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And that is the functions of the major scale.No. An interval is the musical relationship between two voices (notes). That's it. In ascending order: unison, minor second, major second, minor third, major third, perfect fourth, diminished (flatted) fifth, perfect fifth, minor (flatted) sixth, major sixth, minor (flatted) seventh, major seventh, octave, minor (flatted) ninth, major ninth, augmented (raised) ninth, major tenth (usually just tenth), eleventh, sharp eleventh, twelfth (almost never named), minor thirteenth, and major thirteenth. These relationships always apply and have no need of being referenced to any scale. The properties that distinguish scales and modes are the intervals between adjacent notes. The "standard" modes all have two half-step (minor second) intervals, with the remaining intervals being whole steps. Other modes have varying numbers of half step intervals, and some of the other intervals are raised seconds (minor thirds).
In a different context these names are functionally incorrect at best and misleading which is why they are only used to describe an abstract interval by reference to the major scale.