Generator form manipulation: Difference between revisions

A [[User:Cmloegcmluin/Defactored_canonical_form|canonical mapping form]] is an important standard to have as community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]].

If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches.  

For example, the canonical form of [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|5-limit meantone]] is {{vector|{{map|1 1 0}} {{map|0 1 4}}}}, and form has generators with sizes of approximately an [[octave]] and a [[perfect fifth]], respectively. But any pitch system constructed using an octave and a perfect fifth could also have been constructed using an octave and a [[perfect fourth]], because the perfect fourth is the [[octave complement]] of the perfect fifth. Specifically, any pitch we reached previously with a perfect fifth could be instead reached by going up an octave and down a perfect fourth. So in situations where we're approaching 5-limit meantone as a pitch system constructed by an octave and a perfect fourth, we might prefer to have the mapping in that form, which looks like {{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}.  

As a further example, we might prefer only to use generators that are approximations of primes, so we'd like meantone's mapping in the form where the generators are an octave and a perfect twelfth, or [[tritave]] (3/1). This works for a similar reason: anything we could have reached with a perfect fifth we could also reach by moving up a tritave and down an octave, and that form looks like {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.  

|{{vector|{{map|1 1 0}} {{map|0 1 4}}}}

|{{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}

|{{vector|{{map|1 0 -4}} {{map|0 1 4}}}}

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 section 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.  

== generator size manipulation tricks ==

=== first trick: change one generator by the size of another generator ===

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

Let's demonstrate this trick on the most recent example we looked at: meantone as generated by an octave and a tritave. If we begin with meantone in canonical form, {{vector|{{map|1 1 0}} {{map|0 1 4}}}}, where the generators are an octave and a perfect fifth, then we know we need to increase the second generator by the size of the first generator, because a tritave is equal to a perfect fifth plus an octave. So in terms of our variables, we must change <span><math>G_2</math></span> so that it's <span><math>G_2 + G_1</math></span>. According to our trick, then, we must replace <span><math>R_1</math></span> with <span><math>R_1' = R_1 - R_2</math></span>. And so <span><math>R_1'</math></span> is found like this:

R_1 \\

-R_2 \\

R_1'

And then simply replace <span><math>R_1</math></span> = {{map|1 1 0}} with <span><math>R_1'</math></span> = {{map|1 0 -4}} in the mapping, changing {{vector|{{map|1 1 0}} {{map|0 1 4}}}} to {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.

It may be counterintuitive at first that in order to change the size of a generator we must make a change to a mapping row ''other than the one which corresponds to that generator'' (in this example, we changed the size of of the ''second'' generator by changing the ''first'' row). But there is a way to train our intuition on this effect. Think of it like transferring jobs. Remember that in order to know the size in cents of a generator, we cannot look exclusively at that generator's mapping row out of context; we need to know the entire mapping, because the mapping rows all work together to determine how the temperament works. So we can think of every mapping row like a workforce that outsources some of its work to the other rows; they're an interconnected system of workforces. And so when we subtract one row from another, we're in effect saying that the row being subtracted from is going to do less of the row being subtracted's work. So when we subtract <span><math>R_2</math></span> from <span><math>R_1</math></span>, what we're saying is that whatever work the first generator was doing for the second generator, it will no longer do that work anymore, so the second generator will need to take care of that work itself; and that's why <span><math>G_2</math></span> becomes the size of <span><math>G_2</math></span> plus <span><math>G_1</math></span>.

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

=== second trick: negating a generator ===

Fortunately this second trick is very easy. All we need to do in order to negate <span><math>G_a</math></span> (the generator size, e.g. in cents) is to negate each of the terms of the mapping row <span><math>R_a</math></span>. So, we would simply change <span><math>R_2</math> = </span>{{map|0 1 4}} to <span><math>R_2'</math></span> = {{map|0 -1 -4}}, thereby changing the mapping from {{vector|{{map|1 1 0}} {{map|0 1 4}}}} to {{vector|{{map|1 1 0}} {{map|0 -1 -4}}}}<ref>To extend the workplace analogy (if we really like, though I doubt it's helpful here), we could say that we changed R_2's jobs so that they now do the exact opposite of what they used to do, so if we want them to accomplish the same thing as they used to, we have to have them undo their work.</ref>.  

The second step of achieving a generator with the size of a perfect fourth should be familiar: we need to increase our negative perfect fifth to a perfect fourth, and so we need to add one octave, and therefore we need to change <span><math>R_1</math></span> to be <span><math>R_1 - R_2</math></span>, so we end up with:

R_1 \\

-R_2 \\

R_1'

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

=== beyond rank-2 ===

These two tricks should enable us to attain any valid generator sizes we may wish for a given temperament. And these tricks work for any rank<ref>past 1, anyway; of course, rank 1 temperaments are somewhat inflexible in their single generator's size.</ref>, not only rank 2 like we've looked at thus far with 5-limit meantone examples. To be clear, for a rank <span><math>r</math></span> temperament, adding or subtracting <span><math>R_a</math></span> from another generator will only affect the size of <span><math>G_a</math></span>.  

For example, [[Marvel_family#Marvel|7-limit marvel]]'s canonical form is {{vector|{{map|1 0 0 -5}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}, with generators of an octave, tritave, and [[pentave]] (5/1), in that order. We can change that second generator from a tritave to a perfect fifth by decreasing <span><math>G_2</math></span> by <span><math>G_1</math></span>, which we know by the first trick means we add <span><math>R_2</math></span> to <span><math>R_1</math></span>, producing {{vector|{{map|1 1 0 -3}} {{map|0 1 0 2}} {{map|0 0 1 2}}}}. Helpfully, this trick has no effect on any other generators that were not involved, which in this case is just the size of the pentave, which was the one remaining generator out of the three in this temperament.

=== avoiding enfactoring ===

To avoid enfactoring the mapping, i.e. introducing a common factor in one of the rows, you simply avoid ever replacing a row with a multiple of that row. You 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.  

== mingen form ==

A number of generator size targets have been described, such as [[UDP#Generalizing_to_Arbitrary_MOS.27s:_Bright_and_Dark_Generators_.28Chroma-Positive_and_Chroma-Negative.29|chroma-positive generators]], "musician's form", and probably more will be described in the future<ref>Such as forms using [https://mathworld.wolfram.com/LLLAlgorithm.html LLL] to get "a simple set of generators", per a post by Graham Breed on Facebook [https://bit.ly/3AhDYP8 here].</ref>. The generator size target we'll be demonstrating here may not have been rigorously described elsewhere, or named<ref>It is the form used by [[Graham Breed]]'s regular temperament finding scripts hosted here: http://x31eq.com/temper/net.html, at least for rank-2 temperaments.</ref>, but these purposes, we'll be calling it minimum-generator, or mingen, for short. Mingen form is when each successive generator is less than half the size of the previous generator.<ref>Like octave reduction combined with octave inversion, since you can't just add or subtract half octaves until it's between 0 and 600 cents. You have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.</ref><ref>You could always find a smaller and smaller generator by going negative, so this assumes positive generators.</ref>

 

 

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

!<span><math>p</math></span> (¢)

!<span><math>g</math></span> (¢)

!current <span><math>p</math></span> vs. <span><math>g</math></span>

!desired new <span><math>g</math></span>

!required <span><math>r_1</math></span> change

!required <span><math>r_2</math></span> change

|{{vector|{{map|1 0 -4}} {{map|0 -1 -4}}}}

|g < −p

|g + p

|r₁ − 2r₂

|{{vector|{{map|1 1 0}} {{map|0 -1 -4}}}}

|−p <= g < −p/2

|p + g

|r₁ − r₂

|{{vector|{{map|1 2 4}} {{map|0 1 4}}}}

|−p/2 <= g < 0

|−g

|−r₂

|{{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}

|0 <= g <= p/2

|g

|{{vector|{{map|1 1 0}} {{map|0 1 4}}}}

|p/2 < g <= p

|p - g

|r₁ + r₂

|−r₂

|{{vector|{{map|1 0 -4}} {{map|0 1 4}}}}

|p < g

|g - p

|r₁ + 2r₂

|−r₂

=== Wolfram Language implementation ===

The below code essentially works through the input matrix M two rows at a time, beginning with the first two rows. Each pair of rows is "fixed" so that the second row is less than half of the first row. The same set of changes and potential recursions as described in the table in the previous section is used for each pair of rows. When the sizes of the generators is computed, a weighted Frobenius tuning is used for its computational frugality and reasonableness.  

=== limitations ===

 

Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators.<ref>This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.</ref>