This ratio is the FACT......He was probably discovered by accident in room tuning with EARS or by someone who knows the importance in all field of this historical proportion in art and science....
There is no accident in using 38% (really 37.5% of 3/8th, or 6/8th). The room node lengths of any two walls is a complete waveform and multiples of that. 37.5% is simply where the originating waveform and its reflection cancel or reinforce each other out the least.
The golden ratio, unfortunately, is not purely math as you are no doubt aware. It is far too the proverbial square peg that people try to put into a round hole. In this case it is 1.67, and just happens to be close to 1.62. If it was 1.7 or 1.55, people would still try to equate it to the so called golden ratio.
You see this what I consider sillyness applied to 16:9 aspect ratio for TV (movie) but this is again another false analogy. 16:9 was a technical trade-off. No more, no less. Our vision limit is 1.4:1 for black and white, about 2-2.2:1 for color discrimination (hence why widescreen movies are like this), and close to 1:1 for central vision.
You may wish to read this: https://www.fastcompany.com/3044877/the-golden-ratio-designs-biggest-myth