Looking at the diagram I constructed, certain patterns are revealed, wholly consistent with the harmonic understanding of the common practice period.
The first thing I notice was the relationship between F (IV) and B (VII), the characteristic tones. This is the tritone, the most dissonant in our system. There was a time when composers were not even allowed to use this interval. This covers the entire gamut without gaps.
The mi2 and M7, which are inversions of each other, are the same color and same in number, They dovetail to cover the entire gamut with no spaces and no overlapping.
The M2 and mi7, mi3 and M6, also the same color, but their pattern gets a little more complicated.
Even though intervals sharing the same classification (quality and number) are the same distance, there are subtleties. A metaphor of a 12 inch ruler comes to mind. An inch is an inch is an inch. But in the context of tonality - not so. The inch between 2 and 3 is not the same as the inch between 9 and 10. The mi2 between E and F, is not the same as B and C. C is the tonic and a resolution. Another metaphor, the mile going to work in the morning is not the same mile coming home.
Another observation is most melodies stay within a close range and move by the smaller intervals. Probably because the human speaking voice stays within a fourth in normal speaking.
So what does all this mean for a songwriter? Pattern is energy. The better one understands the patterns, the more one is able to manipulate them. All music can be transposed to any key; intervals are the essence of melodic movement.
The diagram: These are the simple intervals confined to an octave. It's based on an equally tempered scale. Unisons are absent, but certainly do have their own significance. The vertical axis is a major diatonic scale in the key of C. Reading left to right are the incrementally widening intervals growing by half steps. I opted for roughly two and a half octaves to see if I could reveal any patterns that may have been obscured by the limitations of only one octave. Too many octaves may have been more thorough, but harder to read
Melodic movement is about intervals, and intervals are relationships. The tones themselves are sensually relevant, but the same tone repeated over and over again would have to rely on the 'time feel' to be appealing. Tones may have flavor, but it's the intervals that provide the sustenance. Like so many things in life, it's not what you see, but what you don't see.
Illustration of simple intervals
Any time I delve into writing about music theory I always come back to a quote by William Wordsworth - '”we murder to dissect”. Music is a living, breathing thing. To extract one element and analyze it requires separating it from all other elements. That being said, lets pretend we have the dead corpse of music on the table. We have removed the heart beat, the life giving pulse of time. We have removed the skeleton, musical form that encapsulates an entire piece. One by one surgically divide the organs from its host. We're left with Definite Pitch, the perception of frequency. (but don't leave now, the gruesome parts over.)
This diagram show how intervals dovetail to fill the entire gamut.