Tukukkk

I have been reading around many sources, but it seems there is not a full standardized way in naming intervals, some sources say that such a thing as a diminished first exist.

So I started making an interval chart and realized something, intervals have two variables, the type (1st, 2nd, 3rd, ...) and the quality (diminished, minor, perfect, major, or augmented). But not all the qualities are to be used by any type.

Types 1,4,5 can use diminished, perfect, or augmented.

Types 2,3,6,7 can use diminished, minor, major, or augmented.

Intervals that are more than an octave apart are called compound intervals, but I don't know if the same rules apply??

So I ended up with something like this:

Now substituting for the Note C I got:

With this chart I think is easy too see all intervals, but I am not convinced of the validity of all of them, for example, I have never heard someone talk about a diminished tenth interval.

So, do you think this is a valid way to get to any single interval for a note?, expanding this more would be valid?, or is there a limit on interval naming?

Your chart looks correct, with one exception: in your bottom chart, the diminished 11th should be 15 semitones from C, not 12.

In theory, every interval is possible and valid, with the exception of flagrantly wrong intervals like a "major fourth." With that said, there is a point of diminishing returns: it's almost always pointless to go through the trouble of identifying something as an augmented 27th, when we may as well just call it an augmented 6th. If we really want to get specific, we can just call it a compound augmented sixth.

Which brings me to something that may save you some time: The Rule of 7. This is just a convenient way to translate between simple (intervals within and octave) and compound intervals. If a third is expanded by an octave, it becomes a (3 + 7 =) 10th. When I mentioned a 27th earlier, that's just a compound version of a (27 - 7 - 7 - 7 =) 6th. Knowing this rule prevents your table from continuing on indefinitely.

Lastly, you will occasionally encounter a doubly augmented or double diminished interval. C up to Gbb, for instance, is a doubly diminished fifth. It's rare, but you will occasionally encounter it.

The categorysation of intervals as you did is one point. Your list is almost perfect.

Another is the notation respectively the signing. The manual of Band in a Box contains a vaste list of chords. (Pdf will follow)

The validation is another question. All may be valid but not all are used. Someone will be the first one but this would not be a new creation.

Many composer use enharmonic change or interchange and ignore the correct writing ( as use in popmusic or the sheets of a single instrument part. Bartok was a purist in correct writing.