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An interval is classified by its consonance or dissonance: as an open consonance (unison, perfect fifth, octave), a soft consonance (major and minor third and sixth), mild dissonance (major second and minor seventh), sharp dissonance (minor second and major seventh), ambiguous (perfect fourth), or restless (tritone).
How are minor, major, and augmented ninth, eleventh, and thirteenth intervals classified with respect to consonance and dissonance?
This seems like pigeon-holing for pigeon-holing's sake, but basically a 9th will be as a 2nd, an 11th as a 4th and a 13th as a 6th. Those are basic major intervals.
Minors will be classified as the same as basics - ♭9 as m2. ♭11 surely doesn't need anything! ♯9 must have the same classification as m3, &sharp11 as tritone.
I don't think that the classification would differentiate between, say, an aug.4th and a dim.5th, as it's the sound in question here, not what the interval happens to be called.
It depends on the musical context, but I tend to use the major ninth as a very consonant interval. This arises from its interpretation as two stacked fifths, i.e. I see it as an open (or, Pythagorean) consonance. Of course this can be somewhat negated if it clashes with other notes, i.e. if you have an octave and/or tenth around, the ninth will be at least mildly dissonant to those.
The minor ninth, by contrast, sounds pretty dissonant by itself.
The pure 11th sounds rather more dissonant than the pure 4th IMO.
For the 13ths I'm not sure, I've never considered how these differ from the sixths.
Do you really want to know this?
I can tell you what I've found if you don't fear the answer and you can read this if you are not afraid of getting crazy:
Interval weight according to Hofmann-Engl 2004
Tone weight after location in accord to Hofmann-Engl 2004
Formula for basic weighting according to Hofmann-Engl 2004
Weight for the> root candidate
As in Terhardt's model, the minor triad is also interpreted here as the sixth, seventh and ninth partial of the overtone series:
Weight for the> root candidate
etc. etc. you'll find even more abstract formulas with which I can't deal anything.
As you can see these formulas pretend to measure the weight for the root candidate of one tone and he will continue for all other tones and then we should measure the weight of all the 3rds summing up to the 16ths. Do you really want to know it. No!
https://www.gmth.de/zeitschrift/artikel/446.aspx
this article is in German but it is easy to read and understand by google translation:
"With the first part of the lesson instruction Hindemith formulated the attempt to analyze also music, which lies outside of the scope of function or step theory. The same claim is made by Balsach and Hofmann-Engl. By contrast, Terhardt and Parncutt seem to limit themselves to tonal music, although Parncutt at least suggests the possibility of extending his theory. Hindemith's model contradicts the conventional interpretation of clearly tonal chords common in pop and jazz music. Terhardt's theory comes to more plausible results, but has difficulties to determine the root of the minor chord. The problem consists not so much in the assumption of a "secretive" fundamental tone, but rather in the interpretation of a minor 3rd as a natural epitome to the supposed fundamental tone, which clearly contradicts the pure mood. The strengthening of the real Basston made by Terhardt tries to mitigate this shortcoming. Parncutt "saves" the traditionally accepted root of the minor chord, as on the one hand, like Hindemith, heavily weighting the fifths and fourths in the chord and, on the other hand (in the 1988 model), as in Oettingen, giving common partial tones as justification for including minor tenors in the basic tone determination , Of course, Parncutt does not solve the problem of interpreting the minor 3rd as a natural epitome to the supposedly 'virtual fundamental tone', but merely reduces it through the postulated numerical values. It is not clear what significance the calculated weights, which are, according to Parncutt, the great advantage over Terhardt, actually have for basic tone determination, since they are relatively again by the distinction between> root supports detractors <. Parncutt further reinforces these tendencies in his revised model of 1997 by weighting the fifths contained in the chord even more, but in return he upgrades the bass tone. The prevailing tonality seems problematic because it presupposes the major-minor tonality."
My answer is: Try to do this classification yourself. You will learn a lot by doing this. You will learn the names, the differences, the elements, and you will train your ear. You might also go to a school teacher and ask whether you can make an survey in his singing class and play the list of chords and let evaluate the pupils the grade of dissonance or consonance. If I were still music teacher I would be very interested in this research.
