Re: Middle Paths

>
> Paul said :
>
> Perhaps I misunderstood you before, Margo . . . but I don't see it as
> a sufficient condition, either. For example, Pythagorean is eventone,
> but it is a JI tuning system, hence not a "middle path" between JI
> and ET.
>

Hi,

I've still been banging away at Jackys conundrum. The basic JI
thing to be approximated is

1/1 11/10 11/9 11/8 11/7 11/6

and as I interpret "middle path" we have some scale which can
approximate it, perhaps on a few degrees simultaneously, and
maybe has other niceties (being MOS, etc...) without being the
entire et.

Since I obsess about 31 all the time, my "solution" was a tetrachordal
non-MOS scale

2222.23 23 222223

which does feature the sequence at one point. (The sequence in
31et is formed by the step sequence 455674, its position is marked
by the dot).

Since then, I've devolved that solution further. It is still a middle
path, though it is sneaking closer to full 31et. The following 24-note
MOS

2112111 2112111 2112111 211

contains the sequence starting at nine points.

2.1 1.2 1.1 1 2.1 1.2 1 1.1 2.1 1 2 1 1.1 2.1 1

Upon strict MOS transposition (by a sharp in this case)

2.1 1.2 1 1.1 2.1 1 2 1 1.1 2.1 1 2.1 1.2 1.1 1

six of the sequences remain the same and three are replaced with
new keys. This is a sort of interesting behavior akin to
deciding where the endpoints are on a linear tuning. The
core keys in the middle remain the same, but the keys at
the endpoints change. The fact that three change here may be
due to a 3D behavior going on, but I'll let one of the
lattice-sticians deal with that.

Oh... while thinking of parent MOS for the 24-out-of-31
system I bumped into 3333333331 among others, which
is a MIRACLE. This child also has the obvious parent MOS
of 4545445. Some other interesting diatonic parents of
this scale are

3333333334 <- MOS and best 10et in 31
333 444334 \
333 443434 \_*
333 434434 /
3333 44434 /
444 44434 <- MOS and best 8et in 31

* The nine tone scales are all related to the
343434343 MOS. For people into thirds, this scale is
pretty fun. The variations provide some tradeoffs by
adding "in tune" fifths and greater variety in the thirds
at the expense of the amount of septimal minor thirds
and major thirds.

I hope this letter doesn't show up twenty times as I
went through some Yahoo-hell with it last night and am
resending a re-edited version from work.

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> >
> > Paul said :
> >
> > Perhaps I misunderstood you before, Margo . . . but I don't see
it as
> > a sufficient condition, either. For example, Pythagorean is
eventone,
> > but it is a JI tuning system, hence not a "middle path" between
JI
> > and ET.
> >
>
> Hi,
>
> I've still been banging away at Jackys conundrum. The basic JI
> thing to be approximated is
>
> 1/1 11/10 11/9 11/8 11/7 11/6
>
> and as I interpret "middle path" we have some scale which can
> approximate it, perhaps on a few degrees simultaneously, and
> maybe has other niceties (being MOS, etc...) without being the
> entire et.

Or being a subset of an ET.

> Since then, I've devolved that solution further. It is still a
middle
> path, though it is sneaking closer to full 31et. The following 24-
note
> MOS
>
> 2112111 2112111 2112111 211
>
> contains the sequence starting at nine points.
>
> 2.1 1.2 1.1 1 2.1 1.2 1 1.1 2.1 1 2 1 1.1 2.1 1

I don't think this was mentioned in Gene's very recent post on 31! It
seems to be saying that a 9/31-oct. generator has an 11-limit
complexity of only 15. Is this right, Gene?

Paul wrote:

> I don't think this was mentioned in Gene's very recent post on 31! It
> seems to be saying that a 9/31-oct. generator has an 11-limit
> complexity of only 15. Is this right, Gene?

Well, *I* make that a complexity of 19, so worse than meantone. Isn't it
the usual neutral third scale?

Graham

--- In tuning@y..., graham@m... wrote:
> Paul wrote:
>
> > I don't think this was mentioned in Gene's very recent post on
31! It
> > seems to be saying that a 9/31-oct. generator has an 11-limit
> > complexity of only 15. Is this right, Gene?
>
> Well, *I* make that a complexity of 19, so worse than meantone.
Isn't it
> the usual neutral third scale?

I guess it should be -- but since Robert Valentine seemed to be
saying that the utonal hexad appeared in nine positions in the 24-
tone MOS, I reckoned a complexity of 15. Maybe R.V. counted wrong?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., graham@m... wrote:

> > Well, *I* make that a complexity of 19, so worse than meantone.
> Isn't it
> > the usual neutral third scale?

> I guess it should be -- but since Robert Valentine seemed to be
> saying that the utonal hexad appeared in nine positions in the 24-
> tone MOS, I reckoned a complexity of 15. Maybe R.V. counted wrong?

This is 24+7, and it inherits the not-that-good 5 and worse 7 of the
dreaded 24-et. However, it has a good 3; the mapping I get is
2,8,-11,5,-1, which would mean complexities of 2,8,19,19,19,19,21 up
to the 15-limit. I didn't talk about it because it seemed to me I
should cut the discussion off at some point. The generator, about
16/13, is a neutral third, two of which give us a fifth, and we have
scales 5454544 and 1441441444.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., graham@m... wrote:
>
> > > Well, *I* make that a complexity of 19, so worse than
meantone.
> > Isn't it
> > > the usual neutral third scale?
>
> > I guess it should be -- but since Robert Valentine seemed to be
> > saying that the utonal hexad appeared in nine positions in the 24-
> > tone MOS, I reckoned a complexity of 15. Maybe R.V. counted wrong?
>
> This is 24+7, and it inherits the not-that-good 5 and worse 7 of
the
> dreaded 24-et.

Not sure what you mean -- it exploits the stunningly good 5th
harmonic and 7th harmonic of 31-tET, at least the way Robert V. was
using it. Rather than inheriting the deficiencies of its one parent,
24-tET, these deficiencies are nicely canceled out by the
deficiencies of the other parent, 7-tET . . . wouldn't you agree,
Gene?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > This is 24+7, and it inherits the not-that-good 5 and worse 7 of
> the
> > dreaded 24-et.

> Not sure what you mean -- it exploits the stunningly good 5th
> harmonic and 7th harmonic of 31-tET, at least the way Robert V. was
> using it. Rather than inheriting the deficiencies of its one
parent,
> 24-tET, these deficiencies are nicely canceled out by the
> deficiencies of the other parent, 7-tET . . . wouldn't you agree,
> Gene?

That's how it works for intonation, but not how it works for
generators, where the errors, now mostly in the same direction,
reinforce. Here's a table of errors in relative cents for 24, 7 and
their sum, 31, for the first three odd primes:

-4 -9 -13
27 -25 2
-38 35 -3

The rc for 31 is the sum of the values for 24 and 7, and the errors
tend to cancel, reflecting the fact that 31 is a good et.

Here's a similar table; now the first column is 7 times the relative
error for 24, the second is -24 times the relative error for 7, and
the third column is again their sum:

-27 227 200
192 608 800
-264 -836 -1100

Whereas before they cancelled, now they must reinforce by the very
fact that they previously cancelled; the result is that the badness
of 24, and the complimentary badness of 7, shows up in the table of
generators. While I was putting the blame on 24, you can see that the
contribution of 7 (as is usual for the "black keys" of the pair) is
even larger; however the distinction is not that relevant because of
the fact that they tend to reinforce anyway.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > This is 24+7, and it inherits the not-that-good 5 and worse 7
of
> > the
> > > dreaded 24-et.
>
> > Not sure what you mean -- it exploits the stunningly good 5th
> > harmonic and 7th harmonic of 31-tET, at least the way Robert V.
was
> > using it. Rather than inheriting the deficiencies of its one
> parent,
> > 24-tET, these deficiencies are nicely canceled out by the
> > deficiencies of the other parent, 7-tET . . . wouldn't you agree,
> > Gene?
>
> That's how it works for intonation, but not how it works for
> generators, where the errors, now mostly in the same direction,
> reinforce.

I don't understand this. What does this mean?

> While I was putting the blame on 24, you can see that the
> contribution of 7 (as is usual for the "black keys" of the pair) is
> even larger; however the distinction is not that relevant because
of
> the fact that they tend to reinforce anyway.

Again, I don't understand. I see the errors canceling, not
reinforcing. Please explain.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > That's how it works for intonation, but not how it works for
> > generators, where the errors, now mostly in the same direction,
> > reinforce.

> I don't understand this. What does this mean?

That's what the two tables I presented were intended to show. Here
they are again:

-4 -9 -13
27 -25 2
-38 35 -3

This is relative cents for 24, 7 and 31; we see that the value of
27 rc sharp for the 24's 5 is canceled by the -25 rc flat of the 7's
5, giving us a mere 2 rc sharp in the 31-et. Similarly, the 38 rc
flat of the 24's 7 is canceled by the 35 rc sharp of the 7's 7,
giving us a 7 only 3 rc flat in the 31 et.

Now the second table:

-27 227 200
192 608 800
-264 -836 -1100

The first column is 7 times the previous first column, and the second
column is -24 times the previous second column; the third column is
again the sum. We see that now 24 contributes 192 to the sum, and 7
608--these are in the *same* direction, because the -24 has changed
the sign as well as made the absolute value larger. The sum is 800,
meaning 8 generator steps to get to 5 using the 24+7 generator. The
errors of 24 and 7, rather than cancelling, have reinforced to give a
generator 4 times farther out than that of the 3. Similarly, for 7 we
again have the two values not cancelling, but reinforcing--the -264
of the 24 et, added to the -836 of the 7 et, gives us -1100, or -11
generator steps in the opposite direction to that of 3 and 5, because
the inconsistent direction of the tuning in the 24 and 7 ets has led
to a corresponding inconsisteny in the direction of the generators,
an hence an increased complexity.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > That's how it works for intonation, but not how it works for
> > > generators, where the errors, now mostly in the same direction,
> > > reinforce.
>
> > I don't understand this. What does this mean?
>
> That's what the two tables I presented were intended to show. Here
> they are again:
>
> -4 -9 -13
> 27 -25 2
> -38 35 -3
>
> This is relative cents for 24, 7 and 31; we see that the value of
> 27 rc sharp for the 24's 5 is canceled by the -25 rc flat of the 7's
> 5, giving us a mere 2 rc sharp in the 31-et. Similarly, the 38 rc
> flat of the 24's 7 is canceled by the 35 rc sharp of the 7's 7,
> giving us a 7 only 3 rc flat in the 31 et.

Right -- this is an application of what we've called the "relative
error theorem" since before you joined.

> Now the second table:
>
> -27 227 200
> 192 608 800
> -264 -836 -1100
>
> The first column is 7 times the previous first column, and the
second
> column is -24 times the previous second column; the third column is
> again the sum. We see that now 24 contributes 192 to the sum, and 7
> 608--these are in the *same* direction, because the -24 has changed
> the sign as well as made the absolute value larger. The sum is 800,
> meaning 8 generator steps to get to 5 using the 24+7 generator.

WHOA . . . how'd you get the rabbit out of the hat, Gene? This is
pretty mathemagical!

> The
> errors of 24 and 7, rather than cancelling, have reinforced to give
a
> generator 4 times farther out than that of the 3. Similarly, for 7
we
> again have the two values not cancelling, but reinforcing--the -264
> of the 24 et, added to the -836 of the 7 et, gives us -1100, or -11
> generator steps in the opposite direction to that of 3 and 5,
because
> the inconsistent direction of the tuning in the 24 and 7 ets has
led
> to a corresponding inconsisteny in the direction of the generators,
> an hence an increased complexity.

And this presumably has something to do with the "tendency" or
something like that which you posted a while back, an "angle" for
each ET which had something to do with whether the errors in the
primes were in the same direction or not . . . ?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Right -- this is an application of what we've called the "relative
> error theorem" since before you joined.

It can't be a theorem without at least a statement, so I'm writing it
up and will post it tomorrow on tuning-math when I'm finished.

> WHOA . . . how'd you get the rabbit out of the hat, Gene? This is
> pretty mathemagical!

This is why I didn't finish--I wanted to state and prove a complexity
theorem to go with it.

> And this presumably has something to do with the "tendency" or
> something like that which you posted a while back, an "angle" for
> each ET which had something to do with whether the errors in the
> primes were in the same direction or not . . . ?

That would be one application of the tendency business--of course I
like that mostly because it is black magic involving some rather deep
mathematics which ends up being simple to compute.