Re: 17, 23, 41 and 353-limit scales

Hi Dave,

Notice anything just a little bit odd about this very 353 limit scale?
107/101, 55/49, 201/169, 223/177, 263/197, 239/169, 295/197, 227/143, 301/179, 351/197,
353/187, 399/199,

N.B. necessarily has to have a detuned octave!

This one should be scheduled for prime time listening
1/1 107/101 257/229 151/127 349/277 311/233 41/29 421/281 173/109 523/311
547/307 421/223 3947/1973
or perhaps it is already, near enough anyway.

What about this 23 limit scale, which has gothic antecedants?

1/1, 23/22, 85/76, 55/46, 5/4, 91/68, 95/68, 136/91, 25/16, 117/70, 161/90, 144/77, 2/1

Here's a 17 limit scale that should sound very sweet
1/1, 273/256, 9/8, 832/693, 1024/819, 4/3, 128/91, 3/2, 1331/832, 693/416, 416/231,
512/273, 2/1,

and something very conspicuously missing from these 41 limit 12 and 17 tone scales

12-tone
1/1, 475/451, 512/455, 493/416, 529/418, 589/442, 289/203, 740/493, 779/493, 832/493,
697/392, 902/475, 2/1,

17-tone
1/1, 475/451, 91/82, 512/455, 493/416, 437/350, 529/418, 589/442, 406/289, 37/25, 740/493,
779/493, 308/185, 832/493, 697/392, 560/299, 221/112, 2/1,
(clue: They have very nice fifths!)

Here is an 11-limit 19 tone scale
1/1, 28/27, 15/14, 49/44, 81/70, 6/5, 56/45, 128/99, 75/56, 25/18, 121/84, 121/81, 99/64,
45/28, 5/3, 140/81, 88/49, 28/15, 27/14, 2/1,

and an 11-limit 31 tone scale
1/1, 45/44, 22/21, 77/72, 35/32, 28/25, 8/7, 90/77, 176/147, 11/9, 5/4, 225/176, 98/75,
147/110, 175/128, 7/5, 10/7, 256/175, 220/147, 75/49, 25/16, 8/5, 18/11, 147/88, 77/45,
7/4, 315/176, 64/35, 144/77, 375/196, 88/45, 2/1,

Actually, I've done a little javascript page which happens to be very good at making them.
Follows on from a suggestion Dan made, which turned out to be surprisingly easy to do and
reasonably efficient.

More of that in a minute, when it's finished.

Robert

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> Hi Dave,
>
> Notice anything just a little bit odd about this very 353 limit
scale?
> 107/101, 55/49, 201/169, 223/177, 263/197, 239/169, 295/197,
227/143, 301/179, 351/197,
> 353/187, 399/199,
>
> N.B. necessarily has to have a detuned octave!

I see that, except for the last note, it _is_ 12-tET, since no
interval is more than 1.2 cent different from a theoretically perfect
12-tET. But with the last note it goes up to 5.3 c.

It's "odd" because no factor is even, which explains the necc. detuned
octave.

I see you checked out my "19-limit" scale. No interval off by more
than 4 c and only using primes 2,3,5,17,19 with 5,17,19 having no
power greater than 1, and 3 having no power greater than 2.

My point of course is that just because it can be described as
19-limit, doesn't make it JI.

> This one should be scheduled for prime time listening
> 1/1 107/101 257/229 151/127 349/277 311/233 41/29
421/281 173/109 523/311
> 547/307 421/223 3947/1973
> or perhaps it is already, near enough anyway.

Yes. Another "odd" one. 12-tET again (this time within 1.0 cent, or
0.6 cents if you fix the octave).

> What about this 23 limit scale, which has gothic antecedants?
>
> 1/1, 23/22, 85/76, 55/46, 5/4, 91/68, 95/68, 136/91, 25/16, 117/70,
161/90, 144/77, 2/1

Ok. 1/4-comma meantone within 2c.

> Here's a 17 limit scale that should sound very sweet
> 1/1, 273/256, 9/8, 832/693, 1024/819, 4/3, 128/91, 3/2, 1331/832,
693/416, 416/231,
> 512/273, 2/1,
...

Whew! I've had enough. You or someone else will have to explain the
others.

> Actually, I've done a little javascript page which happens to be
very good at making them.
> Follows on from a suggestion Dan made, which turned out to be
surprisingly easy to do and
> reasonably efficient.

Ok. But what's the point? Apart from making "recognise the scale" type
puzzles?

-- Dave Keenan

Hi Dave,

> Ok. But what's the point? Apart from making "recognise the scale"
type
> puzzles?

I see, of course, naturally you will think I wrote it in response to
your scale.

Especially since I also forgot to add any explanation at all to the
"calculator" page.

Actually it was the other way round. I'd had it in mind to do it for
about a week, since Dan Stearns suggested the idea.

I'd just taken a few days off from FTS to catch up on other things,
and started on it when I saw your e-mail about the scales, and
realised it would make ones in that family easily.

Here's the reason for it.

http://homepage.ntlworld.com/robertwalker/site_ex/robertwalker/ratio
s_with_factors.htm

Try 250 cents using the continued fraction algorithm
1/1, 7/6, 15/13, 52/45, 119/103, 766/663,

0.0 266.8709 247.7411 250.3039 249.9807 250.0026

Okay, by the continued fraction algorithm result, 7/6 is closer than
any other approximation with larger quotient.

However my other applet finds

1/1, 4/3, 5/4, 6/5, 7/6, 15/13, 37/32, 52/45, 67/58, 119/103,
528/457, 647/560,
cents:
0, 498.045, 386.314, 315.641, 266.871, 247.741, 251.344, 250.304,
249.73, 249.981, 250.013, 250.007,

Notice it is finding many more ratios.

N.B. I've found a bug in that page, so it wasn't getting the best
answers before - anyone whose been trying it out, please have
another go, sorry.

Also found that I used ..\ instead of ../ in the html which Netscape
won't recognise - affects the "show tree" link.

I've just uploaded the fixed versions.

The continued fraction algorithm is leaving out 4/3, 5/4 and 6/5,
which are successively better approximations to 250, all of them
better than
1/1.

So clearly, the continued fraction method doesn't find _all_ the
best ratios, just some of them. It's a fast algorithm for finding
close
rationals, but very gappy.

But we aren't that worried about speed, and more interested
in getting a reasonably complete list of all the close rationals to
an interval.

So, what about a slower algorithm that just tests all the
denominators in turn,
and for each one, works out what the denumerator must be.

For instance, let's try 1.15535... as the decimal, and quotient 17,
we want denum/17 ~= 1.15535... , so denum ~= 17*1.15535 = 19.64...
so clearly we want a number between 19/17 and 20/17.

Testing each quotient in that way will be fast enough with small
numbers for the quotient.

That's exactly what I did. For details, see comments in the code. It
is coded for an interpreted language - one would do it slightly
differently in a compiled language, but difference isn't significant
as the optimising prob. makes less than a factor of two difference
in speed.

Javascript isn't speedy, and I expect in a native language it would
be at least ten times faster, maybe more. But, it's fast enough.

Factorising
1/1, 4/3, 5/4, 6/5, 7/6, 15/13, 37/32, 52/45, 67/58, 119/103,
528/457, 647/560,

gives
1/1 2^2/3 5/2^2 2*3/5 7/(2*3) 3*5/13 37/2^5
2^2*13/(3^2*5) 67/(2*29) 7*17/103 2^4*3*11/457 647/(2^4*5*7)

So this method isn't particularly good for finding n-limit ratios
for n small.

To deal with that, I added an extra test of whether the denumerator
or denom. is n-limit.

It is really slow, one would think, as it just tries to factorise by
repeated division, and returns false if it fails, true if it
succeeds.

However, and this is the nice thing, had the thought to do the
factorisation test _last_.

I.e. first see if the ratio is closer than any previous one found.
If it is, then apply the factorisation test. Only a tiny percentage
of the ratios will be so close, with the percentage getting less
as one moves to higher quotients. So the slow
factorisation test is applied very occasionally.

With that idea, whole thing became very practical.

(actually I did a version of this before that I'd forgotten about,
and even posted it to the TL, but it would have been a bit too slow
for javascript probably).

Now try for 11 limit approximations to 250 and we find
1/1, 4/3, 5/4, 6/5, 7/6, 55/48, 64/55, 81/70, 140/121, 231/200,
1155/1000, 1331/1152,
0, 498.045, 386.314, 315.641, 266.871, 235.677, 262.368, 252.68,
252.504, 249.471, 249.471, 250.044,

Try for 5-limit ones and we get
1/1, 4/3, 5/4, 6/5, 9/8, 32/27, 125/108,
and so on.

.................................................

I also did another program using the continued fractions method
sometime back that found very close approximations of the form
a^n/b^m, which demonstrated a Kronecker's theorem result.
Applying Kronecker's theorem to the logs shows that you can
get arbitrarily close to any number by suitable choice of m
and n provided that a has at least one prime factor that doesn't
divide into b, and vice versa.

In particular, works with a, b prime.

This gave results like:
"
Successive approx. to 7/6 in form 5^m/3^n

1/1 5^7/3^10 5^9/3^13 5^11/3^16 5^24/3^35 5^39/3^57 5^153/3^224
5^1138/3^1667
5^1295/3^1897 5^2437/3^3570 5^3579/3^5243

Values in cents
0 484.65 351.41 218.17 303.1 254.8 268.08 266.02 267.62 267.17
266.71
"
These numbers are large. For instance,
5^7/3^10 = 78125/59049

However it uses the continued fraction algorithm, this time on the
logs of the denum and denom rather than the numbers themselves, or
something likt that, if I remember correctly.

So, was very fast to run, but one expects gaps.

Let's try to fill them with the applet

Going to the "cents to ratios and back" page, we find
7/6 = 266.8709056037376 cents

Now going to the "ratios with factors" page, and trying this,
with 3 and 5 as the prime factors, we get
729/625 = 266.475,
to within .396,

Factorising:
3^6/5^4

.................................................

Now that the bugs have been sorted out pretty much, imagine it may
be useful to newbies, and old hands alike.

Hope so anyway, and thanks Dan for the suggestion!

Now, for your "solutions":

Yes, Yes, Yes,

However

> This one should be scheduled for prime time listening
> 1/1 107/101 257/229 151/127 349/277 311/233 41/29
421/281 173/109 523/311
> 547/307 421/223 3947/1973
> or perhaps it is already, near enough anyway.

missing something.

Rest are pretty easy, clues for them all, but I'll leave it for
anyone who wants to try.

Thought, maybe a bit of fun, that's all.

Robert

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_19586.html#19626

> --- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> > Hi Dave,
> >
> > Notice anything just a little bit odd about this very 353 limit
> scale? 107/101, 55/49, 201/169, 223/177, 263/197, 239/169, 295/197,
> 227/143, 301/179, 351/197,
> > 353/187, 399/199,
> >
> > N.B. necessarily has to have a detuned octave!
>
> I see that, except for the last note, it _is_ 12-tET, since no
> interval is more than 1.2 cent different from a theoretically
perfect 12-tET. But with the last note it goes up to 5.3 c.
>
> It's "odd" because no factor is even, which explains the necc.
detuned octave.
>
> I see you checked out my "19-limit" scale. No interval off by more
> than 4 c and only using primes 2,3,5,17,19 with 5,17,19 having no
> power greater than 1, and 3 having no power greater than 2.
>
> My point of course is that just because it can be described as
> 19-limit, doesn't make it JI.
>

Thanks to Dave Keenan and Robert Walker for the interesting tuning
puzzles! Lots of interesting "philosophical" ramifications to all of
this... but, hopefully, I'm smart enough to stay out of THAT
quagmire...

I think it's great to create a program that can make this kind of
"tuning spam..."

Frankly, it might be nice to have a "puzzles" section of the FAQ.
There's nothing like the moment when one finally "gets it," if one
does...

Thanks again!

________ _____ _____ ___
Joseph Pehrson

well i don't seem to be getting messages from tuning list?
maybe it is a sign1

--- In tuning@y..., kraiggrady@a... wrote:

/tuning/topicId_19586.html#19679

> well i don't seem to be getting messages from tuning list?
> maybe it is a sign1

Hi Kraig...

Did you check your "delivery options" under "preferences" on the
Yahoo egroups website?? I thought you had completely "unsubscribed"
before... Maybe you're not set to receive e-mail (??)

______ _____ _____ ____
Joseph Pehrson

it appears that earthlink is having trouble with its relay. so anything sent to me is not getting through either from this list or otherwise. Please be
advised that any response if one is even necessicery i cannot do. this i am reading directly from the yahoo site. I guess this means I just have to go
back and write music. seeyallsoonnoworlater !!

>
> Hi Kraig...
>
> Did you check your "delivery options" under "preferences" on the
> Yahoo egroups website?? I thought you had completely "unsubscribed"
> before... Maybe you're not set to receive e-mail (??)
>
> ______ _____ _____ ____
> Joseph Pehrson

Robert Walker wrote,

<<Actually it was the other way round. I'd had it in mind to do it for
about a week, since Dan Stearns suggested the idea.>>

Yeah, very nice Robert! I somehow missed this one and hadn't seen this
yet.

Indeed, this is exactly what I had in mind, and so far as I can tell
at the moment, it seems to work great too.

thanks!,

--Dan Stearns

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> Hi Dave,
>
> > Ok. But what's the point? Apart from making "recognise the scale"
> type
> > puzzles?
>
> Here's the reason for it.
>
> http://homepage.ntlworld.com/robertwalker/site_ex/robertwalker/ratio
> s_with_factors.htm

It doesn't really answer my question. It's very clever, but why would
one want to approximate a tempered scale by ratios? Seems sort of
back-to-front. Why not use a just scale to start with, where the
harmonic relationship between the notes (i.e the lattice) has been
considered in its design?

Regards,
-- Dave Keenan

Hi Dan,

> Yeah, very nice Robert! I somehow missed this one and hadn't seen this
> yet.
>
> Indeed, this is exactly what I had in mind, and so far as I can tell
> at the moment, it seems to work great too.
>

Great, thanks!

Definitely more to come of this sort....

Robert

Hi Dave,

> It doesn't really answer my question. It's very clever, but why would
> one want to approximate a tempered scale by ratios? Seems sort of
> back-to-front. Why not use a just scale to start with, where the
> harmonic relationship between the notes (i.e the lattice) has been
> considered in its design?

There is no particular aim for it. It is meant to be open ended, a
general calculation tool that anyone may find useful.

If you want to find the nearest ratio to a measured scale in
cents, then the continued fraction algorithm gives a succession
of closer intervals, but though all the entries are closest
possible for the size of quotient, it is also rapidly converging,
and has many gaps.

This method will show many more intervals, and for example,
one can look to see what the measured scale has in the way of
low quotient 7-limit or 5-limit ratios, or whatever, as approximants.

One can vary the prime factors, and generally, have a lot more
variables to play around with to find the ratios than one does
using the ordinary continued fractions method.

My original reason for getting interested last year was because
I was wondering what there is to the notion of a 5-ness,
7-ness, or 11-ness of an interval.

This finds, say, 5-limit ratios close to a 7-limit one, so
one can explore the way one can gradually move from a 5-limit
ratio to a 7-limit one.

Original reason for this particular one is because Dan Stearns
suggested the idea, and pointed out the gap in the continued
fractions approximants, which I hadn't given much thought
to before,

Robert

Dave Keenan wrote,

<<why would one want to approximate a tempered scale by ratios? Seems
sort of back-to-front.>>

Not really, or rather only from a certain perspective... but luckily
there's more than one perspective <!>.

While different from the approach Robert is using here, the
"equaltone" posts that I put up a while ago were an attempt on my part
to frame equal tunings as a series akin to undertone and overtone
series. Call this an aesthetic approach to theory if you must, but it
is in line with what I do musically, and I think it's not irrelevant
that the two -- music and theory -- try to get to know each other. At
the very least I find it pleasing that one underscores rather than
undermines the other!

Anyway, as you probably know by now I use a lot of equal tunings. And
an RI approach similar to the "morphing series" method that I've used,
or the "like prime" approach that Robert's sieve enables, allows one
to inject some new shades into a given equal tuning. This I find quite
useful for what should be obvious reasons.

Now I myself don't call these things whatever-limit JI and such... and
I, like you I'm sure, would say that there's a good argument why
they're not best described in these terms most of the time. But I have
really warmed to Margo's "RI", and I think it is a very good term for
a lot of what I do in this regard...

And for what it's worth, a lot of folks have been interested in
finding good ET to JI conversion methods over the years. So just the
spirit of answering an interesting question answers your "why would
one want to approximate a tempered scale by ratios" question to my
mind! However, I have other reasons, and hopefully I've been able to
make at least some of them clear here.

--Dan Stearns

Robert Walker wrote,

<<Definitely more to come of this sort....>>

Great, can't wait to see them! Let me know if there's anyway I can
help with the two and three-term, Fibonacci and Tribonacci ideas...
BTW, adding some sort of a GCD function so that the cents to ratios
calculator will always give the ratios in their reduced form might not
be a bad idea too?

thanks,

--Dan Stearns

--- In tuning@y..., "Robert Walker" <robert_walker@r...> wrote:
> If you want to find the nearest ratio to a measured scale in
> cents, then the continued fraction algorithm gives a succession
> of closer intervals, but though all the entries are closest
> possible for the size of quotient, it is also rapidly converging,
> and has many gaps.

> Original reason for this particular one is because Dan Stearns
> suggested the idea, and pointed out the gap in the continued
> fractions approximants, which I hadn't given much thought
> to before.

And thanks Dan, for explaining your application.

Robert,

The gaps between the convergents can be filled in by the
"semi-convergents" as described in
http://depts.washington.edu/pnm/CLAMPITT.pdf

Of course in that paper it's used to find rational fractions of an
octave (melodic/logarithmic), whereas here we're using it for
frequency ratios (harmonic/linear).

Here's an Excel spreadsheet implementation for frequency ratios.
http://dkeenan.com/Music/CentsToRatios.xls

Yours doesn't seem to generate all the semi-convergents.

Regards,
-- Dave Keenan

Dave Keenan wrote,

<<Yours doesn't seem to generate all the semi-convergents.>>

Thanks for the spreadsheet Dave (nice as always), and I do remember
someone pointing out these "semi-convergents" before. But what
Robert's calculator is doing, or one of the things it can do, is to
sieve out *only* the next smallest ratio with a closer approximation
than the one preceding it (set "max quotient" to some suitably big
number, the "tolerance" to zero, and leave the "primes" sieve empty).
So naturally this wouldn't give all the semi-convergents.

I had a (fairly tedious) way to do this before, but this here zippitty
split calculator is pretty darned slick I think.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Dave Keenan wrote,
>
> <<Yours doesn't seem to generate all the semi-convergents.>>
>
> Thanks for the spreadsheet Dave (nice as always), and I do remember
> someone pointing out these "semi-convergents" before. But what
> Robert's calculator is doing, or one of the things it can do, is to
> sieve out *only* the next smallest ratio with a closer approximation
> than the one preceding it (set "max quotient" to some suitably big
> number, the "tolerance" to zero, and leave the "primes" sieve
empty).
> So naturally this wouldn't give all the semi-convergents.

Ok. That makes sense.

> I had a (fairly tedious) way to do this before, but this here
zippitty
> split calculator is pretty darned slick I think.

Yes. Very nice. Especially the prime-factor filtering.

Robert,

An optional power limit on each prime factor might be a good addition.
Let zero stand for infinity, and an empty list is the same as a list
of infinities.

I don't see it as very useful to operate on a whole scale and only see
a single ratio for each degree, because that is only tending to relate
each degree back to the tonic. I'd be more interested in the multiple
interpretations (or puns) available for each degree. But I can of
course still do that, one degree at a time.

-- Dave Keenan

Hi Dave

> An optional power limit on each prime factor might be a good addition.
> Let zero stand for infinity, and an empty list is the same as a list
> of infinities.

> I don't see it as very useful to operate on a whole scale and only see
> a single ratio for each degree, because that is only tending to relate
> each degree back to the tonic. I'd be more interested in the multiple
> interpretations (or puns) available for each degree. But I can of
> course still do that, one degree at a time.

Thanks. I've added a text field to show all the approximations for
each note of the scale, up to 32 notes.

Also it now finds upper convergents, lower, both, or closest only.

Robert

Hi Dan

> Great, can't wait to see them! Let me know if there's anyway I can
> help with the two and three-term, Fibonacci and Tribonacci ideas...
Will do.

> BTW, adding some sort of a GCD function so that the cents to ratios
> calculator will always give the ratios in their reduced form might not
> be a bad idea too?
Thanks, I've done that.

Robert