# Logarithms and octaves

Corresponding Wikipedia article: Logarithm

## Definition

Let a line segment is divided into two parts with any ratio (golden or not). Then a particular part is divided with the same ratio, and so on infinitely. In the example below, a line segment is divided into halves, and then its left part is divided:

The produced segment length sequence $$y_i$$ is a simple geometric progression: $\frac{y_{i+1}}{y_i}=a$ $\frac{y_{i+n}}{y_i}=a^n$ For this example the common ratio $$a=2$$.

This segment system is self-similar, i.e. its parts are similar to the whole, and this whole can be a part of a larger whole. The division of each partial segment as a whole produces a self-similar fractal tree. The figure below shows two levels of this tree:

The progression $$y_i$$is replaced by a continuous function $$y(x)$$: $x=i+\frac{i_1}{N}+\frac{i_2}{N^2}+\dots$ $y(x)=y_i+\frac{y_{i1}}{N}+\frac{y_{i2}}{N^2}+\dots$ The function $$y(x)$$ is defined for an infinitesimally small change of an argument $$\Delta x$$ as a simple linear equation: $\frac{y(x+\Delta x)}{y(x)}=1+k\Delta x$ $\frac{y(x+n\Delta x)}{y(x)}=(1+k\Delta x)^n$ The unique property of this function derivative is a consequence from its definition: $\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\to 0}{\frac{y(x+\Delta x)-y(x)}{\Delta x}}=ky(x)$ Let some interval between two integer values $$(i,i+1)$$ of an argument $$x$$ is divided into n equal parts, and: $\Delta x=\frac{1}{n}$ If $$n\to\infty$$, then $$\Delta x$$ becomes infinitesimally small, and the progression common ratio, according to the second remarkable limit, is an exponential function: $a=\frac{y_{i+1}}{y_i}=\frac{y(x+1)}{y(x)}=\lim_{n\to\infty}{\left(1+\frac{k}{n}\right)^n}=\lim_{n\to\infty}{\left(1+\frac{k}{n}\right)^{\frac{n}{k}k}}=\mathrm{e}^k$ The natural logarithm is an inverse function of an exponential function: $k=\ln ⁡a$ The logarithm derivative is also defined according to the second remarkable limit: $\frac{\mathrm{d}\ln ⁡x}{\mathrm{d}x}=\lim_{\Delta x\to 0}{\frac{1}{\Delta x}\ln⁡\left(1+\frac{\Delta x}{x}\right)}=\lim_{\Delta x\to 0}{\frac{1}{x}\ln⁡\left(1+\frac{\Delta x}{x}\right)^\frac{x}{\Delta x}}=\frac{1}{x}$ The logarithm invention has allowed a fractional irrational index of power, which has no such geometric interpretation as has the square or the cube. The fractals and the logarithms also lead to the fractional spatial dimensionality.

## Logarithmic scale

The logarithmic scale requires any conventional reference point $$y_0=y(0)$$, to convert the relative values into the absolute physical values: $y_i=y_0 ⁡a^i$ $y(x)=y_0 ⁡a^x=y_0 ⁡\mathrm{e}^{kx}$ $\frac{\ln(y/y_0)}{\ln a}=\log_a{\frac{y}{y_0}}$ The octave is an interval of the oscillation frequency scale, which is divided into octaves in a geometric progression with a common ratio 2. The binary (to the base 2) logarithm can define an octave number by its integer part, and its fractional part defines a tone inside the octave scale, which is repeated cyclically from one octave to another.

The logarithms are related to the various laws of the living nature:

Logarithmic scale Description
Sound pitch The reference point is any conventional frequency. The octave is defined unambiguously. It’s known a few tone scale modes inside the octave: the pentatonic, diatonic and chromatic scale.
Sound duration The information is contained not in the music tempo or in the speaking rate, but in a ratio of the sound durations. The scale of the musical note values is a geometric progression with a common ratio 2.
Sound volume (loudness) The reference point is an average hearing threshold. A quiet sound seems to be louder due to the logarithms. The sound pressure spatial decay is nearly proportional to the square of the distance to a sound source, but feels like a linear dependence.
Brightness A dark light seems to be brighter due to logarithms. The brightness spatial decay is nearly proportional to the square of the distance to the light source, but feels like a linear dependence.

## Golden ratio

The relationship of a natural logarithm with the Fibonacci sequence and the “golden numbers”$\ln\sum_{i=0}^{N-1}{F_i}\approx\frac{N}{2}$

$$N$$ $$\sum_{i=0}^{N-1}{F_i}$$ $$\ln\sum_{i=0}^{N-1}{F_i}$$ error
3 4 1,386 8%
4 7 1,946 3%
5 12 2,485 0,6%
6 20 2,996 0,1%
7 33 3,497 0,3%
8 54 3,989 0,3%
9 88 4,477 0,5%
10 143 4,963 0,7%