Yahoo Tuning Groups Ultimate Backup tuning two-dimensional three-term BP scales

The Bohlen-Pierce Lambda scale is:

1/1 25/21 9/7 7/5 5/3 9/5 15/7 7/3 25/9 3/1

0 302 435 583 884 1018 1319 1467 1769 1902
0 133 281 583 716 1018 1165 1467 1600 1902
0 147 449 583 884 1032 1334 1467 1769 1902
0 302 435 737 884 1186 1319 1621 1755 1902
0 133 435 583 884 1018 1319 1453 1600 1902
0 302 449 751 884 1186 1319 1467 1769 1902
0 147 449 583 884 1018 1165 1467 1600 1902
0 302 435 737 870 1018 1319 1453 1755 1902
0 133 435 568 716 1018 1151 1453 1600 1902

Generalized this as a [5,4] scale (where P = 1:3) allows for some
interesting weighted Tribonacci interpretations.

Here's a Phi weighted Lambda where the two-term index is converted
into a three-term index (the examples are given in the nearest BP
Lambda arrangement analogous to a two-dimensional lattice, not to the
nearest odd numbered identity):

0--1468--1034---600
/ \ / \ / \ / \
/ \ / \ / \ / \
332--1799--1365---931---497

0 332 497 600 931 1034 1365 1468 1799 1902
0 166 268 600 702 1034 1136 1468 1570 1902
0 102 434 537 868 971 1302 1405 1736 1902
0 332 434 766 868 1200 1302 1634 1799 1902
0 102 434 537 868 971 1302 1468 1570 1902
0 332 434 766 868 1200 1365 1468 1799 1902
0 102 434 537 868 1034 1136 1468 1570 1902
0 332 434 766 931 1034 1365 1468 1799 1902
0 102 434 600 702 1034 1136 1468 1570 1902

Here's a sqrt(2)+1 weighted conversion:

0--1459--1016---573
/ \ / \ / \ / \
/ \ / \ / \ / \
260--1718--1275---832---389

0 260 389 573 832 1016 1275 1459 1718 1902
0 130 313 573 756 1016 1199 1459 1642 1902
0 184 443 627 886 1070 1329 1513 1772 1902
0 260 443 703 886 1146 1329 1589 1718 1902
0 184 443 627 886 1070 1329 1459 1642 1902
0 260 443 703 886 1146 1275 1459 1718 1902
0 184 443 627 886 1016 1199 1459 1642 1902
0 260 443 703 832 1016 1275 1459 1718 1902
0 184 443 573 756 1016 1199 1459 1642 1902

Note the many near JI interrelations -- including many nearly just
5ths and 8ths.

As the three-term conversion is a [1,4,4], the series would be
1,4,4,9,17,30. So the Yasser like Tribonacci Lambda would be the
following 9-out-of-30 scale:

0----23----16-----9
/ \ / \ / \ / \
/ \ / \ / \ / \
4----27----20----13-----6

0 254 380 571 824 1014 1268 1458 1712 1902
0 127 317 571 761 1014 1205 1458 1648 1902
0 190 444 634 888 1078 1331 1522 1775 1902
0 254 444 697 888 1141 1331 1585 1712 1902
0 190 444 634 888 1078 1331 1458 1648 1902
0 254 444 697 888 1141 1268 1458 1712 1902
0 190 444 634 888 1014 1205 1458 1648 1902
0 254 444 697 824 1014 1268 1458 1712 1902
0 190 444 571 761 1014 1205 1458 1648 1902

--Dan Stearns

🔗Todd Wilcox <twilcox@patriot.net>

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, "Todd Wilcox" <twilcox@p...> wrote:

http://www.egroups.com/message/tuning/17371

> D. Stearns wrote:
>
> > The Bohlen-Pierce Lambda scale is:
> <Snip a whole bunch of charts and lattices and such>
>
> So, I've spent most of today trying to find out how to tune a
Sarangi, and this list is fascinating to me, but I'm just a little
behind on some of the lingo. Can anyone out there recommend a good
place to learn how to understand some of the more advanced things
that are discussed here?
>
> Todd Wilcox

Well, of course, Dan Stearns contributions are some of the most
important parts of this list, but I also am still a little foggy as
to the exact derivation of the CHROMATIC BP scale... other than the
thought that it might be "ideosyncratic..."

___________ _____ ____ _
Joseph Pehrson

🔗D.Stearns <STEARNS@CAPECOD.NET>

Todd Wilcox wrote,

<< Can anyone out there recommend a good place to learn how to
understand some of the more advanced things that are discussed here?
>>

For specific info on the Bohlen-Pierce scale see:

<http://members.aol.com/bpsite/index.html>

For general lingo deciphering, try Joe Monzo's "Definitions of Tuning
Terms":

<http://www.ixpres.com/interval/dict/index.htm>

Probably the best advice I could give though is to just be patient and
always ask specific questions if you have any... I know it took me
quite a while to get ion a groove with the list, and some things just
seemed to take time.

--Dan Stearns

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

D. Stearns wrote:

>> The Bohlen-Pierce Lambda scale is:
>><Snip a whole bunch of charts and lattices and such>

Todd wrote,

>this list is fascinating to me, but I'm just a little behind on some of the
>lingo.
>Can anyone out there recommend a good place to learn how to understand some
>of the more advanced things that are discussed here?

>Todd Wilcox

Todd, your best bet at this point is to go through the archives of this list
and do some intensive reading. As for Dan Stearns' stuff, I find his 'lingo'
to be sometimes difficult to understand, even after he patiently explains
it. If anything I've posted (and perhaps I've posted more lingosity than
anyone else on this list) confuses you, I believe I can guarantee you a
patient and comprehensible explanation, starting from square one . . . just
ask . . .

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17388

> Joseph, the _tempered_ chromatic BP scale, or the Pierce scale, 13
equal steps per 3:1, is not "idiosyncratic" at all . . . it's simply
by far the best simple equal division of the 3:1 for approximating
frequency ratios of odd numbers up to 9. Meanwhile, the _JI_
"diatonic" and "chromatic" Bohlen scales can be thought of as Fokker
periodicity blocks when 3:1 rather than 2:1 is used as the interval
of
equivalence . . . all this is explained rather well at
http://members.aol.com/bpsite/scales.html.

Thanks, Paul... I understand this in general. I was just having a
problem with the explanation on that page as to how he specifically
derived the ratios. Kees was helping some, but he, also, seemed a
little mystified by a bit of the Bohlen process...

In any case, I will look at it again... and thanks for the help!!

______ _____ _____ _
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joe Monzo <MONZ@JUNO.COM>

Hi Paul,

Your "Gentle Introduction" is accessible via a link from my
dictionary entry for Periodicity Blocks:

http://www.ixpres.com/interval/dict/pblock.htm

The actual "Gentle Introduction" begins here:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

-monz

---------

On Thu, 11 Jan 2001 20:41:30 -0500 "Paul H. Erlich"
<PErlich@Acadian-Asset.com> writes:
> Joseph Pehrson wrote,
>
> >Thanks, Paul... I understand this in general. I was just having a
> >problem with the explanation on that page as to how he specifically
>
> >derived the ratios.
>
> If you mean his "idiosyncratic" method, then I will defer to Heinz
> himself .
> . . but if you mean the Fokker periodicity block method, you might
> want to
> search for my original posts on the subject . . . I thought Monz had
> put
> them up on his webpage at one point, but I can't find them now . . .
> Monz?

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17395

> Joseph Pehrson wrote,
>
> >Thanks, Paul... I understand this in general. I was just having a
> >problem with the explanation on that page as to how he
specifically
> >derived the ratios.
>

> If you mean his "idiosyncratic" method, then I will defer to Heinz
himself .

That is what I meant... but I see he is posting to our very list!!!

Perhaps there will be further info. later in my reading!

> . . but if you mean the Fokker periodicity block method, you might
want to search for my original posts on the subject . . . I thought
Monz had put them up on his webpage at one point, but I can't find
them now . . . Monz?

Yes, I found them again at:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

The way to "get there from here" is through the "periodicity block"
dictionary entry.

It's a fascinating subject, and I reviewed it again...

Thanks!

________ ______ __ __
Joseph

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Dan,

Thanks for all the details, which I'll work through.

Certainly getting there in terms of understanding what you are
doing!

> Yes a simple proof would be neat, though I can't see any reason why it
> wouldn't always work. (I'd also assume that a scale with an even

In many practical situations, if you try something a dozen or more times
and it always works, you can be pretty sure it is always going to work.

However, in mathematics, there are a number of things that work in
nearly all cases, but fail for a few rather unusual ones. One likes
to know if a method is of that type, or if it always works. Finding
when it fails to work, if it does, often suggests new ideas and
ways of proceeding too.

Sometimes also a method works for anything one can do easily
in practice, but fails for very large numbers (say). In which case,
one wants to know that too.

--------------------------------------------------------------------------

For a very simple, and famous example: Fermat often omitted his proofs,
and also made many conjectures, some of which were later proved.

He conjectured that every number of the type 2^(2^n) + 1 is prime
This is the sequence
1, 3, 5, 17, 257, 65537,...
All these numbers are prime. But the next number in the series is
4294967297

which isn't prime:
4294967297 = 6700417*641

http://www.utm.edu/research/primes/glossary/Fermats.html

This has practical significance. One can show that any polygon with a
prime number of sides which can be constructed by ruler and compass
using the methods described by Euclid - which means, with some very
specific restrictions on how the tools can be used - has 2^(2^k) + 1
sides for some k.

http://www.math.pku.edu.cn/stu/wsxy/sxrjjc/wk/Encyclopedia/contents/ConstructiblePolygon.html

In fact, all of these polygons can be constructed by ruler and compass,
and have been done, even the one with 65537 sides, by one
enthusiastic mathematician in Goettingen.

4294967297 is too large to be of much practical significance for
constructing a polygon with this many sides. But nevertheless
the problem of whether there are any more Fermat primes is
really fascinating to a mathematician. No more have been found,
though one conjecture is that there are infinitely many of them!

See
http://www.math.pku.edu.cn/stu/wsxy/sxrjjc/wk/Encyclopedia/contents/FermatNumber.html
"Eisenstein (1844) proposed as a problem the proof that there are
an infinite number of Fermat primes (Ribenboim 1996, p. 88), but this
has not yet been achieved."

--------------------------------------------------------------------------

The only way to find out if your method is of this type, or one
that always works for all generators and Myhill scales,
and for any scale however large, is to prove it.

This particular problem is a rather interesting and fascinating
one for a mathematician (perhaps too much so!)

It's really nice to see why the Wilson 17 note scale
construction works for 1/1 5/4 3/2, and your conjecture
really clarifies why the alternating generator method works
for some generators, and not for others.

> wouldn't always work. (I'd also assume that a scale with an even
> number of notes can only be trivalent if equally tempered.)

Sorry, what is this second assumption exactly? What do you
mean by equally tempered here?

I'm sure I'll have lot's of questions after working through your
post.

Thanks,

more later...

Robert

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Dan,

Thanks for all the details, which I'll work through.

Certainly getting there in terms of understanding what you are
doing!

> Yes a simple proof would be neat, though I can't see any reason why it
> wouldn't always work. (I'd also assume that a scale with an even

In many practical situations, if you try something a dozen or more times
and it always works, you can be pretty sure it is always going to work.

Sometimes also a method works for anything one can do easily
in practice, but fails for very large numbers (say). In which case,
one wants to know that too.

--------------------------------------------------------------------------

For a very simple, and famous example: Fermat often omitted his proofs,
and also made many conjectures, some of which were later proved.

He conjectured that every number of the type 2^(2^n) + 1 is prime
This is the sequence
1, 3, 5, 17, 257, 65537,...
All these numbers are prime. But the next number in the series is
4294967297

which isn't prime:
4294967297 = 6700417*641

http://www.utm.edu/research/primes/glossary/Fermats.html

http://www.math.pku.edu.cn/stu/wsxy/sxrjjc/wk/Encyclopedia/contents/ConstructiblePolygon.html

In fact, all of these polygons can be constructed by ruler and compass,
and have been done, even the one with 65537 sides, by one
enthusiastic mathematician in Goettingen.

--------------------------------------------------------------------------

The only way to find out if your method is of this type, or one
that always works for all generators and Myhill scales,
and for any scale however large, is to prove it.

This particular problem is a rather interesting and fascinating
one for a mathematician (perhaps too much so!)

> wouldn't always work. (I'd also assume that a scale with an even
> number of notes can only be trivalent if equally tempered.)

Sorry, what is this second assumption exactly? What do you
mean by equally tempered here?

I'm sure I'll have lot's of questions after working through your
post.

Thanks,

more later...

Robert

🔗D.Stearns <STEARNS@CAPECOD.NET>

Robert Walker wrote,

<< Sorry, what is this second assumption exactly? What do you mean by
equally tempered here? >>

Just that a scale with an equal number of notes can only be trivalent
if the scales midpoint utilizes P/2. And this of course can only be
accomplished if the scale is an equal division of P. Further I'd also
think, and this is really little more than a gut feeling type guess,
that a tribonacci series created from a three-term [a,b,c] index where
the 4th term is an even number probably has to have a 6th term that is
an equal number as well to create a trivalent m-out-of-n.

<< I'm sure I'll have lot's of questions after working through your
post. >>

Great, they're sure to help me as well.

--Dan Stearns

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

Hi Dan,

> Just that a scale with an equal number of notes can only be trivalent
> if the scales midpoint utilizes P/2. And this of course can only be

By way of example, here's a trivalent scale with an even number of notes (6)

1/1 49/48 7/6 4/3 49/32 7/4 2/1

Does your remark apply to it, and if so, how?

It's generated using
1/1 4/3 7/4
i.e. generators
4/3 21/16

reduced into octave, which I think in your notation means P = 2 - have I
understood correctly that by periodicity you mean the interval into which
the scale is reduced?

Notes are generated in this order:

0 5 3 1 4 2 (0)
(I.e. 49/48 is the 5th note generated, etc)

Which note is the scale's midpoint? 4/3? What is P/2, and
how does it utilise it.

I must be missing something as I can't make out what you are saying yet.

Robert

🔗D.Stearns <STEARNS@CAPECOD.NET>

Robert Walker wrote,

<< here's a trivalent scale with an even number of notes (6)

1/1 49/48 7/6 4/3 49/32 7/4 2/1

Does your remark apply to it, and if so, how? >>

Hi Robert,

It's not trivalent. Notice that it has only two unique intervals at
the 4th -- 4/3 and 3/2 -- and not three.

<< Which note is the scale's midpoint? 4/3? What is P/2, and how does
it utilise it. >>

As you said by "P" I mean the periodicity or the interval in which the
scale is reduced by. So by P/2 I mean the logarithmic half of that
value. So if P = 1:2, then P/2 = 1:2^(1/2). If P = 1:3, then P/2 =
1:3^(1/2), etc., etc.

So what I was musing before was that this should pretty simply mean
that no rational scale with and equal number of notes can be
trivalent!

So here's my bit of numerical intuition from the last post once again:
A tribonacci series created from a GCD reduced three-term [a,b,c]
index where the 4th term is an even number probably has to have a 6th
term that is an even number as well to create a trivalent m-out-of-n.

Do you see what I'm saying, or where I'm going here now?

Here's an example where I'll alter the transformation from a two-term
index into a three-term index so as to make the above condition
happen...

Using the two- to three-term conversion, a [1,5] index becomes a
[1,1,4] index where abbbbb becomes accbcc. But this much like the
example you gave is not trivalent, and note that 1, 1, 4, 6, 11, 21,
... does not meet the above condition. However, by changing accbcc
into acbbcc you turn the [1,1,4] into a [1,2,3] where an equidistant
m-out-of-n derived from 1, 2, 3, 6, 11, 20, ... gives

0 120 360 540 720 960 1200
0 240 420 600 840 1080 1200
0 180 360 600 840 960 1200
0 180 420 660 780 1020 1200
0 240 480 600 840 1020 1200
0 240 360 600 780 960 1200

And this is trivalent.

Now this would seem to me to solve the problem by introducing the
half-octave to the scales midpoint -- i.e., P/2. But I've yet to
really dig in and look at this to the point where I can spell the rule
right out. However, I do know that it works and that it works in the
generalized sense.

Is any of this any clearer?

thanks,

--Dan Stearns

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

I wrote,

<< Are you quite sure about that, Dan? We've discussed many scales
with a half-octave as one of the generators and yet which are not
subsets of any equal temperament. >>

Dan Stearns wrote,

>No, I'm really not "quite sure", only pretty sure -- but give me a
>counter example of the type your talking about and we'll see what we
>see!

Well, just about any scale from
http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm would do, but
in particular I can bring up the pentachordal decatonic scale using one of
the "paultone fifths" I recently posted about:

0
107.8676
215.7352
384.2648
492.1324
600
707.8676
876.3972
984.2648
1092.1324
(1200)