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 Post subject: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Sun 24, 2019 11:14 pm 
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How exactly does one derive the Laplace transform for some arbitrary equalization curve for use in LTSpice?

For equalization curves with three inflection points (like RIAA, the Columbia Lp curve, NAB, the London Lp curve) the curve is defined according to the formula in my first attachment. Where ω is the frequency, L(ω) is the gain/loss in decibels at frequency ω, and the τ terms are the time constants for the three inflection points. From this formula I was able to find the inverse, and then the Laplace transform, and it works in LTSpice.

Tau 1 is equal to the treble time constant, Tau 2 is equal to the turn-over time constant, and Tau 3 is the bass shelving time constant. For RIAA, these would be 75, 318, and 3180 microseconds respectively. For the Columbia Lp, these would be 100, 318, and 1590 microseconds respectively, and so on...

The form of the Laplace for use in LTSpice for three inflection points is:

laplace [(1 + Tau_2*s) *(1+ Tau_1*s)] /(1 + Tau_3*s)

...and indeed I can get this to work in LTSpice for any curve with three inflection points and it perfectly matches known inverse curves found elsewhere.

What I can't seem to get to work are curves of the second type, with no shelving time constant and only two inflection points (see my second formula posted). What is the correct form of the Laplace for inverse curves of this second type?

For example, I would like an inverse network for the Columbia 78 curve, defined using voltage controlled voltage sources in LTSpice, which has a turn-over time constant of 530 microseconds and a treble time constant of 100 microseconds, and no defined bass-shelving. Any help here would be immensely appreciated; I'm at my wits' end.

Thank you in advance,
Ben


Attachments:
File comment: Formula for equalization curve with three inflection points.
Shelving_Formula.png
Shelving_Formula.png [ 19.88 KiB | Viewed 1633 times ]
File comment: Formula for equalization curve with two inflection points (no bass shelving).
Without_Shelving_Formula.png
Without_Shelving_Formula.png [ 21.75 KiB | Viewed 1633 times ]
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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Mon 25, 2019 2:14 am 
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The issue turned out to be an issue of my understanding of the underlying physical meaning behind the voltage controlled voltage source and the Laplacian that defines it. You can have a transfer function with only two inflection points, but my issue was that one of the inflection points gave a limit for gain as frequency approached zero of infinity, which is a non-physical result. No circuit can give infinite gain. Duh. Should have had a V8...

Anyway, the solution is to simply treat the early 78s as having an equalization curve of three inflection points, two explicitly defined, and one defined by the lowest frequency the cutting head was capable of recording, which forms a sort of shelving function. This eliminated the "infinite gain at DC" problem. This un-intentional shelving will vary from cutting head to cutting head and also with the other equipment in the recording chain, and thus won't be the same from disc to disc.

Now the issue is, where do I place the shelving frequency for a "typical" electrically recorded 78 from 1938 to the end of the 78 era? Is there meaningful information at 20 Hz? 30 Hz? 40 Hz? Or does the meaningful information cut off even earlier at say 60 or 70 Hz?


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Mon 25, 2019 4:10 pm 
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Just my opinion here, you can safely discard anything below 30Hz UNLESS you are making archival transfers. In that case you must, must include any and all audio information above 20Hz or even lower. For archival storage, no possible audio material should be discarded even if it seems unrecoverable today. Future recovery or reconstruction may be possible. A 30Hz cutoff should, I think, exclude mechanical rumble without removing material of possible interest.

I've worked with old recordings, 78s as well as magnetic film recordings (which led audio state of the art for a couple of decades), attempting to "remaster" them to stand with equal footing next to modern recordings. When the only master recording is from a 78 record, you do your best to make them sound full-range. I don't recall ever hearing anything except errant rumbles below about 60Hz. It is difficult to boost what is not there. Some microphones of the day could easily capture sound in the lowest octaves and the amplifying equipment (usually adapted from radio) could pass that along, but speakers did not reproduce much below 40Hz so no one could hear it anyway. I recall from when I was in my 20s, an old disc mastering engineer passed along some of his techniques. To help ensure a record would be playable by everyone he routinely hi-passed the audio fed to the cutter amps. By removing everything below 50Hz he limited excursions of the cutting stylus, which had two important benefits. First, by not wasting power on frequencies no one would hear, more energy could go into the audible material (volume above the noise floor). Second, large groove swings from "useless" low frequency junk was eliminated which makes the groove more easily trackable by a playback stylus.

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Mon 25, 2019 11:53 pm 
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I've looked at some other references and they seem to align very closely with what you've suggested. One source said that the lowest frequency cut during mastering would be about 40 Hz for the reasons you described. Another reference stated that RCA Victor and Columbia both filtered off anything below 40.64 Hz (E1 in the old A4 = 434 Hz standard), again, for reasons of not wasting cutting amplifier power modulating a groove at frequencies that the reproducing equipment couldn't replicate anyway, and to make the disc easier to track.

I think setting the shelving at 40 Hz would make for a fair compromise and more pleasant playback with less rumbling from the turntable, arm/cartridge resonances, etc being passed along to the amplifier. I'm not archiving anything; I'm simply playing back my classical 78s for my own enjoyment.

I would like to build a 5879 based phono preamplifier with split passive equalization curves for the various recording characteristics, thus my original question. The 5879 is noisier than a 12AX7 or 12AY7, yes, but less fussy with regard to stray Miller capacitance considerations, and also with regard to source impedance issues when setting up the passive filters. Aside from that, the average early Lp or late 78 has enough background noise of its own to render any contribution from the pre-amplifier essentially inaudible. Just because one can achieve super low SNR doesn't mean that's always sensible in my opinion. In this case in particular, other considerations win out, as they should.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 12:29 am 
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Wild, semi-educated guesses:

fraction of active members who have some idea of what the Laplace transform is: <5%

fraction of that group that remembers how to use it to solve problems: ~10% (Thus < 0.5% of all active members)

I'm in the first group only---learned LaPlace in one short course at UCLA about 52 years ago.....now it's all gone....

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"Measure voltage, but THINK current." --anon.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 1:24 am 
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pixellany wrote:
Wild, semi-educated guesses:

fraction of active members who have some idea of what the Laplace transform is: <5%

fraction of that group that remembers how to use it to solve problems: ~10% (Thus < 0.5% of all active members)

I'm in the first group only---learned LaPlace in one short course at UCLA about 52 years ago.....now it's all gone....


I remembered how to work out the Laplace transform. My issue was simply not applying it correctly to the problem at hand. If I had stepped back for a minute and considered what the behavior was at 0 Hz I would have solved the problem in a split-second.

The 78 curves of interest only define the treble and turn over time constants, not the bass shelving time constant, but it's there. It just isn't obvious until you consider a cutting head amplifier trying to deliver an infinite voltage at DC to a cutting head.

Technically, the three inflection point variant is also flawed in that the inverse network can't give pre-emphasis that rises to infinite gain at an infinitely large frequency. In practice this wasn't defined though, as the cutting head itself would peter out at some very high frequency just as it would at some very low frequency.

Looking at the evolution of equalization curves over time is actually fairly instructive. In the 78 era, we define only two points, because the shelving and high frequency roll off are "taken care of" by the equipment itself. Then we define 3 points with the introduction of the Lp because the bass capabilities of the new format and newer cutting heads demand that some set standard exist. Then we introduce a very low frequency pole in the mid-1970s to deal with the excessive gain at infrasonic frequencies because that portion of the curve had been ill-defined, and then finally it has become common to also introduce a pole at very high frequencies to deal with the ill-defined portion of the pre-emphasis curve in inverse networks. During this process, all of the curves except for the RIAA curve have died out, so they aren't "updated" past their useful life. Each leap in technological advancement brings about new problems that are resolved by simply better defining the equalization curves. All five of these points exist for a 78 curve, but three are entirely arbitrary, thus my initial confusion.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 1:48 am 
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benman94 wrote:
I remembered how to work out the Laplace transform..

I don't doubt that---I was simply trying to point out that your audience here may be rather limited.....

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"Measure voltage, but THINK current." --anon.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 3:31 pm 
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40Hz should give excellent results. I wasn't aware of a formalized cutoff spec but the research you have done correlates well with my experiences.

I was also not aware that A=432Hz was "a thing" these days, but after reading a few audio websites...yikes! lots of discussion about how music at that relative pitch aligns with the universe whereas A=440 is discordant. Who knew? As a former oboe player (oboe is the orchestral instrument that traditionally sounds a tuning note at the start of a performance) I carried tuning forks for 440, the norm, and 442 which I don't think I ever used. A=432Hz? No doubt that would dramatically alter the overtones of every pitched instrument. Who's got time for this stuff?

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 4:37 pm 
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pixellany wrote:
benman94 wrote:
I remembered how to work out the Laplace transform..

I don't doubt that---I was simply trying to point out that your audience here may be rather limited.....


It's not something I do everyday, and I'll admit to having to consult a few references for a refresher. I think there's a tendency to only remember what one uses on a daily basis as a general rule. For example, I'm in the same boat with some of the more cumbersome techniques of integration. I'm sure if I thought about it long enough I could use a rationalizing substitution to eliminate a radical, but why waste the time when I can consult a table? I pushed 90% of the less useful mathematics out of my head a long time ago...


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Tue 26, 2019 4:47 pm 
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richfair wrote:
40Hz should give excellent results. I wasn't aware of a formalized cutoff spec but the research you have done correlates well with my experiences.

I was also not aware that A=432Hz was "a thing" these days, but after reading a few audio websites...yikes! lots of discussion about how music at that relative pitch aligns with the universe whereas A=440 is discordant. Who knew? As a former oboe player (oboe is the orchestral instrument that traditionally sounds a tuning note at the start of a performance) I carried tuning forks for 440, the norm, and 442 which I don't think I ever used. A=432Hz? No doubt that would dramatically alter the overtones of every pitched instrument. Who's got time for this stuff?


The old French standard was A=435 or A=434 depending on which reference you choose to believe, and this became extremely common for most of Europe and a handful of orchestras in the US. This dropped away for the most part by 1939 when there was an international agreement to set A=440, but there were straggler orchestras that still tuned lower. And there were also plenty of groups that tuned higher. I believe to this day the New York Philharmonic tunes to A=442.

As for A=432, that's all new-age non-sense.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Feb Wed 27, 2019 11:03 pm 
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Occasionally long threads about concert pitch standards come up on the bass guitar forum I hang out on from time to time. The universe knows nothing of a second or a Hertz, one pitch aligns with it just as well as any other, but you will never convince some people of this. You can however sell them any 10 to 100 nF paper in oil or Sprague orange drop caps you are willing to part with for $50-$100 a pop! And they will be happy!!

The people who actually benefit from A<440 are singers who struggle to reach the top end of the range for the part they are trying to sing. It can be the difference between an acceptable tone and failure for them.

I used to know all about transforms, Laplace, Fourier, etc. But 30 years of using computers who never forget and never make mistakes have bleached all that out of my brain, I fear.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Fri 01, 2019 4:45 pm 
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benman94 wrote:
richfair wrote:
40Hz should give excellent results. I wasn't aware of a formalized cutoff spec but the research you have done correlates well with my experiences.

I was also not aware that A=432Hz was "a thing" these days, but after reading a few audio websites...yikes! lots of discussion about how music at that relative pitch aligns with the universe whereas A=440 is discordant. Who knew? As a former oboe player (oboe is the orchestral instrument that traditionally sounds a tuning note at the start of a performance) I carried tuning forks for 440, the norm, and 442 which I don't think I ever used. A=432Hz? No doubt that would dramatically alter the overtones of every pitched instrument. Who's got time for this stuff?


The old French standard was A=435 or A=434 depending on which reference you choose to believe, and this became extremely common for most of Europe and a handful of orchestras in the US. This dropped away for the most part by 1939 when there was an international agreement to set A=440, but there were straggler orchestras that still tuned lower. And there were also plenty of groups that tuned higher. I believe to this day the New York Philharmonic tunes to A=442.

As for A=432, that's all new-age non-sense.


I'll throw in my 2 cents worth. I used to have very sensitive perfect pitch. I'd joke that I could hear the dominant pitch of a firecracker. It was most sensitive when I was a musician and hung around live music. That reinforced it and made it more bulletproof. Now, over decades, my perfect pitch accuracy has diminished. Interestingly, recorded music doesn't reinforce it like live music does. Yet some things come back very strong. As I write this I can hear in my head an oboe playing A-440 and even a clarinet playing A-440. If it's A-441 or A-439, I can still hear the difference. But when I audioize (my word similar to visualize) a trumpet playing A-440, the pitch is more vague and I don't feel as much confidence that it's right on. Maybe it has something to to with the fuzzier timbre, and maybe that's why orchestras use an oboe as the first reference.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Mon 25, 2019 5:42 am 
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Greetings to Macrohenry and the Forum:

I have a dumb question for you. An oboe has a very distinctive sound... to me anyway. (My wife is principal second violin for a symphony orchestra and as I attend all her concerts, I hear a live oboe putting out an A-440 frequently.) To me, an oboe sounds quite different from a signal generator driving an audio amp at 440 Hz. However, if you look at the waveform of an oboe on an oscilloscope, it is a near perfect sine wave. My ears aren't that subtle (I don't think) so how come I perceive a different sound?

Thanks for any thoughts,

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Tue 26, 2019 2:27 am 
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Jthorusen wrote:
However, if you look at the waveform of an oboe on an oscilloscope, it is a near perfect sine wave. My ears aren't that subtle (I don't think) so how come I perceive a different sound?

I highlighted and bolded your words "near perfect." Near perfect is not perfect enough when the goal is sine wave purity.

Any distortion that makes a sine wave impure is quite audible and has more leverage on the ear than seems intuitive. I can look at sine waves all day that look perfect on my scope, yet I can hear different timbres. Full disclosure, I have hundreds of hours of experience with this as since 1998 I've made Brian Wilson's sine wave instruments used on Good Vibrations. That requires a very pure tone, and even the original wasn't perfectly pure.

There are several ways to electronically produce a tone free of harmonics. When one spends hours at a time trying to achieve a perfectly pure tone, the ear becomes pretty sensitive to distortion. What would be fun would be to overlay the oboe's sine wave over a pure sine wave.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Wed 27, 2019 12:39 pm 
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full disclosure: I'm fundamentally ignorant about music, but I do know some physics and logic.

My guess is that "perfect pitch" simply means really good hearing and memory. Suppose you learned that A is 440, and then you start hearing 439, it's going to sound off---but you would soon get accustomed to it. But maybe perfect pitch also just means the ability to stay "in tune" with the other musicians.....?

With regard to harmonics, it is not surprising to me that one could hear harmonics that were not easily visible on a scope. First, a scope is not normally a very precise instrument. Second, consider the dynamic range of the human ear---120dB, according to this reference:
https://www.learndigitalaudio.com/dynam ... amic-range

This is sound pressure level (SPL), and I'm not sure if that relates to voltage or power**---either way, I think a distortion of a tenth of a percent---or much less--- is very much audible, but probably not apparent on a scope trace.
So--yes--not all "sine waves" are created equal......



**Let's assume power: 10dB is a power ratio of 10:1, 20dB is 100:1, and 120dB is 1E12:1
For relative voltage, take the square root. This means that, if you hear a "sine wave" at the threshold of pain, you'll hear a harmonic that is at the fundamental voltage over 1E6---i.e. a distortion of 0.0001 percent.

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Wed 27, 2019 10:37 pm 
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Jthorusen wrote:
Greetings to Macrohenry and the Forum:

I have a dumb question for you. An oboe has a very distinctive sound... to me anyway. (My wife is principal second violin for a symphony orchestra and as I attend all her concerts, I hear a live oboe putting out an A-440 frequently.) To me, an oboe sounds quite different from a signal generator driving an audio amp at 440 Hz. However, if you look at the waveform of an oboe on an oscilloscope, it is a near perfect sine wave. My ears aren't that subtle (I don't think) so how come I perceive a different sound?

Thanks for any thoughts,
A "near perfect" sine wave means there's significant distortion. Eyes aren't very good at 'seeing' distortion, that's why we have ears. :wink:

Look at these two scope traces. I bet you'd call both 'near perfect' but the second has 5% second harmonic distortion.
Attachment:
h1.gif
h1.gif [ 3.45 KiB | Viewed 1235 times ]

Attachment:
h2.gif
h2.gif [ 3.42 KiB | Viewed 1235 times ]


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Thu 28, 2019 12:15 am 
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Greetings to Flipperhome and the Forum:

I think you have demonstrated the matter adequately. I saw the scope trace on a Youtube video. By inspection, it looked pretty good to me. It would seem however, that I can detect the difference, even with my tired old ears.

Thanks,

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Thu 28, 2019 4:32 am 
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It's interesting to note that there is still no general agreement as to what percentage of non-linear distortion is audible. As a rather general rule, Harry Olson suggested in the late 1940s and early 1950s that there wasn't much point in reducing THD at the amplifier below about 1%, as the non-linear distortion of even the best horn loaded loudspeakers was at least a factor of 2 to 5 higher than that. Yet, in some of his other articles and books, he suggests that even a "small fraction" of 1% might be audible when using an amplifier with a Class B output stage or pentodes. Then, when discussing his push pull parallel 6F6 amplifier, he seems to suggest that anything below 0.75% THD in the amplifier must be inaudible. Then he reversed course again in the mid-50s and suggested that the real limit of audibility lies "somewhere between 0.5% and 1% THD" in an amplifier under ALL conditions, as again, the loudspeaker must be the dominant source of distortion.

It's somewhat disheartening that Olson would never nail down a hard and fast limit to point to, as he was such an authority, and carried out what are still among the best psycho-acoustic experiments regarding what is, and is not, audible in the electronic reproduction of sound. It's understandable though, as the limit of audibility will naturally vary somewhat from person to person, and perhaps even between one's left and right ears.

I personally don't believe that under any reasonable, real world condition a distortion of 0.0001% is going to be audible. Under a majority of conditions, anything less than 1% is probably not going to be audible, or at least not bothersome, but I can't say that I have anything other than a hunch and a deference to the authority of Olson to back that up.


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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Thu 28, 2019 1:26 pm 
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My pea-brain is saying:

Harmonics is what makes 2 instruments sound different. Do a double-blind test to see if people can hear the difference between--eg--two similar guitars. Then have a distortion analyzer or spectrum analyzer "listen" to each.

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 Post subject: Re: Questions about Laplace transforms in LTSpice.
PostPosted: Mar Thu 28, 2019 3:56 pm 
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pixellany wrote:
My pea-brain is saying:

Harmonics is what makes 2 instruments sound different. Do a double-blind test to see if people can hear the difference between--eg--two similar guitars. Then have a distortion analyzer or spectrum analyzer "listen" to each.


That's essentially what he did, but instead of using two different instruments, he used one instrument, and specially designed aluminum and steel panels with thousands upon thousands of small perforations to acoustically vary the perceived THD at the listener's position, and also the apparent frequency response. It worked to a degree. He was able to determine without a doubt that when non-linear distortion is held sufficiently low, all listeners preferred wide-bandwidth sound. He was also able to show that beyond a certain threshold of distortion, listeners found a distorted narrow-bandwidth sound to be less irritating that a wide-bandwidth sound. His results regarding distortion itself were inconclusive. Many people had no ability whatsoever to distinguish between even 5% THD and very low THD (today those people would probably buy cheap Chinese SET amplifiers), others easily could and had a very definite preference for the low distortion. The problem was, and is, how low is low enough for all possible listeners? Obviously an ideal 0% THD would fulfill that requirement, but the question probably doesn't have a practical answer.


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