Yahoo Tuning Groups Ultimate Backup tuning Method for finding the closest rational

Hi everybody,

While extracting the info from the waves proved very difficult, the end result itself is very simple. If x = interval, we simply take

x*k/(x - 1), where k = 1,2,3,... and round to get our numerator p,
k/(x - 1) and round to get our denominator q, and of course take p/q.

These will give our *k'th* wave convergent. The strength of intervals depends on both the lowness of the convergents and the distance of x from p/q. The 12 tempered fifth for example gives {3/2, 3/2, 3/2,...} all the way and the distances from 3/2 are very small. Therefore it is a very good approximation. As an example, I've attached a file showing the first 10 wave convergents for the 12 Tet intervals.

Hope this comes in handy

Rick

🔗rick <rick_ballan@...>

Just to clear a problem up before it begins, the 1st wave convergents give all the very *simplest* intervals like the 8ve, fifth etc...Therefore, more remote (or 'tonally ambiguous') intervals will be bad approximations as the distance from them will be large. The major seventh or 2^11/12 for e.g. gives

{2,2,2,9/5,11/6,13/7,15/8...}.

This says that this interval doesn't reach a good approx till the 7th wave convergent 15/8 which appears relatively late in the wave cycle. Its first 3 are very badly tuned 8ves, its 4th is a badly tuned dominant 7th etc...This makes sense because, from a tonal point of view, intervals above 3/2 are either inversions of those below or are meant to appear as simpler intervals from other chord degrees. Of course a C Ma7 chord for e.g. in an ET, the B note is the maj third from the fifth degree and the fifth from the major third i.e. 15/8 = 3/2 x 5/4 = 5/4 x 3/2.

I'm in the middle of making a graph of this. Once you get to see it you'll begin to see what I mean.

Rick

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Hi everybody,
>
> While extracting the info from the waves proved very difficult, the end result itself is very simple. If x = interval, we simply take
>
> x*k/(x - 1), where k = 1,2,3,... and round to get our numerator p,
> k/(x - 1) and round to get our denominator q, and of course take p/q.
>
> These will give our *k'th* wave convergent. The strength of intervals depends on both the lowness of the convergents and the distance of x from p/q. The 12 tempered fifth for example gives {3/2, 3/2, 3/2,...} all the way and the distances from 3/2 are very small. Therefore it is a very good approximation. As an example, I've attached a file showing the first 10 wave convergents for the 12 Tet intervals.
>
> Hope this comes in handy
>
> Rick
>

🔗rick <rick_ballan@...>

Oh, and I forgot to mention what is probably the most important point, that out of the list of p/q's there is only one value that is closest to the given interval x. This tells us what the interval *is*. For the ma7 eg, this p/q is 15/8.

As far as I'm concerned, this settles the problem of how to relate large coprimes or irrationals to JI intervals and vice-versa. It gives, I think, a good argument that phantom pitch comes from physical properties of the waves, at least in some part, and not solely on "how the human ear/brain apparatus happens to be constructed". After all, I figure that everything from the production and motion of sound to the electronics of recording must rely heavily on some type of one-one transference b/w waves. At any rate, it is now arguable.

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Just to clear a problem up before it begins, the 1st wave convergents give all the very *simplest* intervals like the 8ve, fifth etc...Therefore, more remote (or 'tonally ambiguous') intervals will be bad approximations as the distance from them will be large. The major seventh or 2^11/12 for e.g. gives
>
> {2,2,2,9/5,11/6,13/7,15/8...}.
>
> This says that this interval doesn't reach a good approx till the 7th wave convergent 15/8 which appears relatively late in the wave cycle. Its first 3 are very badly tuned 8ves, its 4th is a badly tuned dominant 7th etc...This makes sense because, from a tonal point of view, intervals above 3/2 are either inversions of those below or are meant to appear as simpler intervals from other chord degrees. Of course a C Ma7 chord for e.g. in an ET, the B note is the maj third from the fifth degree and the fifth from the major third i.e. 15/8 = 3/2 x 5/4 = 5/4 x 3/2.
>
> I'm in the middle of making a graph of this. Once you get to see it you'll begin to see what I mean.
>
> Rick
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > Hi everybody,
> >
> > While extracting the info from the waves proved very difficult, the end result itself is very simple. If x = interval, we simply take
> >
> > x*k/(x - 1), where k = 1,2,3,... and round to get our numerator p,
> > k/(x - 1) and round to get our denominator q, and of course take p/q.
> >
> > These will give our *k'th* wave convergent. The strength of intervals depends on both the lowness of the convergents and the distance of x from p/q. The 12 tempered fifth for example gives {3/2, 3/2, 3/2,...} all the way and the distances from 3/2 are very small. Therefore it is a very good approximation. As an example, I've attached a file showing the first 10 wave convergents for the 12 Tet intervals.
> >
> > Hope this comes in handy
> >
> > Rick
> >
>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

Rick> "Oh, and I forgot to mention what is probably the most important
point, that out of the list of p/q's there is only one value that is
closest to the given interval x. This tells us what the interval *is*.
For the ma7 eg, this p/q is 15/8."

Perhaps my larger question is...are we measuring what's the best ratio in terms of "closest" to the original interval, most resolved sounding, or both?
To me 11/6, which appears less in your "convergents list", sounds a fair deal more resolved than 15/8. It also begs the question if you were testing for, say, 22/15, would the most common/"ultimate" convergent be 3/2 IE "better" or something more like 22/15 or 13/9 IE "closer".

🔗genewardsmith <genewardsmith@...>

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > Oh, and I forgot to mention what is probably the most important point, that out of the list of p/q's there is only one value that is closest to the given interval x. This tells us what the interval *is*. For the ma7 eg, this p/q is 15/8.
>
> How do you know where the list stops? Why is the major 7th 15/8 and not 17/9?
>
Hi Gene,

Well to answer the second question first, the reason is they are not really convergents. This is why I renamed them (or at least nick-named them) 'wave convergents'. They are defined by rounding in the way I gave previously. However, from what I can tell so far all of the traditional convergents seem to be there for the rationals.

To answer the second, if for convenience we reckon intervals against 1 instead of taking a/b, then the 1st wave convergents terminates at interval 3, the second at 5, 3rd at 7...the (p - q)'th at (2(p - q) + 1). I'll post graphs of the results as soon as I troubleshoot a few graphing problems.

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Rick> "Oh, and I forgot to mention what is probably the most important
> point, that out of the list of p/q's there is only one value that is
> closest to the given interval x. This tells us what the interval *is*.
> For the ma7 eg, this p/q is 15/8."
>
> Perhaps my larger question is...are we measuring what's the best ratio in terms of "closest" to the original interval, most resolved sounding, or both?
> To me 11/6, which appears less in your "convergents list", sounds a fair deal more resolved than 15/8. It also begs the question if you were testing for, say, 22/15, would the most common/"ultimate" convergent be 3/2 IE "better" or something more like 22/15 or 13/9 IE "closer".
>
Basically Mike, now that I've got the model in place its time to see where it takes us (the consequences). And I haven't had time to check everything in the detail it deserves yet. In fact these were the first irrationals.

But you can check for yourself. The formula is xq - p for interval x and WC p/q. With x = 2^11/12 and p/q = 11/6 we get xq - p = 0.32649...while 0.101989...for p/q = 15/8. So 15/8 is definitely closer. These were the 5th and 7th WC's. The 8th is 17/9 and is even closer at -0.01026...which I didn't learn till right now. From what I can tell so far, there seems to be some type of play-off between how early the WC appear in the series - they do in fact correspond to earlier cycles - and closness to rationals. Therefore I now have to modify my previous statement.

I'll post a graph as soon as I can. It'll be much clearer then.

Rick

🔗rick <rick_ballan@...>

Ok I posted a file 'Wave Convergent graphs' just to show you the type of results I'm getting. I can't get Mca either to solve for the zeros of the function or draw a straight vertical line for each interval, which would make things much easier, so sorry about that. (who'd think that Mca can't plot x == ? Very strange).

🔗Mike Battaglia <battaglia01@...>

Rick,

How do you pick which wave convergent from the list gets to be "the"
one for some equal tempered interval x? For example, both 9/12 and
10/12 go back and forth between a number of intervals. 10/12, for
example, goes back and forth between 7/4, 9/5, and 16/8, and all of
these are in the general "vicinity" of the 2^(10/12). Which one is
"the" interval here?

How about for 6/12? That's equidistantly placed between 7/5 and 10/7.
Which one of those is "the" interval here?

-Mike

On Sat, Jun 19, 2010 at 12:43 AM, rick <rick_ballan@...> wrote:
>
>
>
> Hi everybody,
>
> While extracting the info from the waves proved very difficult, the end result itself is very simple. If x = interval, we simply take
>
> x*k/(x - 1), where k = 1,2,3,... and round to get our numerator p,
> k/(x - 1) and round to get our denominator q, and of course take p/q.
>
> These will give our *k'th* wave convergent. The strength of intervals depends on both the lowness of the convergents and the distance of x from p/q. The 12 tempered fifth for example gives {3/2, 3/2, 3/2,...} all the way and the distances from 3/2 are very small. Therefore it is a very good approximation. As an example, I've attached a file showing the first 10 wave convergents for the 12 Tet intervals.
>
> Hope this comes in handy
>
> Rick
>
>
>

🔗rick <rick_ballan@...>

Notice also that intervals b/w higher harmonics appear closer to unison toward the left. The triangles appear smaller because there is less manoeuvring room b/w intervals. The largeness of the simpler intervals (to the right) show that there is much more room for error. OTOH, the larger convergents give tighter triangles.

The formula for the max value of x for the (p - q)th convergent is x(max) = [(p - q)/(q +/- 1/2)] + 1. (Whether this is inclusive or the cut off point (discontinuity) I don't know).

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > Well to answer the second question first, the reason is they are not really convergents. This is why I renamed them (or at least nick-named them) 'wave convergents'. They are defined by rounding in the way I gave previously.

The numbers you get by rounding in the way you describe include the semiconvergents and hence convergents of the usual definition. After pruning the list for best approximations, you end up with the same best approximations you would have gotten from the semiconvergents list, you just check more of them. If you keep putting up 17/9 until something better appears on your list, which I thought was your plan, then you get a whole boatload of 17/9s.

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Rick,
>
> How do you pick which wave convergent from the list gets to be "the"
> one for some equal tempered interval x? For example, both 9/12 and
> 10/12 go back and forth between a number of intervals. 10/12, for
> example, goes back and forth between 7/4, 9/5, and 16/8, and all of
> these are in the general "vicinity" of the 2^(10/12). Which one is
> "the" interval here?
>
> How about for 6/12? That's equidistantly placed between 7/5 and 10/7.
> Which one of those is "the" interval here?
>
> -Mike
>
Hi Mike,

Well we need to strike some type of balance between the lowness and regularity of the WC's against the smallness of R. (I suspect there is a formula for this but Mathematica won't give me the zeros for the rationals so I'm finding it hard to gauge this for every instance).

2^10/12, for example, goes back and forth between 7/4, 9/5, and 16/8, and all of these are in the general "vicinity" of the 2^(10/12).

7/4 will appear at every 3rd WC, 9/5 at every 4th and 16/9 at every 7th. This has to do with the fact they correspond to periodic cycles within the wave.

How do you pick which wave convergent from the list gets to be "the"
> one for some equal tempered interval x?

Well, the fourth and fifth check out in all departments so there is no tonal ambiguity there. Finding "the" interval presents no problem. OTOH these ambiguities are mainly arising for intervals more remote from the tonic, which is not completely unexpected (b5, and those above the fifth). From standard harmony we know that they can be arrived at either as inversions of the 'simpler' intervals in the lower half (e.g. min 6 is an inverted maj 3) or as simpler intervals from other degrees (e.g. maj 7 is maj 3 from fifth). Looking at the maj 7 as an inversion of the semitone as 18/17 we get 17/9 which is in the list. Or as 5/4 from 3/2 we get 15/8 which is also there. And these are very close to the tempered interval x with a small R (I gave an example in the graphs). The fact that it's first three are the 8ve 2 also indicates the tonal weakness of this interval. For these, R is large, meaning that it can be seen as a badly tuned 8ve.

But to answer your question about finding "the" interval, I'd found that for *rational* x there was one element that was closest and gave smallest R. But I hadn't checked this for irrational x and found that its not that straightforward, unfortunately. If you see something I've missed please let me know.

Rick

>
> On Sat, Jun 19, 2010 at 12:43 AM, rick <rick_ballan@...> wrote:
> >
> >
> >
> > Hi everybody,
> >
> > While extracting the info from the waves proved very difficult, the end result itself is very simple. If x = interval, we simply take
> >
> > x*k/(x - 1), where k = 1,2,3,... and round to get our numerator p,
> > k/(x - 1) and round to get our denominator q, and of course take p/q.
> >
> > These will give our *k'th* wave convergent. The strength of intervals depends on both the lowness of the convergents and the distance of x from p/q. The 12 tempered fifth for example gives {3/2, 3/2, 3/2,...} all the way and the distances from 3/2 are very small. Therefore it is a very good approximation. As an example, I've attached a file showing the first 10 wave convergents for the 12 Tet intervals.
> >
> > Hope this comes in handy
> >
> > Rick
> >
> >
> >
>

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > Well to answer the second question first, the reason is they are not really convergents. This is why I renamed them (or at least nick-named them) 'wave convergents'. They are defined by rounding in the way I gave previously.
>
> I don't see how you can avoid getting 17/9 as a "wave convergent" without making an arbitrary choice somewhere along the line.
>
Well you probably missed some earlier posts. I derived these from finding the *closest* periodic extrema of the original wave at times t = (2(k_1)+1)/2(a+b) to those of the envelope t = k_2/(a-b). By taking round [ak_2/(a-b)] = p and round [bk_2/(a-b)] = q we obtain k_1 = p+q and k_2 = p-q. The WC's are then p/q. This might seem like a 'jump' but there's months of work behind it (which I won't restate now). I'll write it all up properly when I'm finished.

🔗Andy <a_sparschuh@...>

🔗Jeff Graham <grahams828@...>

David Benson has an online book with a chapter on continued fractions. Google it up.

--- On Thu, 6/24/10, Andy <a_sparschuh@...> wrote:

From: Andy <a_sparschuh@...>
Subject: [tuning] Re: Method for finding the closest rational
To: tuning@yahoogroups.com
Date: Thursday, June 24, 2010, 3:19 PM

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:

> 2^10/12, for example, goes back and forth between
> 7/4, 9/5, and 16/8, and all of these are in the general "vicinity" > of the 2^(10/12).

Hi Rick,

simply try out
http://www.wolframalpha.com/input/?i=continued+fraction+2^%2810%2F12%29

oder if you do understand some german:
http://primzahlen.zeta24.com/de/online_kettenbruch_rationale_approximation.php

bye
Andy