It’s essential to understand the difference between overtones and harmonics for one who wants to understand orchestration from a physical point of view. The easiest to understand are harmonics, which was originally a mathematical term associated with a harmonic series, in which each term has a whole number ratio to all other terms. Overtones rather are associated with a physical system and the term describes the different ways a vibrating system vibrates.
The piano is a good example. Its sound contains a fundamental and overtones. The overtones are close to, but not harmonic. This is because no real string is infinitely flexible, complications because of string length and tension, and the fact that the piano tone is a transient, not a steady periodic motion. But bear with me, and you will understand where I'm going.
Thus “overtones” can be characteristic of a single tone, with different frequency components making up the tone. Another example of overtones would be to excite a steel piano string with, say a periodic magnetic field. If you did it at the frequency of the fundamental, the string would vibrate at that frequency, and there would be very little other vibratory components – no overtones. Then you could excite it at the first overtone frequency that occurs during the conventional vibration. Again, the string will vibrate at that overtone frequency, with no fundamental and very little other components. You can excite the string at any frequency, but if that frequency doesn’t match the fundamental or some overtone, the string will vibrate with little amplitude. This is called resonance. Stay with me.
Any steady, periodic tone must have overtones that are harmonic. If that weren’t the case, the vibration would not be periodic. That’s because the motion must repeat exactly every time, meaning all overtones must also repeat. It’s mathematically impossible for the overtones to repeat as a multiple of the fundamental if they are not some whole number ratio of the fundamental frequency.
We come to the difference between the piano and the violin. The piano tone is not steady or periodic. It’s a transient, made by striking the string, and there's no requirement that the vibration repeat exactly for all cycles. It doesn't, because of physics, not mathematics, and in addition the amplitude of all components during one cycle is less than the corresponding amplitude in the previous cycle. With the steady periodical vibration of other instruments, such as the violin, the vibration is "forced" or maintained by the action of the bow. That explains the periodic nature of the sound. Energy is steadily being transferred from the bow to the string.
So we have an example of the same vibrating sound source – a string – behaving in very different ways. With the violin, the tone contains overtones that are harmonics, and with the piano, the overtones are not harmonic. Of course, the violin can be played in transient ways, like other instruments can, using various techniques other than a steady draw with the bow.
To summarize, all musical tones that are steady and periodic have overtones that are harmonic, and all musical tones that are transients have overtones that are not harmonic, although some can be very close to harmonic.
What does this mean for orchestration? Well, first, I point out that some here seem to think that dissonance – non-harmonicity – is always bad. That’s not true. The dissonance of a cymbal is a favored sound. The 20th century jazz bands of Stan Kenton and Maynard Ferguson are now forever in orchestral annals of examples of how to use dissonance to blood-boiling acoustic effect. Then we also have the later orchestral composers of the modern age. Also, the transient sounds of bells and chimes contain non-harmonic overtones, and it's an art to tune them for maximum effect. The "bigness" of the sound of church bells is enabled by non-harmonic partials. Another example is the equal tempered scale, which has enormous advantages, but some people actually prefer it over other "natural" scales because of the inherent dissonance it provides.
It’s worth looking with more detail at the “upper partials” of a musical tone in the example of the piano. As I mentioned, because the strings are not perfectly flexible, and because string vibrations in some cases produce tensile fluctuations that begin to be at the level of string tension, the overtones in the transient tone are not harmonic, but “stretched,” meaning that the overtones are consistently larger than what the harmonic calculation would predict. And that characteristic makes the piano a beautifully sounding instrument. The stretched overtones mean that individual strings can be more easily discerned from each other and from other instruments, so that chords are vibrant and well identified. Compare that to the steady chords played by an accordion, which are harmonic, due to the steady nature of the sound source. With the accordion, chords are muddled, mushy, because of the exact harmonics. Say you play two notes an octave apart on the accordion. The upper note plays the exact same overtones as the lower note, with the only difference being their fundamentals. It’s very difficult to discern the two notes, and when you could, it’s usually because one note is out of tune.
I could explain much more about the beautiful piano tone and stretched harmonics, the effect of three strings playing together, and the interaction with the sound board. All of that involves overtones in very complicated ways. But I’d rather make corrections to the OP’s comments.
“We all know that an instrument's timbre is uniquely determined by its harmonic series.” This is incorrect because not all instruments have overtones that are a harmonic series. Additionally, the questioner may be confused in believing that we can identify the source of a musical tone only by its timbre. We also need to hear the “start transient.” There are huge differences in start transients between the piano and violin, the accordion, and the wind instruments. The percussive start of the piano – or guitar – tone help us identify the instrument. In fact, there have been scads of experiments wherein people cannot tell the difference between musical tones from different instruments when the start transient is eliminated. Even with instruments that we know have significantly different overtone/harmonic structure. Many people think the clarinet contains ONLY odd harmonics, though that’s not accurate, and when its musical tone is played without a start transient, it’s easy to confuse it with the other steady tones.
I’d like to draw closer to what the OP wants to know, but there are too many misconceptions and the questions don’t really make sense. The OP seems to believe that a harmonic overtone in one instrument can clash with a harmonic overtone in another instrument. Why so?
“if the relative loudness of the harmonic series of two instruments are different (of course the notes in it are the same), then when they play the same interval, one of them might sound more consonant than the other.”
Why would one harmonic frequency be more consonant than another? Or why would the additional volume when a third harmonic of one instrument is added to a harmonic of another? It's pretty much by definition that any harmonic is consonant with any other harmonic. I suggest that the OP needs to understand the difference between overtones that are harmonic and overtones that are not. The simplistic view by the OP is also evident in the steady tone spectrum examples shown. These are steady periodic tones that are not often found in actual music. In practice, there are many dynamics and transients in the playing of any instrument and thus very few tones conform to the information contained in those Fourier spectrums.
The OP is worried that a predominant third harmonic of the trumpet might clash because, “when that third harmonic happens to be a very unpleasant note after combining with the rest of the orchestra, trumpets should not be used.” Is the OP wondering about the clash only when the trumpet player is assigned “a very unpleasant note”? I can’t make heads or tails out of what the OP wants to know.
“Question: Are there any theories about what I mentioned above? Is it an important thing to consider in Orchestration?”
I suggest the OP re-write this question, using hopefully a better understanding of the role of overtones and harmonics in music. Roughly speaking, and I’m not an orchestral composer, my guess is that composers attempt no theoretical applications concerning overtones, but rather rely on their experience of the total sound of each instrument and how they blend together in a very dynamic environment. It's bottom line experience and it's very subjective. It would surprise me greatly if composers consulted graphs of steady-state Fourier series during composition.
