic, as stated earlier, always requires addability, which is a more complex property involving linear dependence. | ||
Merging does not ''necessarily'' involve linear dependence. Linear dependence only matters for merging when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]]. | |||
=<math>g_{\text{min}}=1</math>= | =<math>g_{\text{min}}=1</math>= | ||
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For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{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}}. | For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{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 | 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: {{bra|{{vector|1 0 0}} {{vector|0 1 0}}}}, and so that corresponding mapping is {{ket|{{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=== | ||