Footnote: Tuning to the
Harmonic Overtone Series

What are harmonic overtones?

When a string, tube, or reed is excited to periodic vibration, it produces a tone consisting of a fundamental tone at the lowest, primary frequency of vibration, and also a set of harmonic overtones. Other shapes such as disks and bells produce non-harmonic overtones. The fundamental and overtones, taken together, are known as partials.

The frequencies of harmonic partials are integer multiples of the fundamental tone. That is, the relationship between the frequencies of harmonic partials can be described as the set

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc. or as a subset of them.

Listen to a MIDI file
of that series through 12, where 1 = A0 (55 Hz), with the low A sustaining as a drone for reference.

A "natural" trumpet, then, can theoretically play any and all of the tones that are at integer multiples of the lowest frequency with which its tube is capable of resonating.

Normally, it is difficult to make a brass instrument resonate at its fundamental. Good sounding tones generally begin at the 2nd or 4th partial, allowing production of the tones at integer multiples 4 through 24 or so of the fundamental, with sufficient pressure usually becoming difficult to produce those beyond about 24. If a trumpet's fundamental tone is A1 (110 Hz), the frequencies it could reasonably produce would correspond to the notes

A3 - C#4 - E4 - G4 - A4 - B4 - C#5 - D#5 - E5 - F#5 - G5 - G#5 - A5 - Bb5 - B5 - C6 - C#6 - D6 - D#6 - Eb6 - E6

D# and Eb are different notes in this case; at close, but different, pitches.

Listen to a MIDI file
of that series, up and back down, with an A2 drone for reference. The series then repeats without the drone, this time sustaining the notes so that up to 8 of them overlap and sound together.

Notice how they all resonate together very strongly, even the ones that are closely spaced, instead of clashing as equal tempered tones would. That's one of the beauties of Just Intonation: a very high degree of consonance. The upper ones may even blur together into a single tone. You may hear the fundamental tone, too, even though it is not being played per se. The effect is similar to a single saw tooth wave playing through a swept resonant filter, like on old analog synthesizers.

In any event, that is the range and set of tones one is has to work with with when composing for non-valved trumpets, which are the only kind that would be found in Middle-earth. Brass players can also "lip" the pitch a little above and below those pitches, allowing somewhat more tempered tunings, but the pitch relationships sounded in the second MIDI file are the ones that sound naturally. Gabrielli and Monteverdi are two Renaissance composers whose excellent use of these trumpets is well worth studying.

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Written by David J. Finnamore
Orlando, FL, USA

Page last updated January 2, 2000