I believe that Phi is not the magical ratio that drives nature, but that it reflects the real driver, which is harmonics – generated according to whole-number ratios.
Phi, also known as the Fibonacci number, is a ratio that appears throughout Nature. Last year’s twig growth vs this year’s twig growth, one joint on your finger compared to another, the relative size of cells of a nautilus’ shell, the way that weather patterns evolve. It’s essential to the fabric of the universe. And being that this web-site is dedicated to the notion that vibration is essential to that fabric, you’d think the two would go together. But Phi doesn’t appear to be harmonic at first glance.
Phi is not a whole number ratio – it’s 1.618; determined by starting with a number and then creating a new number by adding up the two previous numbers. So, for example, start with 1:
1 plus 1 is 2
1 plus 2 is 3
2 plus 3 is 5
3 plus 5 is 8
5 plus 8 is 13 – and you keep going. As you keep going, the ratio of the current number and the one before it (e.g. 13/8) gets more and more precise according to the definition of Phi.
So, your normal Phi/Fibonacci series starting with 1 would look like the table below. Did you know that the ratio of the length and width of DNA is 34:21 angstroms? That’s the ratio of the 9th to the 8th position below. Amazing stuff.
But what does 1.618 give us? Yes, it’s the “Golden Mean”; yes if you position objects in your painting to that proportion people will like it more; or have your plastic surgery done with this in mind – but can we get beyond the aesthetics and into a scientific understanding of what is underlying Phi?
Now, you don’t have to start the Phi series with 1. You don’t have to start with numbers at all – you could start with tomatoes, or rain drops, or wind – but numbers are a good way of quantifying our World, and it just so happens that music can be measured with numbers – specifically, the number of vibrations per second (Hertz, or Hz) for a musical note.
As you’ve probably noticed from this web-site, I’ve identified a strange phenomena around the notes of B-flat and F, specifically at the frequencies of 7.2 Hz (B-flat), and 5.4 Hz and 10.8 Hz (octaves of F), and their harmonic sequence. (See my video of this audio interference pattern, here.)
So, at first, I tried just multiplying one of my frequencies by Phi, e.g. F at 10.8 Hz x 1.618 = 17.4744 Hz. But the result isn’t in my musical scale at all. The nearest frequency is my D at 18 Hz. So, I couldn’t figure out what do with Phi for a few years.
So then I thought, what happens if, instead of starting with 1, we start the Fibonacci series using the lowest frequency in my scale, which is a B-flat sub-octave at 0.9 Hz, and see if this pattern gives us the frequencies from the harmonic series I worked out earlier. Here’s the result in the table below.
Note first of all that the Phi ratio for each position below is the same as In the table above, where we started with 1. It’s still a Fibonacci series based on adding up the last two numbers, but we’ve seeded it with 0.9 instead of 1.
The first six positions of Phi give me an Octave, two Fifths, a Third, and another Octave. So far, so musical!
Then at the seventh position, we have a remainder: the Phi series has generated a tone of 11.7 Hz. But 11.7 Hz isn’t part of our harmonic series. You can scan through my harmonic series below and take sub-octaves of these numbers (by dividing by 2, and 2 again, etc.) and none of them come to 11.7. A frequency that is close is 14.4 Hz – which is another B-flat (0.9 Hz x 2 x 2 x 2 x 2 = 14.4 Hz).
Okay, what’s the difference between the tone that the Fibonacci series generated (11.7 Hz) and the closest harmonic (14.4 Hz):
14.4 minus 11.7 = 2.7 Hz.
And what is 2.7 Hz? It’s an F from our harmonic series! (10.8 Hz divided by 4). So, the gap generated by the Fibonacci series and our closest note in the B-flat harmonic series is itself one of the notes from the harmonic series! They say, “the devil is in the details”. But I say, “God is in the gaps!”
A “difference” can be above or below the Fibonacci frequency. In this analysis, we chose 14.4 Hz because it’s the closest frequency in our harmonic series above the Fibonacci frequency. But it works just as well if you choose a frequency from our harmonic series below the Phi frequency: 10.8 Hz is our harmonic-series F, below the Phi frequency of 11.7 Hz. So the difference between these two is:
11.7 Hz minus 10.8 Hz = 0.9 Hz.
And 0.9 Hz happens to be our fundamental B-flat frequency again!
So, whether you over-shoot or under-shoot, where there’s a gap between the frequency that the Fibonacci series lands on and our “closest living harmonic”, that gap will itself be one of our harmonics.
Starting our Fibonacci series with a “seed” number of 0.9 Hz, that gap between the 8th and 9th position we talked about for DNA is an Eb to a Bb. The building block of life is made of music!
You’ll also notice that at position 19 in the table above, the lemma (the gap) doesn’t resolve to one of our harmonics (indicated by a big red X). In this case we have a gap of 125.1 Hz. This isn’t one of the harmonics from our harmonic series either. But our nearest one is 129.6 Hz (our C). So, in this case, we have a “double-lemma” (is that a “dilemma”? – ha ha!) where the gap is 4.5 Hz, which is a very low sub-octave of D.
Here’s someone’s YouTube video of a zoom through a visual fractal. So, as in a pictorial fractal, when you get to a little gap which you think might be something new (a disaster, a tear in the fabric of space-time!) – you realize it’s just another instance of the pattern you’ve encountered before – repeating like ripples in water. It appears that Nature uses gaps to maintain the harmonics of Nature without it becoming overpowered by its own resonance.
Western Science has always had a problem with gaps. You can read my exploration of the harmonic “gap” in the cycle-of-fifths here. It was because of this gap Western music adopted Equal Temperament so that all musical keys were made equally un-harmonic in order that the same piece of music could be transposed to any key – as though the frequencies themselves didn’t matter – just the relative pitches. It’s like saying it doesn’t matter how long a day or a week are so long as a week is 7 times longer than a day. Which is absurd. We live on a planet which spins at a certain rate which is foundational to all the frequencies of nature. The good news is that this also shows how we could restore our civilisation to its natural resonance, “lemmas and all”!
So, I think of Phi differently. Not as a magical construct which guides Nature – as though 1.618 was some template emeshed into quantum physics; but more like Phi as a pea-pod full of harmonics. Perhaps the nautilus is singing away happily to himself and builds the next cell of his shell so it resonates with its predecessor.
Harmonic frequencies are generated in whole-number proportions to the original vibration – and these harmonics propagate further harmonics, with gaps that are themselves further harmonics. If there were no gaps in this harmonic propagation, then everything would be so spot-on harmonically that the universe would resonate itself into oblivion. But, with the gap (which is a much quieter and fainter version of the same harmonic series) the harmony and energy is efficiently propagated without the World shaking itself to bits: Little curlicues of harmony spin off and create their own little harmonic microcosms. And all energy is inter-meshed with all other energy – which might explain something about Quantum Entanglement!
So, Phi is a way for us to measure from the outside, what’s going on inside – which is fractal harmonic propagation.
Some people might regard the early ratios of Phi (positions 1-6 above) as inaccurate – i.e. the early ratios of 1, 2, 1.5, 1.66666 are not 1.618. But these ratios precisely generate the octave, fifth and third – without lemmas – in the first 6 positions of the Fibonacci series. Phi doesn’t “get more accurate” – it’s always “doing its job” – accurately reflecting the next harmonic, or the next harmonic plus a lemma, where needed.
It would appear that it always does this, although I’ve stopped at position 26, but I expect it goes on forever. Someone can write a computer programme to check! As a matter of fact, to render the fractal harmonic propagation as a fractal image/video using my frequencies would be an awesome exploration. Someone should try it 🙂
Perhaps as a metaphor for the Phi fractal propagation of harmonics, we could think of a multi-voice choir which is set off with the first sound. The sound builds with different members of the choir singing along with harmonies. After a while though, some of the choristers begin emitting strange, erratic noises and it seems like all is lost but, in combination with the sounds the rest of the choir is making, these aberrant sounds create “difference notes” – the difference between the “wrong” sound and the “right” sounds – combining to create a third note which is harmonic. If you’ve ever done throat singing with a group of people, you may have witnessed these strange whistling sounds which no-one is making but are the sum of the sounds they are making collectively.
Another interesting area where “difference notes” come into play seems to be the Solfeggio where the difference created by playing two or three of these notes simultaneously is generally spot-on with my harmonic series, even though the Solfeggio notes taken one at a time are not musical and cannot be used to construct a melody. See my exploration of this, here.
So, Nature is propagating harmonics in a Fractal way: there can be a gap/lemma between the generated Fibonacci tone and the note generated from the B-flat harmonic series – but that gap is itself a member of the B-flat harmonic series.
Being as I believe B-flat and F frequencies of 7.2 Hz, 5.4 Hz and 10.8 Hz are resonant still points against a background vibration of our World, then a Fibonacci series like the one above may be an important insight into how energy is propagated and how our World is structured. I would propose that Nautilus shells and all the other reflections of Phi that we can see are actually built on this underlying common resonance – and that’s how it “knows” to construct itself that way, because that’s how the energy resonates and propagates. Perhaps this also relates to how bubbles in water “know” how to construct themselves into perfect spheres the way they do, as Buckminster-Fuller once remarked.
Phi appears to be a symptom of the underlying fractal harmonic propagation of energy – indicating that matter is really a harmonic extension of music. And music – harmony – is what keeps the universe together.
2 thoughts on “Phi – Fibonacci is harmonic”
That is a great discovery, thank you for sharing it..
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Have you noticed that the cycle of fifths is governed by the Fib series?
3: Pythagorean Frame 12:9:8:6 – octave, fifth and fourth
5. Chinese pentatonic
7-8 (Fib value lies between) diatonic scale
12-13 (Fib value 12.7 lunar cycles per year) chromatic scale
21 – here it is starting to run out, but there are logical limits to the circle of fifths at 17, 19 and 21 elements – cf Hothby’s 16 or 17 notes, and Zarlino’s 19-note keyboard. Above this point chaos kicks in so that larger Fib numbers are incompletely realized (like upper reaches of periodic table of elements): e.g. Fib numb 34 incompletely realized by 31-ET, Fib number 55 incompletely realized by 53 commas in the octave.
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