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Geoffrey Romer
Works at Google
Attended Harvey Mudd College
Lives in Seattle, WA
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Geoffrey Romer

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I wonder how many more "look at what Juniper ate" stories we're going to accumulate.
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Geoffrey Romer

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I'm not sure if I'd consider that third one "almost right"
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Kael asked what color Uranus was, and I told him I thought it was a sort of blue-green.

I should have chosen my words more carefully.
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Everybody in America should read this article.
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I just discovered I used the wrong kind of wire for all the yellows.
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No problem.  Be aware, too, that there's nothing that says that you have to have plated through-holes.  The cheapest of the cheap 2-layer boards simply etch the copper traces on each side, then drill holes through the middle of the round trace ends.  That kind of board has an insulating (fiberglass) middle, and without the plating there is no conductivity from one side of the board to the other.  To achieve the intended connectivity, you would have to solder both sides.  Like you said, with stranded wire, sometimes you can rely on  gravity from the top joint to feed the bottom joint at the same time, but it's better to solder both sides manually when the through-hole is unplated.

But like I said, you don't have to worry about that with this board.
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Geoffrey Romer

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"Oh, sure," they told me in Zurich, "you'll have no problem making that one-hour connection. They're in the same terminal." What they omitted to mention was that Heathrow Terminal 5 is split into three buildings about half a mile apart, and you have to clear the world's slowest security checkpoint just to change gates. Plus, the first leg was delayed 20 minutes.

On the other hand, the gate agents complimented me on how quickly I made it, when I staggered up to the gate. Inefficient, but polite; how very British.

I feel sorry for the woman who's about to spend 8 hours sitting next to my sweaty self...
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Good for you. I missed that same connection last time. Ended up driving home from Vancouver at midnight.
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Geoffrey Romer

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"Daddy, can I play with markers? I won't eat them."
"Hang on, Juniper."
"Can I please play with markers? I won't eat them!"
"How about crayons? Do you want to color with crayons?"
"No, Daddy! I will eat them!"

She had a point. I gave her markers.
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Everybody should read this article too. Don't be misled by the mundane headline; the article is urgent and electrifying.

(I swear I won't keep playing the everybody-read-this card)

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"And even if we didn't it's not hard for the government to manufacture an infraction.  They just need to ask you a question they already know they answer to, if you give an incorrect answer."

Hence the 5th amendment right to not answer the questions and the 6th amendment right to have counsel present.
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(Previous post in this series: https://plus.google.com/105863569125529379717/posts/cgbHQe5h5rW)

OK, with the preliminaries out of the way, let’s talk about actual music. Specifically, musical instruments. 

The sound of a musical instrument is traditionally divided into four phases. The attack is the period from the initial silence to the point of maximum amplitude, and is usually very fast. The decay is basically just the transition from the attack to the following phase, the sustain. During the sustain, which can last for many seconds, the sound remains basically the same except for perhaps a gradual decrease in amplitude. The sustain can gradually fade into silence, or the musician can intervene to cut it off, leading to the release, another quick phase where the sound of the sustain is damped back into silence.

The attack and decay are typically the most sonically complex phases, and the most sensitive to various details of how the instrument is constructed and played. In the decay, most of that complexity is filtered out, so that the sustain consists of a relatively simple set of partials (see previous post), giving it a “purer” sound. The long, controllable duration of the sustain, as well as its musical “purity”, makes it central to understanding how different musical sounds relate to each other, i.e. to understanding musical systems. The frequencies of those partials don’t depend much on the idiosyncratic details of the instrument; they mostly depend on the shape of the vibrating object that drives the sound.

Musical instruments are classified according to that shape using something called the Hornbostel-Sachs system:
Chordophones produce sound from vibrating strings, held under tension at both ends. Examples include the piano, guitar, and violin.
Aerophones produce sound from vibrating air, typically confined in a column. These can be further classified according to whether the column is open at one end (e.g. clarinet) or both (e.g. flute), and whether the column is cylindrical (e.g. flute), conical (e.g. oboe), or flared (e.g. most brass instruments).
Membranophones produce sound from vibrating membranes, held under tension around their perimeter. In other words, drums (and kazoos, but they don’t really count).
Idiophones produce sound from the vibration of a rigid body. Examples include xylophones, bells, gongs, and cymbals.
Electrophones produce sound from an electrical signal, typically via a speaker (an electromagnet driving a metal diaphragm). I’m going to mostly ignore these, since they’re too recent to be relevant to the music systems you and I are familiar with, and as a group they have no distinctive musical properties other than their virtually limitless flexibility.

The chordophones are the simplest, physically and mathematically. Waves in a string, much like waves in the air, can be treated as the sum of a bunch of sine waves with different wavelengths,  which propagate up and down the string at a speed that’s determined by the physical properties of the string. Since the speed is fixed, the frequency of the motion (the number of waves that pass a given point in a given time) is inversely related to the wavelength: if you cut the wavelength in half, you double the frequency. The frequency matters because when the string vibrates at a given frequency, it creates vibrations in the air at the same frequency, and that’s the sound we hear.

[All of this would be much clearer with pictures, or better yet animations, or better yet interactive widgets with animation and synchronized sound. But these posts are taking long enough as it is, and G+ won’t let me embed those things anyway]

When a wave reaches a fixed end of the string, it reflects off of it and travels back up the string, and when a moving sine wave combines with its own reflection, it produces a standing wave. The standing wave is a sine wave with the same wavelength and frequency, but it doesn’t appear to propagate down the string; instead, the peaks and troughs of the wave move up and down in place, (peaks becoming troughs and troughs becoming peaks) while the midpoints between the peaks and troughs remain fixed. Since it’s a sine wave, the waves are equally spaced, and so those fixed points (called nodes) are also equally spaced. The ends of the string have to be nodes, because they’re physically held in place, so the wave has to consist of an integer number of peak/trough regions (called antinodes) of equal length.

This means the standing waves can’t have just any wavelength: the wavelength has to be 1/n times the twice the length of the string, for some positive integer n (the factor of two comes from the fact that a complete cycle of a sine wave contains 2 nodes). Since frequency is inversely related to wavelength, this means the frequencies of all the standing waves in the string have to be some integer multiple of the fundamental frequency, which is the frequency of a standing wave with no nodes other than the ends of the string.

So a given vibrating string can’t produce sound at any frequency, but only at a fixed set of frequencies, i.e. partials. Remember I said last time that we tend to perceive sounds as musical if they have high amplitude at only a discrete set of partials? This is why plucked strings sound musical. When the partials of a sound are integer multiples of some fundamental frequency, the partials are called harmonics.

Interesting side note: +Kurt Dresner informs me that there’s a guitar technique called “playing the harmonics”, where you touch a string somewhere along its length after plucking it. The touch damps out all the harmonics except the ones that have nodes where you touched (because your finger didn’t block their motion). The higher the frequency of the harmonic, the more likely it is to have a node near your touch (because higher frequencies have more nodes overall, so this tends to preferentially damp out the lower frequencies, producing a higher pitch.

Although the physics of open-ended cylindrical aerophones like flutes are superficially totally different from chordophones, they turn out to be mathematically nearly identical. The waves are waves of pressure in air instead of displacement in a string, and they are transverse instead of longitudinal (parallel to the direction the wave moves, instead of perpendicular to it), but that makes little mathematical difference. The open ends of the cylinder act as fixed-pressure nodes, leading to exactly the same structure: a fundamental frequency with harmonics at integer multiples of the fundamental.

Things get a little more interesting as you move to other types of aerophones. If a cylindrical aerophone is closed at one end, like a clarinet, that end acts as a pressure antinode. This doubles the effective length of the instrument, since the physical length of the instrument is only half the distance between nodes of the fundamental. However, it blocks the even-numbered harmonics, because they would have a node at the closed end. Curiously, for a conical aerophone like an oboe, the narrow end acts as a node, as though it were open-ended, even though it’s technically closed. The mathematics of flared aerophones, such as most brass instruments, are considerably more complex.

I have a conjecture that the mathematical resemblance between chordophones and aerophones relates to the fact that they are both fundamentally one-dimensional instruments, in which waves propagate back and forth along a single linear axis.

Membranophones, of course, are two-dimensional, and their mathematical behavior is quite different. They can still be understood in terms of combinations of discrete standing waves, but the waves are two dimensional, and no longer simple sine curves. The shapes and frequencies of the standing waves depend on the shape of the membrane boundary, so there are almost as many different spectra as there are shapes of membranes. Whether there are exactly as many, i.e. whether you can always figure out the shape of a drum from its spectrum alone, turns out to be a neat mathematical problem (see http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum for more).

Idiophones are even more mathematically complex, because their spectrum depends on the three-dimensional shape of the instrument body. This complexity also gives the instrument designer great flexibility to alter the partial structure of the instrument. For example, xylophone bars are shaped to have the first two partials at 3 and 6 times the fundamental frequency, whereas marimbas and vibraphones are tuned to 4 and 10 times the fundamental. More complex tricks are possible, such as bells that produce different notes depending on where you strike them. The key thing to notice about idiophones and membranophones is that their partials are rarely if ever harmonics; they don’t automatically occur in fixed integer multiples as the partials of chordophones and aerophones do (Side note: I would be very interested to know if there’s a membrane shape whose partials are harmonic, and if so, what it is).

Next time, I’ll talk about consonance and dissonance, and how they relate a musical system to the partial structure of the musical instruments it’s used with.
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Obligatory link, if Kurt hasn't already pointed you at it -- Vic Wooten playing an "Amazing Grace" solo on the bass using only harmonics: Victor Wooten amazing grace
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I just gave an oscilloscopes 101 tutorial to a couple of 5-year-olds. I think it went pretty well, but I didn't realize until later that Kael had successfully conned me into violating the screen-time ban I imposed on him.
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Geoffrey Romer

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The sword is too much. You lost me there.
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For the first time in over two years, I am fully caught up with Doctor Who. 
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Education
  • Harvey Mudd College
    Computer Science, 1999 - 2003
  • UCSD
    Computer Science, 2003 - 2006
  • Lick Wilmerding High School
    1995 - 1999
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Software Engineer
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    Software Engineer, 2006 - present
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The Naan is wonderful, and generously portioned, and all the food is excellent. Physical space is cold and slightly shabby. Pet peeve: no changing table in men's restroom.
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