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Most trumpets' low C♯ and D are too low, and need correcting by pulling the 1st of 3rd valve.

My first understanding was that the distance between tones in the lower register is larger than between tones in the higher register, therefore the lengthening of the tube needed in the low register.

However, if all the tones in the lowest register (C down to F♯ with 3 ledger lines under the staff) are correct, the harmonics of this tones should also be correct.

My new explanation is that if you tune the valves so that middle F♯ (fingering 2), F (fingering 1), and E♭ (fingering 23) are tuned, then fingerings 13 and 123 are going to be too high.

This would imply that the whole harmonic series would be too high, including the lowest F♯. My feeling is that the lowest F♯ is in tune, while the C♯ (same fingering 123) is not. This does not make sense, maybe the low F♯ is also high?

Any explanation? Why is low C♯ so high but not low F♯? Is it even so?

If my reasoning holds, then low E (12) should also be a little high, if F (1) and F♯ (2) are in tune?


Edit:

I wanted to put numbers on the 1+2 fingering not being in tune if the 1 fingering and the 2 fingering are.

Frequencies of the trumpet's low C and downwards, in equal temperament based on A at 440 Hz:

C:  261.626 Hz
B:  246.942 Hz
B♭: 244.082 Hz
A:  220.000 Hz

Their period:

C:  3.8223 ms
B:  4.0495 ms
B♭: 4.0970 ms
A:  4.5455 ms

And their wavelength (assuming 343 m/s):

C:  1.3110 m
B:  1.3890 m
B♭: 1.4153 m
A:  1.5591 m

And in procent of the reference (C):

B:  105.95 %
B♭: 107.19 %
A:  118.92 %

So if valve 2 is in tune for B (5.95 % length increase), and valve 1 is in tune for B♭ (7.19 % length increase), their sum (valve 1 and 2 down) gives a total length increase of 5.95 + 7.19 = 13.14 %. The theoretically tuned A should have a length increase of 18.92 % instead, so fingering 1+2 makes it a bit sharp.

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Regarding your edit: Are you sure it's pure addition? Adding percentages like that usually doesn't work. I don't know the specifics of your instrument though. –  Matthew Read May 23 '11 at 3:00
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Adding percentages should work if using a common reference (here the length for C). If you add 3 % of 100 (that is 3) to 6 % of 100 (that is 6), it's the same as adding 9 % of 100 (9). This stops working if you have 3 % increase, then additionally 6 % on top of it. In that case the first added 3 % get into the input of the 6 %, which they don't do if the reference is common. Regarding the specifics of the instrument, it's just adding tube length. –  Gauthier May 23 '11 at 6:35
    
Makes sense, I just didn't know that the valves function the same together as they do apart. –  Matthew Read May 23 '11 at 15:13
    
@Ben Alpert: what happened to the formatting? It looks like you edited to display guitar tabs, I supposed that this was done automatically. But it does not make more sense... –  Gauthier Sep 1 '11 at 12:14
    
@Gauthier: I simply merged in your "answer", as it belonged better as an edit to the question. –  Ben Alpert Sep 1 '11 at 16:41

5 Answers 5

up vote 3 down vote accepted

Rotary and piston valve brass instruments are constructed with two valve slides in tune (corresponding to the length of the instrument), being the 1st and 2nd. The fingerings "open," 2nd and 1st are typically in tune, but as soon as you begin combining valves, things no longer add up. It would seem that if the 1st and 2nd valve slides are engineered to play in tune, the combination of 1 and 2 would as well; however, once you start depressing valves you shorten the length of the instrument. With the instrument shorter, the length of the slides no longer works with the new length. It's a bit of a mind job, but if you're really into the mechanics, check out some doctoral/professional/well-researched articles on the harmonic series and brass instruments.

Basically though, this is what pans out in respect to horns (given that the player tuned each slide first...) and trumpets.

0 - good, 2 - good, 1 - good, 12 - slightly sharp, 13 - very sharp, 123 - extremely sharp

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I like this concise answer. See the edit in my question (I put it as an answer put it was moved back), the problem is that increasing the wavelength for lowering a note should be done in percent of the reference wavelength. Increasing tube length in valve instrument is done with constant lengths instead. –  Gauthier Dec 9 '11 at 11:08
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This is a good answer, except it's backwards. Depressing the keys lengthens the total tubing being played, instead of shortening it. –  naught101 Mar 10 '12 at 2:18
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@naught101: Yes, depressing valves lengthens the total tubing. Perhaps he means that the more valves you press, the shorter the total length will be compared to ideal length for the tone to play. –  awe Jul 30 at 11:56
    
Ah yeah, that makes sense, since the valves just add to the length, instead of multiplying it.. –  naught101 Jul 31 at 6:49

I play tuba, and this implies there as well. The case is that the low tones that uses 13 and 123 are not really ideal fingerings for those tones. The reason you feel like the lowest F# (123) is sounding right, is that it is easier to adjust the tone with your lips in the lower register, so it is not the fingering that is correct, it is your adjustment with the lips that makes it right... So if you pull the 3rd valve, you will get the tones on fingerings 13 and 123 right, but the tones using 23 will be wrong (Eb and G#).

In my early days, I played a tuba with only 3 valves as you have on the trumpet where I ran into these issues. The compromise was to have a pull on the 3rd valve, but this is really not ideal.

Now I play a tuba with 4 valves. The 4th valve makes me able to get better match for these problem tones, as I can use fingering 4 instead of 13, and 24 instead of 123. This makes the tones sound just right, and I have also right tuning for Eb and G#, as I don't need the pull on 3rd valve.


Edit:
The problem with four valves arise when I go lower down the register when I use more than one valve in addition to the fourth. This register is not even possible to reach with only three valves, but the intonation problems are actually increased, as it is another one more valve in the combinations. The worst combination is the low C# that actually use all 4 valves.

Some 4-valve instruments come with compensating system to solve this, where the 4th valve tubing routes back through the first 3 valves so that when the 4th valve is used in combination with any other(s), air can automatically be detoured through extra compensating loops.

For more details on this see this brilliant article on The Four-Valve Compensating System which also describes the issue with using more than one valve in detail.

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I used to play 4-valve tuba and 3-valve sousaphone, and always missed having the fourth valve (the the tuning slides were all stuck - school instrument :)). BTW, there are also four-valve trumpets, though rare. –  Michael May 20 '11 at 12:59
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Trumpet players pull the third valve "live" on D and C#. You don't need to tune your valves for an optimal tuning on all tones, if you can move one of them on the fly (trombone players used to make fun of me for moving my ridiculously little slide on some tones :) ). –  Gauthier May 22 '11 at 10:40

You are quite correct that a trumpet is out of tune when more than one valve is depressed. This is unavoidable, as the mathematics just don't add up. For instance, valve 1 is the correct length to lower the open note by two semitones; valve 3 is the correct length to lower the open note by three semitones. But if valve 1 is already down, then adding valve 3 requires lowering the existing note (not the open note) by 3 semitones - it is just a little short.

On smaller instruments (e.g. the trumpet), it is expected that the player can "lip" these notes into tune. On larger instruments, the problem is severe - lipping is just not possible due to the natural resonance of such a long column of air.

I think that sometime in the late 1800s, manufacturers of large instruments perfected the system of "compensating" tubing. If say valves 1 & 3 were pressed, then a small extra length of tubing was added to make the note exactly 5 semitones flatter than the open note. This invention led to the explosion of brass bands, and the introduction of the tuba as a chromatic orchestral instrument. Better class instruments also sprouted a fourth valve, that could be independently tuned (equal to 1st + 3rd valve plus compensation). It also led to the brass band E flat and B flat bass being nicknamed "plumber's nightmare"...

Unfortunately these innovations came too late for Beethoven - imagine what he could have done with a modern orchestral brass section!

I can't find an authoratative reference, so please treat this answer with a grain of salt...

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There are a few different factors at work here, and they hold true for all brass instruments:

  • All of the different partials you can buzz for any one fingering combination fall into the harmonic series, which behaves according to the laws of physics. Equal temperament has very little to do with the laws of physics, and more to do with practicality. As a result, different partials are going to require different adjustments in order to sound in tune with the rest of contemporary western tonality.

  • Lower and lower pitches require more and more tubing. Same physics concept as looking at a guitar fretboard.

  • Different instruments are going to have different constructions and tuning tendencies. All valves are tuning compromises, and there do exist instruments constructed so that low C# and D play in tune without the aid of a slide adjustment. At the "very expensive trumpet" level, you will find manufacturers who adjust the tunings of individual notes by playing with tube diameter at key vibration nodes throughout the instrument's bore.

As others have mentioned, playing trombone at a high level will give you an idea of just how much extramechanical adjustment valve players must do with their embouchure. The 7th partial, for example, is used by trombone players with very sharp slide positions, but valve players don't even touch it (they just skip to the 8th).

So the end-all of it is that each partial is going to have different tuning tendencies. It's become traditional in the trumpet world to compromise low D and C# to help the tuning of the rest of the instrument. So, it's not that lower partials need more adjustment (lower notes do need more tubing, but that is inherent in the length of each valve combination), it's just that that partial in particular and the one below it are both a bit wonky. There's a reason trumpet rarely plays at the bottom of its range: the resonance is not the same and the tubing diameter ends up constricting both the tone and pitch. Less consistency between instruments in the bottom two notes is the result—I've played instruments that both require and don't require adjustment on those notes.

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I'm not really a brass player, though I used to be a band director and have some idea about brass acoustics.

The issue with multiple-valve combinations being too sharp is that the valve system is a compromise. Pretend for a moment that you have a straight trumpet with no valves (so, pretty much a bugle). Of course, you can play pitches in the harmonic series with it. If you want to lower the fundamental pitch by an octave, you need to double the length of the instrument. If you want to lower the fundamental pitch another octave, you need to double the length again.

With that in mind, say the trumpet is one foot long. If you add another foot, you lower the fundamental pitch one octave. To lower the fundamental pitch another octave, though, you now have to add two feet, not one. So, the lower you go in pitch, the greater the length of tube required to make the same interval happen.

Now, go back to the valved trumpet. If you push down the second valve, you lower the fundamental pitch a half step (approximately, but I'm getting to that). If you push down the first valve (alone), you lower the fundamental pitch a whole step. If you hold down the first valve and push down the second, you extend the tube the same length you did when pushing down the second valve from "open" position -- but the actual "ideal" amount the tube's length should increase is farther from the second to third semitone below the fundamental than it would be for the first semitone below the fundamental. Therefore, with no adjustments of any kind, the combination of valves one and two would be sharp. The problem is compounded when you go even lower.

Now, I said "without adjustments." I don't have a reference source for this offhand, but on a standard three-piston-valve trumpet, I believe the first and second valves are designed to be five cents flat, while the third valve is designed to be twenty-three cents flat. This design partially compensates for the effect I have described without making the notes that use fewer valves too flat to be usable. This is the compromise of which I wrote at the beginning of this answer. If you play (without adjusting your embouchure) a note with the first and second valves down, then play the "same" note with just the third valve, you will notice that the note with just the third valve is lower than the note with the first two. The fourth valve on a four-valve instrument works similarly to replace valves one and three, though I do not know the number of cents offhand.

So, combinations of valves two and three are reasonably in tune, but one and three is not sufficiently lower than two and three to make the semitone descent. Therefore, one and three is a sharp combination, which is why you pull out the third-valve tuning slide. Pushing down all valves is even worse, so you may need to pull out the first slide as well (or not, depending if you need to).

Complicating the matter further is that the curvature of the trumpet takes the harmonic series out of tune a little bit. On certain trumpets, this effect is greater than others, but notably, the fifth harmonic (written E, generally) can be quite flat. Sometimes, then, using one-and-two is recommended so that you are basing the pitch off the sixth harmonic (G) instead.

One last thought: if you want a visual of the increase in distance as notes descend, find a trombonist. The seven "traditional" positions correspond to the combinations of three valves that you are used to, but positions six and seven are much farther apart than positions one and two. Or, go look at a guitar's fretboard, since it's the same idea I am describing: each fret is a semitone, and the frets are farther apart where the effective string length is greater.

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Thank you for the long answer. I do understand that due to the logarithmic nature of intervals vs. wavelength, the sum of tubing for one tone down and for one half tone down won't be the correct sum of extra tubing for 1.5 tone down. That is what I meant with: if F# and F are in tune, the sum of the extra tubing won't make E exactly in tune. Nice that you could quantify the length compromise in cents! And a very interesting comment on the harmonic series being out of tune. This question arised from a bass trombone slide position chart, actually. –  Gauthier May 22 '11 at 11:02

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