To quote from John Tyndall in Sound, 1893:
"A rod fixed at one end can vibrate as a whole, or can divide itself into vibrating segments separated from each other by nodes. In this case the rate of vibration of the fundamental tone is to that of the first overtone as 4:25, or as the square of 2 to the square of 5. From the first division onwards the rates of vibration are proportional to the squares of the odd numbers, 3, 5, 7, 9, etc. With rods of different lengths the rate of vibration is inversely proportional to the square of the length of the rod. A rod fixed at both ends and caused to vibrate transversely divides itself in the same manner as a string vibrating transversely. But the succession of its overtones is not the same as those of a string, for while the series of tones emitted by the string is expressed by the natural numbers, 1, 2, 3, 4, 5, etc., the series of tones emitted by the rod is expressed by the squares of the odd numbers, 3, 5, 7, 9, etc." This sounds true. A rod with one end fixed will most certainly have different vibrational modes than a string with both ends fixed. There is a very important difference between 'harmonics' and 'partials'. A 'harmonic' is a multiplying of the fundamental by a factor of 2 or its multiples.
There are 'super harmonics' which are multiples of 2 and there are 'sub harmonics' which are progressive 'halvings' of 2. He is using different terminology here. In both music and physics, you will have the 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x, 11x, etc. harmonics all present in most all musical instruments. The difference in tone between the instruments is determined by their relative strength. And they are all really harmonics, even if they do not necessarily sound "harmonic". Most will actually sound harmonic, as long as you don't have a preponderance of the really odd ones like 7x, 11x, 13x, 17x, etc, which do not fit into our musical scale in any form. 'Partials' on the other hand are all other derivatives of the fundamental that are not divisible by 2 such as a fifth, third, seventh, etc. There are 'super partials' as well as 'sub partials'. All of these partials lie between the harmonic ratios throughout the Scale of Octaves just as though they were musical ratios.
Orthodox science recognizes the former but denies
(or very rarely acknowledges) the latter in engineering circles. Again,
a difference in terminology, I think. Science and engineering are very
aware of the non-harmonic energy you can add to waveforms. Ever heard of
a fuzz box? And think about all of the microprocessor-based special effects
you can get for a band now. They have done a lot of work in that area.
But I seldom hear the term "partial". I usually hear non-harmonic, or fractionally
related, like 10/7 of the fundamental. In our SVP work:
THE DIFFERENCE IS ALL IMPORTANT, ABSOLUTELY ESSENTIAL
AND FUNDAMENTAL.
To give just one brief for instance: A sine wave form is composed of harmonics only. A triangle or saw tooth wave form has both types of ratios. The former sounds smooth and even as from a flute. The latter sounds rough and brash as from a trumpet. A sine wave has NO harmonics, or sub-harmonics, or partials, or anything else. It is the one pure waveform that is not the composite of other waveforms. A square wave, and all other waveforms that are symmetrical above and below the zero line (or "middle line"), is composed of the fundamental and a near-infinite series of odd harmonics. No even harmonics there. A triangular waveform has both even and odd harmonics. The flute is the instrument that produces a note closest to a pure sine wave. The violin produces a triangular waveform, so it has gobs of harmonic energy in the sound, and sounds shrill. Their originating premise (math) is different, their tones are different and their effects on other things are different. In other words, the quality of the sound or vibration is determined to a great degree by its particular chord of ratios, i.e., collection of discrete harmonics and partials. To go one further the vibrating string above mentioned is an enharmonic vibration because its chord of vibration or vibration signature contains all numbers (harmonics and partials). While a flute or the resonator such as we are making for the dynasphere are harmonic tones because its tone is derived of harmonic ratios of the fundamental. Again, a matter of terminology. He doesn't seem to like to consider any multiples except powers of 2 to be harmonic. Everybody else I ever learned from considered all exact multiples of the fundamental to be harmonics.
* Terrance Hodgins *