As for the horizontal interval, it will be a melody which will be a horizontal succession of notes.
As for the vertical interval, it will be hamony which will come from touching vertically related notes.
Intervals will be divided in a way of type and distance.
Distance will be made of unison, 2nd, 3rd, 4th, 5th, 6th, 7th and 8th. octove.
Type will be divided into major, minor, perfect, augmented, diminished.
Perfect intervals will be unison, 4th, 5th and octove.
In the major scale, all the intervals except the intervals between 3rd and 4th, and 7th and octove is made of whole tone. On the other hand, in the minor scale, all the intervals except the intervals between 2nd and 3rd, and 5th and 6th is made of whole tone. Either of major scale and minor scale will respectively 12 types.
The music world looks more myterious, but fees more mathematical.