Article re Tractrix Horn

[WMcD]

Here is information on the development of the original Klipsch tractrix horn. This is the one which went in the Forte II, Chorus II, and Quartet. I'm sure the later ones are different.

Gil

[WMcD]

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Sorry, it took me a while to get the graph to post. And it is a bit large. Naturally those are my annotations to a set of graphs from the article.

The article is important because it gives us data from Klipsch on the "Tractrix." Equally important is that it gives us information on the exponential.

Please recall that I've long harped about how the Klipsch exponentials work. This is not entirely that they are exponential. Rather it is the matter the the mouth for the exponential is about 3 times wider than high. You have that aspect ratio. Naturally the overall AREA at the mouth is set by the exponential equation. Let me say the mouth area is small relative to the tractrix mouth area.

There is also something to the angles of the walls. But I'll mention that later.

The article explains that the tractrix AREA is bigger at the mouth. So we have more area to play with.

The major issue here is the resulting radiation patterns. The graph shows these.

I must say the article is disappointing in that the pictures on page 1 and page 2 only show the dimensions in the horizontal . . . there with dry ice smoke. The good authors do not discuss the effect of the vertical size; but it apparent in the graphs.

Both of these horns are oriented with the bigger axis in the horizontal plane. Call it landscape. This is the orientation in the Forte and the later Corwalls.

We see that the geometries of the walls and mouth size in the horizontal plane are similar in the two horns. Look at page 1 and 2. To the extent the tracrix has magic, it is not magic here.

Looking at the graph, we see the coverage angle versus frequency in the horizontal (and landscape orientation), are the same too. The mouth size is too small to control coverage at low frequencies. However, up around 1.8 kHz, they both narrow down to 60 degrees and this is held up the frequency range. Hence both designs have equal merit.

The result is that, with this landscape orientation, there is little off axis radiation toward the walls of our rooms. This is because the pattern has narrowed to 60 degress except at the low end.

Of course if we rotate the horn by 90 degrees (as with the vertical Cornwall, making it portrate) then there is little radiation toward the floor and ceiling. So you pick your orientation and what is to be avoided.

The most striking difference between the two designs is in the vertical radiation pattern. Again, unfortunately, the article does not give a comparison of the geometries which cause this. However, the so called Tractrix horn in the Forte II is much taller. This is possible because of the rapid increase in AREA at the mouth.

Let us look at the difference in vertical radiation.

First, the exponential.

The first intuitive thought is that it should react as if there is a flashligh back at the driver projecting through the pattern of the mouth. I.e. the the vertical angle of radiation is narrow.

But this is not true at all. We see that the vertical pattern is 180 degrees for most of the test frequencies. (Probably the horn tested was in flat panel, and this is as far off axis as tested.) This means there is no narrowing and the sound is being sent toward the floor and ceiling. Eventually it does narrow.

Look at the tractrix.

In contrast, the coverage angle narrows much sooner as we go up in frequency. Therefore the tractrix with its taller mouth (we have more AREA to allow this) is keeping sound off the ceiling and floor.

This is again contrary to the flashlight analogy. If that held, the radiation through a tall mouth of the tractrix would be more broad. In reality, it is not.

This is good in that we can now have a narrow horizontal dispursion, and also a somewhat better vertical dispursion, at least in this design.

As I've written before, Don Keele describes why this works in his constant directivity horns, which have to do with angles of conical horns and mouth size. The good folks at Klipsch found that the tractrix design could be implemented in a way to follow his analysis.

I'll try to answer questions. The important thing is that we have some hard data to look at.

Best,

Gil

[sunnysal]

Facinating Gil! thanks for posting all this...it leads me to wonder that why, when faced with the facts about shortcoming, klipsch did not work on improving the horn design? or did they?

Have you been following any of the work by Dr. Earl R. Geddes on horn design? a friend of mine over at peavy recommended this guy as a true maverick genius in horn design (sounds little like PWK no?) let me know what you think, http://www.gedlee.com/

warm regards, tony

[edwinr]

Thanks, Gil. Interesting reading. Pardon the question, but the article states that they had problems with the tractrix horn, I presume when they were testing the design from 600hz up, and moved the the frequency response of the design up to 7khz. If I've read that right. From this frequency the tractrix is a successful midrange transducer and was used in subsequent designs. Is the exponential design 'better' at lower/mid frequencies? For example in the Klipschorn the exponential midrange horn operates from around 450hz.

[colterphoto1]

Thanks bunches Gil, nothing better than a lengthy 'white pages' read! I'm considering changing my Cornwall horns out to Tractrix design. Do you have any experience on this mod or know anyone who has done it successfully? It might also take changing out the mid driver and possibly crossover points, so I don't want to get in over my head.

Michael

[WMcD]

To answer edwinr. I don't have the article before me.

I believe they were saying they were having problems with the pure tractix which has a round cross section. Dr. Bruce Edgar had a number of articles in SpeakerBuilder about his correspondence with Mr. Voight, and then Bruce's work to build tractrix based designs.

My understanding is that horns with round cross sections have a problem with directivity. As frequency goes up, the dispursion pattern narrows. Since it is round, that means it narrows in all axis. That goes for exponential and, from what little the article says, the tractrix too.

The only round design which keeps a constant coverage angle is a cone. Except that a final rapid flare is necessary to make that work, as found by Don Keele. Unfortunately, conical horns don't present a good load at low frequencies. Keele solved that with a hyperbolic section at the throat. He therefore perfected constant directivity horns. He has three patents on various designs . . . now expired.

I wouldn't say that the three Klipsch engineers failed to perfect anything. As they point out, much of the design process is to find a good compromise.

One thing which I've also harped upon is that all drivers have a problem. Eventually, the high end output starts to roll off. And horns can help.

All horns except the true constant directivity types start with a wide radiation pattern and then narrow down.

I've not seen it descibed in the classical books but a narrow beam pattern almost always equates to on- axis "gain", at least when compared to the broad pattern. This is well known in antenna design.

Here the flashlight analogy holds true. It is actually the use of a parabloic reflector. None the less, in the Maglight we can vary the width of the beam by twisting the head.

At optimum, the filament of the bulb is at the focus of the parabolic reflector and the beam is narrow. We have an intense result in a narrow pattern.

We can twist the head an a mechanism moves the filament out of the optimum point. Now we have less intense light over a broader area.

Naturally the little old light bulb is always putting out the same energy.

So you can see that if the horn can narrow the pattern just when driver output is falling, this compensates for driver roll off. So sometimes non constant directivity is a blessing.

If you have a constant directivity horn and the driver response is falling . . . you've got a problem. However, various forms of electrical equalization (boost the treble input) can solve this. It can be in an active or passive crossover. Naturally in a passive, we're really just attenuating the lows, rather than boosting the highs . . . but same effect.

Tony, thanks for the heads up. I'll put his book on the Santa list.

- - - -

As an aside for the technically intersted.

The original tractix design was based very much on the geometry of the wave being propagated down the horn. The interest was the geometry of the side wall and not the area of the expansion . . . though naturally they are related. The "tractrix" is the distance of the side wall from the center axis.

However, arguably, we can have horn designs which say: a pure tractix would have this AREA at this point even if the side walls are not the tractix curve from the main axis. So my area is that of a tractrix. Does that make it a tractrix?

On the other hand, the exponential design starts with the area of the horn as we go along the length. There is the e to the xM power equation. However it is just like a bank account with some interest rate. Say every year you get 5% interest. So it grows to 1.05 of the previous starting point, every year. Eventually the savings account will double; and double again. PWK called this doubling distance Lamed (Hebrew letter L). Sorry if that spelling is not correct. Celtic Catholics don't get to go to Hebrew school. Smile.

I just point out that if you find the exponential equation frightening, but do like banking, you're doing the same thing. Lamed is like the Rule of 70.

I looked at the K-400 design and found that the equation for the side wall displacement from the axis is also an exponental function. Wow.

PWK was quoted as saying you make an exponential horn with the area equation and a boundry rigid enough to hold the air. "Then you just have to figure out what shape to make it." So he was very much into the many shapes which can result from the area equation.

In view of the above, I cringe a bit when anyone writes either exponental or tractrix to describe a horn. There are many variations to both.

Gil

[edwinr]

Thanks again, Gil. Interestingly, I 've just read a review of a 'full range' horn speaker named Ferguson Hill model FH001. The horn is constructed of some sort of perspex and the driver appears to be a lowther type. Frequency range is quoted as 150hz to 20khz, so a subwoofer capable of extending to 200 hz would be required. The reviews I've read seem positive. Out of interest I've downloaded a pic. It's not the best but will give you an idea. This supports your view that there is more to horn speakers than non Klipsch enthusiasts would have you believe.

[WMcD]

I'm going to bump this article up again, In my view the tractrix midrange in the Forte II etc. looks a lot like Don Keele's design (but he seems to have re-invented the tractrix by his combination of a matching section, conical section, and rapid flare). It also performs like a CD.

Ha, Maron says you can use smoke rings to test a horn. It looks like Klipsch was experimenting with dry ice smoke.

I'll say it for the last time this week: You all really, should read the Don Keele papers.

Gil

[PrestonTom]

I'm going to bump this article up again, In my view the tractrix midrange in the Forte II etc. looks a lot like Don Keele's design (but he seems to have re-invented the tractrix by his combination of a matching section, conical section, and rapid flare). It also performs like a CD.

Ha, Maron says you can use smoke rings to test a horn. It looks like Klipsch was experimenting with dry ice smoke.

I'll say it for the last time this week: You all really, should read the Don Keele papers.

Gil

Gil, I had a look at this article yesterday. Thanks for posting it.

My confusion, which is evident in the "other" thread is in part based on my limited viewing of the tractrix horns. I only have some photos of the Klipsch versions and the Edgar versions.

I had forgotten that Klipsch used tractrix in their non-commercial equipment as well. When I think of Klipsch gear for the home it usually boils down to the "Heritage" series. My view of the world is limited at times.

I will dig deeper into the Keele patents. I have read his AES manuscript (Whats so sacred ...) That in itself is a wealth of information.

-Tom

[ZAKO]

My smoke ring comment was to point to this reprint in Audio/ March 1991...apparently it was dry ice... I wish the article was more extensive.

[bodcaw boy]

To answer edwinr. I don't have the article before me.

I believe they were saying they were having problems with the pure tractix which has a round cross section. Dr. Bruce Edgar had a number of articles in SpeakerBuilder about his correspondence with Mr. Voight, and then Bruce's work to build tractrix based designs.

My understanding is that horns with round cross sections have a problem with directivity. As frequency goes up, the dispursion pattern narrows. Since it is round, that means it narrows in all axis. That goes for exponential and, from what little the article says, the tractrix too.

The only round design which keeps a constant coverage angle is a cone. Except that a final rapid flare is necessary to make that work, as found by Don Keele. Unfortunately, conical horns don't present a good load at low frequencies. Keele solved that with a hyperbolic section at the throat. He therefore perfected constant directivity horns. He has three patents on various designs . . . now expired.

I wouldn't say that the three Klipsch engineers failed to perfect anything. As they point out, much of the design process is to find a good compromise.

One thing which I've also harped upon is that all drivers have a problem. Eventually, the high end output starts to roll off. And horns can help.

All horns except the true constant directivity types start with a wide radiation pattern and then narrow down.

i have measured some "true constant directivity types" horns not really hold constant coverage patterns but still have constant directivity while i have measured some non-"true constant directivity types" horn measure alot better in the constant coverage arena and by definition of constant directivity, should have been called cd horns. so what is the goal?

I've not seen it descibed in the classical books but a narrow beam pattern almost always equates to on- axis "gain", at least when compared to the broad pattern. This is well known in antenna design.

Here the flashlight analogy holds true. It is actually the use of a parabloic reflector. None the less, in the Maglight we can vary the width of the beam by twisting the head.

At optimum, the filament of the bulb is at the focus of the parabolic reflector and the beam is narrow. We have an intense result in a narrow pattern.

We can twist the head an a mechanism moves the filament out of the optimum point. Now we have less intense light over a broader area.

Naturally the little old light bulb is always putting out the same energy.

So you can see that if the horn can narrow the pattern just when driver output is falling, this compensates for driver roll off. So sometimes non constant directivity is a blessing.

If you have a constant directivity horn and the driver response is falling . . . you've got a problem. However, various forms of electrical equalization (boost the treble input) can solve this. It can be in an active or passive crossover. Naturally in a passive, we're really just attenuating the lows, rather than boosting the highs . . . but same effect.

Tony, thanks for the heads up. I'll put his book on the Santa list.

- - - -

As an aside for the technically intersted.

The original tractix design was based very much on the geometry of the wave being propagated down the horn. The interest was the geometry of the side wall and not the area of the expansion . . . though naturally they are related. The "tractrix" is the distance of the side wall from the center axis.

well kinda of true....the geometry of a single wall is called "schiele's anti-friction curve". the tractrix equation calls for area expansion.....

However, arguably, we can have horn designs which say: a pure tractix would have this AREA at this point even if the side walls are not the tractix curve from the main axis. So my area is that of a tractrix. Does that make it a tractrix?

if it follows the area expansion dictated by the equation, then yes it is a tractrix. the area equation, exponential, hyperbolic, conical, etc describe area expansions. they offer no advice on tying up degrees of freedom. so as paul used to say, you can ask 10 different people to design a 500 hz, exponenetial horn and you will get ten different horns. and all ten horns would be exponential horns. why is that; not enough information. but if you throw in, let's say, 90 x 40 coverage, you begin to narrow the degrees of freedom done and all ten horns begin to start to look similar.

On the other hand, the exponential design starts with the area of the horn as we go along the length. There is the e to the xM power equation. However it is just like a bank account with some interest rate. Say every year you get 5% interest. So it grows to 1.05 of the previous starting point, every year. Eventually the savings account will double; and double again. PWK called this doubling distance Lamed (Hebrew letter L). Sorry if that spelling is not correct. Celtic Catholics don't get to go to Hebrew school. Smile.

I just point out that if you find the exponential equation frightening, but do like banking, you're doing the same thing. Lamed is like the Rule of 70.

I looked at the K-400 design and found that the equation for the side wall displacement from the axis is also an exponental function. Wow.

PWK was quoted as saying you make an exponential horn with the area equation and a boundry rigid enough to hold the air. "Then you just have to figure out what shape to make it." So he was very much into the many shapes which can result from the area equation.

In view of the above, I cringe a bit when anyone writes either exponental or tractrix to describe a horn. There are many variations to both.

Gil

have a blessed day,

roy delgado

[bodcaw boy]

I'm going to bump this article up again, In my view the tractrix midrange in the Forte II etc. looks a lot like Don Keele's design (but he seems to have re-invented the tractrix by his combination of a matching section, conical section, and rapid flare). It also performs like a CD.

that's like waving a red flag in front of an old pussycat like me......

it may look like keele's design but it is far from it. don't get me wrong. i have alot of admiration for don. he is truly one of the worlds living legends. but as he and i have discussed before, the sole goal of constant directivity was constant directivity. and he meant it to be that it implied constant coverage. but somtimes that is not the case. it also useful to mention that don's track down constant directivity land was inspired and fueled by the k-5j horn. and in case some people didn't know, don worked a klipsch for awhile. while at klipsch he measured the k-5j and became fascianted with its ability to control coverage in the horizontal.

now back to the comment, it may look like it but it is not constructed using the formula for constant directivity; exponential for loading, conical for pattern control and hyperbolic for launching the wave. my whole concern with this was the other things that were given up by just focusing on coverage. i thought that it could be achieved without having some many discountinuties. that is what the modification to tractrix turned out to be.

Ha, Maron says you can use smoke rings to test a horn. It looks like Klipsch was experimenting with dry ice smoke.

I'll say it for the last time this week: You all really, should read the Don Keele papers.

Gil

have a blessed day,

roy delgado

[jorjen]

Roy,

Sent you a PM.

Jordan