You're asking musicians about identity relations… most of us have relations & can identify them, when sober. Apart from that, I don't think musicians can help you with maths. For those of us musicians struggling to separate our identity from our reflexive relations I'd suggest another round of drinks, make mine a double ;-)

It's prob more accurate to posit that beauty is proportional to the quantity (or lack of) and variety (or lack of) of the tones and pauses themselves, as opposed to the identity relations of them. Which brings me to your second reference - The beauty of a melody COULD be lost by rearranging the tones and pauses... also, it might not be (and I think you mean to say "through time" there and not, "throw time.") There's no reason to think the beauty will be lost just as there's no reason to suspect that otherwise applying the same relations in one pretty melody to another song guarantees beauty.

You may be interested in checking out the work of musician/philosopher Leonard Meyer, especially his book Emotion and Meaning in Music. I'll also add that we typically discuss four properties of sound: in addition to your pitch (often called frequency) and duration, we also add intensity (volume) and timbre.

:-) @Tetsujin I have also asked at the Mathematics Stack Exchange. They said that it is a music theory question. Thank you for the links. Greetings!

@Tetsujin that some or even most musicians are unfamiliar with identity relations does not imply that all are.

@phoog, draw me a venn diagram… :P

@Richard but intensity and timbre do not define a melody, do they? A melody does not become a different melody when played on a different instrument or at a different volume. In fact, absolute pitch does not even define a melody, but only relative pitch, since a melody retains its identity when transposed to a different key. Come to think of it, even the durations of the notes can be modified to a certain extent. Nikola: what about beautiful or non-beautiful performances of the same melody? Are you concerned only with melody in the abstract?

@phoog I suppose that non-beautiful performances of a melody mean that pitch and duration values of its tones and pauses are changed, so, yes, I am concerned only with a melody in the abstract

@StojkoskiNikola well a non-beautiful performance might be so because of timbre or volume rather than pitch or duration. Furthermore, pitch variations that might contribute to a lack of beauty may be fairly small, within the range of variation that distinguishes one tuning system from another. Which tuning system renders a melody more beautiful will be a matter of opinion.

@phoog Ok then, let's hold to a melody in the abstract

Each tone has two properties: pitch and duration; each pause has one property - duration. That's too simple an abstraction to be useful if considering beauty. Pitch can vary within a sounded note, and intensity and timbre can vary over the course of a note too. These aspects account for some of the subjective beauty of the melody. Additionally, unaccompanied melodies are very rare - much of the effect of real-world melody is due to the chord progressions they are set against.

@topo Reinstate Monica This is very useful. Can you give me examples of unaccompanied melodies considered beautiful? I have thought of ringtone melodies.

@StojkoskiNikola an example I can think of is the Last Post. But many people would consider it very much less beautiful played by a sinewave or other simple waveform compared to being played by a good bugler, because the atypical timbre wouldn't bring about the associations that the bugle sound would.

@StojkoskiNikola an example of topo's point: take a melody that is considered (subjectively) beautiful. Examine the underlying harmony and rhythm (even most ringtone melodies have accompaniment,) and one discovers that it is the way that all of the pieces fit together that creates the subjective beauty. Leave the melody the same and change the accompaniment, perhaps the beauty is lost (perhaps not.) I see you edited your question a bit but, I'd still maintain that taking a melody and playing it in 'random time order' MIGHT affect its beauty... might not. That is sort of what soloing is...

@Tim Burnett - Bassist I cannot agree with the claim that playing the tones in 'random time order' might not affect its beauty. imagine, for example, how would it sound if odd and even tones swap places.

What do you mean by odd and even tones?

"how can we determine the quantity and variety of identity relations [...]" - What do you mean? There are several possible methods for determining that, but undoubtedly, you wouldn't be satisfied with just any method of determining it; you're looking for a method which you will find suitable. But what are your criteria for a suitable method? If you don't tell us your criteria, the best we can do is make wild guesses as to what you're looking for.

I'm sure what you're asking makes sense to you, but I think somewhere within the post you lost me. As far as I can tell, beauty is very subjective within music, so if I had to guess, the assumption "beauty is proportional to the quantity and variety of the identity relations that the melody structure contains" is most likely introducing the problems. Not everyone considers 12-tone serialist compositions to be the most beautiful melodies, and there are also many other factors weighed by each individual when judging beauty, some of which may be subconscious and none of which are universal.

@Tanner Swett, I suppose that the most simple method is most suitable. Could you please tell me more about those possible methods.

@StojkoskiNikola The simplest method is to say, "All pieces of music have infinitely many identity relations, and they have an infinite amount of variety." But I'm guessing that you don't consider that a suitable method. If that's right, then can you explain what makes it unsuitable?

@Tanner Swett Sorry, I don't understand you. How do you mean infinitely?

@StojkoskiNikola By "infinitely many identity relations," I mean that if you were to make a list of all the identity relations, then the list would be endless.

@Tanner Swett How can be infinitely many identity relations between a finite number of entities?

Uhhh... the same way the finite entities 0-9 can represent infinity through combination, variation, repetition, and arrangement

@StojkoskiNikola Well, imagine that you have three identical notes, numbered 1, 2 and 3. Note 1 is identical to note 2 (call this fact A). Note 2 is identical to note 3 (call this fact B). Note 1 is identical to note 3 (fact C). Fact A is equivalent to fact B (fact D). Fact B is equivalent to fact C (fact E). Fact A is equivalent to fact C (fact F). Fact D is equivalent to fact E (fact G)... and so forth, without end.

This is a very interesting thought. Honestly speaking I was thinking about leaving this thread (what a bad idea). I was aware that human reason has the ability to relate between relations but never thought of going further and talk about "relation between relations between relations" and so on. There must be a solution! I guess that human reason must have certain limits for abstraction. Thank you for your beautiful thought @Tanner Swett. Please tell me more about other possible methods you were mentioning.