Natural Harmonic Series
Below are the notes of the natural harmonic series (or partials) for a trumpet. Number 1 is the fundamental and all the successive harmonics are derived from this. The number of each harmonic is the proportion of the fundamental wavelength which gives this pitch, for example:- the 7th harmonic has a wavelength 1/7 of the fundamental. Like shortening a string to get a higher note.
Unfortunately this simple mathematical relationship doesn't produce a scale, or in some cases what we now expect as musical intervals. In particular the 7, 13 and 14th harmonics are very flat, and the 11th very sharp. Click here to hear the
harmonic (50K) series as pure tones.

Below is the harmonic series represented in musical notation.

Harmonic series
This problem of tuning was known of since the time of Pythagoras and resulted in a system of temperaments which finally lead to the invention of equal temperament around the time of Bach. Equal temperament divided the octave into 12 equal semi-tones, rendering all the notes 'out of tune' (intervals within an octave are non consonant - they give beats) - but equally. This meant that all keys would sound the same. Until equal temperament was widely adopted, different tunings (temperaments) were used for each group of (2-3) keys. Stray too far from the key you were tuned for and the intervals between certain notes became very unmusical. These non-equal temperaments allowed a proportion of the notes to be in perfect consonance with each other, a dimension to music which is now lost in modern performance. This is a sound file (189K) which has a 3rd (C-E) and a 5th (C-G) tuned to natural temperament (perfect consonance - no beats) and then equal temperament. The 3rd produces very fast beats, corresponding, approximately, to a low C, the 5th very slow beats. The quality of sound reproduction on individual systems may affect how easily the beats can be heard.

The strangeness to the modern ear of non-equal temperaments has lead to the use of hole systems on the natural trumpet to correct intonation, particularly of the 11th and 13th harmonics. Whereas the players of the time would have probably corrected intonation by lipping the notes. It should be pointed out that the original players would not have had to correct the intonation of individual notes as much, in the temperaments of the time, as is necessary in modern equal temperament.


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