The astronomer Galileo Galilei wrote: ”Mathematics is the alphabet with which God has written the universe”.

I wanted to know more about this language used during the creation.

So I’ve study with much interest mathematical proportion showed by nature noticing especially the golden mean relationship Phi=1,6180339... because it’s present everywhere in nature, beginning from the atom going up to the galaxies, representing the divine proportion of universal harmony. This relationship represents even the beauty of natural harmonic geometries.

When the creation is talking to us using a mathematical language and if we can affirm, that the most used proportion is Phi=1,6180339… , the golden mean section, the question is, why this proportion is not present in the scale of western music when we consider that the music should be the highest artistic expression of natural harmony? The official answer is, that a string of a music instrument divided by the proportion of Phi never would match with the natural spectrum of overtones based on fractions of little natural numbers like 2/1, 3/1, 3/2, 4/3. Accords or music intervals according to this fractions would be considered harmonic and graceful by our ears. Under a physic profile, the Phi-Interval is very far away from corresponding with numbers based on fractions of little numbers and because of this fact, who goes to built up a Phi-based music system is an obstacle by the opinion of the “experts” in the field of musical harmonics. Is it possible to create a Phi based music system? It does make sense from a musical point of view?

The answer is yes if we are able to jump the obstacles of superficial evidence entering the miraculous connections that the number of golden mean section offer. The first element to take care of is, that every music system has to be an exponential system (on the keyboard of a piano every 12 keys we get music notes with double frequency) because our hear- sense is based on a logarithmic scale.

In a very simple way, we can use the powers of Phi to get a series of Phi-based numbers following an exponential curve

Now we have to fill the spaces between a Phi-power and the next because under a musical view the Phi-interval is almost large and we need some more keys in between to compose music. Under a mathematician view, this means that we have to divide the space of a Phi-Interval in a natural number of parts that we can decide freely. For example, we can divide this space into 9 parts using 1,618(n/9) where n is a natural number, even negative. For n=9 the exponent is 9/9=1 getting Phi1=1,618. For n=13 happens an interesting thing: Phi13/9=2,003876 a value very near of 2 that is corresponding to the harmonic music interval that doubles the frequency of a base frequency called Octave in traditional music.

This little example shows, using Phi as a base for an exponential music system, that we can obtain a proportion that respects the natural over-or undertones of a vibrating string but we have to go beyond the single interval of the octave.

If we choose to divide the Phi interval into 7 parts, we obtain an optimal connection with the number 3 (3 times the frequency of basic sound) and con number 2 because of Phi16/7=3,0039 e Phi10/7=1,9886. In addition, we get connections based on the combination of 2 and 3: 2/3=0,666 e 3/2=1,5 corresponding to natural harmonic music interval of the vibrating string. The system with 7 parts in a Phi-interval permits us to get these music-intervals 2, 3, 1.5, 0.666 in an approximated manner.

Would it be possible to get the exact numbers of natural harmonic intervals of a vibrating string? Observing in an accurate way the powers of Phi, we notice that it’s possible to obtain every natural number exactly by adding powers of Phi remembering that traditional harmonic intervals are based on fractions of little entire numbers. More little are the natural numbers creating the fraction(1,2,3,4..) , more harmonic sounds the music interval like ¼, ½, 1/3, 2/3, 3/4,4/3, 3/2, 2/1, 3/1, 4/1.

Here we see how to built the first 4 numbers by adding Phi-Powers:

We can use the technique of adding Phi-powers to get natural numbers with absolute precision, no approximation e we can use these precise numbers systematically instead of the approximated values of the curve getting by Phi(n/7) creating a perfect system of embedding all the harmonic intervals of music with perfect Phi-based logic for every music note of the system. The difference of the exact frequencies from the original curve Phi(n/7) is less than 1%.

Obviously, we can represent the values of the Phi-system like a golden mean spiral that would be like a Nautilus.

In his book “432 Hertz: the music revolution. Golden mean tuning for biological music”, Riccardo Tristano Tuis writes: “If we could hear the music based on the golden mean spiral it would be in a certain manner the music for life, on a biological level but even on a perceptive level, because it would use the same math of both”. Tuis continues citing LaRouche from the Schiller Institute:”There is nothing mysterious or mystic around the introduction of the golden section as an absolute value of the life process” in reference to music. He writes:” The perfect music scale (the moderated scale is it not) is the one with the proportions of the frequencies of the music notes one from another based exactly on the golden section with the intonation register based also on it”. In the same book Tuis public a music scale based on 12 notes per octave but he doesn’t found the perfect Phi interval for all the music notes. In every case, it is appreciable to see his honest effort to search the truth about universal music. In the last version of the Phi based music system, we have the natural frequencies indicated by Tuis like 432Hz, 288Hz, 216Hz, 144Hz, 72Hz according to this choose.

After discussing the harmonic intervals of the Phi based music system, we have to examine the Phi interval closer. We understand that it’s possible to create the harmonic intervals by adding powers of Phi but the Phi-interval itself is pleasant? Considering the harmonic laws of overtones related to the vibrating string, the Phi-interval should be horrible for the hearer but in praxis, it isn’t. At the contrary, it’s very pleasant e we will try to find an explanation. For this, we will take the seeds of the sunflower like an example. The positions of the seeds were chosen to fill out the whole area of the circle without leaving empty spaces. Beginning in the center of the circle, turning around, at which angles of the whole circle of 360° we have to position the seeds to fill out the circle at best? In order to avoid empty space, a single seed should never happen exactly behind another one respect of the center of the circle creating beams like in a bicycle wheel because the space between a beam an another one is growing from the center to the border. Using angles based on fraction composed by little numbers we would get beams inevitably. The beam-like distribution of the seeds is corresponding to a harmonic music interval based on a fraction of a natural number multiplied with 360°. In this way, the “seeds” will match exactly one behind another but this kind of distribution is not indicated to fill out the whole area of the circle with a maximum number of seeds. To position the seeds, the sunflower uses the so-called golden angle of 360°/Phi2=137,5077°. In this way, a seed will never happen exactly behind another one. Transforming something like this in music intervals we will expect a horrible sound but it isn’t for the same reason why the positioning of the sunflower seeds is not ugly but highly beautiful with his embedded spirals turning clockwise and counterclockwise and you cannot stick your easily from it because the beauty is so fascinating. The same happens to hear the Phi interval.

Another kind of explanation more technical would be the following: The Phi proportion divides the time axis in a fractal manner creating infinitely all the powers of Phi itself and the human ear recognize the perfect repeating of these values. Probably our brain calculates the sums of the powers of Phi creating perfectly natural numbers harmonic to our ears. To the Phi interval we can add another note that corresponds to another power of Phi or a natural number or a fraction of natural numbers pushing the keys on our Phi-intonated piano keyboard (because they are part of the music system we could do so), our brain recognized the perfect embedding of this musical agreement of all these values interpreting it “harmonic” even if this kind of harmonic is fractal-like nature and not like the beams of a bicycle.

The following figure shows how the duration of the Phi power based oscillations are creating other durations corresponding to powers of Phi and the creating the natural numbers of 1,2 and 3. You can observe the frequent presence of embedded powers of Phi in a fractal manner on time axis creating the same nodes. Of course, the same principle is valid for frequencies that have the inverse value of oscillation duration.

Phi-4=0,145898

Phi-3=0,236068

Phi-2=0,381966

Phi-1=0,618034

Phi-1+Phi-2=Phi0=0,618034+0,381966=1,000000

Phi1=1,618034

Phi1+Phi-2=1,618034+0,381966=2,000000

Phi2+Phi-2=2,618034+0,381966=3,000000

This kind of research is limited only to the artistic sector of music? Absolutely no. The musician and book author Alessio Di Benedetto, says that “we are dipped into an infinitely oscillating field, like countless music harmonies going out from a single basic sound. From this field, we recognize only the frequencies near to us”.

At this point, we have to begin to discuss the String theory. In few words, this theory says that all that exist is pure energy that is vibrating. It depends on the way the energy is vibrating at a certain frequency if there would be a manifestation of a force or a subatomic particle. In other words, the string theory is talking about the symphony of creation and give us an idea of the music of creation. The creation shows a lot of golden mean proportions so we will aspect to find the correspondent music intervals in the music of creation.

In the moment, the String Theory is the favorite candidate to be able to predict and explain with the same physical model the 4 forces in the universe (electromagnetism, gravity, a strong and weak force of nucleus) and the favorite for the Theory of All. To reach this goal, the String Theory needs mathematical connections (theory of numbers including prime numbers). The fact, that in this theory all is vibrating give us the idea of a universal symphony using always the same music scale. Which one? In the String Theory connections of Phi with harmonic relationships are leading to important results. The mathematician Michele Nardelli used the numbers of the Phi based music system in the String Theory and he got very interesting results. We don’t wonder about that the genius Creator of the universe based the creation on a fractal Phi music system with a big sense of beauty and arts.

I studied for more than 10 years in order to get a comfortable music scale that contains the harmonic overtones together with the Phi intervals. In the past years I made a lot of attempts to get a scale with the maximum of connections in order to obtain a music scale that gives the maximum of combinations to a music composer as the systems mentioned above but only recently I use a scale with 36 keys for an octave where the Phi proportion is exactly at key no. 25. This system is much more powerful as the systems I used in past and I think, that it’s the definitive systems to be used to compose music. The musicians used 9 commas for every key. The last Phi system uses 3 commas (36 notes instead of 12) and it’s compatible with the 9 (3x3) commas.

I figured out a music scale with the tables to intonate the keys incent containing the maximum of harmonic connections and Phi proportions. I think, that this kind of music scale is more complete in order to play a natural music based on the same mathematics as creation.

Christian Lange

12 november 2013

Golden Music System Phi

Golden Music System Phi (n /7) and mathematical connections between prime numbers and "landscape" of string theory.