Misokinesia

Misokinesia (2022) explores the long-term timbral effects of strictly organized spectra. It consists of six Stases - stationary sounds, generated by multiple sine wave oscillations. Their individual frequencies are calculated according to specific ratios, based on well known numbers from science and the arts. Due to the long-term listening experience, the resulting timbres are internalized.

Golden:

Theodorus:

Plastic:

SQRT2:

PI:

12root2:

Harmonics & Intervals

Sounds of most musical instruments consist of spectra with so called harmonics, which are - besides minor deviations - integer multiples of the fundamental frequency. This results in what we perceive as harmonic sounds with a perfect fusion of the partials to a single sound. Basic waveforms, like saw tooth or square wave can also be approximated by Fourier series with exact integer overtones. However, due to stiffness, some instruments' sounds, like those of bells, chimes and thick strings, to contain significant inharmonic content.

The means of electronic music, allowed the precise composition of timbres by additive synthesis with single oscillators. This approach has for example been studied by Karlheinz Stockhausen. For his composition Studie II he used the arbitrary interval of \(\sqrt[25]{5} \approx 1.0665\) to define the ratio of partials in a spectrum.

Starting from this arbitrary choice of an interval for spectral composition, Misokinesia explores the nature of overtone series, based on a fixed ratio between \(N\) neighboring partials:

\begin{equation*} \frac{f_n}{f_{n-1}} = \frac{f_{n+1}}{f_{n}}, 2 \leq n \leq N-1 \end{equation*}

Realization

Ten analog oscillators have been used to construct the spectra manually. For each interval, the lowest partial has been chosen heuristically, trying to create a sound which emphasizes the interval's properties. Many resulting timbres have significant low frequencies and thus require adequate reproduction systems. For large intervals, partials may even lie below or above the audible range but are still kept in the mix.

A one hour take was recorded for each interval, thus incorporating drift and other long term inaccuracies of the oscillators. In contrast the possibility to create perfect stationary timbres with digital means, this approach delivers sounds with fluctuations, which have been kept at a possible minimum.

The relative gains of the partials were tuned by ear, with the aim to create the most comfortable blend for long term listening, without hiding single frequencies.

Intervals

Different prominent numbers, among them transcendental ones and scientific constants are used to define the partial intervals and thus the individual parts of Misokinesia. Ideal candidates are greater than one but not too large, in order to allow enough partials within the audible range.

Golden

The golden ratio has been explored in many aesthetic applications since the antiquity. It is defined by two numbers with their ratio being the same as the ratio of their sum to the larger of the numbers:

\begin{equation*} \frac{a+b}{a}= \frac {a}{b} \end{equation*}

The solution results in:

varphi = frac{1+sqrt{5}}{2} = 1.618

The lowest frequency was set to 20Hz, resulting in the following partial structure:

F0: 20.00 Hz, F0: 32.36 Hz, F0: 52.36 Hz, F0: 84.72 Hz, F0: 137.08 Hz, F0: 221.80 Hz, F0: 358.89 Hz, F0: 580.69 Hz, F0: 939.57 Hz, F10: 1520.26

Theodorus

Theodorus' constant is the square root of 3, with various applications in geometry and trigonometry:

\begin{equation*} \theta = \frac{f_n}{f_{n-1}} = \sqrt{3} = 1.73205 \end{equation*}

The lowest partial was set at 30Hz, resulting in the following ten partial frequencies:

F0: 30.00 Hz, F0: 51.96 Hz, F0: 90.00 Hz, F0: 155.88 Hz, F0: 270.00 Hz, F0: 467.65 Hz, F0: 810.00 Hz, F0: 1402.96 Hz, F0: 2430.00 Hz, F10: 4208.88

Plastic

The Plastic Number is the real solution of the cubic equation:

\begin{equation*} {\displaystyle x^{3}=x+1.} \end{equation*}

The solution is perceptually close to the perfect fourth (\(\frac{4}{3} = 1.333\)), thus resulting in a rather pleasant beating pattern:

\begin{equation*} {\displaystyle \rho ={\sqrt[{3}]{\frac {9+{\sqrt {69}}}{18}}}+{\sqrt[{3}]{\frac {9-{\sqrt {69}}}{18}}} = 1.3247} \end{equation*}

Plastic's spectrum starts with a partial at 40Hz:

F0: 40.00 Hz, F0: 52.99 Hz, F0: 70.20 Hz, F0: 92.99 Hz, F0: 123.18 Hz, F0: 163.18 Hz, F0: 216.17 Hz, F0: 286.37 Hz, F0: 379.36 Hz, F10: 502.54

SQRT2

The square root of 2 is

\begin{equation*} \sqrt{2} = 1.414 \end{equation*}

In musical intervals, this resembles the tritone (\(\frac{45}{32} = 1.406\)), resulting in a familiar yet dissonant timbre. For SQRT2, the lowest partial is located at 25Hz, leading to the following partial frequencies:

F0: 25.00 Hz, F0: 35.36 Hz, F0: 50.00 Hz, F0: 70.71 Hz, F0: 100.00 Hz, F0: 141.42 Hz, F0: 200.00 Hz, F0: 282.84 Hz, F0: 400.00 Hz, F10: 565.69

PI

$\pi$ is one of the best known and widely used mathematical constants. The transcendental number is defined as the ratio between a circles circumference $C$ and its diameter $d$:

\begin{equation*} \pi = \frac{C}{d} = 3.14159 \end{equation*}

With a value greater than 3, $\pi$ is large enough to make it impossible to fit ten partials within the audible range, when used as a partial interval. Sticking with ten partials, the lowest frequency is thus set at 0.5Hz:

F0: 0.50 Hz, F0: 1.57 Hz, F0: 4.93 Hz, F0: 15.50 Hz, F0: 48.70 Hz, F0: 153.01 Hz, F0: 480.69 Hz, F0: 1510.15 Hz, F0: 4744.27 Hz, F10: 14904.55

12root2

The twelfth root of two corresponds to a semitone in equal temperament:

\begin{equation*} \sqrt[12]{2} = = 1.05946 \end{equation*}

This minor second is a very dissonant interval, thus resulting in harsh timbres when used for arranging ten partials. The very small ratio between two neighboring partials results in a narrow spectrum, spanning less than one octave. For the chosen lowest frequency of 30Hz, the resulting sound has no significant high frequency content and a distinct beating pattern is the dominant characteristic:

30.00 Hz, F0: 31.78 Hz, F0: 33.67 Hz, F0: 35.68 Hz, F0: 37.80 Hz, F0: 40.05 Hz, F0: 42.43 Hz, F0: 44.95 Hz, F0: 47.62 Hz, F10: 50.45