Douglas Blumeyer's RTT How-To: Difference between revisions

We call such a matrix a '''comma basis'''. The plural of “basis” is “bases”, but pronounced /ˈbeɪ siz/.

Try one. How about the hanson comma, {{monzo|6 5 ₋6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{monzo|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this is technically called “Gaussian elimination”. Feel free to work through any other examples yourself.

=== null-space ===

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But these matrices, the intersections of two of them at once, are another level of specialness. We can do even better with them. There’s actually a mathematical function which when input any one of these matrices will output {{monzo|{{val|19 30 44}}}}, thus demonstrating their equivalence with respect to it.

First, transpose the matrix. That means the first column becomes the first row, the second column becomes the second row, etc.

</math>

Now the null-space function, to take you from {{monzo|{{val|19 30 44}}}} back to the matrix, is pretty much the same thing, but a bit simpler. No need to transpose or reverse. Just start at the augmentation step:

Again, we find ourselves in the position where we must reconcile a strange new representation of an object with an existing one. We already know that meantone can be represented by the vector for the comma it tempers out, {{monzo|-4 4 -1}}. How are these two representations related?

We can work this one out by hand too:

</math>

It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gaussian addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{val|12 19 28}} is exactly what you get from summing the terms of 5-ET {{val|5 8 12}} and 7-ET {{val|7 11 16}}: {{val|5+7 8+11 12+16}} = {{val|12 19 28}}. Cool!

Let’s review what we’ve seen so far. 5-limit JI is 3-dimensional. When we have a rank-3 temperament of 5-limit JI, 0 commas are tempered out. When we have a rank-2 temperament of 5-limit JI, 1 comma is tempered out. When we have a rank-1 temperament of 5-limit JI, 2 commas are tempered out.<ref>Probably, a rank-0 temperament of 5-limit JI would temper 3 commas out. All I can think a rank-0 temperament could be is a single pitch, or in other words, everything is tempered out. So perhaps in some theoretical sense, a comma basis in 5-limit made out of 3 vectors, thus a square 3×3 matrix, as long as none of the lines are parallel, should minimally represent every interval in the space.</ref>

So far, everything we’ve done has been in terms of 5-limit, which has dimensionality of 3. Before we generalize our knowledge upwards, into the 7-limit, let’s take a look at how things one step downwards, in the simpler direction, in the 3-limit, which is only 2-dimensional.

* A way to think about this is we’re losing one generator so now one of the primes is expressed only in terms of the others; When we move by a 2, we lose 4 points for 5; And when we move by a 3 we gain them back

* This is probably the place to finally talk about non-octave periods? Unless that’s forced to come up back in the wedgie section