Meantone & co

Here are four more:

81/80

Map:

[ 0 1]
[-1 2]
[-4 4]

Generators: a = 20.9931/50; b = 1

badness: 108
rms: 4.22
g: 2.944
errors: [-5.79, -1.65, 4.14]

Nothing left to say about this one. :)

2048/2025

Map:

[ 0 2]
[-1 4]
[ 2 3]

Generators: 14.0123/34 (~4/3); b = 1/2

badness: 211
rms: 2.613
g: 4.32
errors: [3.49, 2.79, -.70]

A good way to take advantage of the 34-ets excellent 5-limit harmonies
is two gothish 17-et chains of fifths a sqrt(2) apart.

78732/78125 = 2^2 3^9 5^-7

Map:

[ 0 1]
[ 7 -1]
[ 9 -1]

Generators: 23.9947/65 (~9/7); b = 1

badness: 346
rms: 1.157
g: 6.68
errors: [-1.1, 0.5, 1.6]

393216/390625 = 2^17 3 5^-8

Map:

[ 0 1]
[ 8 -1]
[ 1 2]

Generators: a = 31.9951/99 (~5/4); b = 1
Works with 31,34,65,99,164

badness: 251
rms: 1.072
g: 6.16
error: [.602, 1.506, .904]

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 2048/2025

> A good way to take advantage of the 34-ets excellent 5-limit harmonies
> is two gothish 17-et chains of fifths a sqrt(2) apart.

I should have added that this is the diaschismic temperament.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are four more:
>
> 81/80
>
> Map:
>
> [ 0 1]
> [-1 2]
> [-4 4]
>
> Generators: a = 20.9931/50; b = 1
>
> badness: 108
> rms: 4.22
> g: 2.944
> errors: [-5.79, -1.65, 4.14]
>
> Nothing left to say about this one. :)
>
>
> 2048/2025
>
> Map:
>
> [ 0 2]
> [-1 4]
> [ 2 3]
>
> Generators: 14.0123/34 (~4/3); b = 1/2
>
> badness: 211
> rms: 2.613
> g: 4.32
> errors: [3.49, 2.79, -.70]
>
> A good way to take advantage of the 34-ets excellent 5-limit
harmonies
> is two gothish 17-et chains of fifths a sqrt(2) apart.

This is my interpretation of the Indian scales.

>
>
> 78732/78125 = 2^2 3^9 5^-7
>
> Map:
>
> [ 0 1]
> [ 7 -1]
> [ 9 -1]
>
> Generators: 23.9947/65 (~9/7); b = 1
>
> badness: 346
> rms: 1.157
> g: 6.68
> errors: [-1.1, 0.5, 1.6]

Would you call this one a "unique facet of 65-tET"? Is this the kind
of thing that Graham's searching pairs of ETs is likely to miss?

>
>
> 393216/390625 = 2^17 3 5^-8
>
> Map:
>
> [ 0 1]
> [ 8 -1]
> [ 1 2]
>
> Generators: a = 31.9951/99 (~5/4); b = 1
> Works with 31,34,65,99,164
>
> badness: 251
> rms: 1.072
> g: 6.16
> error: [.602, 1.506, .904]

Magic. Rather than continuing, Gene, could you start over going from
lowest badness to highest badness?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Magic.

Not Magic; another 5/4 system.

Rather than continuing, Gene, could you start over going from
> lowest badness to highest badness?

This way puts similar systems together; why do it the other way?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Magic.
>
> Not Magic; another 5/4 system.
>
> Rather than continuing, Gene, could you start over going from
> > lowest badness to highest badness?
>
> This way puts similar systems together; why do it the other way?

To get a sense of whether I can accept your kind of "flatness" in
this context.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Magic.
>
> Not Magic; another 5/4 system.

Is this better or worse than Magic in the 5-limit?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > This way puts similar systems together; why do it the other way?
>
> To get a sense of whether I can accept your kind of "flatness" in
> this context.

I don't get it--how will mixing the apples with the oranges do that?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > This way puts similar systems together; why do it the other way?
> >
> > To get a sense of whether I can accept your kind of "flatness" in
> > this context.
>
> I don't get it--how will mixing the apples with the oranges do that?

I just want to get the sense that some fixed value of badness is
going to remove unimportant ones while retaining all the important
ones. For example, I'd definitely set tbe badness cutoff high enough
so that meantone, augmented, and diminished, even if they had errors
as large as in 12-tET, make it in.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Would you call this one a "unique facet of 65-tET"? Is this the kind
> of thing that Graham's searching pairs of ETs is likely to miss?

No, he'd be bound to hit it if he did things right.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> No, he'd be bound to hit it if he did things right.

Graham is more likely to include things I wouldn't than vice-versa--his methods can include more than one version of the same system, so that you have for instance meantone, and another meantone with a seemingly useless doubling of the generator steps. There are at least three version of kleismic on his list--one the usual, one with doubled generator steps, and one with a half-ocatave period.

In-Reply-To: <9vsgv8+d425@eGroups.com>
Gene wrote:

> Graham is more likely to include things I wouldn't than vice-versa--his
> methods can include more than one version of the same system, so that
> you have for instance meantone, and another meantone with a seemingly
> useless doubling of the generator steps. There are at least three
> version of kleismic on his list--one the usual, one with doubled
> generator steps, and one with a half-ocatave period.

Yes. That's an interesting, uh, feature of my algorithm for going from
ETs to linear temperaments.

A normal 5-limit kleismic can be generated from 19- and 34-equal:

>>> h19 = temper.PrimeET(19, temper.primes[:2])
>>> h34 = temper.PrimeET(34, temper.primes[:2])
>>> (h19&h34).mapping
[(1, 0), (0, 6), (1, 5)]

The defining wedge product is

>>> (h19^h34).complement().flatten()
(-6, -5, 6)

One example where it goes wrong is 15- and 23-equal

>>> h23 = temper.PrimeET(23, temper.primes[:2])
>>> h15 = temper.PrimeET(15, temper.primes[:2])
>>> (h23&h15).mapping
[(1, 0), (0, 12), (1, 10)]

the wedge product is

>>> (h15^h23).complement().flatten()
(12, 10, -12)

and so simplifies to be equivalent to the above. In this case, it's
obvious from looking at the generator mapping that something's wrong,
because it has a common factor of 2.

The other strange example is 4&34:

>>> h4 = temper.PrimeET(4, temper.primes[:2])
>>> (h4&h34).mapping
[(2, 0), (6, -6), (7, -5)]

for the same reason

>>> (h4^h34).complement().flatten()
(-12, -10, 12)

Here, though, there's nothing obviously wrong with the resulting
temperament.

So are these behaviours a problem? They follow from the generator and
period being chosen ahead of the mapping. That's probably correct,
because this is supposed to be an extension of the Scale Tree(s). They
won't usually get to the top of the list, because they have a higher
complexity than the equivalents. But it does mean temperaments will be
underrated if they appear in this way. I'm hoping that any set of ETs
containing a pair with torsion will also contain a pair without torsion.
Also that it'll be far less common in the higher limits. If not, we can
always get to those temperaments by using wedge products. With the
current implementation, at least, I expect that would slow things down.

Graham

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > No, he'd be bound to hit it if he did things right.
>
> Graham is more likely to include things I wouldn't than vice-versa--
>his methods can include more than one version of the same system, so
>that you have for instance meantone, and another meantone with a
>seemingly useless doubling of the generator steps. There are at
>least three version of kleismic on his list--one the usual, one with
>doubled generator steps, and one with a half-ocatave period.

Aren't these all valid systems to consider? I wouldn't want to leave
any stone unturned.

--- In tuning-math@y..., graham@m... wrote:

> Yes. That's an interesting, uh, feature of my algorithm for going from
> ETs to linear temperaments.

If you want, you can get rid of it by implimenting torsion extermination.

> >>> (h15^h23).complement().flatten()
> (12, 10, -12)
>
> and so simplifies to be equivalent to the above. In this case, it's
> obvious from looking at the generator mapping that something's wrong,
> because it has a common factor of 2.

I would simply divide out by the gcd.

> >>> (h4^h34).complement().flatten()
> (-12, -10, 12)
>
> Here, though, there's nothing obviously wrong with the resulting
> temperament.

There's nothing obviously right about it either. Again, it could be eliminated by dividing by the gcd, and would we really lose anything we want? This sort of thing is exactly analgous to the 24-et version of 5-limit, [24, 38, 56] which started this whole torsion thing. You have a kleisma not in the kernal, but its square *is* in the kernel, if you try to produce a temperament starting from the comma side, and that wouldn't work, just as it won't work to try to get the 24-et from the torsion commas. We therefore have temperaments not classified in terms of wedgies if we allow these.

> So are these behaviours a problem?

We've seen that torsion seems to lead to confusion of one kind or another, and we don't get any positive value from it I can discern.

In-Reply-To: <9vto8g+6cu4@eGroups.com>
Me:
> > Yes. That's an interesting, uh, feature of my algorithm for going
> > from ETs to linear temperaments.

Gene:
> If you want, you can get rid of it by implimenting torsion
> extermination.

Yes, like this:

>>> (h23^h15).complement().linearTemperament().mapping
[(1, 0), (0, 6), (1, 5)]

Me:
> > >>> (h15^h23).complement().flatten()
> > (12, 10, -12)
> > and so simplifies to be equivalent to the above. In this case, it's
> > obvious from looking at the generator mapping that something's wrong,
> > because it has a common factor of 2.

Gene:
> I would simply divide out by the gcd.

But you also said that 15&23 only defines the generator, not the
temperament. So applying both rules means 15&23 and h15&h23 are
melodically incompatible, which looks wrong to me.

> > >>> (h4^h34).complement().flatten()
> > (-12, -10, 12)
> >
> > Here, though, there's nothing obviously wrong with the resulting
> > temperament.
>
> There's nothing obviously right about it either.

What do you mean? It's a linear temperament consistent with 4- and
34-equal in the 5-limit. Exactly what you asked for.

> Again, it could be
> eliminated by dividing by the gcd, and would we really lose anything we
> want?

The mapping is [(2, 0), (6, -6), (7, -5)]. The GCD is 1. So dividing by
it doesn't change anything. You do lose the exact half-octaves, which are
a common property of 4- and 34-equal. So the result wouldn't be what you
asked for.

> This sort of thing is exactly analgous to the 24-et version of
> 5-limit, [24, 38, 56] which started this whole torsion thing.

Well, we're keeping torsion in periodicity blocks, aren't we?

> You have
> a kleisma not in the kernal, but its square *is* in the kernel, if you
> try to produce a temperament starting from the comma side, and that
> wouldn't work, just as it won't work to try to get the 24-et from the
> torsion commas.

We're not going from the comma side, so that's not a problem.

> We therefore have temperaments not classified in terms
> of wedgies if we allow these.

Yes, we do.

Me:
> > So are these behaviours a problem?

Gene:
> We've seen that torsion seems to lead to confusion of one kind or
> another, and we don't get any positive value from it I can discern.

The 24 note periodicity block produced by a comma and diesis is the
traditional 22 shrutis plus two extra notes. That's good enough for me.
The confusion only comes in when you try and generate a temperament.
Where the ETs have torsion, the temperament comes out no problem.
Although it's over-complex, and there'll be confusion if you try to
calculate a periodicity block.

Graham

--- In tuning-math@y..., graham@m... wrote:

> The 24 note periodicity block produced by a comma and diesis is the
> traditional 22 shrutis plus two extra notes. That's good enough
for me.
> The confusion only comes in when you try and generate a
temperament.

Naah . . . I would say it's already confusing beforehand. Why would a
comma squared be considered an equivalence if a comma itself isn't?