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Has anyone tried to describe the "feel" that the various notes in a scale have? For instance, in any scale, the tonic, root, or 1st degree, could be described as something like calm, centered, or grounded, but what about the 2nd, 3rd, 4th, 5th, etc.?
For example, in a major scale, maybe it would be something like:
You are talking about the qualia problem of musical scales. There is a narrow field in musicological research investigating this, Zuckerkandl (1956) in his classic work stated that the "feeling" inherent to scale degrees is all about the "direction" in which they point. Arthur (2018) conducted a study in which participants where asked to write down their associations with specific scale degrees, just look at the table on the last page in the PDF, where the most frequently mentioned associations are collected. David Huron (in Sweet Anticipation, 2006) did a similar experiment and found the following common responses: 
The difficulty here is that, perceptually speaking (in contrast to music theory) it is not clear where the tonic as the reference point of such a qualia framework is. In chordally conceived Western music, the momentary chord root often perceptually dominates over the root of the key, so that the perceived qualia are often a mixture of the scale degree related to the root of the key and that related to the root of the momentary chord.
