When I say twice as loud, that is an approximation, purely based on my own personal perception.
I am not using any measurement tools. And I do not assume the person who asked for help may be using measurement tools.
For more exactness, the exact difference in power between the balanced output and the single ended, in watts, varies. It is not an exact ratio, and this ratio of how much the balanced output provides, in comparison with the single ended - on a specific headphone, it also depends on the input impedance of the load - i..e.
I recall some headphone amplifiers, that I recall reading their specs, most likely one of the Chord dongle DACs - but it could have been some other brand, where the watts on the balanced vs the single ended were exactly the same.
Here is an example from the Topping DX1 version 2 (single ended is in the middle column, balanced outputs are on the rightmost column. As you can see, the ratio of max output in watts, at a specific load, varies quite a bit, from 115/29 @ 600 Ohms, to 730/800 @ 16 Ohms.
In my dongle DAC which has both balanced and single ended headphone outputs, I tend to use a compensatory gain of 7.5 dB gain, which I set by ear, in my playback software,when I move an IEM on the same cable (a modular cable, with interchangeable connectors for 3.5mm SE and 4.4mm balanced) from balanced to SE.
Sorry to have to redirect you, your quote of 10dB as the doubling gain, is not exactly accurate cos it assumes you are listening to a specific frequency of 1Khz tone.
From google search
The Perceptual Rules of dB in Audio
Because human hearing responds to sound logarithmically rather than linearly, the relationship between physical sound measurement and psychoacoustic perception relies on a few core benchmarks:
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1dB: The smallest detectable change in volume that a highly trained listener can perceive under ideal conditions.
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3dB: A clearly noticeable and commonly accepted threshold for an audible difference in volume. It requires a literal doubling or halving of acoustic energy.
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6dB: A physical doubling or halving of the sound wave’s amplitude, but only perceived as a fraction of “twice as loud”.
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10dB: The psychoacoustic threshold of twice (or half) the perceived loudness. To achieve this, audio systems must increase physical power by \(10\) times
More info at the Youtube video below
https://youtu.be/080I7w1pr34?si=YuqxKlj-rSYcBWkS
Why Perception Varies
Perception is not a flat, one-size-fits-all scale. The “doubling” rule applies most consistently to mid-range tones around \(1\text{ kHz}\) and at standard listening volumes. At softer listening levels, a smaller jump (e.g., \(2\) to \(3\text{ dB}\)) can trick the brain into perceiving a doubling of volume.
Additionally, human ears are less sensitive to extremely high and low frequencies at quieter volumes, as outlined by the Fletcher-Munson curves (equal-loudness contours). As a result, the perceived dB range for “twice as loud” shifts dynamically depending on the frequency of the sound and the environment you are listening in.
**A doubling of the sensed volume (loudness) is equivalent to a level change approximately between 6 dB and 10 dB.
Why, cos loudness is phychoacoustic. Further reading at the link below.**
So in conclusion, when I said a balanced output, in my case, sounds twice as loud, that was correct, cos I use, based on listening, a gain of 7.5dB to correct the drop in perceived loudness, which is between 6dB and 10dB, that is correct in my own case, and for the input impedance(i.e the headphone amplifier load) of the IEM I use most of the time, and to me sounds this difference in gain, sounds about twice the loudness.
But as you can see from the Topping DX1 version 2, that gain, or any attempt to match loudness, would not possibly incorrect for the Topping DX1 version 2, cos at every load ( headphone or IEM input impedance), one would have to listen to determine what sounds like the appropriate compensating gain, between the SE and balanced outputs.
While I used my own example which needs about 7.5dB to double the loudness, the more important point is that loudness typically varies between the SE and balanced and that when comparing these two outputs on the same device, there is a need to compensate for the variation in loudness, in the signal chain, to match the actual listening levels, and as I said in the previous post, to avoid the deception of the louder signal sounding better.
In some headphone outputs, there is no difference in wattage between the SE and the balanced>
I bothered to check, and indeed I am correct, in the case of the Chord Mojo 2, this is so.
From google.
The updated version of the Chord Mojo 2 features both a 3.5mm single-ended and a 4.4mm balanced headphone output. Despite the physical difference, both outputs share the exact same maximum power output and internal analogue signal path. [1, 2]
Because Chord Electronics derives the 4.4mm connection’s signal from a single-ended internal architecture, you do not get the traditional balanced power boost (e.g., doubling the voltage). [1]
