This ingenious device, designed by Herman von Helmholtz XR (1821-1894), was the very first sound synthesizer: a tool for studying and artificially recreating musical tones and the sounds of human speech.
Suppose I sing the word ‘car’ and then on the same note sing ‘we’. The two vowel sounds will be similar in so far as they have the same pitch G , yet they have a clearly distinct sound quality, or timbre G . What is it that accounts for this difference, and the timbres G of musical sounds in general? Helmholtz set out to answer this very question in the mid nineteenth century, building on the work of the Dutch scientist Franz Donders (1818-1889).
Helmholtz showed that the timbre G of musical notes, and vowel sounds, is a result of their complexity: just as seemingly-pure white light actually contains all the colors of the rainbow, clearly defined musical notes are composed of many different tones. If you play the A above middle C on an organ, for example, the sound you hear has a clearly defined “fundamental” pitch G of 440Hz G . But the sound does not only contain a simple “fundamental” vibration at 440Hz G , but also a “harmonic series” of whole number multiples of this frequency G called “overtones” (e.g., 880Hz G , 1320Hz, 1760Hz, etc.). Helmholtz proved, using his synthesizer, that it is this combination of overtones at varying levels of intensity that give musical tones, and vowel sounds, their particular sound quality, or timbre G .
How the synthesizer works
Helmholtz’s apparatus uses tuning forks, renowned for their very pure tone, to generate a fundamental frequency G and the first six overtones which may then be combined in varying proportions. The tuning forks are made to vibrate using electromagnets and the sound of each fork may be amplified by means of a Helmholtz resonator with adjustable shutter operated mechanically by a keyboard.
By varying the relative intensities of the overtones, Helmholtz was able to simulate sounds of various timbres G and, in particular, recreate and understand the nature of the vowel sounds of human speech and singing. Vowel sounds are created by the resonances G of the vocal tract, with each vowel defined by two or three resonant frequencies G known as formants. When we say or sing ‘a’ (as in ‘had’), for instance, the vocal tract amplifies frequencies G close to 800Hz G , 1800Hz and 2400Hz amongst others. When we require a different vowel sound, the muscles of the throat and mouth change the shape of the vocal tract, producing a different set of resonances G .
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
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.
In May of 2002 composer Karlheinz Stockhausen conducted a music contest online. He submitted 12 sound samples, mostly very long examples of various forms of white, pink and other forms of “colored” noise. Along with these samples were very specific instructions of how the excerpts of sound would be transposed onto the formula that he provided. The “formula” provided was Sagittarius from his Tierkreis (Zodiac) suite. The problem that I saw was how to take standard sheet music and apply pitchless samples into something that could be recognized. After some thought, I imported small chunks of the provided noise samples into my Akai S 6000 sampler. I used the sharpest high Q filter that I could find and created unique samples for every pitch that would be used in the piece. For example, for A 440 I created a very sharp filter that would only let noise pass through at or around 440 Hertz. After some trimming and other small edits I stored and mapped the samples across a standard keyboard. I then took the score provided by Stockhausen and manually entered it into Finale. Once the score was complete I played the exported MIDI file in the S 6000 sampler to produce the final audio file which I called “Sagittarius Formula”. Professor Stockhausen granted me “First Runner Up” for the submission of this piece.
For Those Interested, Here Are The Original Instructions Sent By Stockhausen.
Original Instructions as submitted by Professor Stockhausen
To whom it may concern
1. as a “formula” one page with the notation of SAGITTARIUS.
1. This part of my ZODIAC lasts normally 25.4 sec. with the tempo quarter note = 85. For composing a piece of about 3 1/2 minutes, it should be stretched 8 times slower. Inside these 3 1/2 minutes, various transpositions in durations and pitches can be juxtaposed and superimposed. Use ritandandi and accelerandi.
2. All the pitches with their durations should be used at least once in the original form.