Is it a proven mathematical equation?
Is it black and white and carved in stone and agreed by 100% of all people?
If not, then it’s his opinion.
It’s 100% not rocket science or neurosurgery or medicine.
This is not an accurate statement at all. I think you should take a little time to learn about this science and the ones you compare it against. I am a scientist, I study human behavior, and my work straddles medicine, psychology, and human perception. My expertise in designing studies encompasses all of these. Then there is audio, more a hobby, but one I've taken very seriously and have real academic credentials for. Audio is physics.
It is a mistake to suggest that human preference studies are mere opinion as they misunderstand what is being studied. Further, it is a mistake to assume that these other sciences are as concrete as you suggest.
Is pain management studies merely opinion since they cannot directly equate taking a pill with a direct measure of pain (such as nerve receptors leading to the firing of electrical impulses caused by pain)? No it is not opinion, because the methods used are highly objective and rigorous.
Toole's work is not his opinion because he used a scientific method of inquiry that minimized sources of bias. He and his colleagues have openly published their work in 3rd party peer-reviewed journals. It is a requirement of published research that you openly share everything you did such that any other knowledgable scientist could recreate what you did.
You asked if there is a formula that underlies what he says. The answer is yes. You seem to think no, but I suspect you never read his published articles. They absolutely created a mathematical formula used to indicate if a person will like a speaker or not based on its anechoic measured performance. They compared numerous algorithms to find out which combination of measured performance led to the highest prediction of preference. For example, they found that response smoothness and bass extension were the most important indicators in the formula, but not the only indicators. Sean Olive pointed out to me in a recent email exchange that the smoothness of the response off-axis is a more important indicator of sound quality than is the rate at which it falls. In other words, having a flat DI is more important than having an elevated DI.
What I think you don't like about this work is the notion that we can mathematically predict, from a probabilistic standpoint, what sounds good. I believe, from past comments, that you believe that sound quality is a personal thing. You are correct in that what matters most is what sounds good for you. The problem is that is not science and the assumption that everyone's perception of good sound is unique for them is flawed. Humans are far more similar than different and our tastes are no different. This isn't marketing, nobody is designing speakers with a particular sound and then using marketing to influence what people like. That would be a silly thing to do, as there is no benefit to treating a speakers frequency response like a fashion trend.
What Toole and colleagues did was use a rigorous understanding of human physiology and research (much of it done by other people, not themselves) around psychoacoustics to then understand how that plays into what we like in the sound of a speaker. This was tested in double-blind manners to ensure they were not biased. A lot of that work started and was done at the NRC, not Harman, and the NRC had absolutely nothing to gain from their research. Harman allowed (and still allows) it's research team to operate separate from the marketing and development of products. They don't have scientists invent a new product, and then ask the research team to prove its best. It goes the other way. They use their research to discover what leads to the best technical performance. They use the products that come out of that to then test their hypothesis. When its confirmed, the products remain in production and they share the results. When it doesn't, it becomes an aspect of published research, but obviously not something that remains in production or is touted as a great innovation by marketing.
One thing to remember is that Toole, Olive, and Welti are all very careful about sharing opinion or speculation. When they make a bold claim, it is almost always backed up by rigorous scientific evidence and supported by a sound understanding of human psychoacoustics.
It's also important to know that most of the best and most innovative research in medicine and physics today is also probabilistic. I know because I do some of this work. One of the hottest new medical trial designs for evaluating the effectiveness of a treatment is known as a Bayesian Adaptive Trial. I modified that approach for use in evaluating the effectiveness of social support policy programs. These use both Bayesian statistics as well as the general concept of Bayesian inference to allocate individuals to different treatment arms based on the probability that they will have a successful outcome. An algorithm is used to make predictions at the point of randomization (they are randomized with a probability bias toward the most effective option) and once that algo basically makes near perfect predictions, we can stop the trial and know how to make a diagnosis and treatment that works. This approach has led to major breakthroughs in cancer treatment. I fully expect it will also lead to major breakthroughs in how we educate our children, what services we make available to families in need, and how we address public health crisis.
Lot's of speaker manufacturers develop products based on nothing more than strongly held opinions. Companies like Harman, Kef, and RBH are not examples of that, they rely on rigerous human perception studies.
