The angular nature of curves

The angular nature of curves { ... or why we HEAR things that we can NOT hear ... }

My DIY speakers cover the whole range of audible frequencies. The whole range of frequencies that I can hear. But what about the frequencies that I can ***NOT*** hear? As it turns out, the "audible" range of frequencies from a loudspeaker system is simply ... just not good enough.

Do you like bats? Keep them as pets, maybe?
Do they like to listen to your music?
Here is an idea that you may wish to implement for the sake of your pet bats.

Or even for the sake of Yourself, if you come to think about it.

Let's concentrate for a moment on those frequencies that we can NOT hear.
Wouldn't it be nice if you had the reassurance that those frequencies are “also there” ?
Just for the sake of it ?
Sounds like Heresy ?
Overkill ?

Yes, definitely heresy and overkill ... that is what you are probably thinking right now.

But for me - not necessarily.
Indeed - I need them.
I want them.
I wnat them to be *** there *** .

OK, OK, .....

Come now, agreed.
You can *** NOT *** hear a separate, single sine-wave tone, one that ranges within the 20 kHz to 40 kHz region.
That is a fact.
Especially if you are older than 4 months old.
It is a fact which I shall not even try to contest at this point.

But there is a “small caveat” to the aforementioned.

Actually, you *** ARE *** able to hear the small little differences in the sound, a “coloration” of the high frequencies.
A coloration that takes place, if the music material, the "signal", if you will, is impaired, limited by the “slew rate”,
or slope rise / fall time, or "lack of quickness of change" within the music transients as represented by your speaker system.

Especially those transients, very "dense" pieces of your music, ones that contain lavish amounts of high frequency content.
If your speakers do not faithfully convey, or "pass on", those ultra high order frequency harmonics that are contained within your music material,
if they can't convey the “transients”, then this will result in an effect that you will be hearing a “coloration” .... or in other words, an “unnaturalness” of your music.

The lack of those higher order frequencies simply translates to transient slopes that are not "as steep" as they should be.

Funny thing about this "slew rate" thing. People tend to kill themselves within the power amplifier domain, so as to boast that the amplifier is "as fast as lightning", "quick on transients", etc,
which literally translates into a very high value of the maximum slew rate limitation, popularly (and erroneously) simply called "slew rate" which is expressed in Volts per Microsecond.
Yes, the parameter literally means, what is the maximum speed of Volts-change, or ... how fast can you amplifier change the voltage on it's output from one extreme to another.
Indeed, within reason, the bigger this value, the better (although there are strings and trade-offs attached to that).

But how come we end up with these ultra fast amplifiers, and fall into complacency ?
What about the speakers ?

Similarly, as when an amplifier, which is lacking those higher order frequencies, produces a signal, the transients of which are slow, sluggish, and simply not "fast" enough,
the same simple logic would suggest, that a similar issue also applies to the speakers. Speakers are also a part of the signal path, and a very important one.
As the saying goes: the weakest link in the chain is the crucial one.

But how is this so, that these very fast rise and fall times have anything to do with ultra-sonic frequencies?
And what has this do to with pet bats ?


Your music is not a trivial sine wave.
If we were to listen to sine waves, then OK, you are right, I rest my case. Fully agreed.
There would be no point in talking about the 20 kHz to 40 kHz frequency range, as we indeed have no chance of hearing anything above 20 kHz.

But … the thing is, ... I am an Audiophile, and not a Frequency-Generator-O'Phile.
Listening to sine waves is very boring.
I have yet to meet an audiophile that listens to a frequency generator, and not to music.
(... with the not-so-rare exception of some piece of audio equipment placed on a workbench, under repair or under test).

I much prefer listening to music, rather than to frequency generators.

It just so happens, that as opposed to a single tone, a single sine wave, normal music DOES indeed contain very steep edges in some places of the graph, as looked upon within the time domain.
But let's skip the "normal music" case, which is ephemeral and hard to visualize.

Lets concentrate on a much simpler example and analyze it:

A compound signal, consisting of three sine waves.
A superposition of 1). a "base" frequency F1, and 2). it’s third harmonic F3, and finally, 3) its fifth harmonic, the F5 frequency.

Such material resembles a square wave.
If you do not believe me – simply throw such a superposition of three sine waves on the screen of your oscilloscope, and you shall see.

The thing is, superpositions of a base frequency with a fair proportion of it’s natural higher order harmonic frequencies actually may result in "close to vertical" edges within the signal that you are listening to.

Lets assume that your base frequency is: F1 = 8 kHz.
Three times eight gives you the value of your third harmonic frequency: F3 = 21 kHz.
Five times eight gives you the value of your fifth harmonic frequency: F5 = 40 kHz.

What follows is a picture depicting three graphs, related to the frequencies as stated.
The picture is sourced from a very interesting article about the "Fourier Transform of the Square Wave", which happens to be a subject that is very much related to what I am talking about here.
So in the case you have the urgent need to dig into this deeper, and have the spare time, please browse through this following link, just for starters:

But returning to the picture:

What you see in the picture above is three signals, as represented in red, and consisting of “overlapped” component frequencies.
In simple words: the red line constitutes the "sum", or "addition" of the two thin black lines that constitute the component frequencies.

Now, ... please look at the UPPER graph.

The “widest” frequency, a single black “sine wave” – is the base frequency F1.
It is easy to identify, as it only has one single "top hump".
But on this very same graph, the one on top, you also see a smaller, “third harmonic” frequency F3, also in black, and it is superimposed upon the base frequency F1.
How do I know that it is the “third” harmonic? Easy. Simply count the number of humps on it.

Now, having these two, we "add", or superimpose them. The results are in red. What you see in the color red is the result of such a summation, or superposition, if you prefer.
The red line, which is the sum of the two frequencies F1 and F3, starts to resemble a … square wave.
Now, you may say that this is a long shot, that it is hugely "imperfect" on that upper picture, but the “first impression”, a "first resemblance" of a square wave is already there.

Now, ... please look at the MIDDLE graph.

On that Humpty-Dumpty mediocre square wave, as copied from the top graph, we now superimpose yet another frequency, an additional harmonic, and mainly the 5-th harmonic, F5.
Both the copied signal, as well as this new addition, the F5, are presented in the color black.
Now, do you see what happened?
Again, we add up the two, and present the resulting signal line in the color red.

The red line starts to resemble more closely the “ideal” square wave.
It is still not perfect, but the resemblance is much better.

Please note how the leading and trailing edge of this “imperfect square wave” become more and more steep, as we gradually add the higher harmonic frequencies.
The steepness of the rising and trailing edge of this "square wave" is much more pronounces, as was the case on the upper graph.

Now, ... please look at the BOTTOM graph.

On the bottom graph, yet again, we make a black copy of the red line as obtained on the middle graph, and again, we add yet another harmonic, yet a higher one, and mainly the 7th harmonic, F7.
Again, you can see how the sum of these signals, the resulting red signal line, is “almost” an “ideal” square wave.

But there is a vast difference between “ALMOST” and “IDEAL”. This difference consists of all the rest of the harmonics, of all the harmonics that are yet MISSING.

So, now you may rightfully ask: So, ... how many harmonics are still missing, actually ?

The answer is: INFINITY.

If you were to strive to achieve an IDEAL square wave, you would need to provide an infinite number of additional odd-order harmonics.
In other words: you would need to provide an infinite bandwidth.
A bandwidth with no end.
You would have to add an infinite number of higher harmonic frequencies.

So, now you get the picture? Look at the whole of the above "in reverse".

If your loudspeaker system is not capable of conveying an infinite bandwidth (oops, that is not technically possible), or at least the ultra-high frequency content, say of at least up to 40 kHz (more = better), then the “coloration” of your base F1 frequency, and the “way” it "looks" and “sounds”, will be just a “bit different”. It will be lacking the “steepness” of those square wave edges,
the edges that NEED to be faithfully represented from time to time.

And it is this tiny “bit of a difference” that is simply speaking just not good enough. Without the 40 kHz (at least!) of bandwidth, faithfully represented by your ultra-high frequency tweeter, super-sonic tweeter, if you prefer, your musical “square wave” response shall rather resemble the sluggish "top graph", and not the "bottom one".
Your music will be just as that top graph: slow, colored, flimsy, without the air, without the details.

This is why the title of this text is "The angular nature of curves".
We started of with seemingly innocent sine waves, each and every one of them with nice curves and graceful looks,
but end up with an almost angular signal, with ultra steep slopes and very short rise and fall times.

And this is just a superposition of a FEW frequencies. Now, imagine the mess that you have there in the case of a real music signal.

As with any audio project, the moment that you think that you have just “finished” it …
... that is more often than not exactly the moment, when you come up with your “yet another wild idea”.

And so was the case this time.

Just as I finished my Jensen 1071 DIY project, as per design of Mr., I came up with this idea.
A Modification.
An ultra-high-frequency-tweeter.

Something for the Bats. Ribbon tweeters … or any other super-sonic tweeter technology, of your personal preference.

For the sake of a simple experiment, I purchased, after thinking about this subject for some time, some extra transducers: super sonic ribbon tweeters.
Just as before, I purchased them at, and they arrived promptly, before the Englishman in New York was able to pronounce: “ChrzÄ…szcz brzmi w trzcinie”.

You may ask: why “ribbon”?

Good question. It can be any other super-sonic tweeter technology, one that is capable of running up to 40kHz without significant distortion.
Well, the thing is, the ribbon tweeters were simply readily available and obtainable within a convenient source at hand,
one that I have previously tested and knew that the guy's deliver timely and with no frills.

Besides, the mass of the ribbon within such a ribbon tweeter is very, very small.
This indeed translates into very low THD distortion, especially in the higher frequency ranges.
At the same time, the ribbon tweeters have a very high efficiency, in the order of 95 to 100 dB, some models even more.

If we make a crossover for such very high frequencies, as I am considering to obtain here, the signal on the other side of the cross-over may actually turn out to be very weak.
So this pairs well with the high efficiency of the ribbon as such.
Actually, there might even be a need to attenuate them slightly, but having said that,
we know how to build the (probably) best non inductive resistor in the world, so that is absolutely a non-issue here.

I attached a photo of these new super-sonic capable ribbon tweeters. They are already in action. My listening test is ongoing.
All the bats are sitting (hanging?) in the living room and enjoying Sting with his "Englishman in New York" thing …

And I herewith confirm: YES, there is an audible difference in the way the sound … sounds.

The whole upper midrange, as well as the very high frequency range has received a whole bucket-load of more “air” …
This is just but a preliminary, crude, out-of-the-box setup, a hasty “lets hook them up ASAP and listen to the resulting sound" kind of setup.
Little matchboxes serve as super-sonic tweeter stands, and crocodile jumper cords perform the function of a crude hookup cable.
Not to mention the very crude crossover to accompany them, which is a rude 6dB / octave high pass filter, with a cut-off frequency, which is presently set to 12 kHz.

But the effect, the *** AIR *** .... yes, it is *** THERE ***.

I like the results. I am far from saying that the original Jensen 1071 design is less than perfect. It is indeed a graceful sound, very balanced. very good.
We are simply talking about personal tastes and preferences here. I simply like them just a bit better with the super-sonic extension thing.

OK, so this is just a crude setup, yet another case of work-in-progress.
A long way to go, from this current state, to the final add-on boxes version, with the carpentry,
the woodwork, the varnish, the crossover, setting up correct phase relationships, measurements and stuff.

But the *** Wow !!! *** effect is definitely there.

I am far from saying that I chose the best super-sonic extension solution, the best solution that is out there. I made a wild guess and am happy with the results.
Obviously, there are some even better options out there, in terms of super-sonic tweeters. With better dispersion characteristics, etc., etc.
As always, it basically boils down to how much you want to splash out on such a super-sonic extension.
But in the general terms - now you know what you are hunting for, so I wish you some happy hunting.

And in case of any doubts - consult the issue with your pet bats.