The trumpet is microtonal by its very nature because different pitches, which are made by overblowing a length of brass tubing, produce a harmonic series and are therefore not equally tempered:

In the example, pitches are shown as sounding in open position on a C trumpet. Open position refers to there being no valves depressed: the trumpet player changes the pitch by reducing the aperture of the embouchure, resulting in the air pressure increasing as the pitches change through the series. Bold numbers refer to pitches higher than equal temperament, italic numbers smaller than equal temperament. The pure, beatless intervals of the harmonic series (within the 8 or 9 harmonics available on the trumpet) are very slightly larger, in the case of the perfect fifth, or quite substantially smaller, in the case of the major third and minor seventh than their equal temperament counterparts.  It is noteworthy that the harmonics (numbered above the notes) do not begin with the fundamental, but the second harmonic: on the trumpet, the fundamental can only be sounded as a pedal tone, which is more a special effect than a pure tone (this is due to a complex combination of factors due to the shape of the bore of the tube, the nature of the valves and the shape of the bell). To create the full range of pitch possibilities, further harmonic series are generated by using the valves to lower the fundamental (represented by the second harmonic).  The various combination of valves allow further series up to an augmented fourth below the open series:

However, the maths of this doesn’t add up in the way one would expect. Of the last four valve combinations above, all except the 3rd valve series produce pitches that are sharper than would be expected, with the 1, 2 & 3 series being nearly a quarter-tone sharp.  Essentially, the more valves that are added, the sharper the resultant pitches are. To counter this tendency, modern trumpets are fitted with two sliding ‘triggers’ which adjust the pitch downwards. One is attached to the first valve slide and can lower the pitch by up to a quarter-tone and the other is attached to the third valve slide, which can lower the pitch by around a semitone (depending on the pitch and the instrument).

What general implications are there about pitching on the trumpet which we can underline from the above? If we examine all the normal pitch-producing possibilities of the trumpet we find that here is only one possible fingering for middle C and all but one of the pitches below it (the exception being A because this can be played with the 3rd valve or 1 + 2 valves) but, for example, four versions of the high B flat/A sharp, all of which produce a slightly different pitch due to the fact that they emanate from different harmonic series or because they use higher or lower-sounding valve combinations or both.

These sharper or flatter alternative fingerings can be used in combination with the valve slides to produce quarter-tones by exaggerating their natural tendency. In other words, the inclination of any particular fingering to be slightly high or low can be widened by the triggers and, on occasion, some adjustment of the lip, jaw or tongue can be made, as in standard tuning adjustments (note that the triggers only lower the pitch).

The following serves as an example of the practical application of this idea:

The first E quarter-sharp has three versions which use one or other of the valve slides or both in combination.  Valves 1 + 3 (7th harmonic) produce a pitch around 31 cents lower than F, the valve slides are used to lower further to 50 cents. The second E quarter-sharp (the fourth pitch above) uses valve 1 which is a 6th harmonic, producing a rounded F: with the first valve slide at full extension, this produces a quarter-tone lower. The pitch emanating from valves 1 + 2 + 3 is a 7th harmonic E but as this series is almost a quarter-sharp, an adjustment with the lip brings the pitch into tune. Finally, valves 1 + 2 (a 6th harmonic, the pitch E) is already slightly sharp which again makes an adjustment with the lip possible.

The actual choice of fingering is naturally dependent on context, so it is always helpful to have alternatives. Stephen Altoft and I examined these possibilities in full to make the Quarter-Tone Fingering Chart as part of our microtonal trumpet project. We generated 274 fingering combinations from which we disregarded 63 for reasons of poor tone or poor tuning or there just being far better alternatives.

This chart was made for a three-valve standard trumpet.  Later in this series, I will look at the arguments for a four-valve microtonal trumpet and some of the historic and current manifestations of this idea.