When my students learn what intervals are in music, they first learn the interval name, a second, third, fifth, and so forth, before they learn the kind, major, minor, perfect, and so on. It is one of those strange things about music theory that a number represents some distances between notes, while a word represents other distances. It is a curious method of measurement. Imagine if some inches were bigger than other inches, or some centimeters were bigger than other centimeters. Yet that is what we have in music. Some seconds are bigger than other seconds, some thirds bigger than other thirds, etc. The bigger seconds are called major, and the smaller ones are called minor. The bigger fifths aren’t call major, they are called perfect, and the smaller fifths aren’t called minor, they are called diminished. Who ever thought of this strange and confusing way of measuring musical distances? It must have been someone who was more interested in the sound of the intervals than in the look of them on paper. Fifths and octaves have a purity to their sound, and a lack of color and richness that the other intervals have. Like an exceptionally cut diamond, the pureness of a perfect fifth or octave is the exception, and makes these intervals worthy of being called perfect, for they are as perfect and pure as two musical tones can be.
The perfect fourth, on the other hand is purer than major intervals, but less perfect than the perfect fifths and octaves. The perfect fourth is really the black sheep of the perfect interval family. For much of music history, composers have treated it as a dissonant interval, resolving it as seconds and sevenths are, and allowing them to be run in parallel, which fifths and octaves (in four-part writing) are not. Fourths have a great desire to become thirds, and we can recognize them by our desire to hear them succeed.
That leaves us with the major intervals. Major sixths are the widest of intervals that are major, and sound just fine harmonically. Thirds also sound good harmonically, but are much smaller than sixths. Seconds are dissonant and small, while sevenths are dissonant and big, so a big interval that is dissonant can be recognized as a seventh, while a small interval that is dissonant can be recognized as a second. Every dissonance has a resolution we have come to expect and want through our experience listening to music. While these expectations help us recognize intervals, they depend on a tonality having been established. A melody that ascends a perfect fourth from ^5 to ^1 is at rest when it arrives at the tonic, but a melody that ascends a perfect fourth from ^7 to ^3 is not at rest when it arrives at the mediant, but demands further motion toward the tonic. Yet both instances of a perfect fourth can be recognized as such as much by the tension or release it creates as by the simple distance between the notes.
With these and other differences in, consonance, sonority, and harmonic function, the odd labeling of intervals begins to make sense. Minor second are more dissonant than major seconds. Perfect fifths are more pure and consonant than major sixths, and major sixths exist between ^1 and ^6 in major the tonality, but not in the minor tonality. The difference in their sound is a matter of tonal context, not just theoretical abstraction. It is enlightening to hear a single interval in the context of many keys. In fixed do, do to mi is tonic in do major, but sub dominant in sol major. It is the root and third of the tonic in do major, but the third and fifth of the tonic in la minor. This too is an important context in which to hear and recognize intervals. Trained this way, the placement of keys on a piano becomes unimportant to the student who has learned intervals through thorough ear training. Encountering intervals on an instrument becomes one of recognition and application, instead of the basis for a non-aural understanding. Any clarification the instrument brings to aural understanding is good, but it is not the foundation of that understanding. .