Yahoo Tuning Groups Ultimate Backup tuning Kees van Prooijen please!

As I explained in a personal reply to Paul I just didn't have the time yet.
I promise to try to get to it this weekend.

I just received a message from my ISP that they have had a disk crash.
Everything should be OK again now.

http://www.kees.cc/tuning/s235.html
http://www.kees.cc/tuning/s2357.html

For the time being here is the same thing along the lines of the
Pierce-Bohlen scale
http://www.kees.cc/tuning/s357.html

Kees
------------------------------------------
Kees van Prooijen
email: kees@kees.cc
web: http://www.kees.cc

----- Original Message -----
From: Paul H. Erlich <PErlich@Acadian-Asset.com>
To: <tuning@onelist.com>
Cc: <kees@kees.cc>
Sent: Thursday, September 30, 1999 11:36 AM
Subject: [tuning] Kees van Prooijen please!

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Kees van Prooijen wrote,

>For the time being here is the same thing along the lines of the
>Pierce-Bohlen scale
>http://www.kees.cc/tuning/s357.html

This should be of great interest to Heinz Bohlen. Essentially this means
that, if you extend the tritave (3:1)-equivalent (5,7) lattice of notes in a
rather natural way (I'll let Kees make this precise), you won't find a
better approximation of an equal division of the 3:1 with more than 13 notes
until you get to 271 notes. Heinz did in fact conceive his 13-tone scale in
JI (and it's almost certainly a periodicity block in the tritave world), so
this is right up his alley. Kees' results also imply that for the
octave-equivalent (3,5) lattice, 34 is the first to improve on our usual 12,
and for the octave-equivalent (3,5,7), 99 is the first to improve on
Fokker's 31.

🔗Kees van Prooijen <kees@xxxx.xxxx>

Here is something:

http://www.kees.cc/tuning/perbl.html

------------------------------------------
Kees van Prooijen
email: kees@kees.cc
web: http://www.kees.cc

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Kees van Prooijen recently referred us to the results of his search for
> "good" (in what sense he hasn't fully explained) periodicity blocks. The
web
> pages he referred us to have been taken down. I hope this is not
permanent!
> I, for one, look forward to understanding what he did.
>
> -Paul
>
> > You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.
>
>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

From http://www.kees.cc/tuning/perbl.html:

>My interpretation of good periodicity blocks is not only that they do not
contain smaller intervals than the >defining unison intervals, but also that
they do not contain intervals only expressible in larger integers then
>those unison intervals.

That makes perfect sense.

>A measure of this 'expressibility' of a 5-limit interval (m,n) can be
defined as:

>m*log(3) + n*log(5)

Before I get to other objections, wouldn't that mean that, if m and n have
opposite signs, the 3-term and the 5-term could cancel one another out,
leaving a very complex interval with a very small measure?

Or did you really mean

|m|*log(3) + |n|*log(5)?

(which would be an octave-equivalent version of Tenney's harmonic distance;
I've explained how Tenney's formulation makes sense when you include factors
of 2 but doesn't when you ignore them)

Your picture indicates that you are more clever than either of those
alternatives -- in fact your picture implies that the region of interest,
when translated to my triangular lattice, is effectively a hexagon, with all
vertices equidistant from the center! In other words, this agrees perfectly
with the metric I advocate when the 5-limit is specified as the standard of
consonance! So I think you have a good intuition for a complexity measure
here but failed to explain it correctly. Can we try again?

>'1:3 tritave' (is that your term Paul?)

Pierce called the 1:3 of the Bohlen-Pierce scale a 'tritave'.

>To extend this in the third dimension for 7-limit in the octave is slightly
more complicated.

It seems, once you get the logarithmic relation right, your region of
interest would effectively be a distorted cuboctahedron, with all vertices
equidistant from the center. Again, this would agree perfectly with my
7-limit metric.

>One of the reasons that a number of well-known possibilities does not occur
in this list is the following: there >is no account of the fact that, when
the two errors, before they are taken to be absolute values, have equal
>signs, they compensate each other partially for the representation of the
major sixth 3:5. If we let this play >along in the combined error
calculation as follows, we get:

> f3 + log(3)/log(5)*f5 + log(3)/log(5)*f(5/3)

Considering the major sixth with equal weight as the major third seems
totally consistent with the pictures you drew, while not considering it at
all would be consistent with looking at rectangular regions in the lattice.
Did you notice how close these results, and the 7-limit equivalents, are to
the results of your periodicity block search -- while the results which
don't consider the major sixth are much less close? Are your "f" errors
measured in terms of the step size of the tuning? What if you used absolute
error size? And how about, instead of adding the three errors, taking the
sum-of-squares, or just taking the maximum?

This is very exciting stuff (to me) -- I hope Paul Hahn is also able to
participate too.

By the way, Kees14.png does not show up on my browser.

🔗Kees van Prooijen <kees@xxxx.xxxx>

From: Paul H. Erlich <PErlich@Acadian-Asset.com>

>
> From http://www.kees.cc/tuning/perbl.html:
>
> >A measure of this 'expressibility' of a 5-limit interval (m,n) can be
> defined as:
>
> >m*log(3) + n*log(5)
>
> Before I get to other objections, wouldn't that mean that, if m and n have
> opposite signs, the 3-term and the 5-term could cancel one another out,
> leaving a very complex interval with a very small measure?
>
> Or did you really mean
>
> |m|*log(3) + |n|*log(5)?
>
> (which would be an octave-equivalent version of Tenney's harmonic
distance;
> I've explained how Tenney's formulation makes sense when you include
factors
> of 2 but doesn't when you ignore them)
>

Of course you're right that I was wrong. I was to hasty writing that down.
It should be:

|m|*log(3) + |n|*log(5) when the signs of m and n are equal
max( |m|*log(3), |n|*log(5) ) when they are opposite.

Equivalently in higher prime limit cases, add the terms with eqal signs and
take the max of the two results.

> Are your "f" errors
> measured in terms of the step size of the tuning?

Yes, they are.

> What if you used absolute error size?
Absolute error size as the measure for the fitness of an ET is
scientifically, ethically and morally obscene and I refuse to discuss that
:-)

> And how about, instead of adding the three errors, taking the
> sum-of-squares, or just taking the maximum?
Those are reasonable suggestions. To be honest, I don't even remember what I
actually used as a norm. You must realize I am digging all this up from at
least 15 years ago.

> By the way, Kees14.png does not show up on my browser.
You didn't miss miss much. Do you have an old browser (you can tell me
privately, it's a bit OT)
I guess I'll play safe and transform those png's into jpeg's

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Kees wrote,

>Of course you're right that I was wrong. I was to hasty writing that down.
>It should be:

>|m|*log(3) + |n|*log(5) when the signs of m and n are equal
>max( |m|*log(3), |n|*log(5) ) when they are opposite.

>Equivalently in higher prime limit cases, add the terms with eqal signs and
>take the max of the two results.

Good. Then what you're really using as your measure is the odd limit of the
ratio (more precisely, if x is the exponential of your measure, then the
ratio is what Partch would call a "ratio of x"). Good for you!

>> What if you used absolute error size?
>Absolute error size as the measure for the fitness of an ET is
>scientifically, ethically and morally obscene and I refuse to discuss that
>:-)

Well, the smiley would indicate you're joking, but if you're serious, I
totally disagree. Sure, one wants to penalize higher ETs for their
complexity, but it is up to the musician to decide on the right way of doing
that, and making it directly proportional to the number of notes is purely
arbitrary.

Now, Kees, would you explain your algorithm in a little more detail, such as
why you list 25:27 when it is larger than two of the unison vectors already
found? Finding it doesn't change the list of two best UVs, which is all you
need for the 5-limit, right?

Thanks,
Paul

🔗Kees van Prooijen <kees@xxxx.xxxx>

----- Original Message -----
From: Paul H. Erlich <PErlich@Acadian-Asset.com>
To: <tuning@onelist.com>
Sent: Tuesday, October 05, 1999 9:42 AM
Subject: RE: [tuning] Kees van Prooijen please!

> Well, the smiley would indicate you're joking, but if you're serious, I
> totally disagree. Sure, one wants to penalize higher ETs for their
> complexity, but it is up to the musician to decide on the right way of
doing
> that, and making it directly proportional to the number of notes is purely
> arbitrary.

Let's say I'm partly serious. I certainly don't agree that making an error
measure relative to the step size is arbitrary. For instance it's provable
that it's exactly this error that is minimized by using continued fractions
to create ET's for a single limit scale. See:

Prooijen, Kees van. "A Theory of Equal-Tempered Scales", Interface vol. 7
no. 1, June 1978, pp. 45-56.

>
> Now, Kees, would you explain your algorithm in a little more detail, such
as
> why you list 25:27 when it is larger than two of the unison vectors
already
> found? Finding it doesn't change the list of two best UVs, which is all
you
> need for the 5-limit, right?
Certainly you're right and that's a good question and I don't really know
why I did that. Looking into the code, I kept the three smallest intervals,
only taking the two smallest into account for the periodicity blocks. I
probably wanted to keep track of the intervals that just fell out, being
'almost interesting'.

Paul, thanks for your interest in my archeological efforts.

Kees

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>Let's say I'm partly serious. I certainly don't agree that making an error
>measure relative to the step size is arbitrary. For instance it's provable
>that it's exactly this error that is minimized by using continued fractions
>to create ET's for a single limit scale.

Of course, but a mathematical procedure doesn't justify a musical result; it
has to be the other way around.

Now, let me go back to a post I made on Wed 9/15/99 with title: "RE: Fokker
periodicity blocks from the 3-5-7-harmonic lattice", which was before you
joined in I believe. There were a couple of typos in the post, but it
appears I was trying to do pretty much the same thing that you were trying
to do in the 3-D lattice. I wrote, "If we start with the simplest (defined
by distance on the triangular lattice) unison vectors, and then replace the
largest one with the next simplest, and repeat the process . . ." I got the
list of unison vectors from Fokker's paper, so the overall list doesn't
conform to the boundary shape we agree should be used for any given level of
complexity, but that only becomes an issue at the inner and outer edges. I
totally screwed up, though, because the list I used was truncated to include
UVs smaller than 35 cents and I started with the largest rather than
simplest ones, or something like that. Towards the end, my results start to
agree with yours: here is the end of the list of unison vectors (with ratios
translated from matrix notation) and scale sizes that I posted:

| 1 -3 -2|
| 3 -1 -3| = 0
| 2 2 -1|

(6144:6125, 1728:1715, 225:224)

| 1 -3 -2|
| 1 0 3| = 31
| 2 2 -1|

(6144:6125, 1029:1024, 225:224)

| 1 -3 -2|
|-1 -2 4| = 31
| 2 2 -1|

(6144:6125, 2401:2400, 225:224)

| 1 -3 -2|
|-1 -2 4| = 0
| 0 -5 2|

(6144:6125, 2401:2400, 3136:3125)

| 1 -3 -2|
|-1 -2 4| = 99
|-7 4 1|

(6144:6125, 2401:2400, 4375:4374)

| 8 1 0|
|-1 -2 4| = 171
|-7 4 1|

(32805:32768, 2401:2400, 4375:4374)

Anyway, let me ask you this: Considering the 5-limit case for now, how do
you know that there won't be an interval in the scale that is smaller than a
unison vector? Even if the unison vectors both are the smallest within the
half-hexagonal "boundary", the parallogram defined by those vectors may
extend beyond the boundary, and may happen to include one or more smaller
unison vectors. (Flipping the sign of one of the unison vectors will lead to
a parallelogram that extends over a different part of the boundary, and
again one or more unison vectors may lie in that region.) Isn't that so?