More errors and complexities

My treatise on prime-weighted errors and complexities is ever expanding, and might even approach being a real treatise when I'm finished. It's now at 11 pages so as non-12 types you might want to have a look before it expands any further.

http://microtonal.co.uk/primerr.pdf

More introduction now, wedgie complexity, some examples and stuff.

Graham

Graham Breed wrote:
> My treatise on prime-weighted errors and complexities is ever expanding, > and might even approach being a real treatise when I'm finished. It's > now at 11 pages so as non-12 types you might want to have a look before > it expands any further.
> > http://microtonal.co.uk/primerr.pdf
> > More introduction now, wedgie complexity, some examples and stuff.

In equation (5), you introduce u-sub-i (let's say u_i for simplicity) as the absolute value of t_i - h_i, but it appears that to get the correct values for table 2 using equations (9) and (11), u_i needs to be (t_i - h_i) without the absolute value.

Beyond that, I'm not entirely sure that I understand everything, but it looks like the TOP RMS formula for rank-2 temperaments is pretty simple, and even with rank-3 temperaments you only need to invert a 3x3 matrix (if my understanding of the dimensions of these things is correct). An example might be helpful there (meantone for instance, or something else that would go with your 19-ET example).

Herman Miller wrote:
> Graham Breed wrote:
> >>My treatise on prime-weighted errors and complexities is ever expanding, >>and might even approach being a real treatise when I'm finished. It's >>now at 11 pages so as non-12 types you might want to have a look before >>it expands any further.
>>
>>http://microtonal.co.uk/primerr.pdf
>>
>>More introduction now, wedgie complexity, some examples and stuff.
> > > In equation (5), you introduce u-sub-i (let's say u_i for simplicity) as > the absolute value of t_i - h_i, but it appears that to get the correct > values for table 2 using equations (9) and (11), u_i needs to be (t_i - > h_i) without the absolute value.

Um ... yes ... it should be d_i instead of u_i in equation 9.

> Beyond that, I'm not entirely sure that I understand everything, but it > looks like the TOP RMS formula for rank-2 temperaments is pretty simple, > and even with rank-3 temperaments you only need to invert a 3x3 matrix > (if my understanding of the dimensions of these things is correct). An > example might be helpful there (meantone for instance, or something else > that would go with your 19-ET example).

The formulae are simple, and it'd be nice if there were a way of making them simple to understand. As it is, this PDF is getting longer and longer. I will certainly add examples along with a glossary and trying to make sure everything's understandable as you read it in sequence.

Yes, it's a 3x3 matrix you have to invert for rank 3 temperaments. In general, you invert an RxR matrix for a rank R temperament. You can do that by hand if you don't want to use a library, but remember that the library functions for least squares optimizations have less rounding error (or there's a problem with one of my equations).

Graham

"So 8:1 is allowed to be three times as
out of tune as 7:1 when it�s only a little bit bigger!"

Here you're already talking about error being proportionate
to size, without explaining why it should be so.
I'd love to see this expanded:
"The rationale is that prime numbers are weighted according
to the probability that they will occur in composite ratios."

Just under equation 6, I think you mean "weighted error of the
ith prime".

-Carl

Carl Lumma wrote:
> "So 8:1 is allowed to be three times as
> out of tune as 7:1 when it�s only a little bit bigger!"
> > Here you're already talking about error being proportionate
> to size, without explaining why it should be so.
> I'd love to see this expanded:
> "The rationale is that prime numbers are weighted according
> to the probability that they will occur in composite ratios."

No, you've taken the end of the paragraph that tries to explain it:

"""
A prime-based error measure would give an overall error for the temperament by
treating all u_i equally.
This gives unreasonable results in practice.
For example, the intervals 2:1 and 7:1 are counted on an equal footing,
so an error of 1 cent in 2:1 is treated as badly as an error of 1 cent in 7:1.
But an error of 1 cent in 2:1 leads to an error of 3 cents in 8:1.
So 8:1 is allowed to be three times as out of tune as 7:1 when it's only a
little bit bigger!
To give more flexibility, each prime error is given a different weighting when
calculating the overall error for a temperament.
"""

I talk about weighting when I get to weighting.

> Just under equation 6, I think you mean "weighted error of the
> ith prime".

Okay, fixed that.

Graham