Non octave over / under scale

Hi Dan,

I've just found how to generalise your over under technique
to non octave scales with the same number of notes to a
non octave as the octave version.

http://tunesmithy.co.uk/uo_non_oct.htm

Long live the non octave scale!!

Robert

Thanks Robert,

This is really wonderful and exactly what we were talking about last
night. I didn't think you could do it so quickly, great job!

Something very useful to note is that the algorithm decreases in
accuracy the higher the period, but by the same token it increases in
accuracy--and to a really astonishing degree--the smaller the period,
even if the rounding of x is quite severe such as to the nearest whole
number.

One suggestion I'd like to make is that the optimum x, which you've
already included, actually be the initial default for the x at he top
of the page. This way you can tweak that if you like, but it initially
gives you the series at its optimum bend value... and this would be
both relevant and quite handy as it keeps you from having to scroll
down and cut and paste every time you tweak the period and the number
of notes. Other than that it's freakin aces!

(Now if we could only get it to give each x its corresponding Scale
Tree!)

thanks a million,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, June 26, 2002 5:29 PM
Subject: [tuning] Non octave over / under scale

> Hi Dan,
>
> I've just found how to generalise your over under technique
> to non octave scales with the same number of notes to a
> non octave as the octave version.
>
> http://tunesmithy.co.uk/uo_non_oct.htm
>
> Long live the non octave scale!!
>
> Robert
>
>
> ------------------------ Yahoo! Groups
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Thanks Robert,
>
> This is really wonderful and exactly what we were talking about last
> night. I didn't think you could do it so quickly, great job!
>
> Something very useful to note is that the algorithm decreases in
> accuracy the higher the period,

maybe gene or dave or robert could suggest a way of improving this,
maybe a "second order term" in the approximation -- should be
mathematically interesting if nothing else.

i'm not intending at all to say that there's anything "wrong" with
dan or anyone else using this the way it is, of course . . .

Hi Robert,

If I could make a couple more suggestions--I wish you could have the
number of decimal places value for the upper x adjustable (rather than
rounded) like you do the bottom one, and that you'd give the max
difference as a large (i.e., with many decimal places) number, as that
is one number that I'd like to see exactly what's going on to a very
high degree of accuracy. Conversely I think it would be clearest if
you gave both the series' cents examples as you do the et (i.e.,
rounded top the nearest whole number).

The Scale Tree examples are not at all what I had in mind, maybe we
could try and work it out off-list later?

Thanks as always!

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, June 26, 2002 8:33 PM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> > Something very useful to note is that the algorithm decreases in
> > accuracy the higher the period, but by the same token it increases
in
> > accuracy--and to a really astonishing degree--the smaller the
period,
> > even if the rounding of x is quite severe such as to the nearest
whole
> > number.
>
> Yes indeed - I've just tried out 5/4 as the scale repeat and got
very close to n-et.
>
> > One suggestion I'd like to make is that the optimum x, which
you've
> > already included, actually be the initial default for the x at he
top
> > of the page. This way you can tweak that if you like, but it
initially
> > gives you the series at its optimum bend value... and this would
be
> > both relevant and quite handy as it keeps you from having to
scroll
> > down and cut and paste every time you tweak the period and the
number
> > of notes. Other than that it's freakin aces!
>
> > (Now if we could only get it to give each x its corresponding
Scale
> > Tree!)
>
>
> Those are easy changes as it so happens
> (about a dozen lines of code or thereabouts) - done
>
> http://tunesmithy.co.uk/uo_non_oct.htm
>
> If anyone has just been browsing that page
> recently, old version may be in your cache
> - in Windows anyway, exit from internet explorer
> (all of its windows if you have several web pages
> on the go at once) and start it up again and
> reload the page and you should see the new
> changes)
>
> Robert
>
>
>
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Hi Dan,

Okay, I've done that, and renamed the text area to
"Neighbouring Scales"

I never understood the connection with the scale
tree so we can work that out off-list.
So long as it is algorithmically related
to the scales, should be easy to do.

http://www.tunesmithy.co.uk/uo_non_oct.htm

Robert

Hi Robert,

There's still a bug in the boat somewhere... try checking a 2-tone
series with a 2/1 period and say 50 decimal places of x... now check
the max error result... it should be zero!

Also the cents values in the scale examples are sometimes rounded and
sometimes not, could you have them as no decimal places all the time?
I think that's much cleaner there.

thanks,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, June 26, 2002 9:13 PM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> Okay, I've done that, and renamed the text area to
> "Neighbouring Scales"
>
> I never understood the connection with the scale
> tree so we can work that out off-list.
> So long as it is algorithmically related
> to the scales, should be easy to do.
>
> http://www.tunesmithy.co.uk/uo_non_oct.htm
>
> Robert
>
>
> ------------------------ Yahoo! Groups
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Thanks Robert,

This is set up nicely now, but there still seems to be some
bugs--check the et cents value for a 2-tone scale with a 2/1 period
for example.

take care,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 27, 2002 7:20 AM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> > There's still a bug in the boat somewhere... try checking a 2-tone
> > series with a 2/1 period and say 50 decimal places of x... now
check
> > the max error result... it should be zero!
>
> Ok fixed, - it was rounding errors, which you always get in high
> precision decimal point type calculations on a computer -
> the value shown was e.g. 2.2737367544323206e-13
> which means 2.2737367544323206*10^-13
> but the e-13 was hidden because I made the text field too small
> to show it. Anyway I've rounded it to max of 12 decimal places, so
> should show 0 now.
>
> http://www.tunesmithy.co.uk/uo_non_oct.htm
>
> > Also the cents values in the scale examples are sometimes rounded
and
> > sometimes not, could you have them as no decimal places all the
time?
> > I think that's much cleaner there.
>
> What's happening here is that it shows a maximum of one decimal
place,
> rather than always showing one place. So if the decimal value is 0
it
> leaves it out.
>
> I've changed this to show a fixed number of decimal places - that
> also helps when comparing the two sets of cents figures
> as they line up nicely. Also set the default to show no decimal
> places so you'll see whole numbers of cents.
>
> Robert
>
>
>
>
>
>
> ------------------------ Yahoo! Groups
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Hi Dan,

> This is set up nicely now, but there still seems to be some
> bugs--check the et cents value for a 2-tone scale with a 2/1 period
> for example.

Yes.

I was rounding the likes of 1199.9999
by just truncating it at the decimal point.

Now I round it normally, and then if it
has no trailing zeroes, and a decimal point
is desired with trailing zeroes add them in
at the end.

- fixed.

Robert

Hi Robert,

Okay, I really like everything now and I haven't noticed any more
bugs, but I think you should probably nix the Scale Tree section. I
think it's superfluous as it now stands, and unfortunately I'm not
going to be able to work on a way to code what I want it to do from
the U-->O algorithm right now, and I'm soon to go off-line (more on
that later.)

Thanks a ton and sorry to be such a bother.

take care,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 27, 2002 11:55 AM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> > This is set up nicely now, but there still seems to be some
> > bugs--check the et cents value for a 2-tone scale with a 2/1
period
> > for example.
>
> Yes.
>
> I was rounding the likes of 1199.9999
> by just truncating it at the decimal point.
>
> Now I round it normally, and then if it
> has no trailing zeroes, and a decimal point
> is desired with trailing zeroes add them in
> at the end.
>
> - fixed.
>
> Robert
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
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Hi Dan,

> Okay, I really like everything now and I haven't noticed any more
> bugs, but I think you should probably nix the Scale Tree section. I
> think it's superfluous as it now stands, and unfortunately I'm not
> going to be able to work on a way to code what I want it to do from
> the U-->O algorithm right now, and I'm soon to go off-line (more on
> that later.)

Glad you like it!

I don't think it need be too arduous to explain - the
tree itself is derived from four numbers, e.g. for
x = 3.5 they are 0/5 and 7/7, you've told us,
so one just needs to know how you get those numbers
from x and that would be all that is needed to do the
programming. Or, how you derive x from the scale tree
if you do it the other way round.

Of course I'd also like to know how your tree works and what
the connection is with the scales themselves.

From what I've seen of your work before it probably involves
some kind of sideways look at the numbers so that the pattern
suddenly falls into place, but I can't see it yet at all.

I'm sure others are left wondering too. So I hope you can
take this thread up again at some point. But, if you want
to leave us wondering, that's okay of course!

BAck to work finishing FTS. I've enjoyed coding your
algorithm I must say.

Also might be interesting to say a bit about why I find
these ones particularly interesting. The thing is that
by being between overtone and undertone series they
can be used as partials to make new timbres which
are in between harmonic and inharmonic timbres.
I find these intriguing.

I did some experiments on that a while back and
in fact use one of them as one of the fractal tunes
that are included with FTS.

Robert

Hi Robert,

The basic tree, or JI to ET, idea was simply this--for any given x
there is only *one* fraction of the period that is always a 3/2; a
4/3; a 5/3; a 5/4; a 7/5; a 8/5; a 7/4, add infinitum. So every x
defines its own

1/1 2/1
3/2
4/3 5/3
5/4 4/5 8/5 7/4

(etc.)

as ET scale degrees.

I used to know how to derive this from the algorithm but I've forgot
(I'm sure it had to do with defining the two terms that seed the tree,
1/1 and the period). It wasn't hard though, and I'm sure once you've
understood what it actually is that I'm doing, or going after here,
you'd be able to quickly parse it out and program it in (though I'm
not sure I generalized this for any non-octave period either).

thanks,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 27, 2002 6:13 PM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> > Okay, I really like everything now and I haven't noticed any more
> > bugs, but I think you should probably nix the Scale Tree section.
I
> > think it's superfluous as it now stands, and unfortunately I'm not
> > going to be able to work on a way to code what I want it to do
from
> > the U-->O algorithm right now, and I'm soon to go off-line (more
on
> > that later.)
>
> Glad you like it!
>
> I don't think it need be too arduous to explain - the
> tree itself is derived from four numbers, e.g. for
> x = 3.5 they are 0/5 and 7/7, you've told us,
> so one just needs to know how you get those numbers
> from x and that would be all that is needed to do the
> programming. Or, how you derive x from the scale tree
> if you do it the other way round.
>
> Of course I'd also like to know how your tree works and what
> the connection is with the scales themselves.
>
> From what I've seen of your work before it probably involves
> some kind of sideways look at the numbers so that the pattern
> suddenly falls into place, but I can't see it yet at all.
>
> I'm sure others are left wondering too. So I hope you can
> take this thread up again at some point. But, if you want
> to leave us wondering, that's okay of course!
>
> BAck to work finishing FTS. I've enjoyed coding your
> algorithm I must say.
>
> Also might be interesting to say a bit about why I find
> these ones particularly interesting. The thing is that
> by being between overtone and undertone series they
> can be used as partials to make new timbres which
> are in between harmonic and inharmonic timbres.
> I find these intriguing.
>
> I did some experiments on that a while back and
> in fact use one of them as one of the fractal tunes
> that are included with FTS.
>
> Robert
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
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Hi Dan,

Yes, once I understand, but I don't understand, sorry.

I haven't got a starting point to work with. Could ask
more questions but as you are taking time off the list it isn't
an appropriate time right now. Maybe later.

Robert

Robert,

One thing you could try is this:

take any x (or a couple of them) in a 2/1 period

then find the n that gives a 3/2

then find the n that gives a 4/3

without reducing the fractions of the period (if they are in fact
reducible, subtract the 4/3 from the 3/2 and now you've got the 1/1

now find the n that gives a 5/3

now, and again, without reducing the fractions of the period (if they
are in fact reducible, subtract the 5/3 from the 3/2 and now you've
got the 2/1

from there you can let the tree grow on its own, and perhaps parse out
how to generalize this not only to all values of x, but also to all
periods

Remember, what we want is a tree of ET fractions of the period, and
not ratios (though they are in fact analogous).

Hope that helps!

take care,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 27, 2002 10:23 PM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> Yes, once I understand, but I don't understand, sorry.
>
> I haven't got a starting point to work with. Could ask
> more questions but as you are taking time off the list it isn't
> an appropriate time right now. Maybe later.
>
> Robert
>
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
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Hi Dan,

That's really interesting.

The missing piece in the puzzle was that your
0/5 etc is a shorthand for 0th note of 5th scale.

Trying with x = 3.5 I find:

3/2 at 7/12

4/3 at 7/17

To subtract you subtract the denominator and deumerator
separately:

4/3 - 3/2 = (4-3)/(3-2) = 1/1
7/17 - 7/12 = (7-7)/(17-12) = 0/5

The 1/1 is of course the 0th degree of any scale.

I find the 5/3 at 14/19

5/3 - 3/2 = 1/1
14/19 - 7/12 = 7/7.

So that gives your 1/1 and 2/1.

Now construct the scale tree for these 0/5 etc
ratios and you get:

0/5 7/7
7/12
7/17 14/19
7/22 14/29 21/31 21/26
7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33

which you say will give the locations in the tree of
1/1 2/1
3/2
4/3 5/3
5/4 7/5 8/5 7/4
6/5 9/7 11/8 10/7 11/7 13/8 12/7 9/5

Checking, for instance 14/39, then indeed, the 14th degree
of the 39th scale is 9/7!!

http://tunesmithy.co.uk/uo_non_octave.htm

So you are saying that these over / under scales are
a kind of re-arrangement of the intervals of the
scale tree. Presumably each interval occurs only
once in any of the scales for any given value
of x.

I wonder which values of x it works for. Trying a few
values I find sometimes you get many repeats of the
ratios, and it doesn't seem to work for those
(or maybe one has to choose the right one of
all the 3/2s and 4/3s in the tree to make it work
??).

For x = 3.5 you get a repeated 3/2 at 14/24
- for x = 3.4 you get a repeats at 17/29 and 34/58

I wonder what happens there? Perhaps the scale tree skips those
repeated positions - that's possible.

One could also well imagine some such result such as
that it works for the first n rows where n gets larger
the closer you get to the best value. But so far
it seems to work for all the values I've tried.

For x = 3.5, I find

3/2 at 17/29
4/3 at 17/41
5/3 at 34/46

so that makes the 1/1 0/12 and the 2/1 17/17
leading to

0/12 17/17
17/29
17/41 34/46
17/53 34/70 51/75 51/63
17/65 34/94 51/111 51/99 70/104 85/121 85/109 68/80

1/1 2/1
3/2
4/3 5/3
5/4 7/5 8/5 7/4
6/5 9/7 11/8 10/7 11/7 13/8 12/7 9/5

and it works!

This would be programmable by searching through the scales
until one finds the 4/3, 3/2 and 5/3, and breaking off hte
search if you don't find them after, say, 100 rows.

If you do find them, check that you get results of the
form 0/n, n/n for the 1/1 and 2/1, and if not, keep
searching perhaps??

Then one could construct the tree and verify it. Would
be nice to know why it works though...

Plenty to think about...

I've added a new field to the page to let one show
a particular degree of the scale, useful if it has many notes
in it.

Robert

HI Dan,

Sorry the second case was for x = 3.4

For x = 3.4, I find

3/2 at 17/29
4/3 at 17/41
5/3 at 34/46

and gave the wrong url.
AS before its
http://tunesmithy.co.uk/uo_non_oct.htm

Robert

Hi Robert,

Hey, it's getting there, here's another suggestion:

If we let the period be a 2/1 and start from x = 3, and then increase
x by increments of .1, we have the following fraction of the octave
3/2s:

3/5 31/52 16/27 33/56 17/29 7/12 18/31 37/64 19/33 39/68 4/7

So it's easy to see from this that the numerators are just simple
sequences of x as whole numbers. This means that the sequence of 2/1s
should be x/x as a whole number, and from there it's easy to figure
the 1/1s.

Let's look at x = 3.6 for example:

0/13 18/18
18/31
18/43 36/49
18/56 36/74 54/80 54/67

(etc.)

Works like a charm. But what we still need to know is how to initially
derive the value of the denominator from the algorithm in some way...
but I imagine a brute force mole of some sort could work it out too.

Let me know if this bit gets you any further--I'll check in between
packing boxes!

take care,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Friday, June 28, 2002 10:20 AM
Subject: [tuning] Re: Non octave over / under scale

> HI Dan,
>
> Sorry the second case was for x = 3.4
>
> For x = 3.4, I find
>
> 3/2 at 17/29
> 4/3 at 17/41
> 5/3 at 34/46
>
> and gave the wrong url.
> AS before its
> http://tunesmithy.co.uk/uo_non_oct.htm
>
>
> Robert
>
>
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--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> The thing is that
> by being between overtone and undertone series they
> can be used as partials to make new timbres which
> are in between harmonic and inharmonic timbres.

that doesn't make sense. if the partials are not in an overtone
series, the timbre is inharmonic -- regardless of whether the
partials form an undertone series, an "equaltone" series, or
whatever . . .

HI Dan,

Yes indeed.

x = 30/10 31/10 32/10 33/10 ... 40/10
3/2 at 3/5 31/52 16/27 33/56 ... 4/7
30/50 31/52 32/54 33/56 ... 40/70

Also try looking to see what is at 32/54 etc - that's
a 3/2 too.

Try multiples of 3, 4 etc:

x = 9/3 10/3 11/3 12/3
3/2 at 9/15 10/17 11/19 12/21

x = 12/4 13/4 14/4 15/4 16/4
3/2 12/20 13/22 7/12 15/26 16/27

x = 15/5 16/5 17/5 18/5 19/5 20/5
3/2 at 15/25 16/27 17/29 18/31 19/33 20/35

x = 18/6 19/6 20/6 21/6 22/6 23/6 24/6
3/2 at 15/30 19/32 20/34 21/36 22/38 23/40 24/42

So now we just need to discover the pattern in the
denominators for the 1/1 - and that is pretty easy,
it's just 5 times the denominator for x.

So that's the pattern. It's not a proof in the
mathematical sense at all yet, but I'm convinced,
I think it must be true.

But why???!!

I'll ponder over it for sure...

I've added a couple of new buttons to the
page now to let one look for a particular value.

http://tunesmithy.co.uk/uo_non_oct.htm

Robert

Hi Robert,

Another quick point here which should help you understand what's going
on a little better...

You wrote,

> I wonder which values of x it works for. Trying a few
> values I find sometimes you get many repeats of the
> ratios, and it doesn't seem to work for those
> (or maybe one has to choose the right one of
> all the 3/2s and 4/3s in the tree to make it work
> ??).
>
> For x = 3.5 you get a repeated 3/2 at 14/24
> - for x = 3.4 you get a repeats at 17/29 and 34/58
>
> I wonder what happens there? Perhaps the scale tree skips > those
repeated positions - that's possible.

What happens is that *every* multiple of a given fraction of an octave
will always be exactly the same ratio. So every value of x should
theoretically give a complete Scale Tree that is unique that to that x
(and, if generalized, to that period).

So what you have is the whole JI universe of ratios, uniquely
articulated to ET scale degrees in giant, unbounded grids--each
relative to the degree of bend and the complexity of x (the simpler
the value of x, the simpler the fractions of the period).

take care,

--Dan Stearns

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Friday, June 28, 2002 10:15 AM
Subject: [tuning] Re: Non octave over / under scale

> Hi Dan,
>
> That's really interesting.
>
> The missing piece in the puzzle was that your
> 0/5 etc is a shorthand for 0th note of 5th scale.
>
> Trying with x = 3.5 I find:
>
> 3/2 at 7/12
>
> 4/3 at 7/17
>
> To subtract you subtract the denominator and deumerator
> separately:
>
> 4/3 - 3/2 = (4-3)/(3-2) = 1/1
> 7/17 - 7/12 = (7-7)/(17-12) = 0/5
>
> The 1/1 is of course the 0th degree of any scale.
>
> I find the 5/3 at 14/19
>
> 5/3 - 3/2 = 1/1
> 14/19 - 7/12 = 7/7.
>
> So that gives your 1/1 and 2/1.
>
> Now construct the scale tree for these 0/5 etc
> ratios and you get:
>
> 0/5 7/7
> 7/12
> 7/17 14/19
> 7/22 14/29 21/31 21/26
> 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
>
> which you say will give the locations in the tree of
> 1/1 2/1
> 3/2
> 4/3 5/3
> 5/4 7/5 8/5 7/4
> 6/5 9/7 11/8 10/7 11/7 13/8 12/7 9/5
>
> Checking, for instance 14/39, then indeed, the 14th degree
> of the 39th scale is 9/7!!
>
> http://tunesmithy.co.uk/uo_non_octave.htm
>
> So you are saying that these over / under scales are
> a kind of re-arrangement of the intervals of the
> scale tree. Presumably each interval occurs only
> once in any of the scales for any given value
> of x.
>
> I wonder which values of x it works for. Trying a few
> values I find sometimes you get many repeats of the
> ratios, and it doesn't seem to work for those
> (or maybe one has to choose the right one of
> all the 3/2s and 4/3s in the tree to make it work
> ??).
>
> For x = 3.5 you get a repeated 3/2 at 14/24
> - for x = 3.4 you get a repeats at 17/29 and 34/58
>
> I wonder what happens there? Perhaps the scale tree skips those
> repeated positions - that's possible.
>
> One could also well imagine some such result such as
> that it works for the first n rows where n gets larger
> the closer you get to the best value. But so far
> it seems to work for all the values I've tried.
>
> For x = 3.5, I find
>
> 3/2 at 17/29
> 4/3 at 17/41
> 5/3 at 34/46
>
> so that makes the 1/1 0/12 and the 2/1 17/17
> leading to
>
> 0/12 17/17
> 17/29
> 17/41 34/46
> 17/53 34/70 51/75 51/63
> 17/65 34/94 51/111 51/99 70/104 85/121 85/109 68/80
>
> 1/1 2/1
> 3/2
> 4/3 5/3
> 5/4 7/5 8/5 7/4
> 6/5 9/7 11/8 10/7 11/7 13/8 12/7 9/5
>
> and it works!
>
> This would be programmable by searching through the scales
> until one finds the 4/3, 3/2 and 5/3, and breaking off hte
> search if you don't find them after, say, 100 rows.
>
> If you do find them, check that you get results of the
> form 0/n, n/n for the 1/1 and 2/1, and if not, keep
> searching perhaps??
>
> Then one could construct the tree and verify it. Would
> be nice to know why it works though...
>
> Plenty to think about...
>
> I've added a new field to the page to let one show
> a particular degree of the scale, useful if it has many notes
> in it.
>
> Robert
>
>
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