Do blind tests really prove small differences don't exist?

[arnyk]

Not true at all. If I can show something mathematically to be true, it is absolute.

Unfortuanately, math is not binding on the real world. From your comments Amir and with all due respect, I sense that somehow that you have never ever built any real world thing from a mathematical model.

I've had the experience of building numerous real world things from mathematical models. Some might say that I've made a career out of building real world things from mathematical models. I've built things from math models I developed, and I've built things from math models that others developed. I've watched others build things from my math models, and I've watched other people build things from other people's math models.

Never, ever has a real world thing performed perfectly identically to the corresponding math model. Sometimes its a complete miss, sometimes its close, and sometimes its so close that it takes your breath away. But it is never, ever has something built in the real world based on a math model performed identically to the math model. I don't think that anybody has ever built something that exactly matched the math model.

So much for your math models, Amir. Math is a wonderful thing, but at its highest levels, its a closed box. Within the box, math is perfect. Outside, not so much. Nothing ever escapes from that box and into the real world completely unchanged. Close, but never ever a cigar!

For example, 16 bit audio will have maximum signal to noise ratio of 98 db.

Spoken like someone who has never measured equipment that runs at 16 bits. In the real world, if you find something that runs with 16 bit data and has an unweighted SNR of even just 93 dB, you are a pretty happy camper! I take it you never heard of dither, Amir. No proper digital system lacks it! It takes its toll on the good old SNR. If you have a real world 16 bit digital system with a measured SNR of 98 dB, the measurement was weighted - probably A-weighted. A-weighting is good for up to a 10 dB improvement in measured SNR.

If you claimed that it is 80, I can absolutely prove you wrong. Ditto if you claimed it is 120db.

Back in the real world 16 bit equipment with 80 dB SNR is very common. If you use the right noise-shaped dither, no less than the inestimable Dr. Stanley Lipshitz tells me that you can have a perceptually-equivalent SNR of 120 dB.

Thanks again for yet another trip through your world, Amir. Unfortunately yoru world would seem to have very little connection with the real world at this point. ;-)

 

[arnyk]

Terryj IMO you touch on something that is not normally discussed in the forums about blind test matching amps.
And that is output impedance with loading from speaker and level matching.
I understand level matching is critical to remove cues and the usual discussion online is that this needs to be 0.2db or better.

This is not exactly true. There are acceptable deviations from the +/- 0.1 dB rule that were first set forth in Clark's JAES paper. The following graph shows those deviations:

This is a scanned copy of the corresponding chart in Clark's JAES article.

The source of this chart is a study of JNDs due to frequency response variations. One of its founding principles is that frequency response variations are more noticable towards the perceptual center of the audio band, and are more noticable the larger the percentage of the audio band they cover. This is, of course common sense. If you wanted to relate this chart to JNDs, JNDs for FR and level differences are typically about 3 times the amount given by the chart.

However in the real world, amps have pretty different output impedance against frequency range-speaker loading, which can also be exacerbated by its type and design of the output stage.
It is rare for moderate priced amps to have an output impedance of 0.005ohms across the whole frequency range, and quite possible comparing two different amps one will be say 0.01ohms and the other up to an uncommon 0.7ohms across frequency for SS amps, in the past the differences would had been much greater and the effect.

Not only that, but the source impedance provided by power amps typically rises at high frequencies, but still within the audio band. In linear amps this is due to the amp's open loop gain decreasing within the audio band and the possibility that there is an small, usually highly-damped inductor in series with its output to improve stability and reliability. In the case of switchmode power amps, the impedance rise at high frequencies can be pretty large.

Anyway using real speakers and music you are talking easily 0.5db differences and even with today's SS it is possible but not common up to 1.7db differences between two amps in parts of the FR.

This is one reason why we recommend level matching by measuring voltages across the loudspeaker terminals. However, it is quite possible that the variations you mention are within the ABX Level and Frequrency Response Matching Criteria set forth above. And if the amp matching is well outside the ABX criteria, we have an explanation for why it sounds different.

 

[Orb]

Arny,
yeah seen that graph before but appreciate it has context to this discussion, can you explain more on what it is showing with the 3 threshold lines please, looking at it this suggests the 0.2db is not needed ever if all 3 lines are relevant.
BTW you usually suggest along with supporting all those discussion on AVSF that all testing MUST be within 0.1 to 0.2db.

On the amps, yeah I understand the various mechanisms and makes sense to highlight for others but this does not remove that good general SS amps can still vary by 0.4db to 1.5db in the frequency range for 8ohm to 2ohm loading. The 2ohm is going to pretty rare I agree, however it is common for there to be somewhere a difference quite above 0.2db.
Going back a few decades the output impedance variations were even greater for the more common SS amps.
Adding Class D or anything requiring a filter in the output stage does seem to exacerbate this, although the latest Class D from Hypex UcD and Primare UFPD are doing a stunning job in removing this issue.
One aspect I may be misunderstanding Arny is matching via voltages at speaker terminals; how does this help to level FR differences caused by output impedance and the speaker?
You are not changing their values even when level matching, which is ensuring gain is equal for the signal output from the comparing amps for a given loudness.
I can show many well designed amps that are well outside the 0.2db mark when considering output impedance and loading, but then this comes back to that chart, which if you have the time to explain it a bit more really would be appreciated as touches on the flip side of my earlier statement that either something is going with ABX or, listeners are even worse with actual music when it comes to descriminating-masking-sensitivity even more problematic.

Still that Clark chart should raise questions about level matching and hearing thresholds so if anyone can explain it thats great.
Thank you
Orb

 

[arnyk]

Arny,
yeah seen that graph before but appreciate it has context to this discussion, can you explain more on what it is showing with the 3 threshold lines please, looking at it this suggests the 0.2db is not needed ever if all 3 lines are relevant.
BTW you usually suggest along with supporting all those discussion on AVSF that all testing MUST be within 0.1 to 0.2db.

There are 4 lines, not 3 - the one marked level relates to the 0.2 dB requirement.

For me the 0.1 dB rule is a one size fits all, if its this good you never have to go any further type rule.

BTW while even good power amps may vary by 0.5 dB from flat in the audio range, often both amps in a test are very similar.

It's about matching, not necessairly about getting perfectly flat frequency response.

In fact most power amp designers design their amps so that they are about 0.5 to 1 dB down at 20 KHz with the specified lowest load resistance.

The 1/3 octave line for example says how big a dip or a rise can be and still be acceptable, if it is 1/3 octave wide.


So you have two amps, one is 0.5 dB down at 20 KHz, the other is 1 dB down at 20 KHz, but this difference only extends over the half-octave centered at 17 KHz or whatever. The chart if interpolated says that varaitions of several dB at 17 KHz should be OK. In fact I've never tested two amps that were so far apart that any additional eq was needed to bring things into line, and I've done some tubed versus ransistor tests.

Before trying to match amps with an eq box, I might experiment with some low value resistors in the speaker lines of the better amp - like 0.1, or 0.33 ohms, maybe up to 1 ohm or more.

 

[Phelonious Ponk]

English, gentlemen?

You seem to be saying that output impedance of amps vary by design and these varying impedances, when faced with variations in the load at the speaker, can result in audible FR response variations in the amplifiers. But I could have gotten that completely wrong. If not, could I have an example? And speak very slooooooowly, please.

Tim

 

[Orb]

English, gentlemen?

You seem to be saying that output impedance of amps vary by design and these varying impedances, when faced with variations in the load at the speaker, can result in audible FR response variations in the amplifiers. But I could have gotten that completely wrong. If not, could I have an example? And speak very slooooooowly, please.

Tim

Yes that is spot on
See no need for us to speak English hehe
This is one reason JA devised the speaker simulation test for Stereophile measurements, however bear in mind that this is a general model and not specific to any actual speaker.
And this is one reason tube amps can be off from neutral as their output impedances are usually much higher unless managed in a way like McIntosh and their autotransformer.
Class D is notorious for this affecting them as well, apart from a few designs such as those I mentioned.
SP tend to review very well designed-sounding amps so it is difficult to see this in operation but the two links provide some insight.
http://www.stereophile.com/reference/60/
Using a worse case scenario for output impedance deviation, so as a modern amp here is the Nad M2:
Look at the figures 1,2 and 3, and associated paragraphs.
http://www.stereophile.com/content/nad-m2-direct-digital-integrated-amplifier-measurements

But, I want to reiterate this is not necessarily a disaster, on average the variation is probably anywhere from 0.1db to 0.4db for good amps sometimes in the sensitive frequency zone and often outside, and who knows whether this is actually audible if going by Clark's chart Arny showed.
On a well designed amp the output impedance figure (depending upon design) will be exceptionally low, as an example the Devialet D-Premier has an output impedance of 0.005 to 0.007ohm across 20hz to 20khz from what I remember.
However most manufacturers and measurements not reflecting changes from 8ohms will show a nice and flat frequency response, its valid until considering that speakers are not a consistent 8ohms, so 2 amps can be stated as flat between 20hz-20khz in terms of manufacturer specs and measurement with 8ohms, but they could differ with real world speakers (and again it also depends upon the speaker as their phase and impedance all differ over frequency range).

Thanks
Orb

 

[Orb]

There are 4 lines, not 3 - the one marked level relates to the 0.2 dB requirement.

For me the 0.1 dB rule is a one size fits all, if its this good you never have to go any further type rule.

BTW while even good power amps may vary by 0.5 dB from flat in the audio range, often both amps in a test are very similar.

It's about matching, not necessairly about getting perfectly flat frequency response.

In fact most power amp designers design their amps so that they are about 0.5 to 1 dB down at 20 KHz with the specified lowest load resistance.

The 1/3 octave line for example says how big a dip or a rise can be and still be acceptable, if it is 1/3 octave wide.


So you have two amps, one is 0.5 dB down at 20 KHz, the other is 1 dB down at 20 KHz, but this difference only extends over the half-octave centered at 17 KHz or whatever. The chart if interpolated says that varaitions of several dB at 17 KHz should be OK. In fact I've never tested two amps that were so far apart that any additional eq was needed to bring things into line, and I've done some tubed versus ransistor tests.

Before trying to match amps with an eq box, I might experiment with some low value resistors in the speaker lines of the better amp - like 0.1, or 0.33 ohms, maybe up to 1 ohm or more.

I think you are looking at this from the most ideal perspective Arny, there are plenty of measurements out there for diverse range of amps that show a reasonable % are also deviating in the sensitive frequency, and reasonably common between 7khz to 15khz, not just the more common at above 19khz.
However we differ on key point; output impedance of amps are never the same, it is wrong to say the average amp has identical output impedance performance of another amp, unless it is very well designed.

Thanks for explaining the graph as it can be tricky to follow without reading its specific paper (which I appreciate needs to be purchased), although the explanation on what the chart shows and how it applies to a real world example could be further expanded Arny.

Cheers
Orb

 

[microstrip]

This week I spent two wonderful days at Harman, including hours of presentation and data on blind testing. Yet on the simple question of how you would treat the side wall by a speaker, there were 2.5 to 3 answers from three world-renowned experts in speakers and acoustics! Are there three versions of the truth or are two of them wrong? (...)

It would be a pity if your experience would become lost in this never ending battle about DBTs. Would you be interested in opening a new thread about your stay at Harman?

 

[Steve williams]

I know that Amir is writing a review for his visit to Harman

 

[amirm]

It would be a pity if your experience would become lost in this never ending battle about DBTs. Would you be interested in opening a new thread about your stay at Harman?

As Steve noted, I will definitely be documenting my experience there and in a dedicated thread. I took so much notes that it will take me a while to put them in a cohesive post. I also have a collection of pictures which I need to clear with Harman before publishing (don't expect any issues but do need to ask).

So look to some or all of it coming out next week, assuming I can finish all of my gardening chores this weekend!

 

[Stereoeditor]

For example, 16 bit audio will have maximum signal to noise ratio of 98 db.

Spoken like someone who has never measured equipment that runs at 16 bits. In the real world, if you find something that runs with 16 bit data and has an unweighted SNR of even just 93 dB, you are a pretty happy camper!

Amir wrote "maximum" S/N Ratio, Mr. Krueger. It appears that you misunderstood him to mean "typical" S/N Ratio. And as you have already acknowledged that Amir has the top-model Audio Precision SYS2722 analyzer and has quote measurements made on it, your comment that Amir "has never measured equipment that runs at 16 bits" is disingenuous at best.

John Atkinson
Editor, Stereophile

 

[arnyk]

English, gentlemen?

You seem to be saying that output impedance of amps vary by design and these varying impedances, when faced with variations in the load at the speaker, can result in audible FR response variations in the amplifiers. But I could have gotten that completely wrong.

You have it right. I prefer to talk about FR variations at the terminals of the speakers.

One of the characteristics of an ideal power amplifier is that its output has perfectly flat frequency response at some point inside it. Between that point and its output terminals we place an imaginary impedance which we call the amplifier's source impedance. Ideally this impedance would be very low, for example like an inch of #12 copper wire. In reality it is usually a little larger, and ranges from less than 0.1 of an ohm to several ohms. Its impedance may not be the same at all frequencies. Typically it increases slightly or greatly with frequency.

The amplifier's source impedance plus the impedance of the speaker cable combine with the impedance of the loudspeaker to form a voltage divider. This results in a non-flat frequency response at the speaker's input terminals. It's almost like a little imaginary equalizer.

The problem with this little imaginary equalizer is that we can't adjust it, and in general it wasn't designed to optimize anybody in particulars stereo system.

If not, could I have an example? And speak very slooooooowly, please.

Imagine an amplifier that has a source impedance of 1 ohm at all frequencies. If you read a proper spec sheet it would say that it had a damping factor of about 8. If you hook a real world speaker up to it, the speaker might have an impedance of 10 ohms at many frequencies, but would rise to 100 ohms at some frequencies. At the frequencies where the speaker's impedance is 10 ohms, there would be an effective loss due to the amplifier's source impedance of about 1 dB which could be audible. At the frequencies where the speaker's impedance is 100 ohms, there would be an effective loss of only about 0.1 dB which would not be audible. The net effect is that an additional set of frequency response variations on the order of 0.9 dB would have been imposed on the speaker as compared to an ideal amplifier.

 

[arnyk]

Amir wrote "maximum" S/N Ratio, Mr. Krueger. It appears that you misunderstood him to mean "typical" S/N Ratio.

Actually, what I read was maximum S/N ratio in the real world. In the real world every proper digital channel is properly dithered, so the theoretical number of 98 dB is not the right answer.

And as you have already acknowledged that Amir has the top-model Audio Precision SYS2722 analyzer

No, I simply haven't acknowledged that. I may have speculated that might be the case, but I don't know for sure what model AP Amir has. All I know is that he claims to have some AP device. I suspect that this equipment, if it exists is not in his house, but is at the audio shop that he founded. His access and frequency of use of it is questionable.

and has quote measurements made on it,

I am unaware of the specifics of that. Without specifics I have no idea what he has measured and not measured. For example AP test sets can be used as voltmeters. Using a voltmeter gives minimal insight into the specifics of discussion.

your comment that Amir "has never measured equipment that runs at 16 bits" is disingenuous at best.

This would be the result of a series of assumptions that you made John, and not related to any statements that I have made.

It is my perception that Amir doesn't do real world testing on audio gear with anything like the frequency of you or I, John. He asserts otherwise, but I have yet to see the talk of someone who is in the gear very often.

 

[amirm]

Unfortuanately, math is not binding on the real world.

It is not? When we remodeled our new house, we wanted to take out two posts supporting our 30 foot plus cathedral ceiling. An Engineer designed a monster laminated 24 inch beam to replace the existing much, much thinner one supporting the roof previously. They chainsawed the house roof in half, and inserted this thing:

Look at the size of it relative to the workers!

I am typing this under that beam. The comfort I have in doing so comes from "math being used in real world" to design the strength of it to make sure it can carry the *maximum* load that is ever subjected to it. So math definitely applies to real world. See more below.

From your comments Amir and with all due respect, I sense that somehow that you have never ever built any real world thing from a mathematical model.

There is no respect in that statement so let's put aside platitudes. What is ironic is that the case I made is 100% supportive of your point of view yet in the mission to argue against anything I say, you took it the other way.

So I will repeat it again. If math can prove something to be inaudible, no one can ever argue against it. Read that again. It says that if we can show something to be inaudible using mathematics, then no amount of subjective data saying otherwise mean anything whatsoever. The case is closed by definition.

As a person who advocates many things in audio is inaudible, you should celebrate that fact, not argue against it.

As to my accomplishments in using mathematical models, I don't care if you acknowledge any of it. It does not matter anyway because the above statement stands on its own. As does the illogical nature of you saying mathematics is not binding on real world.

I've had the experience of building numerous real world things from mathematical models. Some might say that I've made a career out of building real world things from mathematical models. I've built things from math models I developed, and I've built things from math models that others developed. I've watched others build things from my math models, and I've watched other people build things from other people's math models.

Good for you Arny.

Never, ever has a real world thing performed perfectly identically to the corresponding math model. Sometimes its a complete miss, sometimes its close, and sometimes its so close that it takes your breath away. But it is never, ever has something built in the real world based on a math model performed identically to the math model. I don't think that anybody has ever built something that exactly matched the math model.

You are reinforcing the point I made, yet claim to disagree with it. I gave the 98db example of CD audio S/N. That sets the *maximum* specification for the system. That is what the math predicts to be the best performance achievable. Real products may or may not match it. But what we can say with 100% confidence that no one can claim it to be quieter than that.

The above is very useful to settle one class of arguments -- that we can exceed a performance level predicted by mathematics. We cannot. We can underperform it as you say and that leaves us the complicated world of then using subjective data to figure out if that level of underperformance matters.

Let's apply this to jitter. I have said repeatedly that if measured jitter is below 500 ps peak to peak, then its distortion products as a matter of math is below the noise of your digital system at 16 bits and 20 Khz of bandwidth (for periodic jitter). If you then buy a digital system that has a measured jitter of say, 100 ps, then you are *assured* by mathematics to not have jitter be a distortion product you have to worry about. You don't need to read this thread. You don't need to know about ABX or any other blind testing. You don't need to hire a 100 people to go run a test. Nothing. You know the answer already.

Now take a system that underperforms and has a jitter of 5,000 ps. Now the math doesn't help you prove it is inaudible. To be clear, it also doesn't prove it is audible either. You now have to go and run blind test until cows come home to profile every type of jitter in every type of equipment to see what is audible and what is not. After all, you have no a priori knowledge of what jitter exists in my system so you better test the universe or else your testing may not apply to me.

See how useful the math is here? Good news for us is that we don't have to spend hardly anything in grand scheme of things to get under 500 ps of jitter. Do that and math is your friend. Don't and another math, that of infinite permutations, comes to bite your behind!

So much for your math models, Amir. Math is a wonderful thing, but at its highest levels, its a closed box. Within the box, math is perfect. Outside, not so much. Nothing ever escapes from that box and into the real world completely unchanged. Close, but never ever a cigar!

Per above, a smart pragmatist knows how to use mathematics to perform real world jobs. Whether it is to design a beam of a house that is held at two posts 40 feet away yet it doesn't sag, or the computations of jitter, mathematics can help us a lot in understanding our system limitations and design the appropriately.

You have built a lifetime of experience on promoting experimentation. So I get why you go there. Me? I have seen how expensive and challenging it can be to do that right. In that sense, if I can use mathematics to cut out a portion of the required testing area, I do. That is what we do in real world where we have to build things commercially and make money on them. We care about things that have infinite cost and look for ways to reduce that.

Spoken like someone who has never measured equipment that runs at 16 bits. In the real world, if you find something that runs with 16 bit data and has an unweighted SNR of even just 93 dB, you are a pretty happy camper! I take it you never heard of dither, Amir. No proper digital system lacks it! It takes its toll on the good old SNR. If you have a real world 16 bit digital system with a measured SNR of 98 dB, the measurement was weighted - probably A-weighted. A-weighting is good for up to a 10 dB improvement in measured SNR.

So? That has nothing to do with the point I made. If you want to prove me wrong, you have to show that measured performance can exceed the ceiling established by the mathematics. You have said the opposite above.

Back in the real world 16 bit equipment with 80 dB SNR is very common.

There is? That is a worthy of bookmarking next time you repeat PCs have S/N far above that!

If you use the right noise-shaped dither, no less than the inestimable Dr. Stanley Lipshitz tells me that you can have a perceptually-equivalent SNR of 120 dB.

Noise shaped dither indeed can be used to raise the effective S/N in ear's most sensitive frequency range to above what the flat dither would show. Don't look but it is the math that sets that ceiling . BTW, another excellent citation is that of Bob Stuart and his wonderfully written and easy to read AES paper on coding digital audio. It is in our technical library.

Thanks again for yet another trip through your world, Amir. Unfortunately yoru world would seem to have very little connection with the real world at this point. ;-)

Arny, another friendly warning as the moderator to not make this discussion personal. State your case and let people deduct if I live in the real world or not.

 

[mep]

It is my perception that Amir doesn't do real world testing on audio gear with anything like the frequency of you or I, John. He asserts otherwise, but I have yet to see the talk of someone who is in the gear very often.

So Arnyk, where do you perform all of your testing and what test equipment do you have? Can you give us all a little insight into your background?

 

[arnyk]

So Arny, where do you perform all of your testing and what test equipment do you have? Can you give us all a little insight into your background?

I have long had a bunch of traditional electronic test equipment, including a 'scope, meters, analyzers, generators, microphones, dummy loads, etc. I also use computers with audio interfaces as test equipment. I have a degree in engineering and did everything for my master's degree but the thesis project. My schooling was interrupted once by forced induction into the Vietnam war and once again by my wife's pregnancy. I've worked professionally as an automotive engineer and both applications/data base and also systems level IT. When I worked in IT I worked in a variety of roles from management to business analyst to programmer writing in both high level languages and assembly language. I invented ABX testing and did the first ABX test with the first ABX Comparator that I designed and built.

I currently have a small business working with PC hardware and networking.

My current audio project is implementing a subwoofer int my church's sanctuary. I designed the subwoofer enclosure using T/S parameters. I had to update the crossover in that sound system from 2-way to 3-way. I pulled one out of my inventory, and found that it was non-operational. I diagnosed the problem, replaced some capactors and chips, and then tested its performance using a PC as both the signal generator and analyzer. I then implemented the actual installation of crossover, power amp, and loudspeaker. I did some preliminary adjustments by ear with test signals and then during a live orchestra rehearsal. I have yet to set up my microphone and computer analysis system for fine tuning, but the music director is already pleased with the results.

 

[mep]

Is your degree in electrical engineering? So you have a bunch of old test gear and now you mainly use your computer as a test rig like Ethan? No AP gear?

 

[arnyk]

Is your degree in electrical engineering?

The school I attended still does not give out differentiated degrees. You can be anything from a software engineer to a mechanical engineer and you get a BSE. My area of concentration was Systems Engineering which requires completing courses for both an EE and a ME plus software engineering. I spent almost 5 years in undergraduate school plus just under 2 years in graduate school.

So you have a bunch of old test gear and now you mainly use your computer as a test rig like Ethan? No AP gear?

Some of my test gear is old, some is new. My primary high end measurement tools are composed of audio interfaces + computers running analytical software, but I also have HP, Fluke, and Heath test gear.

I was given an AP System One but I don't use it very much because I believe in using the best tool for the job. I typically provide my own test equipment, at least until the client likes what I use and buys his own.

I mostly work at home and at client locations.

 

[mep]

The school I attended still does not give out differentiated degrees. You can be anything from a software engineer to a mechanical engineer and you get a BSE.

What school was that? I have never heard of any engineering univerisity that gives out vanilla engineering degrees. How do you market that degree when you leave school when it can't be defined what type of engineer you are? Normally you come out of school as an electrical engineer, mechanical engineer, chemical engineer, or civil engineer. Those degrees define who you are in the marketplace and the type jobs you will apply for.

 

[Orb]

You have it right. I prefer to talk about FR variations at the terminals of the speakers.

One of the characteristics of an ideal power amplifier is that its output has perfectly flat frequency response at some point inside it. Between that point and its output terminals we place an imaginary impedance which we call the amplifier's source impedance. Ideally this impedance would be very low, for example like an inch of #12 copper wire. In reality it is usually a little larger, and ranges from less than 0.1 of an ohm to several ohms. Its impedance may not be the same at all frequencies. Typically it increases slightly or greatly with frequency.

The amplifier's source impedance plus the impedance of the speaker cable combine with the impedance of the loudspeaker to form a voltage divider. This results in a non-flat frequency response at the speaker's input terminals. It's almost like a little imaginary equalizer.

The problem with this little imaginary equalizer is that we can't adjust it, and in general it wasn't designed to optimize anybody in particulars stereo system.



Imagine an amplifier that has a source impedance of 1 ohm at all frequencies. If you read a proper spec sheet it would say that it had a damping factor of about 8. If you hook a real world speaker up to it, the speaker might have an impedance of 10 ohms at many frequencies, but would rise to 100 ohms at some frequencies. At the frequencies where the speaker's impedance is 10 ohms, there would be an effective loss due to the amplifier's source impedance of about 1 dB which could be audible. At the frequencies where the speaker's impedance is 100 ohms, there would be an effective loss of only about 0.1 dB which would not be audible. The net effect is that an additional set of frequency response variations on the order of 0.9 dB would have been imposed on the speaker as compared to an ideal amplifier.

The input/damping factor/etc is not really important and may confuse when discussing just the characteristic of the output impedance of an amp and impedance interraction with the speaker, although I agree all of it is important when looking to level match (which as I mentioned can only be done at a specific frequency to ensure equal gain, and can only be truly matched if physically modified).
Is it really an imaginary equalizer?
I think you will find this is more a factor with engineering rather than intentional implementation as the behaviour will be unpredictable due to speakers being so different.

Personally I feel it is easier to say that a good SS amps will have an output impedance around 0.1ohm or better through the whole frequency range.
For most it is better to ignore the cable and just look at the output impedance of the amp 20hz to 20khz for static loads at 16ohm 8ohm 4ohm 2ohm, unless you know how the simulated load is achieved as JA has shown in previous ref article.
The key factors are output impedance of the power amp and the impedance of the speaker, however it is also important IMO for the amp to be very stable with its output impedance figure without it changing much over the frequency range and impedance.
Anyway 0.1ohm is good, and getting down to 0.01ohm is IMO getting on SOTA for SS if it is under this for frequency and impedance, and will mean an amp should be consistent no matter what speaker used.

That said, it seems Dan has some interesting circuitry in his Momentum as it has a moderate output impedance of 0.15 to 0.17ohms, and yet it is near as close to flat from 1hz to 20khz, with speaker loading from 1ohm to 100ohm.
I use the term moderate output impedance for that figure as it is one Paul Miller goes with for that measurement, where JA may feel it is still very good.

Further example to show this; look at tube amps and the 1st set of measurements and output impedance comments JA shows at SP:
http://www.stereophile.com/content/...nature-monoblock-power-amplifier-measurements
http://www.stereophile.com/content/rogue-audio-m-180-monoblock-power-amplifier-measurements
The black line is a simulation of what a speaker may look like to the amp, with its swinging phase and impedance through the FR.
If got the time one could try to look at real measurements JA has done for speakers showing phase-impedance and then compare to the output impedance figures of an amp, to get a rough picture.

Anyway this is digressing away from the original point, that the output impedance of amps means average to good ones may have frequency differences of 0.2db up to 1.5db, depending upon the speaker and music, and no level matching can change this behaviour unless physical intervention with additional circuitry.
Thanks
Orb