recording/music theory type question

  • Thread starter Thread starter KingstonRock
  • Start date Start date
K

KingstonRock

PC load letter?
The other day my math teacher, in introducing trigonometry, started talking about sine waves in music. He was saying how different instruments sound different because the sine waves they create are each distorted in a different way (as in mishaped). But I *know* thats not true, because last year I did a research report on sine waves in music and read all about how the the lowest string on a guitar when hit is not only an E, but all of its other 31 audible harmonics. Later, I realized maybe he wasnt wrong because I've zoomed in on a track in sonar, and individual instruments are only represented as one wave, though they originally resonated at many frequencies. If you zoom in enough, I understand you can see evidence of the sampling rate in the steps of the wave, but it still doesnt look like it was a true sign wave, even before a/d conversion.

Here's where the real confusion begins. If you have 10 different guitars, you can play 10 different E's, but they certaintly won't sound the same. If we were to assume that the wood of the guitar, its shape, weight and overall design influenced the expression of those 31 audible harmonics to change the sound from guitar to guitar, than how are those 32 sin waves summed in a recording? It kind of makes sense that a natural summing would occur as the overall effect of all the waves would cause a single osscilating electrical signal within the electronics, as they interacted with the pickups; this would also make sense as it probably occurs at the vibrating membrane of a microphone. This also brings up another question, I've heard a good distortion musically excites the upper harmonics, how is this? It makes sense that it might naturally limit the more predominant lower harmonics, therefore bringing out the upper ones, but an electrical signal is only one wave, its no longer those 32 different notes.

There is a lot of stuff up there, but I guess it really comes down to, How are multiple frequencies respresented in a single wave?

Also, now i see why bouncing to tracks digitally isn't preferrable, because the transistion from maybe 16 different waves to 2 has to be done by math, which always seems to mean rounding , and therefore innacuracies.

I actually have some more questions, but I'll probably be able to answer them myself if I understand this first.

Eric
 
The differences between those guitars (or anything else producing pitched sound, for that matter) is going to be in the relative amplitudes of those sine waves. Remember also that the amplitudes will vary with time, and while some will be pure harmonic freqencies of the root note, others will be fixed resonant frequencies (& related harmonics) of the various physical components of the instrument. Also, you will hear the 'phantom' sum and difference frequencies as these components interact. As you might begin to suspect, just creating a mathematical model of a single note on a single instrument could fill many blackboards. When you start accumulating enough of these constructs to try deriving behavioral formulae about the instrument, you're getting into the realm of the physical modeling synthesis engines that started appearing a few years ago. The processing power required to calculate the sound of notes played in real time is immense, that's why physical modeling synths only started appearing in the late 90s, quite expensive, usually monophonic (or with a very few voices) and not really achieving a convincing sound - though they did a better job than most other synthesis engines.

Scott
 
The only orchestral instrument that approaches a pure sine wave that I'm aware of, is the flute - the other instruments are, as pointed out, full of harmonic content that looks nothing like a sine wave. You got a good look at reality when you zoomed in on the track in Sonar -

If you are lucky enough to have access to an oscilloscope and a Hammond organ, you can experiment firsthand with additive synthesys - hook the scope up to the output of the organ, and alter the drawbars while listening and looking at the scope display - note that the only time you see a sine wave is when only ONE of the drawbars is pulled out. As you pull more of the drawbars out, the waveform gets more complex -

this gets to your main question, "How are multiple frequencies respresented in a single wave? " - The answer to that lies (at least partially) in additive synthesis. Nearly all instruments have harmonics in their characteristic tones, caused by things like the body of the instrument, string harmonics, playing technique modifying string harmonics, etc - since a single electronic signal can only be at one distinct voltage at one time, what you're seeing on the scope (or Sonar waveform) is the instantaneous voltage per unit of time (sample), caused by mixing several different frequencies together.

Your brain is able to take this complex waveform and break it down into its component parts to some degree - which is why you can pull out the 4' drawbar on that Hammond organ and aurally tell that you added 1 or 2 octaves above the 16' drawbar sound - however, if you look at the resultant waveform on a scope, it's not nearly as understandable until you start to see what each drawbar does to the waveform, then it starts to make sense when you see all the little sine waves superimposed on the bigger, lower freq one.

Just looked back on what I wrote, and it probably just makes things MORE confusing - but what the hell, you were confused anyway... :=)
 
actually i pretty much understand what both of you are saying, because what I didnt really think about before is that the same summing that occurs at the microphone is probably almost exactly the same summing that occurs at the ear. When I did a search on google, I read that a huge amount of math and processing can pick out individual sine waves present in the signal inside the computer (look for autotune 4 sometime next year, LOL), while your brain is doing it automatically all the time.

What makes it confusing is trying to imagine all those negative and postive points in the 16 different sound waves at any given instant in a multirack recording combining. It almost seems like regardless of their frequency they would cancel each other out, but luckily they dont. Its also weird how a speaker vibrating in only one way can reproduce all of those frequencies, and that we can even pick out the different ones. Maybe its not so hard to believe since an eq can alter specific frequencies within that one wave without affecting the others too much.
 
Last edited:
Kingston, you metioned how recordings could be "excited" by taming lower order harmonics and letting upper harmonics come out better. A study of the overtone series shows that the more prominence of higher overtones usually equals a more "brash" type of sound, because higher overtones start getting closer and closer in interval, and also notes that are not divisable by the fundamental show up. The dissonance introduced in these frequencies create tones that tend to sound more "harsh" because of how they effect the fundmental frequency via coupling and phase effects.

Pleasing "distortion" in recordings can mean different things based upon what you want in the sound. There are expectations you have for a sound. If they are not met, the sound quality if poor. Fair enough.

It is interesting to note how sampling rate in digital equates into this discussion (I always look for an angle to talk about sampling rates, or which, the 44.1 standard right now is just plain too low for my liking!!!) With the Nyquist filter cutting the frequency response to half the sampling rate, well, think of a cymbal, or trumpet playing it's note, and how upper harmonics are not present in the record. Indeed, people can make claims that microphones and speakers do not exceed 20Hz because their stated specs say they don't, but that is pure ignorance. Those specs relate to the accuracy of the mic and speakers for frequencies that we can hear. But music is made up and effected by higher frequencies that we CAN'T directly hear, but the overtone series also shows that the presence of upper frequencies effects the lower frequencies even though we can't hear the upper frequencies, there are still physical things happening from the coupling and phasing of the the combined frequencies that effect what we can hear. Even with a phase shift of the frequencies above what we hear, those frequencies are still PRESENT, and effect what we hear! Just because the manufacture doesn't list specs for their gear (well, the cheaper one's don't at least...:)) over 20KHz, doesn't mean that the gear isn't capable of producing energy above 20KHz, and possibly doing so with acceptable accuracy!!! Thus, the capability of actually recording these higher frequencies is important, because when they are not present on playback, we lose the full "realism" of the original sound! Even WITH phase issues from poor design, there is a certain amount of phase shift that is acceptable. Many transformers start phase shifting frequencies staring at 16KHz, but still manage to sound very pleasing to the ear. As pleasing as a transformer that is accurate up to 20KHz? Well, maybe or maybe not. Depends on how much content between 16 and 20Khz is important to the sound. If the fundamentals are up there, of course there would be some severe problems with even a 5% phase shift. But if say the fundamental at the input of the transformer is 250Hz, well, a 5% phase shift at 16KHz isn't going to sound too terribly bad!

Here is a paper written that people who claim that high frequency content (stuff above what we can hear) in digital recordings isn't all that important NEED to read:

http://www.cco.caltech.edu/~boyk/spectra/spectra.htm

Rupert Neve claims that 192KHz sampling rates are going to be needed to get digital recording as good as high end analog. This paper, and some others that I could probably dig up support that fully! SSL consoles boast about the ability to pass frequencies up above 100KHz! People mix with NS 10's on those consoles!!! Indeed, the tweeter on a NS10 isn't going to offer much accuracy that high, but the electronics of the console and how they effect stuff we can't hear sure will become evident on the frequencies we CAN hear.

Anyway, fun topic and worthy of exploration in more detail.

Ed
 
KingstonRock said:
He was saying how different instruments sound different because the sine waves they create are each distorted in a different way (as in mishaped).
Click to expand...

Distorted is not the best word to use there. Teachers water down info for students....

KingstonRock said:
There is a lot of stuff up there, but I guess it really comes down to, How are multiple frequencies respresented in a single wave?
Click to expand...

Do some research on superposition, and Fourier series and analysis if you are adventurous. There are a lot of decent java applets out there that illustrate superposition theory.
 
sonusman - after thinking this through, i definitely agree that 44.1 is way to low, I had always figured just cut if off where we cant here it anymore. But then some said the anti aliasing filters were so close to the audible range that they caused 44.1 to sound bad. But now, the effect of those higher frequencies on the summing of the overall sound at the microphone is not being heard when that summing is occuring again at our ears. That makes sense, right?

Also, when I was talking about pleasing distortion i was talking about a distorted guitar amp, though i suppose the same idea is true for tube/tape saturation, which I guess are all really totally different types of compression than the kind that a compressor will give.

I'm gonna read that paper you suggested tonight
 
That is correct Kingston. There was another thread I care not to mention :) where this was discussed a little bit, but for having to do with how humans react to different frequencies. I think this can come into play too.

Distortion is distortion. There really isn't different types of it, it just simply is! Concerning a guitar, this distortion you are talking about is the extreme and obviously intentional. In overall audio production, it is far more subtle, and often NOT intentional by any means.

By the way, distortion where overtones that are divisable by the fundamental tend to sound more pleasing to the ear than overtones that are not divisable by the fundamental. This would account for why a muted trumpet tends to sound less pleasing to the ear that a guitar would playing the same exact fundamental. The muted trumpet tends to have stronger overtones that are not divisable by the fundamental. Imagine a similar type of overtone, or even distortion applied over MANY tracks! That is of course getting a little off course to what I am talking about. But thought I would throw it in anyway.

Ed

Ed
 
sonusman said:
This would account for why a muted trumpet tends to sound less pleasing to the ear that a guitar would playing the same exact fundamental.
Click to expand...

Whew! Glad Miles isn't still around to hear that... :confused:
 
Ed,

> By the way, distortion where overtones that are divisable by the fundamental tend to sound more pleasing to the ear than overtones that are not divisable by the fundamental. <

There's no such thing as an overtone that is not divisible by the fundamental. Think about it!

--Ethan
 
...Hehehe, this is how Math meets Music. Yup, NOW I'm getting interest to hear about FFT theory, Sinus wave, how the synth really works, etc...etc... Something I realy like to learn. I'll keep my eyes on this thread forward... keep goin' boys...

;)
James
 
ecs113 said:
Maybe he meant sensuous pleasure vs. intellectual pleasure
Click to expand...

i'm not sure what that means. why would i have to choose whether listening to Miles is sensuous or intellectual? can't it be both?
 
Perhaps the teacher was referring to the perceptual difference in the even harmonic series and odd harmonic series? Even harmonics tend to be less harsh, and some actually like the sound their addition gives.
 
There's no such thing as an overtone that is not divisible by the fundamental. Think about it!
Click to expand...

I think that he meant divisible into a whole number. Anyhow, I think appeals to nature are specious. Something is musically pleasing on a personal and/or socio-cultural basis. Human beings can train themselves to feel pleasure from pain. Our aesthetics are not tied to our biology.

Having said that, most of what we find pleasing has been inscribed in our culture by analog media. When you want that sound, that's where you need to go to get it. It all depends on what you are trying to accomplish--you may not even want something that sounds "pleasing," or warm. For example, Bjork's last album sounded cold, beautifully so--and digital was instrumental in attaining that sound and was used effectively.

But this is from a guy who just picked up an old Otari deck....
 
VP,

> I think that he meant divisible into a whole number. <

That's not what he said and he seemed pretty clear.

> Having said that, most of what we find pleasing has been inscribed in our culture by analog media. <

I really doubt that! Most of what we find pleasing in a mix -discounting the music itself - is related to the distribution of frequencies. A punchy kick, a biting lead, a warm string sound - all of those are frequency response issues.

--Ethan
 
Eric,

Actually your teacher was wrong. All waves are not just distorted sine waves. The sound waves from a guitar, for example, may look sort of like sine waves, but the waves from say a bowed violin or cello look more like saw teeth. The waves are just the shape they are according to the mechanism that produced them and not mysteriously forged from some prototypical sine wave. That being said, a very smart person back in the early 19th century named Joseph Fourier (pronounced For-ee-ay) did prove that any wave of any shape can be represented as the sum of some number of sine waves having certain frequency, amplitude, and phase relationships. There are instruments called Fourier Transform Spectrum Analyzers that use the mathematics he developed in order to represent waves as a discreet frequencies. Our ears also have a similar ability. Representing sound waves this way is very useful, but remember that those sine waves are not the source of the sound wave, they are simply interpretations of the wave.

Of course, you can also turn it around and use Fourier's principle to synthesize desired wave shapes by adding up various sine waves. In this case you could say that the resultant sound wave is composed of sine waves. But this is a special case.

Probably the reason you see something like sine waves when you zoom in on your mix is because bass and kick drums have a rather sine like character. If you recorded a cello, oboe, and clarinet trio the waves would look very different.

As far as distortion goes, a guitar pedal for example, doesn’t add frequencies. All it does is change the shape of the wave, typically by chopping off the top and/or bottom. Our ears perceive and our instruments can represent this new shape as if harmonics were added to the original wave. But like I said, all the distortion device did was change the wave shape. And all distortions are not necessarily harmonic in nature either. One could distort a sine wave in a wave editor by redrawing it and coming up with, lets say, something when viewed through a spectrum analyzer that looks like the original frequency plus another peak 3.14 times higher in frequency, then another at 0.734 times the original. These are not harmonically related frequencies, but they represent distortion of the original wave all the same.

Hope this actually helped answer your questions.

Thomas

http://barefootsound.com
 
A punchy kick, a biting lead, a warm string sound - all of those are frequency response issues.
Click to expand...

No they are not. They are aesthetic judgments that are then analyzed and described in terms of frequency response. What we find pleasing about analog has been codified by the music made with it. You can't separate the two.
 
littledog said:
i'm not sure what that means.
Click to expand...
I was referring to music not pleasing to the ear but pleasing to the mind.
littledog said:
why would i have to choose whether listening to Miles is sensuous or intellectual? can't it be both?
Click to expand...
Who said you have to choose?
 
You must log in or register to reply here.
Back
Top