High Brass Overtones
The following graphs are Fourier transforms of the audio waveforms of various high brass instruments. These transforms show the fundamental tone and overtones, or partials, that give each instrument its unique sound.
All tones were produced by the same musician (Jonathan Clymer) playing an 'open' G (concert F) on Bb-pitched instruments, except for the last one where the G was produced using the alternate '1 & 3' fingering. Recordings were made with GarageBand on an iMac using the internal microphone. Each recording was exported as a 'High Quality' 44.1 kHz MP3 and SoX was used to convert the MP3 format to a 'dat' file with an output rate of 8820 Hz. The Fourier transform was performed in Excel using 4096 points corresponding to a total waveform time of 464.4 ms. Output from the transform thus yielded a resolution of 2.15 Hz, and a highest observable (Nyquist) frequency of 4410 Hz.
No attempt was made to correct the inherently non-ideal response of the internal iMac microphone. This is likely the largest source of error in this work, and prevents the direct comparison of the amplitudes of the overtones within any particular transform. However the results are of potential use in comparing the relative amplitudes between the different transforms. Hence from the following graphs one cannot conclude that the amplitude of some of the overtones is higher than the fundamental for a particular instrument, but we can reasonably observe that one instrument has lower relative amplitudes for certain overtones than another instrument.
Tones were produced with a cornet, two trumpets and a flugelhorn with 5C mouthpieces of various shanks. For similar work performed in lower brass instruments see this page. The corresponding Helmholtz pitches and approximate frequencies for each overtone are given in the table below.
The first transform on a Yamaha 2310 Bb cornet with a 5C cornet mouthpiece was repeated three times with different parts of the waveform in order to gauge reproducibility of the process. The three transforms look fairly similar, with the largest visual discrepancy occurring in Replication 2, where the relative amplitudes of overtones 1 and 2 were reversed from both Replication 1 and 3. Variations in transforms for different instruments that are larger than the ones seen in these three replications are likely real.
Figure 1. First replicate of a cornet with a 5C cornet mouthpiece.
Figure 2. Second replicate of a cornet with a 5C cornet mouthpiece.
Figure 3. Third replicate of a cornet with a 5C cornet mouthpiece.
The same cornet was played with a flugelhorn 5C mouthpiece instead of a cornet mouthpiece. Note the increase in the first overtone and slight decreases in higher overtones.
Figure 4. Cornet with a 5C flugelhorn mouthpiece.
Next is a Dillon flugelhorn with a 5C flugelhorn mouthpiece. Compared to the cornet (Figures 1-3), the flugelhorn has considerably dampened higher overtones.
Figure 5. Flugelhorn with a 5C flugelhorn mouthpiece.
When a 5C cornet mouthpiece is used with the flugelhorn, there is a somewhat surprising shift toward increased amplitudes in the lower overtones.
Figure 6. Flugelhorn with a 5C cornet mouthpiece.
Two trumpets were evaluated, both with a 5C trumpet mouthpiece. The first is a standard configuration student model Conn Director with a 4.5" bell and a 0.478" bore as measured at the outer diameter of the main tuning slide. Differences between this trumpet and the cornet are subtle.
Figure 7. Conn trumpet with 5C trumpet mouthpiece.
The second trumpet was a Brasswind pocket trumpet with a smaller 3.75" bell, but a larger 0.500" bore, again measured at the outer diameter of the main tuning slide. The instrumentalist noted that this trumpet was more open and blew more freely than the Conn trumpet. Differences from both the cornet and standard-configuration trumpet are more noticeable, with an increased first overtone and substantially decreased 3rd and 5th overtones. This horn appeared to have an even more conical overtone series than the flugelhorn.
Figure 8. Pocket trumpet with 5C trumpet mouthpiece.
One last experiment was performed using a cornet with a cornet mouthpiece, but playing the normally open G with the alternate first and third valve fingering. The longer effective instrument length provides increased lower overtones and relatively reduced higher overtones.
Figure 9. Cornet with 5C cornet mouthpiece with alternate 1&3 fingering.
A simple numeric description of the overtones is given by the ratio of the first moment of the frequency to the zeroth moment, which I call here the Brightness:
where ν is the frequency and Iν is the intensity at that frequency.
The rank ordering of the Brightness of the instruments tested here, going from darkest to brightest is:
Evidently changing the mouthpiece on the cornet had a substantial effect, as much using an alternate fingering, and more than changing to a trumpet or flugelhorn.
One way of showing the differences is to graph the normalized cumulative intensities. In this graph instruments with more energy in the lower overtones are shifted upward.
