Temperament addition: Difference between revisions
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== Arithmetic on non-addable temperaments == | == Arithmetic on non-addable temperaments == | ||
=== 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 {{bra|{{vector|4 -4 1 0}} {{vector|13 -10 0 1}}}} and septimal blackwood {{bra|{{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 perform entry-wise arithmetic as if they were: | |||
<math>\left[ \begin{array} {rrr} | |||
4 & 13 \\ | |||
-4 & -10 \\ | |||
1 & 0 \\ | |||
0 & 1 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
-8 & -6 \\ | |||
5 & 2 \\ | |||
0 & 0 \\ | |||
0 & 1 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(4+-8) & (13+-6) \\ | |||
(-4+5) & (-10+2) \\ | |||
(1+0) & (0+0) \\ | |||
(0+0) & (1+1) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
-4 & 7 \\ | |||
1 & -8 \\ | |||
1 & 0 \\ | |||
0 & 2 \\ | |||
\end{array} \right]</math> | |||
And — at first glance — the result may even seem to be what we were looking for: a temperament which tempers out | |||
# 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 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}}. | |||
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. | |||
=== 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: | |||
<math>\left[ \begin{array} {rrr} | |||
1 & -6 \\ | |||
2 & 0 \\ | |||
-3 & 5 \\ | |||
1 & -2 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
2 & -14 \\ | |||
-3 & 0 \\ | |||
0 & 0 \\ | |||
1 & 5 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(1+2) & (-6+-14) \\ | |||
(2+-3) & (0+0) \\ | |||
(-3+0) & (5+0) \\ | |||
(1+1) & (-2+5) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
3 & -20 \\ | |||
-1 & 0 \\ | |||
-3 & 5 \\ | |||
2 & 3 \\ | |||
\end{array} \right]</math> | |||
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 {{bra|{{vector|-4 1 1 0}} {{vector|7 -8 0 2}}}}. This second result {{bra|{{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 {{bra|{{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 === | |||
In fact, we could even take the same sets of commas and merely reorder them to come up with a different result! Here, we'll just switch the order of the two commas in the representation of septimal blackwood: | |||
<math>\left[ \begin{array} {rrr} | |||
4 & 13 \\ | |||
-4 & -10 \\ | |||
1 & 0 \\ | |||
0 & 1 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
-6 & -8 \\ | |||
2 & 5 \\ | |||
0 & 0 \\ | |||
1 & 0 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(4+-6) & (13+-8) \\ | |||
(-4+2) & (-10+5) \\ | |||
(1+0) & (0+0) \\ | |||
(0+1) & (1+0) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
-2 & 5 \\ | |||
-2 & -5 \\ | |||
1 & 0 \\ | |||
1 & 1 \\ | |||
\end{array} \right]</math> | |||
And the canonical form of {{bra|{{vector|-2 -2 1 1}} {{vector|5 -5 0 1}}}} is {{bra|{{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 === | |||
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 {{ket|{{map|1 0 -4 -13}} {{map|0 1 4 10}}}} and septimal blackwood's is {{ket|{{map|5 8 0 14}} {{map|0 0 1 0}}}}. So what temperament do we get by summing these? | |||
<math>\left[ \begin{array} {rrr} | |||
1 & 0 & -4 & -13 \\ | |||
0 & 1 & 4 & 10 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
5 & 8 & 0 & 14 \\ | |||
0 & 0 & 1 & 0 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(1+5) & (0+8) & (-4+0) & (-13+14) \\ | |||
(0+0) & (1+0) & (4+1) & (10+0) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
6 & 8 & -4 & 1 \\ | |||
0 & 1 & 5 & 10 \\ | |||
\end{array} \right]</math> | |||
In order to compare this result directly with our other three results, let's take the dual of this {{ket|{{map|6 8 -4 1}} {{map|0 1 5 10}}}}, which is {{bra|{{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 {{ket|{{map|97 152 220 259}} {{map|-30 -47 -68 -80}}}} + {{ket|{{map|-95 -152 -212 -266}} {{map|30 48 67 84}}}}.</ref> | |||
=== Summary === | |||
Here's the four different results we've found so far: | |||
<math> | |||
\begin{array}{ccc} | |||
\text{canonical} & & \text{alternate} & & \text{reordered canonical} & & \text{other side of duality} \\ | |||
\left[ \begin{array} {rrr} | |||
-4 & 7 \\ | |||
1 & -8 \\ | |||
1 & 0 \\ | |||
0 & 2 \\ | |||
\end{array} \right] & | |||
≠ & | |||
\left[ \begin{array} {rrr} | |||
-49 & -23 \\ | |||
3 & 1 \\ | |||
19 & 8 \\ | |||
0 & 1 \\ | |||
\end{array} \right] & | |||
≠ & | |||
\left[ \begin{array} {rrr} | |||
-7 & 5 \\ | |||
3 & -5 \\ | |||
1 & 0 \\ | |||
0 & 1 \\ | |||
\end{array} \right] & | |||
≠ & | |||
\left[ \begin{array} {rrr} | |||
22 & 41 \\ | |||
-15 & -30 \\ | |||
3 & 2 \\ | |||
0 & 2 \\ | |||
\end{array} \right] | |||
\end{array} | |||
</math> | |||
What we're experiencing here is the effect first discussed in the early section [[Temperament arithmetic#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 perform it 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>⟩ = {{ket|{{map|14 22 32}} {{map|-1 -1 -1}}}} which canonicalizes to {{ket|{{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 {{ket|{{map|14 22 32}} {{map|-8 -12 -17}}}} which also canonicalizes to {{ket|{{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 arithmetic will work out fine.</ref> | |||
And so we can see that despite immediate appearances, while it seems like we can simply perform entry-wise arithmetic 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. | |||
=== How it looks with multivectors === | |||
We've now observed the outcome when adding non-addable temperaments using the matrix approach. It's instructive to observe how it works with multivectors as well. The canonical multicommas for septimal meantone and septimal blackwood are {{multicomma|12 -13 4 10 -4 1}} and {{multicomma|14 0 -8 0 5 0}}, respectively. When we add these, we get {{multicomma|26 -13 -4 10 1 1}}. What temperament is this — does it match with any of the four comma bases we've already found? Let's check by converting it back to matrix form. Oh, wait — we can't. This is what we call a [[decomposability|nondecomposable]] multivector. In other words, there is no set of vectors that could be wedged together to produce this multivector. This is the way that multivectors convey to us that there is no true temperament sum of these two temperaments. | |||
== Further explanations == | == Further explanations == | ||