Jeff Kaufman  ::  Blog Posts  ::  RSS Feed  ::  Contact

Trumpets Belong in C#

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:
KeyError
C#0.20%
Eb0.26%
Bb0.31%
F0.36%
Ab0.37%
C0.39%
F#0.47%
E0.49%
D0.51%
A0.51%
G0.53%
B0.58%
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:
KeyError
C#0.05%
Eb0.05%
E0.05%
Bb0.12%
Ab0.25%
F0.26%
A0.29%
C0.35%
F#0.45%
G0.49%
D0.50%
B0.58%
What crazy fingerings did it come up with for C#?
NoteNormal FingeringOptimal Fingering
F32-13(sharp 0.07%)2-13(flat 0.04%)
F#32-23(flat 0.09%)2-23(flat 0.04%)
Ab32-1(sharp 0.06%)2-1(flat 0.05%)
Bb32-0(flat 0.04%)2-0(sharp 0.01%)
C43-13(sharp 0.19%)3-13(sharp 0.07%)
C#43-23(sharp 0.02%)3-23(sharp 0.07%)
Eb43-1(sharp 0.17%)3-1(sharp 0.06%)
F43-0(sharp 0.07%)4-13(flat 0.04%)
F#44-23(flat 0.09%)4-23(flat 0.04%)
Ab44-1(sharp 0.06%)4-1(flat 0.05%)
Bb44-0(flat 0.04%)4-0(sharp 0.01%)
C55-1(flat 0.73%)6-13(sharp 0.07%)
C#55-2(sharp 0.20%)6-23(sharp 0.07%)
Eb56-1(sharp 0.17%)6-1(sharp 0.06%)
F56-0(sharp 0.07%)8-13(flat 0.04%)
F#58-23(flat 0.09%)8-23(flat 0.04%)
Ab58-1(sharp 0.06%)8-1(flat 0.05%)
Bb58-0(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.

full post...

Low Trumpets

I picked up the trumpet a few years ago and while I've been making slow and steady progress expanding my range and stamina, it's still a very hard instrument for me physically. About a year later I tried my cousin's old 3/4-size baritone horn, and was blown away by how much easier it was to play. That, combined with really liking how various people used the trombone in contra dance, got me wanting something in this range. A standard baritone has the bell pointed up, however, which didn't seem like a good fit for unamplified outdoor playing, or unamplified playing in general. When I saw a picture of a marching baritone, a bell-forward design, however, I thought "let's get that!". So, one King 1124 later:

Very little research went into this. I was excitied about the idea, I'd played a little on something somewhat similar, and I found one one ebay. It arrived in mid April and a month later it was on the new Free Raisins CD. I'm pretty happy with how it's been working out, but I hear people talking about euphoniums, bass trumpets, flugabones, and I wonder: what are all these? more...

Better Apartment Price Map

The initial model for my apartment price map was good enough for getting a sense of what different areas cost, but it was not an ideal fit for the underlying distribution. Specifically it assumed a linear relationship between number of bedrooms and cost (pretty reasonable) going through the origin (not reasonable). The problem is, a 2br doesn't cost twice as much as a 1br:

Yes, it's approximately linear, but the line doesn't go anywhere near the origin. Another way of saying this is that there's a price-preimum for having your own unit. This is the cost of a kitchen, bathroom, etc. I originally tried to handle this by eyeballing it and saying that we could count apartments as having a number of "rooms" equal to one more than the number of bedrooms, but this isn't right either. If we do a simple linear regression we see an intercept of $1,526 and a slope of $481. The intercept is the cost of a studio or "0br" and the slope is how much each additional bedroom costs $481, on average, over the whole Boston area. more...

Glasses Angles

I see people wearing glasses in a variety of positions:

The closer you wear your glasses to your eye, however, the more of your field of view they cover: more...

Why Reduce Swelling?

Recently I jammed my fingers, and aside from resting them the standard healing advice is to reduce the swelling with ice, compression, elevation, and ibuprofen (advil/motrin). This is surprising, especially the medication. Why would we evolve swelling, a response to a common injury, if it's harmful? I could believe that in some cases you could have excessive swelling and it would be good to reduce that, but in the typical case you would expect swelling to be useful.

A similar argument applies to fevers: the body raised its temperature, so why does it make sense to come in with acetaminophen (tylenol) and bring it back down? Again, if a fever gets high enough we do need to control it, but we should expect letting the fever run its course to be the healthiest option in most cases.

What is it that these systemic medications do that our bodies weren't able to evolve on their own? Or is the modern environment different enough from the one in which we evolved that a response like fever or swelling once was useful but no longer is? Or maybe it's not useful after all; has anyone done an RCT looking at the impact of swelling reduction actions or medication on healing? Can I get some volunteers willing to self-administer hammer blows? [1]

(This is basically an algernon argument, a use of the evolutionary optimality challenge.)


[1] People would first need to administer the hammer, then check to see which treatment to apply. The trauma must be applied blind.

full post...

Weight Loss: How Practical?

There seems to be a basic factual disagreement about the feasibility of weight loss via diet and exercise changes. If someone wants to lose weight, using whatever is the best method, how likely are they to succeed? Friends in the medical community say it's totally practical and success is common. Friends online, several of them into HAES, say people tend to lose only a small amount of weight and then regain most of it over the next few years. If you think losing substantial weight is within the ability and motivation of the typical obese person, and that risk of various diseases (primarily cardiovascular) decrease if you lose weight [1], then it makes sense for medical providers to be trying to get people to lose weight. On the other hand, if you think it's not practical, then medical providers should just focus getting people to be healthier, without trying to get them to lose weight.

Because people tend to regain weight they lose, a study needs to have substantial followup. If you just check back in six months you might find people have lost a reasonable amount, but checking back a few years later they're more likely to be back where they started. So we want at least two years, ideally something like five. Unfortunately it's hard to keep track of people over long periods. They move away, get frustrated, or otherwise drop out of the study. But if the people who drop out of a treatment group are different from the people dropping out of the control group then this could make for misleading results. Some studies report things like only having the weight available for 62% of participants 9 months along: this is clearly not going to work.

The best study I can find seems to be the 2008 Weight Loss Maintenance Randomized Controlled Trial (pdf). They started with 1685 people averaging 213lbs, and brought them through six months of group weight loss treatment. Of these, 25% failed to lose at least 9lbs, required for continuing with the study, and another 14% were excluded for other reasons, leaving 1032 people for the second phase of the study. (Yes, the first phase wasn't randomized, which means we can't take this 25% number too seriously, but since people tend not to just lose weight on their own it's not too bad.) In the second phase they were randomly assigned to three groups for weight loss maintenance: in-person followup, automated internet followup, and a control group ("self directed maintenance"). Then they tracked weight regain for the three groups:

more...

More Posts


Jeff Kaufman  ::  Blog Posts  ::  RSS Feed  ::  Contact  ::  G+ Profile