You are still overlooking SNR loss by significant digital attenuation.
Think about what happens if you lose 1 bit of resolution. It means you lost 1/2 of all possible digital sample levels. When you lose 2nd bit your recording will be coded only by 1/4 of originally available sample value set. With 3rd lost bit it is 1/8 and so on.
Lets count that for -48 dB, 24 bit recording and DAC resolution of 20 bits. Assume that recording is mastered up to 0 dB (that's unfortunately often the reality). It means 2^20 values = 1048576 available values for digital samples. When you lower digital volume to -48 dB, you lose upper bits 8 bits out of 16 (you can roughly imagine it like shifting bits to right, although it is only an estimate view since 1 bit is not exactly 6dB). What remains within DAC resolution (I will assume 20 bits) are bits 9 to 20, together 13. That's 2^13 = 8192 values instead of the original 1048576. Simply said you get much less digital values to encode the same audio content. Therefore that encoding will be less precise and of course S/N ratio is significantly lower.
The 20 to 22 bits measured DAC dynamic range is reality and existence of lower bits than 20th or 22th and even computing volume in higher resolution math than 32 bits does not change anything on the final result. You can perform digital volume level computing on any high precision but when the following delta sigma modulator + D/A stage does not bring higher dynamic range than those 120 to 132 dB then the lowest bits, which are part of your thought computing, are simply lost below DAC noise floor, so their existence doesn't matter and has no effect on analog output.
You cannot increase DAC dynamic range by using more precise digital volume processing. You can count any number of lower bits in digital domain but when they are lost below noise floor they don't influence the analog result.
Higher precision math for volume control is used to compute enough precise final computing result. Digital volume processing means an algorithm consisting of series of arithmetic operations. Each operation may be subject of rounding error and that error raises with number of operations performed. Therefore it is usual that DSP is computed at higher resolution than the resolution of audio data processed so then the rounding error does not affect the input data range (for example 32 bits). That's the only reason of using higher resolution math for DSP.
You're correct but if you read carefully my post, you will see that I've never claimed that the dynamic range is increased. On the contrary, I wrote that SNR is getting worst with digital volume control - which is equivalent to say that dynamic range is lost. Then, I demonstrated that -in my setup- it does not matter because what is lost is anyway way below hearing threshold.
Let me do the same back-of-the-envelope calculations but from a dynamic range perspective. To recap, my amplifier has a 17 dB gain, my speaker are 89 dB SPL at 1 m for 2.83 V. Now, ASR measured the dynamic range of the E50 at 122 dB relative to full-scale. The full-scale for the E50 is about 4 V (that is, 12 dBV). So the full-scale output of my system is 12 + 17 - 9 + 89 = 109 dB SPL at 1 m (which is pretty loud in my book but still somewhat reasonable). Now, based on the dynamic range (from a DAC-limited perspective) we thus have 109 - 122 = -13 dB SPL as the lowest resolvable 'meaningful' sound that my system can produce (by meaningful, I mean above noise level)... pretty close to what I calculated earlier and still totally inaudible. By the way, 122 dB corresponds to about 21-bit ENOB and if you look at all other relevant DAC metrics, the E50 scores always around the 21-bit mark.
What happens if you use digital volume control? Well, this -13 dB SPL figure stays the same but, on the other hand, the 109 dB SPL one is reduced. For example, if I use my usual -35 dB volume settings, my maximum SPL is 74 dB and the dynamic range is thus 74 - -13 = 87 dB. Yes, it has worsened BUT it does not matter as -13 dB is still 13 dB below audibility threshold.
Now, let's use an
ideal analogue volume control. In the same example as above, the maximum SPL would still be 74 dB but as the dynamic ideally stayed at 122 dB, we have now -48 dB SPL as the noise floor. Much better than with digital volume control but we are now 48 dB below audibility threshold, which does not improve sound quality w.r.t. being 13 dB below that same threshold.
So, yes from a pure number perspective it was better to use analog volume control but there is no audible concern whatsoever to use digital volume control in this case. Think of it this way: in both cases, we have an
audible dynamic range of 74 dB (i.e., compared to 0 dB SPL). There are reasons why some of the thresholds in
ASR thresholds are slanted.
There are however cases where using digital volume control can be 'problematic' though. For example, let's assume I was using the E50 connected to the same amplifier but with a highly efficient loudspeaker, say 110 dB SPL at 1 m for 2.83 V. The full-scale output of that system is 12 + 17 - 9 + 110 = 130 dB SPL at 1 m (way too loud for home usage I would say). Now, the lowest resolvable 'meaningful' sound this system can produce is 130 - 122 = 8 dB SPL, which is above the 0 dB SPL audibility threshold. This system is not transparent. To listen at the same typical SPL level, I would have to set the E50 to -61 dB. The dynamic range is then limited to the 74 - 8 = 66 dB. Using analog volume control in that case would be beneficial and increase the theoretically audible dynamic range by 8 dB (that is 74 instead of 66 dB). In that case, were I to decrease the gain of my amplifier to 9.2 dB (the lowest settings of Benchmark's AHB2), I would move the DAC noise floor from 8 down to 0.2 dB SPL. This is close to transparency but because of other sources of noise, I may prefer higher margin and would possibly use analog gain control in that case...
In conclusion, digital volume control can be as audibly transparent as analogue volume control. It however does not only depends on the DAC but rather on the full system. The amount of digital attenuation is only relevant in so far that high digital attenuation followed by high analogue gain (be it amplifier or loudspeaker sensitivity) may be detrimental as the high analogue gain may push the noise floor of the DAC in the audibility range.
I would argue that your last paragraph is the domain of philosophy. In digital signal processing, higher bit depth equates less roundoff errors but also higher resolution - you would use one term or the other depending on the context...
(edited some typos)
