Find three points on a scale using golden ratio

Question

My question came from my musicial part: The note A4 on a piano is 440hz and the note A5 (one octave higher) is 880hz.

On the piano there are 12 notes between A4 and A5 (include).

Im trying to find three notes between A4 and A5 (hence on the scale of 440 to 880) Using the golden ratio: 1.618 (accuracy of three points is enough here).

More specific: There are 5 points on an axis from 440 to 880: A4=440 note1=?, note2=?, note3=?, A5=880

I want to find (note1, note2, note3) Where note2 is: “1.618 closer to note1 than note1 to A4”, note3 is “1.618 closer to note2 than note2 to note1” and A5 is “1.618 closer to note3 than note3 to note2.

Im not sure if one can prove that 3 points like this exist, but I guess we can prove that there are n points (n is natural) like that that exist, right?

How can I find these n’s? And when finding one, how can I find its n notes?

Answer 1

From what I'm understanding, you want the ratio between $d_2$ and $d_1$ to be 1.618, and so on for $d_3$ to $d_2$, and $d_4$ to $d_3$. You basically end up with a system of equations:

Take:

$$d_1=n_1-A_4$$ $$d_2=n_2-n_1$$ $$d_3=n_3-n_2$$ $$d_4=A_5-n_3$$

Where $n_k$ is the $k_{th}$ note and $A_4<n_1<n_2<n_3<n_4<A_5$ and $d_k$ is the distance between $n_k$ and $n_{k-1}$ and $n_0=A_4$ and $n_4=A_5$.

Then, by the golden ratio constraints you've asked for, we end up with:

$$n_3-n_2=(A_5-n_3)\cdot1.618$$ $$n_2-n_1=(n_3-n_2)\cdot1.618$$ $$n_1-A_4=(n_2-n_1)\cdot1.618$$

Solving these taking $A_4=440$ and $A_5=880$, we end up with

$$A_4=440$$ $$n_1=636.7701$$ $$n_2=758.3833$$ $$n_3=833.5459$$ $$A_5=880$$

All in Hz. Each of those are in the golden ratio to the preceding one. You can generalize this as long as you know your initial and final note.

Answer 2

If what you want is the ratio of the frequencies to be $\varphi$ for each of the four ratios then this is impossible.

The interval on the ordinary $12$ tone scale is the twelfth root of $2$, which is $1.05946309436 \approx 1.06$. Four your five note scale you want $$ 2^{1/5} = 1.148698355 \approx 1.15 . $$ That's pretty far from $\varphi \approx 1.618$.

Even just a two note scale gives you a ratio pf $\sqrt{2} \approx 1.4 < \varphi$ .