### Trumpets Belong in C#

October 24th, 2014
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People typically write for trumpets in Bb [1] or nearby keys because that's where their fingerings are simplest. But if you're willing to tune your trumpet for the key you're going to play in, C# is actually the one where the notes will need the least lipping to play in tune.

The valve system of a trumpet is superficially simple: you have three valves, one that lowers pitch by a whole step, one by a half step, and one by a step and a half. For example, the easiest note to play on a trumpet is the 'Bb' you get with all the valves open. From this position the first valve will lower you from Bb to Ab, the second from Bb to A, and the third from Bb to G. To get other notes, however, you're going to need to start combining valves, and that's where the fudging comes in.

The note you get out of a trumpet depends on the length of tubing the air travels through. To make a valve that lowers pitch by a half step, you send the air through 5.95% more tubing. [2] The problem is, after the first valve lowers us a whole step by adding 12.2% more tubing, adding the second valve on top of that only lowers us by 5.30%. But it really gets bad once we add the third valve. If the valves are set at exactly a whole step, half step, and step and a half, then when we put all three in we'll be adding 37% instead of 41%. So as you play the note you adjust with your lips, or you use a third-valve slide to add a bit more tubing.

But what if you don't want to have to make adjustments? What if you want each note to come out as close to where it should be as possible? Then play in C# major and set the tuning slides to make that as in-tune as it will go. I wrote a simulator that figures out the optimal settings for the tuning slides for a given set of notes and computes the remaining error, and here are the twelve major keys in descending order of intonation:

Key Error
C# 0.20%
Eb 0.26%
Bb 0.31%
F 0.36%
Ab 0.37%
C 0.39%
F# 0.47%
E 0.49%
D 0.51%
A 0.51%
G 0.53%
B 0.58%
This is assuming that you use the standard fingerings, but what if we allow the optimization to also choose the best fingering for each note, even if that gives us something pretty weird? Then it can do even better:
Key Error
C# 0.05%
Eb 0.05%
E 0.05%
Bb 0.12%
Ab 0.25%
F 0.26%
A 0.29%
C 0.35%
F# 0.45%
G 0.49%
D 0.50%
B 0.58%
What crazy fingerings did it come up with for C#?
Note Normal Fingering Optimal Fingering
F3 2-13 (sharp 0.07%) 2-13 (flat 0.04%)
F#3 2-23 (flat 0.09%) 2-23 (flat 0.04%)
Ab3 2-1 (sharp 0.06%) 2-1 (flat 0.05%)
Bb3 2-0 (flat 0.04%) 2-0 (sharp 0.01%)
C4 3-13 (sharp 0.19%) 3-13 (sharp 0.07%)
C#4 3-23 (sharp 0.02%) 3-23 (sharp 0.07%)
Eb4 3-1 (sharp 0.17%) 3-1 (sharp 0.06%)
F4 3-0 (sharp 0.07%) 4-13 (flat 0.04%)
F#4 4-23 (flat 0.09%) 4-23 (flat 0.04%)
Ab4 4-1 (sharp 0.06%) 4-1 (flat 0.05%)
Bb4 4-0 (flat 0.04%) 4-0 (sharp 0.01%)
C5 5-1 (flat 0.73%) 6-13 (sharp 0.07%)
C#5 5-2 (sharp 0.20%) 6-23 (sharp 0.07%)
Eb5 6-1 (sharp 0.17%) 6-1 (sharp 0.06%)
F5 6-0 (sharp 0.07%) 8-13 (flat 0.04%)
F#5 8-23 (flat 0.09%) 8-23 (flat 0.04%)
Ab5 8-1 (sharp 0.06%) 8-1 (flat 0.05%)
Bb5 8-0 (flat 0.04%) 8-0 (sharp 0.01%)

The fingerings are actually almost the same. It made four substitutions:

• 13 on the fourth partial instead of the raw third partial for F#4
• 13 on the sixth partial instead of 1 on the fifth partial for C5
• 23 on the sixth partial instead of 2 on the fifth partial for C#5
• 13 on the eigth partial instead of the raw sixth partial for F5

These let it tune the fundamental (Bb and harmonics) sharp by just a little bit more and lengthen the first valve to compensate, and then choose more notes using the first valve.

Overall, the main thing I'm taking away from this is that the intonation issues I'm having playing trumpet and baritone in contra dance sharp keys like D and A is to be expected and requires active compensation by the player. In other words, I should learn to use the third valve slide.

(All this holds for other three valve instruments like the ones I was talking about yesterday.)

[1] Which trumpet players call "C" for historical reasons. In this post (and in life in general) I'm going to be ignoring this and using concert pitch.

[2] Why? Well, if you lower a note by a half step twelve times you need to get the same note an octave down, which means we need to end up with twice as much tubing. Solve for this amount and you get 5.95%. It's the twelfth root of two (1.0595), less one to make it a percentage increase instead of something to multiply by.

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