The number of semi-tones between notes is not sufficient for describing an interval. A C to an F#, for example, has 6 semi-tones and makes an augmented fourth, but the same number of intervals exists from C to Gb, which is a diminished fifth. The change in the type of interval has to do with the relationship between the two notes according to the key.

All intervals exist in sound, but the notation is meant to describe those relationships. Some intervals are impossible or ridiculous to notate.

Say we have Cb to Ab. That is a Major 6th. A minor 6th would be Cb to Abb, which we can still notate. But a diminished 6th, which is quite possible in some keys, is not possible to write with normal notation because it would require an Abbb. Triple flats are not standard, if they are used at all. I have never seen one despite reading some very chromatic music.

The number of semi-tones between notes is not sufficient for describing an interval. A C to an F#, for example, has 6 semi-tones and makes an augmented fourth, but the same number of semitones exists from C to Gb, which is a diminished fifth. The change in the type of interval has to do with the relationship between the two notes according to the key.

All intervals exist in sound, but the notation is meant to describe those relationships. Some intervals are impossible or ridiculous to notate.

Say we have Cb to Ab. That is a Major 6th. A minor 6th would be Cb to Abb, which we can still notate. But a diminished 6th, which is quite possible in some keys, is not possible to write with normal notation because it would require an Abbb. Triple flats are not standard, if they are used at all. I have never seen one despite reading some very chromatic music.