Harmonics

One of my interests is creating projects that demonstrate physics concepts. This is particularly true for physics concepts that are mentioned by Mr. Hubbard in has writings or lectures. One such concept is harmonics. The above representation, obtained from an educational website, illustrates the basic idea here. “Hz” stands for Hertz, the chosen name for a unit of measure of frequency, previously known as Cycles Per Second.

These terms, as far as I know, are borrowed from the world of music, where they have been in use at least since the time of the Greeks, who liked to play around with the mathematics of vibrating strings.

One way of looking at harmonics is the idea that they can be derived by taking a string and dividing it into different numbers of parts that add up to the total length. That gives you a series of whole fractions for different string lengths (periods), and a series of whole number multiples for frequency (or tone). In audio, we usually refer to frequencies rather than wavelengths or periods.  In radio and light, you are more likely to see wavelengths referred to.

The harmonic series illustrated above contains two octaves. An octave is a frequency exactly two times another frequency. In music, octaves are given the same note letter, as they indeed sound like the “same” note.

Traditional music scales are based on whole fractions. It was possible to determine relationships between notes using fractions before we had electronic means to measure frequency. Thus, a traditional musical scale would be made up of a fundamental tone and then a series of chosen higher tones relating to the fundamental by whole fractions of a value between one and two. The most common notes used were sub-octaves of the harmonics of the fundamental tone. Thus: 3/2, 5/4 and 7/4, 9/8, 11/8, 13/8 and 15/8. Many other tones are possible, but it was found – or considered – that these sounded the most musical when played together. Modern tuning systems approximate these notes while creating a scale that makes transposition between keys (scales starting with different fundamentals) much easier.

My project

With my project, I just wanted to demonstrate what several harmonics of a fundamental sound like.

The biggest challenge in generating such tones electronically is to get a pure tone (sine wave). Sine is the name of the function that describes a pure tone. It is a term taken from trigonometry (the study of the properties of angles and circles).

I wanted six sine waves that were exact mathematical multiples of the fundamental. The only practical way to achieve this is starting with digital signals. Those signals can then be built into sine waves using various processes. I was a little nervous about how well this would work, and how easy or difficult it would be to create good sine waves. But it worked out OK. In this design, most of the sine waves are constructed from 16 voltage steps. For the fundamental tone, I made knobs to control the size of the voltage steps. For all the other tones, I used fixed resistors. The basic idea was taken from a magazine article from the 1980s that I had saved in my digital library.

This project works fairly well. The power supply was a little complicated, because I needed four different voltage rails: +10V, +5V, (ground = 0V), -5V and -10V. My fixed-resistor sine wave generators work quite well. The one using variable resistors is a little flaky, but does an acceptable job. I get seven harmonics from this equipment, including the fundamental, and I can mix them in different ratios to get richer sounds.

I built this into a cabinet that was already occupied by an old multimeter I purchased years ago. I decided there would be enough space for it without removing that old meter, so it remains a part of the project. I can even use it to measure the amplitude of the output signal! 