Loudness war, the limit of the 0dB in digital or Analog better than Digital?

By convention, in digital, the 0 dB level is the highest possible level, impossible to go above this “absolute zero”. This is why the values are given by counting down from 0 dB, -10 dB as in the example below.

Audio meter with maximum value at 0dB

From digital point of view, the 0 dB corresponds to the highest possible value according to the number of bits available, here are the examples of these values according to the digital media:

MediaNb of bitsMaximum value for 0dBDynamic
CD16 bits+- 3276890 dB
Hi-Res24 bits+- 8388608138 dB
Master DXD32 bits+- 2147483648186 dB

For the case of the CD, the 0dB corresponds to the digital value +- 32768 (15 bits and 1 bit for the parity +-), and the lowest level corresponds to the computer value 0.

The difference between the highest level and the lowest level of the signal gives the dynamic range. The dynamic range is expressed in dB. You can find more informations about the dynamique here.

When listening to a song, a louder song attracts more attention, which is why we try to have the maximum level for the music. With digital, the first step is to adjust the level to the maximum, i.e. to 0 dB. But, if one is not careful, one raises the level too high, it is then a phenomenon of clipping that occurs. But we always want to sound louder. So, if we can’t raise the maximum level, we’ll raise the average level and compress what could exceed 0 dB with a limiter. This phenomenon has the consequence of reducing the dynamic range. The more you raise the level, the more you reduce the dynamic range, it is this phenomenon which gives the Loudness war.

The examples below will illustrate these clipping and dynamic range reduction phenomena with the use of a limiter, and measure their impact in terms of distortion.
To illustrate the phenomenon, we will use a reference signal of 1 kHz (this is valid for all frequencies):

Clipping: the graph below shows the impact of clipping. From an original signal at 1 kHz with the maximum level at 0 dB, we decide to increase its level by 3 dB.
As the digital signal cannot exceed 0 dB, the result is a red signal with the top of the sinusoid cut off. If there was no 0 dB limit, we should have obtained the green curve.

To further illustrate clipping and measure its impact, we use the reference signal and a gain is applied to the audio signal to try to exceed 0.5 dB from 0 dB.

This unfortunately causes a clipping of the signal (in digital we can not exceed 0dB) and we can see a flattening at the top and bottom of the signal..

The spectrum representation of this 1 kHz signal clipped at 0.5 dB shows many peaks of distortion, these peaks represent the harmonics of the 1 kHz signal, so we speak of harmonic distortion as shown in the spectrum below:

If you increase the sound level a little more, you get more clipping on the signal. The graphs below show the waveform and the spectrum for a clipping of 1 dB and 2 dB:

Clipping is a very brutal phenomenon in digital and generates a lot of distortion. Even if it should never appear, clipping can be found from time to time on recordings that have not been mastered in sound level. To avoid this problem, we use limiters.

Limiter: the graph below shows the impact of using a limiter. Starting with an original signal at 1 kHz whose maximum level is 0 dB, we decide to increase its level by 3 dB using a Brickwall limiter.
The signal obtained in blue does not exceed 0 dB, but the limiter has modified the signal, it is distorted (more rounded at the top and bottom) compared to the original green signal.

To further illustrate the use of the limiter and measure its impact, below is an example with a signal at 1 kHz that should have exceeded the 0 dB by +2 dB but to which a signal limiter has been applied to avoid clipping.

The graphs below show the waveform and spectrum for a +3 dB signal passed through a limiter to bring it down to 0 dB.

The table below represents the distortion produced by clipping or limiting on a 1 kHz signal.
We can see that the values are very high for clipping, 4.6% for 1dB of clipping, limiters reduce this distortion by 2 but it remains high.
The storage (writing on hard disk, storage on CD) of a digital signal does not induce distortion, which is not the case in analog (vinyl, tape). There will be some distortion during Digital to Analog conversion, but it will remain very low (0.005% depending on the DAC used).

 Signal at 1 kHzDistorsion
Clipping 0,5 dB2,17%
Clipping 1 dB4,60%
Clipping 2 dB9,31%
Limiting 1 dB2,54%
Limiting 2 dB4,67%
Limiting 3 dB6,30%

Now we will compare with analog media in terms of distortion and analyze the distorsion in this case.

To begin with, vinyl is a good reference because there should be no clipping during the cutting of the vinyl record so that there are no problems during playback. Here is a 1 kHz signal at 0 dB read from a vinyl.

The 1kHz reference signal read from a test disc, we see a nice shape of the sine wave.

The spectrum representation shows the distortion peaks, which are significant, but also a higher noise floor than digital.

We can see that vinyl at 0dB has a distortion of 1.13% in our example, with a value of 0.25% at -20 dB (below).

Another very common recordable analog medium is the cassette.

The 2 curves show a 1 kHz signal recorded on a tape recorded (Sony TCK 555) at 0 dB with a Dolby C NR noise reduction system:

The reference signal recorded and played back on the cassette deck, shows a slightly rounded shape at the top and bottom of the sinusoids.

The spectrum here shows much more harmonics (1.65%) with more peaks than on the vinyl. The background noise is also higher than on vinyl.

Other examples of signal at 1kHz on a Nakamichi LX3 cassette deck for 0 dB (distorsion 0.46%) and +3 dB levels (distorsion 0.85%).

Another analog medium is reel-to-reel, using 1/4 inch tapes at 19 inches/second 38 cm/s). Here are three examples using a TEAC 3440, a Tascam 38 with and in DBX noise reducer.

For these multi-track tape recorders, only 2 tracks are used.

The table of measures represents all the results in terms of distortions. In digital (blue), if there is no processing with clipping or limiting, the copy of a digital file does not cause any distortion hence the 0% reference.

Summary of measurements
In conclusion:

Digital is the best medium in terms of bandwidth, distortion and noise floor, but it has a constraint to respect, it is not to try to exceed 0 dB or to maximize the sound level with the use of limiters. Because in these cases, the distortion increases quickly and becomes several times higher than analog supports like vinyl, tape and cassette.

Whether on CD, streaming or digital files, the loudness war only makes a very high quality medium inferior to analog media in term of distorsion, that’s the paradox. The problem does not come from the support, but from the music that we put on it and the treatment that is done beforehand, in particular with the mastering to make it sound “loud”. We still hope for a change of this behavior, but despite the growing awareness, it remains a will for the distribution of music.

Measurement condition:

The measurements are given as a guide for the equipment below and may vary depending on the type of equipment and the type of tape or cassette used.

The distortion measurement was made with RightMark Audio Analyzer 6.4.5.

The Brickwall limiter used for the tests is LoudMax version 1.4.

The vinyl measurement was made from the Elipson Ultimate Technical Record, with an Audio Technica AT33PTG2 cartridge and the Xono-based DIY preamp.

The cassette measurement was done on a Sony HF60 cassette, with a Sony TCK555 turntable, 3 heads, in monitoring mode with Dolby C NR, and in playback on a Nakamichi LX3 deck.

The tape measurement was done on a SM900 tape (Recording The Master) with a TEAC A-3440 on 2 of the 4 available tracks, and with a SM911 tape (Recording The Master) with a TASCAM 38 on 2 of the 8 available tracks with and without DBX noise reduction.

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