Generator form manipulation: Difference between revisions

Now clearly all three of these mapping forms look related, and they are indeed, but the exact relationships between them may not be immediately apparent, or how those relationships correspond to the relationships between their generator sizes. The purpose of this article is to demonstrate tricks for transforming from one matrix form to another so that we can make the generators the sizes we want, and along the way we'll look at how the tricks work in order to explain these relationships.  

The most basic trick is this: for a rank <math>r</math> temperament whose mapping has rows <math>R_1, R_2 ... R_r</math>, and corresponding generators with cents <math>G_1, G_2 ... G_r</math>, if we want to increase <math>G_a</math> by <math>G_b</math>, then replace <math>R_b</math> with <math>R_b' = R_b - R_a</math>.

Conversely, if we want to ''decrease'' <math>G_a</math> by <math>G_b</math>, then we replace <math>R_b</math> with <math>R_b' = R_b</math> ''plus'' <math>R_a</math>.

Now let's demonstrate a slightly more complicated example: transforming the 5-limit meantone mapping from its canonical form which uses a perfect fifth to the form which uses a perfect fourth. Again, we know this is possible because the perfect fourth is the octave complement of the perfect fifth.  

And we're done, having found the mapping {{ket|{{map|1 2 4}} {{map|0 -1 -4}}}}.  

[[File:Generator step by step manipulation.png|thumb|600px|A demonstration of how one might transform the size of a generator of a rank-3 temperament.]]

The fact that these tricks have isolated effects on the generator sizes like this makes it straightforward to compose sequences of them, applied one after the other, to attain an incredible variety of valid generator sizes, as you can see in the diagram to the right.

Using these two tricks, you do not have to worry about enfactoring the mapping, i.e. introducing a common factor in one of the rows. This is because neither of these tricks ever involve replacing a row with a multiple of that row; we always replace a row with a combination of at least one each of two different rows, as in trick 1, or with the row negated, as in trick 2.

In order to define the size of the generators, you need to specify a tuning strategy. Though if you have two generators that are close enough that their size ranking depends on the tuning, then you probably have other problems. In any case, the tuning strategy that we'll be using here is T2, because it's decent enough and easy to compute.

The following table shows how to obtain minimal-generator form from various starting positions, by synthesizing the two generator size manipulation tricks explained in the previous section. In this table, the period <math>p</math> is the first mapping row <math>r_1</math> in cents and the generator <math>g</math> is the second mapping row <math>r_2</math> in cents. It uses the simple example of 5-limit meantone.