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On valve brass instruments, a single fingering may be used to play different notes. For example on the trumpet, leaving all valves open (up) allows the player to play the following notes:

C - G - C - E - G - Bb - C - ...

(The first playable interval is a 5th instead of an octave, but that is another question.)

The wave created when playing a low C has a specific spectrum that determines the timbre.

What is the spectrum of the G just above this low C?

  1. the timbre is very similar, so the spectrum should also be very similar. If low C is a weighed sum of the harmonics of the fundamental (C - G - C - E - G - Bb - C - ...), the spectrum of G should have the same components transposed (G - D - G - B - D - F - G - ...).

  2. G is played on the same length of tubing as the low C, therefore the harmonics present should be the ones allowed by just this length of tubing. Then the spectrum of G should be G - C - E - G - Bb - C - ....

These two points of view are incompatible. I have a feeling that 2 is wrong, but I can't see why the tube length of C would allow such a strong D (a ninth above the low C) in the spectrum of G.

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Based on how the overtone series works, I'm pretty sure that number 2 is correct. The timbre is probably similar because the same overtones are sounding. But I'm not nearly knowledgeable enough about the physics of sound or brass instruments in general for this to be a reliable answer. – Babu Apr 12 '12 at 14:20
The high Bb on most trumpets is slightly flat when played with all valves up. So, it is usually fingered with the first key. – Luke Apr 12 '12 at 17:14
@Luke: just as the E is better played 12 than 0 on some models, but that is not the point. The point is that you can play what looks like the components of the harmonics series without changing the fingering. I could have asked about a clarion. – Gauthier Apr 12 '12 at 19:08

1 Answer

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Your first assumption is (mostly) right. Trumpet physics are actually fairly complicated: The basic tube has an open and closed end which by itself would only produce odd numbered harmonics with a quarter wavelength fundamental. However the tapered mouthpiece tapered bell change the harmonics spacing so it gets much closer to the natural harmonic progression.

At higher frequencies the trumpet harmonics spectrum gets fairly dense so there is always a useful harmonic near by when you need it. Also the tapering makes the resonance not super sharp so you can actually move the frequencies around a bit.

So the horn by itself is capable of supporting a fair amount of frequencies. Which ones are actually contained in a specific note depends by the excitation. If the excitation itself doesn't contain a certain harmonic, the horn won't amplify it either. In your example of C the excitation from your lips already contains the harmonic series (C, G, C, E ...) and the horn just amplifies them. Same for the G (G, D, G, B ...). While playing the G the horn is perfectly capable of reproducing an E, your lips just don't excite it.

Another nit-picky remark: It's actually not entirely correct to call the harmonics by note. This is exactly true only for the octaves the other notes are slightly off (harmonic tuning vs. temperate tuning). For example, the 6th harmonic (minor 7) is off by about 2% or 31 cents.

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You mean that the series of notes played with the same fingerings are actually not harmonics, but that the horns are designed in such a way that in only resembles it? I have heard this before, of someone who actually built an instrument to demonstrate it, but I was a bit sceptical to that demonstration. – Gauthier Apr 17 '12 at 9:20
"While playing the C the horn is perfectly capable of reproducing an E, your lips just don't excite it." I suppose you mean "reproducing a D", don't you? – Gauthier Apr 17 '12 at 9:22
@Gauthier: Technically speaking the notes are determined by the mechanical modes. For a string the modes follow nicely a harmonic series. For a tube that's open on one side and closed on the other, the modes have different frequencies (odd harmonics only). The mouthpiece and the bell are used to make make the tube non-cylindrical and so to dial in the modes to match the desired frequencies. The main difference to a string is termination: The string is equally fixed at both ends, the tube is open on one side and closed on the other. – Hilmar Apr 18 '12 at 11:49
@Gauthier: I reference "G" as the root note in my example, not "C". "E" is not a (lower) harmonic of "G". Or maybe I mistyped and someone fixed it already. – Hilmar Apr 18 '12 at 11:53
@Hilmar: I figured you meant G and not C - as it said fron the start - in the example with E, so I edited it. I hope that was ok and a correct interpretation of what you intended to point out. – Ulf Åkerstedt Apr 19 '12 at 23:50

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