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Old 08-21-2010, 03:01 PM   #1
sandt38

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Distortion. Even and odd order, high and low level.

There are a few aspects of distortion that are important and worth discussing. First off, distortion is present in every phase or component of the audio system… and I mean every phase. There is simply no denying that. But there are different types of distortions, and some are considered objectionable, and others desirable by audiophiles as well as audio manufacturers and designers. I also want to include how society has been conditioned (or dare I say brainwashed, to quote a friend) to actually expect or enjoy what was often considered undesirable harmonics. While I was not intending to include this subjective information here, the friend I quoted above (pjhabit) brought some interesting information up in my MP3s, how and why they affect our systems, and system choices write up, and I felt it deserved a bit of play in this article.

The first thing we need to grasp the concept of is transfer function. Transfer function is the representation, in terms of temporal frequency, of the relation between the input and output of a system. What does this mean? Very simply, it is the relationship between the input signal, and the output signal of a system. For our purposes, it is basically the music or the electronic representation thereof (input signal) and what we hear (output signal) through our entire signal chain. From the input signal to the amplifier, in relation to the output signal of the amplifier (the amplified music), or the input signal to the speaker from the amplifier in relation to the sound we hear from the speaker, transfer function is at play everywhere in an audio system.

Distortions or harmonics are changes in the input signal as it goes through its transfer function in comparison to its output signal. In every audio system, through each piece of the signal chain, certain distortions are realized. The signal on the CD is different from what has been read and processed by the HU. The signal entering our RCA cable is different from the signal entering our amplifier. The signal entering the amplifier is different from our amplified signal. The signal entering our speaker wire is different from that entering the tinsel lead, and the signal entering the tinsel is different from the sound we hear… yes, I know I left out DACs, processors, crossover networks, and many other stages the signal follows, but since this is just for illustrative purposes, I think it will suffice.

So clearly, there are many phases in the system that generates distortions. There is just no way to avoid distortions. However, certain distortions are not desirable (notice I did not say pleasing), such as even order high level distortions, and some are actually desirable, such as odd order low level distortions.

Please keep these 2 thoughts in mind…

Even order harmonics result from asymmetrical nonlinearities of a transfer function.

Odd order harmonics result from symmetrical nonlinearities of a transfer function.

Even order harmonics, represent octave multiples of the fundamental note. Remember, we double a frequency to attain its octave. So they thicken the sound, but continue to emphasize the same tone, just with more octaves mixed in, continually reduced in volume as the octaves climb. By adding to the fundamental note, in the form of related frequencies (ie. Octaves of the fundamental note) we are basically making the note warm.

Odd order harmonics are not octave multiples. While we are multiplying the output, we are not multiplying the octave, therefore we are adding harshness, or an overbearing reproduction of an unreproduced note, simply because the added distortions are not related frequencies. So they also will thicken up the sound, but in a much different way because they aren't replicating the fundamental note. Instead they are creating some kind of virtual chord above the note.

Let’s look at how amplifiers overdrive or clip, as this is a very obvious form of distortion and one we can all relate to. Clearly this is not the only form of audible distortion, and I am aware that not everyone has actually heard a tube amplifier as opposed to a solid state amplifier, but I am assuming we have all heard about how a tube tends to sound warm, or inviting. So based upon that assumption, I am going to use amplifier clipping and how it sounds to help explain the audible difference between even and odd order harmonics. In fact, I will also attempt to use the varying levels of amplifier clipping to help explain the sound difference between high order harmonics and low order harmonics. While this may not be an absolute model (as I know some will try to bury the sentiment from a technical standpoint, and they will be correct), I just want to use various levels of overdrive to help explain the audible characteristics of high order or high level harmonics in comparison to low level harmonics from a perspective we are all familiar with. First we need to realize that there is soft clipping, as in moderate overdriving of a tube amplifier, and then there is hard clipping, which we all associate with solid state amplifiers becoming harsh. But hard clipping is not limited to solid state amplification as tubes, when pushed too hard, can definitely attain this square wave even order clipping. Every amplifier driven to hard clipping will generate even order harmonics. But if we learn how they overdrive, we can understand the warmth associated with tube amplifier clipping, and the harshness associated with solid state amplifier clipping.

Let’s look at tube amplification as it begins to overdrive. First, as a tube amplifier starts to overdrive it maintains a sine wave signal, but it starts to squeeze the overdriven portion of the output signal down. This is a function of the analog gain of the tube, which is not a switching device. It basically acts like a shock absorber, as there is a smooth signal, not a switched signal. While the frequencies do not change, the amplitude of the clipped signal starts to shrink; generating less output at the clipped frequency, and the surrounding frequency amplitude starts to climb. At this point the transfer function is no longer linear. It will continue to squeeze the signal down until it reaches hard clipping. At this point, the entire amplitude of the signal is fairly symmetrical and then it starts to drive into the square wave that we associate with clipping. This (pre square wave) is even order distortion, and this distortion causes a warm sound.

Solid state amplifiers overdrive in a completely different fashion. Proponents of solid state amplification often falsely state that distortions based off soft clipping of a tube is inherent no matter the amplitude. They suggest that the nonlinearities occur throughout the tube’s useable range. However untrue, it brings us to how clipping of a solid state amplifier is more linear indeed. The increase in amplitude associated with solid state output remains linear until the unit is overdriven. However, because the solid state amplifier uses switching to increase the signal amplitude, there is no analog “shock absorber” to soften the blow of the clipping, it is just on-off. When the amplifier starts to clip it is immediate, and it is what we termed “hard clipping” above. This is odd order distortion, and this distortion is harsh or grating.

Now let’s get an idea of high level or high order distortions, and low level or low order distortions. I will simplify this as much as possible. Low level/order distortions are harmonics that occur closer to the fundamental frequency. High order distortions are harmonics that occur farther away from the fundamental frequency. That is, if the fundamental frequency is 40Hz, the low level harmonics are occurring at 51Hz, 102Hz, etc. I want to point out that this is a low level odd order harmonic. If it were a even order low level harmonic we would see a fundamental of 40Hz with harmonics at 80Hz, 1200Hz, etc… An example of a high level odd order harmonic of the 40Hz would be 4331Hz, 5421Hz. Please be aware that these are just examples, I am not saying that this would be the characteristic harmonic frequency chain from a 40Hz fundamental.

Remember above I stated that even order harmonics thicken the sound, but continue to emphasize the same tone, just with more octaves mixed in, continually reduced in volume as the octaves climb? Well let’s use this thought to help us get an audible idea of high level/order harmonics. When you have your HU at 75% of its total volume and set your amplifier so that it just starts to clip/distort you have a low level distortion. Now, turn the gain on the amp and you hear more distortion, more harshness. What you are doing here is starting with a low level clip and adding amplitude to that clip. Suddenly you are taking those additional octaves that were reducing in volume, and increasing their volume, as well as adding more octaves through the amplification of those distortions. This is a rough idea of a high level distortion that we have all likely experienced. So now you get an idea of the difference in sound between a low and high level distortion. This again is not really the best representation of high level distortions, but it gives us a rough understanding of how they sound and what they are.

So now that we understand even and odd order distortions, and low and high level distortions, what can we derive from this information? High order harmonics are more audible because they are far from the fundamental frequency (see my write up on MP3s, How and Why They Affect Our Systems, and System Choices), and they are more offensive then low order harmonics. But, even order harmonics are more natural, far less objectionable than odd order harmonics as they do not take away from the fundamental frequency. So what we really would prefer have is an even order low level harmonic. This is from a purist’s standpoint, however… Today’s society is developing a slightly different opinion on what is desirable and what is not desirable…

It all started really with rock guitarists in the 60s and 70s. From my research, the first known, popular, odd order distortion introduction started as an accident…ironically with a tube amplifier. Apparently a tube got dislodged from its seat and the guitarist, Link Wray, liked the way it made his solos stand out. This caught on, and guys like Hendrix and Harrison were poking holes in their speakers and cutting them with razors (remember, even order distortions are asymmetrical, and therefore mechanical damage to the cone will cause an uncontrolled, hence: asymmetrical, distortion) to acquire similar sounds, and getting close to and facing the amps to add aggressive feedback through the pickups. Leo Fender took advantage of this and started developing distortion pedals. So the harsh sound associated with even order became a desirable trait. With the introduction of solid state amplification, it became popularized, as the overall sound of solid state is aggressive, harsh, and grating compared to its tube brethren, adding to the odd order harmonics the musicians were trying to shoot for.

The music industry has fallen into this too. Poor recordings, and over amplification because of high EQ spikes in music causing clipping introduces these even order harmonics… remember, the majority of recordings are done with digital equipment so we see odd order distortions, and by introducing clipping we take low level distortions and as a side effect we increase their amplitude stretching them into high level distortions. There was a paper written about the issues with digital recording vs. tube recording that is worth reading. While it is a very long and technical, it is a great read so hopefully you will take the time to read Tubes Vs. Transistors, is There an Audible Difference?. Since the music industry has been using solid state and clipping signals during mastery, the younger generation has no idea what a true, warm, unclipped, well recorded and mastered score actually sounds like.

So we have been conditioned to accept, and enjoy odd order distortions. Most of you reading this are from a generation that has never heard, hell you have never seen a tube amp. The only sound you are familiar with is the hard solid state sound. You have been associating a bright sound with accuracy and detail, when it is anything but accurate and detailed; it is just odd order harmonics.

I will be doing more tech write-ups at my homepage. If you like them, please look at my profile for a link under the contact info tab.

Last edited by sandt38; 08-21-2010 at 07:46 PM.
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Old 08-21-2010, 06:16 PM   #2
pjhabit

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Hey, I appreciate the props & like what you're doing w/ these write-ups ...but this one needs some revision
I think you've got the even order & odd order thing switched around...
even order distortion = rich/warm (2,4,6 etc. multiples of the fundamental)
odd order distortion = harsh/bright (3,5,7 etc. multiples of the fundamental)
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Old 08-21-2010, 07:41 PM   #3
sandt38

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You are correct. I keep flip flopping the 2. It was explained to me like this:

The even harmonics of 2000 Hz, 4000 Hz, and 8000 Hz are fine; they're perfect octaves of the original note and can add fatness and warmth to the sound. Tube are great for distortion because they naturally generate more of these even harmonics when over driven.

The odd harmonics (generated by a lot of transistor stuff) are where the problem lies. Those 3,000 Hz and 5000 Hz harmonic distortion notes actually clash musically with normal notes found in musical scales.
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Old 08-22-2010, 12:31 AM   #4
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Awesome writeup. I've been needing to read something like this for a while.

I'm really glad you are feeling well enough to make a nice long thread like this.

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Old 08-22-2010, 12:51 AM   #5
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Good write up. I think more accutately, even order distortion is even multiples of the original harmonic (2,4 & 8). Odd order on the other hand aren't necessary odd order (3, 5, 7), they just aren't even order, so an odd order distortion may be 1.7, 3.4, 6.8x. Ie when we hard clip our 40hz sine wave it "morphs" into a 45hz wave (and the 45hz wave's associated harmonics) and if we crank the volume/gain a little more then the 45hz wave morphs into a 50hz wave, 60hz wave, etc.

This is why clipping can be so loud, by clipping we're actually taking advantage of the fact that the higher frequency wave is louder for a given displacement. It's also what makes clipping so easy to spot to the trained ear - when things start getting louder in higher frequencies, but not so much in the lower frequencies, then you are clipping.

Thanks for taking the time Sandt.
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Old 08-22-2010, 03:55 AM   #6
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wonderful , i'm interested in cars and autotools
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Old 08-22-2010, 07:55 PM   #7
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Quote:
Originally Posted by Jimi77 View Post
Good write up. I think more accutately, even order distortion is even multiples of the original harmonic (2,4 & 8). Odd order on the other hand aren't necessary odd order (3, 5, 7), they just aren't even order, so an odd order distortion may be 1.7, 3.4, 6.8x. Ie when we hard clip our 40hz sine wave it "morphs" into a 45hz wave (and the 45hz wave's associated harmonics) and if we crank the volume/gain a little more then the 45hz wave morphs into a 50hz wave, 60hz wave, etc.
I think the non-harmonic distortion you're referring too is intermodulation.

This graph gives a good look at harmonic distortion. You'll notice the fundamental frequency is 500Hz & the even/odd order harmonics are whole number multiples of the fundamental. They aren't necessarily 'octave multiples' (ie, the odd & 6th order), but the overtones that do fall on these octave multiples are usually considered more pleasing. It makes sense because each octave step is musically related to the fundamental. The 5th order overtone is actually 2 octaves plus a major third from the fundamental, so you can think of the 'un-related' overtones as forcing a different tone/chord/frequency to the signal (rather than 're-enforcing' the fundamental)...intermodulation distortion (not shown in the graph) does this as well.
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Old 08-22-2010, 08:49 PM   #8
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Quote:
Originally Posted by pjhabit View Post
I think the non-harmonic distortion you're referring too is intermodulation.

This graph gives a good look at harmonic distortion. You'll notice the fundamental frequency is 500Hz & the even/odd order harmonics are whole number multiples of the fundamental. They aren't necessarily 'octave multiples' (ie, the odd & 6th order), but the overtones that do fall on these octave multiples are usually considered more pleasing. It makes sense because each octave step is musically related to the fundamental. The 5th order overtone is actually 2 octaves plus a major third from the fundamental, so you can think of the 'un-related' overtones as forcing a different tone/chord/frequency to the signal (rather than 're-enforcing' the fundamental)...intermodulation distortion (not shown in the graph) does this as well.
Thanks for clearing that up.
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Old 08-22-2010, 09:11 PM   #9
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Great reseach Sant!!!!!!!!!!!!
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Old 08-22-2010, 09:53 PM   #10
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Walking past the amp clipping part, the natuaral harmonics in the instrument/recording is why I look to build the best flat line speakers I can. It's the harmonics that I seek from the 100-3.5kHz range. I want it all and SO FAR, I can get it with a bi-amped high power solid state set-up. No tube v SS debate here. I just think you can't hear ANY harmonics right with crap speakers to begin with.
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Old 08-23-2010, 01:12 AM   #11
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the most common form of distortion isn't even mentioned here.. inter-modulated distortion..
note that the thread should be labeled THD which is harmonic distortion cause from solid state components...

IMD or inter-modulated distortion comes from the moving components..
im not going to ruin sandts thread but with the technology we have today IMD is more important in high-output and/or high-bandwidth components..

all in all very good writeup reguardless..

the only thing i disagree with is..some audiophiles desire soft clipping like that introduced with tube driven amps.. true audiophiles pay alot of money for over built amps and components(line drivers xover etc) to avoid any type of distortion.. sure the moderen transistors amps do sound different when clipped but as long as the signal is adequately amplified IE high enough slew rate, dampening factor s/n ratio your not going to hear a difference because an inaudible amount of distortion is introduced.. IE noone will prove richard clark wrong because its a FACT the human brain cannot perceive it..

that being said I have to disagree with his methods because he uses precision equipment to set the amps..

in a normal installation its very common to set your gains with a DMM and or a good ear. ill agree set like this as long as you don't exceed this its very hard to hear with good amps but not all amps are built the same and some have latency problems in the xovers, low noise floors, poor dynamic capability and reaching its limits some amps are much easier clipped and noticeable..
my advice is get good equipment use good install techniques and understand what and why your doing something a specific way..
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Old 08-25-2010, 03:50 PM   #12
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Good points have been brought up on IMD in the replies here. Perhaps I will address them in another write-up.

If anyone wants anything else addressed, and I mean anything from a technical standpoint, please go ahead and let it be known. If it is within the realm of my understanding, I will be glad to try to shed some light on it.

If there is any question about the terminology used within these articles, please ask and I will address them as well. My goal in these write ups is to help everyone understand the main topic of the paper, so it doesn't help if you don't understand what amplitude or fundamental note, or octaves are, as they are an important aspect in this write up. I may go ahead and do a glossary of terms or something similar in the long run, but for the time being I am more interested in what we are hearing, which is why I have elected to do these write-ups.

There was a time when this forum was a very good technical resource, and I hope that perhaps these papers will start to bring this type of discussion back. While I am aware that there are many n00bs who don't care about this and they just want to know what sub will play loudest in a particular set-up, there are still guys here interested in what is really going on with our systems. So hopefully this will generate more discussion from a technical standpoint.

So keep the questions and requests coming. I will do my best to answer and fill them in my free time. You can add them to the threads I have been putting in these write-ups, or if you don't want people to know you have a particular lack of understanding of a certain concept (as I know sometimes we don't want to be harassed for not understanding something), please feel free to PM me. I will not disclose any information regarding the user who PMs me these queries.

Thanks for taking the time to read, as I know I am almost as wordy as Geolemon himself.

Last edited by sandt38; 08-25-2010 at 03:54 PM.
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Old 08-25-2010, 05:00 PM   #13
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Regarding the comment by Joel:

Quote:
IMD or inter-modulated distortion comes from the moving components..
im not going to ruin sandts thread but with the technology we have today IMD is more important in high-output and/or high-bandwidth components..


the only thing i disagree with is..some audiophiles desire soft clipping like that introduced with tube driven amps.. true audiophiles pay alot of money for over built amps and components(line drivers xover etc) to avoid any type of distortion.. sure the moderen transistors amps do sound different when clipped but as long as the signal is adequately amplified IE high enough slew rate, dampening factor s/n ratio your not going to hear a difference because an inaudible amount of distortion is introduced.. IE noone will prove richard clark wrong because its a FACT the human brain cannot perceive it..
IMDs play an important role in what we hear today, as they are introduced by solid state amplifiers, through TIM or transitory intermodulation distortions also termed "slew rate". The early effects were noted by audiophiles, but not during tests as they were tested using sine waves. But the engineers didn't test for TIMs as it was previously unheard of until the introduction of solid state amplification. Since there was a single frequency input at testing, they did not notice the effects of TIM, as intermodulation is the sum of, and or the difference of the fundamental frequency. To clarify; the sum is the answer of numbers, in our case the frequencies are the numbers, that are added to one another, and the difference is the answer to numbers subtracted from one another. For example, if we have a sine wave or single frequency of 340Hz, the sum and difference of the fundamental create an even order harmonic. What does this mean? 340+340 is 680, or an octave of the fundamental note. 340-340 is 0, effectively canceling the low fundamental. This distortion would not be discernible as derogatory as it is directly related to the fundamental. Now, when the music was introduced we may see 2 fundamental notes being played at the same time (I know there will usually be more, but this just compounds the TIM and makes the explanation more difficult, but you should get the point with this simple example), for arguments sake let's say the 2 played notes are 340 and 400. Suddenly we have an erratic odd order harmonic, properly characterized as IMD of 740Hz and 60Hz (340+400=740 and 400-340=60). This distortion literally becomes IMD rather than odd order as it is the sum and difference of the 2 frequencies. This was a phenomenon never noted in valve (tube) amplifiers, but it was a factor in solid state amplifiers.

This is another reason why many audiophiles prefer valve amplification compared to solid state. Although admittedly most manufacturers of high end solid states have slew rates that help avoid high order IMDs. Please see the paper above to understand why the high order distortion is more objectionable than a low order distortion.

It should also be noted that IMDs are not all related to moving components. Dissimilar metals, for example: Diodes, and our connections, also add to IMD.

So to state that IMD is important in todays systems as it is comes from moving components is not necessarily wrong, but it is a generalization of what IMDs are and where they occur. They do tend to occur in SS amplification as opposed to tube amplification. This is why you do not see slew rate numbers attached to valve amps, although it could be argued that IMDs are introduced were a solder joint (tin) is used to affix 2 copper cables.

This is some worthwhile information I will include on the IMD paper in the works.

Last edited by sandt38; 08-25-2010 at 05:25 PM.
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Old 08-25-2010, 09:47 PM   #14
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thanks sandt my head is in my as sometimes i was thinking of termporal distortion..
IMD and temporal are still the mos sough after forms of distortion..
althought like i said pretty much any mid to high end SS amps will have inaudible IMD/THD..
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