I don't understand (was: inverse of matrix --> for what?)

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 3:17 PM
> Subject: [tuning-math] Re: inverse of matrix --> for what?
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > For 5-limit, we will only need two unison vectors to define
> > an ET, in this case 55-tET. One of these unison vectors should
> > of course 81:80, the unison vector that defines meantone.
>
> I got two of the commas on my list--one, of course, 81/80, and
> the other 6442450944/6103515625 = 2^31*3*5^(-14).

Thanks for responding to this, but I'm afraid it's all too cryptic
for me, and I don't understand any of it. I'm sure that you've
discussed much of this in tuning-math posts which went over my
head... if you have links to relevant posts, I'd appreciate it.

Now for the specific questions:

> My badness score for the associated temperament is 6590, but some
> of the other commas do better--in particular, 2^47 3^(-15) 5^(-10)
> scores 1378; which hardly compares with the score of 108 for
> meantone and would not make my best list, where I have a cutoff
> of 500, but it isn't garbage.

What's "badness"?

> The period matrix is
>
> [ 0 5]
> [ -2 11]
> [ 3 7]

?? -- what does this mean?

> and the generators are a = 19.98/65 and b = 1/5;

?? -- Why is the generator not 2^(38/65), which is the closest
thing in 65-EDO to a 3:2? What do these numbers mean?

> it really is more of a 65-et system than a 55-et system, and
> scores as well as it does since it is in much better tune than
> the 55-et itself, with errors:
>
> 3: .317
> 5: .228
> 5/3: -.040

By "much better in tune", you mean that 65-EDO is a better
approximation to the JI ratios than 55-EDO? What is the unit
of measurement for these "errors"? Help!

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> What's "badness"?

In this case I was using the rms error in cents of approximating 3,5, and 5/3 times the cube of the rms of the generator steps to reach these, the higher the number the worse, roughly speaking, and hence "badness".

> > The period matrix is
> >
> > [ 0 5]
> > [ -2 11]
> > [ 3 7]

> ?? -- what does this mean?

It has to do with a linear temperament, and gives how 2, 3 and 5 are represented by it.

>
> > and the generators are a = 19.98/65 and b = 1/5;
>
> ?? -- Why is the generator not 2^(38/65), which is the closest
> thing in 65-EDO to a 3:2? What do these numbers mean?

Because it isn't meantone or anything remotely like it. For one thing, it divides the octave into five equal parts as part of the system, which is why it can only work for ets such as 55 or 65 which are divisible by five.

> By "much better in tune", you mean that 65-EDO is a better
> approximation to the JI ratios than 55-EDO? What is the unit
> of measurement for these "errors"? Help!

Cents; and yes, the 65-et is quite a lot better in tune in the 5-limit than the 55-et. The errors in question were actually those of an optimized tuning of the temperament itself, which could equally well temper 55 as 65 notes.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning-math@y...>
> > Sent: Tuesday, December 18, 2001 3:17 PM
> > Subject: [tuning-math] Re: inverse of matrix --> for what?
> >
> >
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >
> > > For 5-limit, we will only need two unison vectors to define
> > > an ET, in this case 55-tET. One of these unison vectors should
> > > of course 81:80, the unison vector that defines meantone.
> >
> > I got two of the commas on my list--one, of course, 81/80, and
> > the other 6442450944/6103515625 = 2^31*3*5^(-14).
>
> Thanks for responding to this, but I'm afraid it's all too cryptic
> for me, and I don't understand any of it. I'm sure that you've
> discussed much of this in tuning-math posts which went over my
> head... if you have links to relevant posts, I'd appreciate it.

This is not hard. These are the two simplest ratios for defining 55-
tET in the 5-limit. You can't find a simpler pair.

>
> Now for the specific questions:
>
>
> > My badness score for the associated temperament is 6590, but some
> > of the other commas do better--in particular, 2^47 3^(-15) 5^(-10)
> > scores 1378; which hardly compares with the score of 108 for
> > meantone and would not make my best list, where I have a cutoff
> > of 500, but it isn't garbage.
>
>
> What's "badness"?

It's some function that penalizes boththe complexity and the tuning
error of a linear temperament. Not too relevant for you here.

> > The period matrix is
> >
> > [ 0 5]
> > [ -2 11]
> > [ 3 7]
>
>
> ?? -- what does this mean?

This means that

the 2:1 is obtained by combining 0 of generator a and 5 of generator
b.

the 3:1 is obtained by combining -2 of generator a and 11 of
generator b.

the 5:1 is obtained by combining 3 of generator a and 7 of generator
b.

> > and the generators are a = 19.98/65 and b = 1/5;
>
> ?? -- Why is the generator not 2^(38/65), which is the closest
> thing in 65-EDO to a 3:2? What do these numbers mean?

This is a linear temperament with a generator of 2^(19.98/65), and an
interval of repetition of 2^(1/5) (instead of the usual 2).

> > it really is more of a 65-et system than a 55-et system, and
> > scores as well as it does since it is in much better tune than
> > the 55-et itself, with errors:
> >
> > 3: .317
> > 5: .228
> > 5/3: -.040
>
>
> By "much better in tune", you mean that 65-EDO is a better
> approximation to the JI ratios than 55-EDO?

Yes.

> What is the unit
> of measurement for these "errors"?

Cents.