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All the audio mixer circuit examples I see online are based off of the summing amplifier:

enter image description here

This is an inverting op amp configuration so the output is inverted. I built a similar circuit and I couldn't tell that the output signal was inverted by listening to the audio...so is there a reason why you would want to de-invert the output of the mixer, by putting in another inverting op amp stage at the output, like in this circuit:

enter image description here

If the audio output sounds the same whether the output signal is inverted or not, why would you ever need to de-invert the output?

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When listening to a single channel, delay or inversion doesn't matter. When the overall audio has multiple signals (stereo has two, for example), then the relative phase or inversion between channels matters. You can invert all channels or none. Inverting some but not others messes up the phase relationship between the channels and the sound position you are supposed to perceive from them won't be correct.

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Imagine mixing 2 channels, say a vocal and a guitar track, through an inverting summing amplifier and recording the output. Of course it would be inverted but you wouldn't hear anything different.

Now, imagine if you wanted to mix a channel of the vocals (recorded at the same time as the 1st recording but from a different microphone) to the previous mix.

What would happen is that most of the vocals would disappear because you would be subtracting rather than adding the second recording.

So if we didn't use the second inverting amplifier, we would need to keep track of how many times something has been mixed to decide if we need to reverse the phase of the signal we want to mix.

This makes things awfully complicated for no reason.

There is also the advantage of being able to add a phase reverse function simply by bypassing the second inverting amplifier. There is a phase reverse button on the top of each channel of most mixers I've ever come across.

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Adding this because it's relevant to audio mixing and routing;

One very good reason to have all inputs and outputs in phase by default is that it is not at all uncommon to process audio signals externally, either using the mixer's send and return paths or by splitting the signal and routing one path through a processor and the other path to the mixer "dry". If there's a phase inversion in your mixer circuit, it will present cancellation issues with any sort of parallel signal routing.

That being said, it can be extremely useful (and sometimes necessary) in the realm of audio engineering to have the option of selectively inverting the phase of a given input/output channel, or group/bus.

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In a stereo mixer, both channels will be inverted, so will retain the same relative phase.

For most waveforms, the envelope changes so slowly compared to the frequency that sounds don't start and end on a recognisable edge. There is no way to tell whether the waveform has been inverted.

There is a school of thought amongst some golden-eared audiophiles that insist that waveform polarity is important for things like drumbeats, where you want the sound to start on a compression wave. You would need to maintain polarity through your entire audio chain to ensure this however, making sure your speakers were the fight way round. Like you, I've never noticed a difference, but at least I know my ears are forged from the finest cloth.

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so is there a reason why you would want to de-invert the output of the mixer, by putting in another inverting op amp stage at the output

Wavelengths of sound through are as shown below: -

enter image description here

So, if you are sitting 1 metre from a speaker that is producing a frequency of about 172 Hz, that sound is inverted compared to the sound emanating right at the cone of the speaker. Can you tell the difference?

However, a frequency of 343 Hz is back "in phase" at 1 metre but if a speaker is producing both tones and you slowly moved away from the speaker would you hear some strange artefacts at various positions?

No you wouldn't - in fact for any compex musical sound all you notice is a reduction in volume as you move away.

Clearly though, for stereo reproduction you want to keep left channel and right channel with the same phase relationship electrically else you will get bass cancellation at a listening position.

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