If we put the harmonic mediant and harmonic submediant together with the diatonic chain, we can clearly show how tones are derived from either a 3rd or 5th harmonic. The 3rd harmonic is shown on the horizontal axis, and the 5th harmonic is on the vertical axis.
Using shifts the letter names also clearly reflect the harmonic derivation of the tones.
This is the beginning of a structural model called a Two-Dimensional Tone Lattice. We begin by filling in spaces between the tones of the central unshifted chain. The partial lattice shows how this chain (Dorian) is expanded symmetrically to include tones derived from the 5th harmonic and subharmonic from the source tone.
This same process is used on other pitches in the diatonic chain to continue filling in spaces between the central tones as we build the lattice.
Notice how the order of the letters above and below the central unshifted chain follow the same fifths order, so that new shifted chains of fifths are apparent, and our shifted tones have accidentals while the unshifted tones are the naturals.
As things are getting a little crowded, we temporarily remove the tones derived so far, and we find the remaining tones which are shifted naturals, which fall very near those of the central chain. They are all shifted by a comma.
This process is continued until we show the remaining tones...
Such a structural model has no intrinsic size limitation, can have many more than just two dimensions, and can be very useful for showing how tones relate harmonically. We have derived only a small diatonic portion of a Two-Dimensional Lattice based on harmonics 1 through 6, giving us only two generating harmonics: the primes 3 and 5. Things get much more interesting when the harmonic relations move past the number 6 into higher prime numbers. Maybe later we will have time to explore the higher harmonics. For the time being, higher numbered harmonics and larger dimensioned lattices are left for you to imagine. There are plenty of amazing things to be found using only the 3rd harmonic, extending the diatonic chain into higher numbers of tones. Such structures are the topic of the next section.
NEXT: Chapter 6 · Superscales