I’m also worried about how much of the harmonics in the music is LOST by the amp?

Of course, I don’t want gross harmonic distortion, but don’t abuse or lose the precious harmonics in the virgin recording either. No way to measure that, though, right? Thats where the ears come in…
The sort of fine detail you are talking about is lost everywhere. From the panel through the power supplies, from the platter to the phono stage, from the amp through the crossovers and internal wires and speaker cabinet and drivers. Every tiny little bit of it. For some reason people want to focus on amp. Or speakers. Or what was the latest one? Oh yeah, stylus. The stylus is the most important. Didn't you know?

Meanwhile, where was it I saw that one guy, everything matters? https://systems.audiogon.com/systems/8367 Right. That's the one. Everything matters. Including, yes, even the amp.   

You can hear it easily. Play your favorite music. Flip off your circuit breakers. Play it again. Hear that? That was just one little bit. Imagine getting rid of that, times 100. https://youtu.be/zZcZ6eJoxeE?t=1
ChiFi...Garage Boom Boxes....Cable Lifters....Magic Dots....Meh!
Every link in an audio signal chain plays its part, however, it is an irrefutable truth that what is lost at source cannot be recovered downstream, Ergo that Rock on a Stick is of no little importance.
Get a great amp not some junk.
While it's easy to measure THD, is it even theoretically possible to measure loss of harmonic information?
OP it’s not hard to amplify sound or even get a pretty low THD. IMD and the ability to go from 20hz to 10khz and ONLY hear (ZERO NOISE), then POP to 10KHZ.. that is what makes an amp dynamic in it’s presentation.

I use to call it tube wind up.. LOL trying to get BASS out of a valve or a lot of the old SS and it NOT sound bloated or rob the HECK out of that "POP" I just spoke of..

Pass, Krell, SS Macs, ect. Heavy PS units.. BIG HEAVY AMP SUCKING PIGS... :-) They only CAN SERVE one thing, because they don't conserve ANYTHING.. POWER and PUNCH Dynamos!!! Sonny Liston KO power..

AWB had a song called "Cut the Cake" (?) That song tells me the story EVERY single time about a GREAT amp.. and a great set up.. If I get to tapping my feet and throw the hat on the floor.. IT’S GOOD!!!

As far a losing information, that is a source issue. Getting it to your ears that’s a different story.. The connection at the source, all the way through YOUR system and THEN is dependent on the last wall it bounced off of. The boundaries are either part of the solution or part of the problem, BEFORE it EVER hits your ears?

The idea of loosing information via source isn’t the issue, getting it to your ears correctly, IS..

Oh it is time to feed the chickens, too..

Come on Dog..

The fine detail and harmonics is normally lost in the speaker and when you find a good one you will realize what you have been missing the whole time and keep an open mind because the good ones are not the new stuff!
But in a long chain of gear, it would be amazing to know what's getting lost where. At least with the final step, comparing speakers to headphones is one option. 
I took down the long answer. This shorter one is better. The amazing thing, what you are looking for that is lost is: music. A little or a lot is lost every inch of the way.

Everything matters. Go and listen. You will see.
speakermaster536 posts07-07-2021 The fine detail and harmonics is normally lost in the speaker and when you find a good one you will realize what you have been missing the whole time and keep an open mind because the good ones are not the new stuff!
I agree that speaker enclosures can be killers. Sound coming from the speakers is terrific, but the reverse sound trapped in the enclosure exerting pressure and interfering with the cone's free movement is something that really needs to be thought through.

I treat all internal surfaces to reduce reflections and use the best absorption filling I can find to convert sound waves to into heat. (Yeah, I know a lot of people rave about compromising the enclosure's physical volume and all that, but I've found it ain't necessarily so, or to put another way - necessarily critical.)

Modern cones and support mechanisms shouldn't be dismissed, so I'm not at all convinced "the good ones are not the new stuff!".
Reference speakers are on average starting at $15-$20 k for a decent true reference speaker on Average considering today $50k is  the mid high end range  for a speaker, there are a few exceptions to the Rule.
one key is a solid cabinet , High quality drivers , and premium Xover
components. Many companies use $30 tweeters and that’s ok for a under $5 k 
speaker ,name brand such as Accuton, Scanspeak ,Seas ,in their upper range are premium and more $$ costly but, much lower in THD . Speakers sold direct 
like Spatial Audio labs ,made in U.S.A  open baffle speaker but in the X series 
all drivers are premium especially the custom Beyma AMT tweeter- upper midband  and under $10 k a steal beating  many speakers  2x+ Their cost.
If we can’t measure the musical information lost throughout the chain, the measurements we do have about what gets added in terms of distortion are a smokescreen.  If you lose 25%, but add negligible distortion, you pretty much still suck.  Right?
I mean, signal to noise ratio would seem to be the most relevant stat in common usage, right?
I mean, signal to noise ratio would seem to be the most relevant stat in common usage, right?
Even when it's below audible levels, I have always wondered if it can contaminate the sound.

A very strong SNR of 120dB is almost certainly beyond audible range, but could be important for the fragile harmonics.  I’m sure there are people here who know this scientifically.
Exactly! Where does the noise come from? It it mostly ripple on the DC supply? Is a massively over engineered supply helpful?

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Try not to worry so much.
Loads more is lost in other parts of the chain.

As you rightly say OP, 120dB below the signal is a looonnngggg way down.  You can demonstrate that to yourself if you have an amp with numbered volume settings at known dB intervals.  You won't hear it, but the full volume signal will blow your eardrums if you're silly enough.
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@audioman58: the venerable Quad 57 can still be considered a reference speaker! They can usually be found for around $2K. I have two pairs.
@doogiehowser Or just barking.
Quads excel In the  mid range especially the older 57, but not in Bass or highs vs $$ speakers 
maggis I have owned a bunch cheap Xover parts,and frame is a disaster unless you buy a quality Mye stand  they flex like a sail
I am referring reference in a 3 way capacity when I am saying a retail speaker $15-20 k is average for a mid fi reference speaker 
by today’s $$ pricing standards when $100 k is considered upper end ,$50k mid reference ,which I cannot afford.sadly .
I'm not 100% certain but I think REW might be able to measure harmonics.
+1 mostly everyone- the entire audio chain matters

If you’re after better sound, chasing harmonic numbers will not tell you if a component sounds good in your audio chain. Also, what sounds good is subjective- some folks like the warmth of tube gear while others seek a different kind sound. You have to use your ears.

Our hearing is much more complex than current science can measure. While good measurements can lead to good sound, good measurements does not equal good sound. You have to use your ears.

Perhaps it’s semantics but we don’t usually evaluate our systems on what we lost like “harmonics”, but rather how well it performs. You’ll likely hear better sound including harmonics on a more transparent audio chain which usually means more expensive components.

If you want to improve the sound of your audio chain, you need to make sure your speaker is revealing/transparent enough to hear what the components are doing upstream. Then you can change/upgrade components to move towards the sound you find most pleasing - by using your ears.
Cable quality plays a much bigger role than most realize
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@kennyc - my point isn’t that measurements trump listening, it’s that the measurements we generally rely on, like THD, are so basic and uninformative.  While less distortion is better, wouldn’t it be great to see a measurement of harmonic richness or degradation?  Whether a speaker, dac, amp, or cable?

@tweak1 - indeed.  My new silver ICs from Darwin cables are recent proof of this!
just. switch from flat wound on your Fender Precision and discover WHY your random setup on the DBA IS WRONG…….
A REAL cello will kill ya……
I’m also “worried” about how much of the harmonics in the music is LOST…
seems like you were concerned with harmonics in your audio chain VS
wouldn’t it be great to see
The hypothetical “Wouldn’t it be nice if….”

The later is a rabbit hole debate with no audio benefit. “Worry” is not the same as “wishful thinking”.  I shouldn’t have wasted time on this pointless endeavor.
I’m also worried about how much of the harmonics in the music is LOST by the amp

I don’t want gross harmonic distortion, but don’t abuse or lose the precious harmonics in the virgin recording either.
Since you are specifically addressing amplifiers, you are correct to be worried. Electromagnetic interference (EMI) is the killer and it's wise to address it as best you can, both internally and externally.

Unused ports can act like antennas with direct paths to sensitive circuits, so that's one external source easily fixed with dummy shielding plugs. Since most equipment have AC supplies, that's the internal source. If you are willing to open the case, in most cases it can be resolved by fitting a grounded shield between the power supply and amplifying circuits.

As others have rightly mentioned, amplifiers are just the start and EMI / RFI should be addressed wherever possible. Shielding power cables can stop the EMI from radiating outside of the cables, so they are well worth considering. Since RFI (Radio Frequency Interference) is everywhere around us in the atmosphere, shield everything and connect it directly to a central earthing point, not the equipment's case.

@pedrob - thanks for the advice. I've always been curious what an extra grounding system would do, for instance. And some shielding plugs has also been recommended. I like that idea too. 
To get high S/N and low THD you need to add quite bit of dB of negative feedback to the circuit. 
This reduces the second harmonic and increases IMD…leaving you with a harsh and sterile sound, stripped of harmonics.  
So basically what I’m saying is you may need a poor measuring tube amp if you want harmonic richness without transistor distortion. 
My recent foray into digital coax cables includes Pangea Premier SE 1.0m, 1.5m, and finally the 1.5m XL, which is taking a good bit longer to break in, but provides a more intimate presentation, after 3 days with lots of replay (~20 hours so far) still missing a bit of HF sparkle, but I am confident it will come around. Best $150 I've spent in a long time. Are more expensive coax cable that much better. I doubt Ill be finding out any time soon 

I don’t want gross harmonic distortion, but don’t abuse or lose the precious harmonics in the virgin recording either. No way to measure that, though, right?
Sure you can.

The thing is, no amp is going to lose harmonic information in the audio range. They will all *add* harmonic information though and because distortion obscures detail, it will subjectively appear that some amps are better at getting the 'harmonic structure' (overtones) right than others.
EMI may well cause secondary and unwanted effects in amplifiers. One source is the speaker cables that are not only information bearers for analog audio signals. They are also RF antennas. The problem is that amplifiers may not have sufficient RF attenuation for RF related EMI fed into the amplifier via the speaker cables. As most amplifiers have an internal feedback signal from speaker output, RF noise can be injected into the negative feedback loop of the amplifier. This may well cause secondary effects for example within the micro-dynamics.
+1 atmasphere
Amplifier design guru Nelson Pass tinkers with harmonics sometimes adding more to his amp designs to get a pleasing sound. 
Numbers only get you so far - it’s the sound that sells. High end audio design is both science and art.
+1 atmasphere
Amplifier design guru Nelson Pass tinkers with harmonics sometimes adding more to his amp designs to get a pleasing sound.
If you are talking First Watt agree 100%. Pass Labs? Not so much. 
High end audio design is both science and art.
Yes. The trick is to design for the ear rather than the spec sheet. They are not the same :)

The spec sheet emphasizes low distortion, But it turns out that the ear will be less sensitive to distortion if the overall distortion has a certain signature: a predominate 2nd and/or 3rd harmonic in sufficient quantity to mask the higher orders. If this is the case then it will be very hard to tell the difference between 0.5%THD and 0.005THD.

If the lower orders are not in sufficient amplitudes, the higher orders (even if the amp has only 0.005% THD) will cause the amp to sound bright and harsh.

You have to know this simple fact when designing amps, if you want to make an amp that sounds like music rather than electronics.
@atmasphere if amps are not losing harmonic information wholesale, are the other components and cables generally more prone to losing it? Or is my worry about lost harmonics not necessary, and distortion the main culprit through the chain? 
If the system has 20Hz to 20KHz bandwidth you're not losing any audible harmonics. So if it seems that you are, something else is afoot, and that something else is distortion. So yes.

For example a thinner, less 'fleshed out' sound might be because the amp has a coloration of brightness and harshness which is common with many solid state amps. This is caused by distortion- the ear assigns a tonality to all forms of distortion and the higher ordered harmonics get harshness and brightness.

Because the ear is sensing brightness, it will seem as if the sound is dry even though the bass is present. This is because the ear 'tilts' your perception depending on what spectrum is perceived as loudest.

When solid state amps exhibit this behavior its usually because they lack the feedback needed to clean up the distortion caused by the feedback itself, which is almost entirely higher ordered harmonics. This is literally why tube amps are still around after all these decades. They don't have enough feedback either, but they generate more 2nd and 3rd harmonic which masks the higher orders- and so sound smoother and often as a result more 'harmonically correct'. 
A 20Hz to 20KHz bandwidth system is an absolute necessity. Just because it provides a full bandwidth doesn't mean there the finer detail and harmonics are being lost, which I believe is the question.

The best approach is to treat your equipment in the same way as sensitive medical equipment.
This subject is basically fantasy football. I was never any good at it. I would always say okay you pick your team players, I pick my stadium, with $3B tax credits, waivers, lease buy-back. The real fantasy is thinking its anything to do with winning games.

With this one the real fantasy is thinking its losing harmonic information. There is some lost, mostly though it is ultra-sonic. Which we already know no one can understand how that matters. So? Get some supertweeters, hear for yourself!

Back to Fantasy Island. The real problem is like Ralph said, harmonics being added. Every component does this. Every single one. Speakers, cables, amps, DACs, interconnects, power cords, floors, racks, walls, on and on. They all vibrate like crazy.

I never really understood just how much this is the case until the last year, as one after another layer of added harmonic resonance was removed and my system became as a result clearer and more natural. The real problem in having a violin, or harmonica, or drum or whatever sound like what it is, it is not the missing harmonics but the ones all these components add.

So when I went to cones a lot of them were gone, but they were mostly shifted to a narrow band- ringing. This imparted a hardness to the sound that in some systems can be mistaken for detail. As things get better and better however it becomes more clear this is not right.

Springs remove the ringing but leave harmonic resonance. Added harmonics. Altered tone. Instruments that no longer sound as they should. Remove these, with just the right amount of damping, wow what a difference! Less harmonics, not more.

Same with speaker cables. Same with interconnects. Amps? Why would amps be any different?

In reality, I mean. In fantasy land anything goes. Fantasy or reality. The choice is yours.
Ones digital cable could be why some have negative comments about SS, and likely class D. I have owned quite a few AB SS amps, as well as class D (and a handful of tube amps & preamps. My current class D which I have owned for close to 2 years is the EVS1200 from Ric Schultz: Tweakaudio.com

Since I first got it I was using an optical cable. I wasn’t crazy about the sound and thought I needed to replace my Audio Alchemy DDP-1 + PS 5, but I don’t have that kind of $$$ sitting around, and so, I needed a plan B. About a month ago, I got the bug to try coax (FINALLY). Being budget minded, I bought 1.0m Pangea Premier SE ($49.95 from Audio Advisors) and started a thread. Someone recommended to get 1.5m, which I did, and though it was harmonically richer, I wasn’t sold on it.

I replaced the 1.5m Pangea Premier SE with the 1.5m Pangea XL, the presentation is much more intimate. I find my body grooving to the music: PRAT. Now, some 50 hours of play time later, I am blown away (especially for $150!). So far, every redbook CD I played I am much much closer to the actual recording: singers and instruments are extremely palpable and when recorded, bass, drums, and piano are very authoritative.

I also like Pangea cables. I got the optical also a while back. I hear way more fidelity when hooking up using the coaxial input. I need to upgrade there next. 

The fine detail and harmonics is normally lost in the speaker

Can also be lost in the type of amp used, when phase shift happens to the 2nd 3rd 4th harmonic structure of the fundamental, mids and highs notes above in this case Class-D >1khz.
As can be seen (red trace) there is 75 degrees!!!!!!! of phase shift at 10khz cause by the switching noise "output filter" in this Class-D amp, and still 40degrees!!!! at 1.5khz. https://ibb.co/cCL1M8k
Most of the magnitude of this phase shift can be remedied by, as Technics did with the SE-R1 to move the switching frequency (and it’s output filter) from 600-700khz to 3 x higher, 1.5mhz instead, and so the phase shift would also be moved 3 x higher up the frequency response scale far more out of the audio band more like good linear amps are.

And the reason why so many linear hiend amps sound so much more natural in the upper mids/highs because as they don’t have these sort of phase shifts well down! into the audio band. As they aren’t trying to "filter out" "Class-D switching frequencies" on their outputs, and their frequency responses go out to more than 100khz with hardly any of this sort of harmonic structure distorting phase shift down into the audio band.

Cheers George
Not that I agree with a lot of stuff he says, I do with this one

Cheers George
Here's an interesting study.

Inaudible High-Frequency Sounds Affect Brain Activity: Hypersonic Effect
@Georgehifi - great link, thank you. The comments below the video from Paul and others are worth reading too.