On The Sensations Of Tone-Helmholtz

“In 1863 Helmholtz published Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (On the Sensations of Tone as a Physiological Basis for the Theory of Music), once again demonstrating his interest in the physics of perception. This book influenced musicologists into the twentieth century.”

Helmholtz double-siren


“Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist who made significant contributions to several widely varied areas of modern science. In physiology and psychology, he is known for his mathematics of the eye, theories of vision, ideas on the visual perception of space, color vision research, and on the sensation of tone, perception of sound, and empiricism. In physics, he is known for his theories on the conservation of energy, work in electrodynamics, chemical thermodynamics, and on a mechanical foundation of thermodynamics. As a philosopher, he is known for his philosophy of science, ideas on the relation between the laws of perception and the laws of nature, the science of aesthetics, and ideas on the civilizing power of science. A large German association of research institutions, the Helmholtz Association, is named after him.”

No sound for the first minute, then sounds like a variety of instruments including a vocoder.

Theory And Examples Of Ring Modulation

Ring Modulators have been around a long time and were very popular on the earliest of synthesizers.  Still popular today, the number of users has grown to include guitar players and others looking for a unique sound. A Ring Modulator needs 2 inputs to produce any output but on most units there is a internal oscillator that will function as one of the inputs. The internal oscillator is usually referred to as the “carrier” and many times can be voltage-controlled from an external source. The ring modulator produces sum and difference frequencies between the interaction of the carrier oscillator and the audio input signal. So if the carrier frequency is 1000 Hz (Cycles per Second) and the audio input frequency is 800 Hz, the Ring Modulator’s output will be 1800 Hz and 200Hz. Depending on the make and model of the Ring Modulator you should not hear the carrier oscillator or the input audio waveform, although in real world use, you may hear some leakage through the unit. Many models will also have a internal Low Frequency Oscillator (LFO) tied into the carrier, this LFO will modulate or change the frequency of the carrier to expand the range of Ring Modulator effects even more. The LFO is used to create slow effects like tremelo or vibrato and may have the choice of several waveforms such as sine, triangle or square wave. Also the LFO should have an “amount” or “drive” control that allows the user to select exactly how much of the LFO effect should be applied to the carrier. Typical frequency range of an LFO may be .1 Hz to 30 Hz. The carrier oscillator may range as low as 1Hz to a high of 3 to 7KHz.

Helmholtz has an entire chapter on the sum and different frequencies in his landmark work, “On The Sensations of Tone”, here is a small excerpt:

“It is the occurrence of Combinational Tones, which were first discovered in 1745 by Sorge, a German organist, and were afterwards generally known, although their pitch was often wrongly assigned, through the Italian violinist Tartini (1754), from whom they are often called Tartini’s tones.”

“These tones are heard whenever two musical tones of different pitches are sounded together, loudly and continuously. The pitch of a combinational tone is generally different from that of either of the generating tones, or of their harmonic upper partíais. In experiments, the combinational are readily distinguished from the upper partial tones, by not being heard when only one generating tone is sounded, and by appearing simultaneously with the second tone. Combinational tones are of two kinds. The first class, discovered by Sorge and Tartini, I have termed differential tones, because their pitch number is the difference of the pitch numbers of the generating tones. The second class of summational tones, having their pitch number equal to the sum of the pitch numbers of the generating tones, were discovered by myself.”

So it was Helmholtz himself that discovered the sum component of the combinational tones.

Here is a chart from his book that describes combinational tones that are generated from various inputs.

Sum And Difference Chart

Sum And Difference Chart

We can use this chart, constructed over 100 years ago to, calculate the output of a Ring Modulator if certain musical ratios are presented to the X and the Y inputs of the modulator. The first interval listed is the octave, but that ratio may not give a very interesting output. So lets try the next interval listed, the Fifth. With the Fifth’s natural frequency ratio of 2:3, the output will have a fundamental frequency that is one octave lower than the lower of the two inputs. This should not sound like the typical output of a Ring Modulator and may be more musically useful to some composers.

This is an example of a vocal sample and sine wave input.  To try to make some valid comparisons of the various sounds, I played a simple C scale for all examples.

Vocal Sample and Sine Wave Input

Vocal Sample and Saw Wave Input

Square Wave and Triangle Wave Input



Ring Modulation of Basic Waveforms with Doepfer A114 Ring Mod;


Helmholtz-Introduction To One of The Greatest Scientific Works In History

Hermann Ludwig Ferdinand von Helmholtz

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In the present work an attempt will be made to connect the boundaries of two sciences, which, although drawn towards each other by many natural affinities, have hitherto remained practically distinct—I mean the boundaries of physical and physiological acoustics on the one side, and of musical science and esthetics on the other. . The class of readers addressed will, consequently, have had very different cultivation, and will be affected by very different interests. It will therefore not be superfluous for the author at the outset distinctly to state his intention in undertaking the work, and the aim he has sought to attain. The horizons of physics, philosophy, and art have of late been too widely separated, and, as a consequence, the language, the methods, and the aims of any one of these studies present a certain amount of difficulty for the student of any other U of them ; and possibly this is the principal cause why the problem here undertaken has not been long ago more thoroughly considered and advanced towards its solution.

Hermann Helmholtz

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