Trumpet Tuning Tendencies Relating to the Overtone Series with Solutions

The fundamental does not exist as a real note on the trumpet. The first member of the harmonic series that exists on the trumpet is the first overtone, or second harmonic. Let’s use the lowest open note (no valves) as the first overtone for discussion. This would be low C on the trumpet. These tendencies apply to the overtone series starting on any valve combination. In relation to equal temperament, the notes in the overtone series are out of tune as follows:

First overtone (low C)         in tune
Second overtone (second line G)    2 cents sharp
Third overtone (middle C)   in tune
Fourth overtone (top space E)    14 cents flat
Fifth overtone (top of the staff G)  2 cents sharp
Sixth overtone (Bb above the staff, played open)  31 cents flat
Seventh overtone (high C)    in tune


These numbers vary based on the trumpet, but the tendencies are the same, except for a rare exception.
Because it is so out of tune, the sixth overtone is never played open; it is played first valve.  Therefore, this is technically an alternate fingering, but has become standard.  Refer to fingering charts for how to play the sixth overtone in each overtone series (in other words, starting on a different valve combination).
To make it easy, here are some other notes that tend to be out of tune on the trumpet, mainly because of the presence of valves, which makes it impossible for the trumpet to be completely in tune.

Low C#, D   very sharp
Low E      sharp
Second space A    sharp
A above the staff        sharp


Here is a summary of the typically out-of-tune notes on the trumpet:

Low C#, D   very sharp
Low E      sharp
Second line G  sharp
Second space A     sharp
Fourth space C# through top space E    flat
Top line F through top of the staff G sharp
A above the staff     sharp


Use slides to fix sharp notes whenever possible.  The third valve slide should be used for the low C# and D.  The first valve slide should be used for all other sharp notes that use the first valve (low E, second space A, top line F, and A above the staff); the first valve slide will probably only need to be moved slightly for these notes.  This leaves the flat notes and some sharp notes (second line G, top line F#, top of the staff G) that one must now “lip” in tune.  This involves directing the air stream upwards for flat notes and downwards for sharp notes.  The challenge then also becomes to play these notes with as beautiful a sound as though you are playing through the center of the horn.

Some other thoughts:

Notes from low C below are so “moveable” that they do not have traditional pitch tendencies and typically slides are not needed.  Notes below the staff tend to be flat because one is relaxing the embouchure so much to reach them.  
Notes above the staff tend to be sharp for younger players as they tend to “squeeze” for these notes, and this pinching makes these notes sharp.
Fatigue also affects intonation on the trumpet, and will affect players differently.  Some go flat when tired, some go sharp.
Temperature affects pitch.  Cold trumpets play flat.  Hot trumpets play sharp.  Temperature also affects intonation exponentially on pitches which are already out-of-tune.  For example, top space E is always flat, but with each decrease of 5 degrees in temperature, this E will get flatter by larger amounts.
Each trumpet has slightly different intonation issues.  Each student needs to learn the tendencies of his/her own trumpet by first putting third space C or third line B in tune with the tuning slide.  When purchasing a trumpet, one should try to pick a horn that puts the notes that are hardest to play in tune closest to in tune (the flat notes and G on top of the staff).