Key of C Major from the Fourier Series on D

Jill: What interval produces phi?? Is it close to an augmented fourth or a minor sixth??? Obviously this is critical, but at different points of your writing I think it is one then the other. Clearly I am not reading accurately, but I don't want to quote you incorrectly.

Also what is the significance of the intervals with least resonance and those intervals with the greatest resonance???

Rich: To your questions, phi=1.618033 is between a minor sixth, 8/5 = 1.6, and a major sixth, 5/3 = 1.666. The reason that I discuss it in relation to a tritone (aug 4th / dim 5th) is due to the location of phi in a 7-step diatonic scale.

The center of resonance (harmonic center) of a resonating tone is the ninth partial in the natural harmonic series. For the tonic C in a C major scale, this is the supertonic D, which is called the ninth. It forms naturally as part of the harmonic series (for fundamental C in the example) and acts as an axis or pivot around which all of the other harmonics balance (or orbit). So, recognizing the ninth as the harmonic axis in an octave, the inverse harmonic center is a tritone away or G# in our example. Thus, {D, G#} act as a harmonic axis of symmetry in the harmonic series and diatonic scale of C.

Now measuring phi upward and downward from G# we land in the gaps between {B, C} and {E, F} which are symmetrically balanced around the harmonic center D. The tense diatonic tritone (and only tritone in the diatonic C major scale) is then {B, F} which we anticipate and recognize naturally as resolving to the tonic major third {C, E}. This is where the relationship of phi to the tritone comes in. The two phi-damping locations in a diatonic scale over an octave are measured from G# or inverse harmonic center of C. The tritone function (driving force in music harmony and human music perception) then oscillates across these two phi-damping locations from {B, F} -> {C, E} -> {B, F} -> {C, E} -> ... Our perception of musical harmony is first and foremost based on recognizing this oscillation process across the phi-damping locations relative to the harmonic axis {D, G#}.

As for my definition of resonance, the major sixth is the most resonant because the wave partials in the harmonic series naturally add in such a way that it is the interval in an octave least damped or deadened by phi. Another way of saying it is energy sharing is least bound or constrained at a major sixth. The first major sixth interval in the harmonic series occurs between the third and fifth partials (or {G, E} for fundamental C}, thus 5:3 = 1.666.

The least resonant interval in a 12-step octave is the inverse harmonic center, which is G# for fundamental C and harmonic center D (see p.124 in Interference). This is because G# is the furthest away from the harmonic center D. But within the 7-step diatonic scale over an octave, the least resonant intervals correspond to perfect fourths from the harmonic center or {D, G} and {A, D}. They are equivalent to perfect fifths from the tonic major third as {C, G} and {A, E}. In traditional music harmony, perfect fifths and fourths are considered the most stable, most calm and most resolved intervals, thus confirming them as the least perceptually resonant intervals in a diatonic scale. Notice that {G, A} balance symmetrically around inverse harmonic center G#.

If you will sit down at a piano and play these intervals while reading this explanation, you will see the symmetry while hearing the tension and resolution as described above.