Sandy's Notes

Sandy, the blind pianist girl who was kidnapped in "See No Evil", listed a lot of frequency/note pairings, and the NCIS team used the information she gave to find her mother.

I will start with a description of the notes Sandy mentioned. If you need a primer on the physics of music, what an "equal tempered scale" is, etc, see the section below on "Physics of Music"

The first few notes Sandy quotes prove she's using the equal-tempered scale: the 44th note is E4 at 329.63 Hz. A5 is at 880 Hz. She said birds were chirping at a G7, the 83rd key, at 3136 Hz. (I won't count off for the .04 Hz: it's mathematically at 3135.96; it's just odd, to me, to give E4 to two places but to round off G7.)

The chirping birds, however, did start my nitpicking: the chirping we heard was a varying pitch (frequency), and had too much fullness (ie, too many other frequencies) to be heard when applying a narrow frequency filter. Eventually, I'll try to strip out the soundtrack, and do an FFT on the chirping, just for my own curiosity's sake.

Next came the train sound: "in the area of A7, ... 4000 Hz". Unfortunately, on the equal-tempered scale, A7 is only 3520 Hz. (4000 is between B7 and C8). Relative to 4000, 3800 is actually about one step down, not "two keys down" like she said. The writer started off right on, but just kept getting worse and worse.

When I started this investigation, I'd misremembered that she said "C7", and thought she was using an A-based octave number, in which case that C7 would be at 4186 Hz, compared to A7 of 3520. The frequencies quoted weren't quite right -- but, assuming starting at that C, and going down two notes, it would be A#/Bb, at 3729 Hz. Interestingly, assuming a harmonically tuned frequency scale based on C, but setting the frequency of that octave's A to 3520, A#/Bb would be 3801, almost exactly where she told them to look. Unfortunately, the path to get there was quite convoluted, and began with something that I remembered incorrectly, so that makes this whole paragraph an unimportant digression, other than my subconscious, which turned her A into a C, apparently really wanted to give the writer's an "out". Too bad I was wrong. :-)

Returning to the actual train sound: I have the same complaint I did for the chirping: the narrow frequency search wouldn't have revealed a full-sounding train whistle, like we heard. Further, the note the train was whistling was nothing like the one Sandy quoted: I've put a pure A#7/Bb7 at 3729Hz into this mp3--WARNING: it's loud and high, so turn down your speakers. A friend suggested Sandy actually heard the squealing of breaks, or other metal-on-metal screeching in the train. Now that I've made this sound file, I tend to agree. Abby then worked magic to isolate just the train whistle based on Sandy noticing the screech.

Physics of Music

Here's a little physics of music theory (and an "I'm sorry" to anyone who wanted to keep music as far away from physics and math as possible: they're intimately related). Every note has a frequency (vibrations per second, in Hertz [Hz]) associated with it. Frequency and wavelength (the physical length of the vibration) go hand-in-hand: the higher the frequency, the shorter the wavelength.

Most musical instruments are either basically air tubes with some way to make the air vibrate, or strings which are caused to vibrate. The length of the tube or string determines what notes can be played. Only the fundamental note (the simplest sound wave that fits in the tube or on the string) and harmonics (integer multiples of the fundamental) can be played on any given tube or string. This harmonic tuning gave rise to what is known as the "Just scale" [1]. By changing the tube or string length (finger holes in woodwinds, valves in many brass instruments, and frets or alternate strings in stringed instruments), you change the fundamental and available harmonics are available, thus giving a wider variety of notes that can be played.

Harmonic tuning is great, until you try to transpose (start on a different note). The frequency relationship between notes aren't quite the same when you shift the starting point, even though there are the same number of note steps between. The frequency is only off by a few hertz, but it's enough for the human ear to recognize, so the music sounds a little "off".

Enter "equal-tempered" tuning: By spacing all the notes of an octave equally across the frequency range, the relationship between any two adjacent notes is always the same, so playing a pattern of notes starting on a C will sound the same, just higher, if you start on an F. Octaves mathematically double the frequency, so each C is twice the frequency of the C below it. Because it's a factor-of-two difference, the "even spacing" is in the power of 2, not in the number of Hertz different. This means that going up one half-step (the change between a "white key" on a piano and the "black key" right next to it, or the change between two "white keys" when there are no "black keys" between) is mathematically multiplying the frequency by 2^1/12.

For whatever reason, the equal-tempered scale was set with A at exactly 440 Hz, and the frequencies for the other notes set accordingly. A number is often subscripted next to the note-letter to tell which octave it is in, though where the "octave" officially changes depends on whom you ask. Usually, the subscript changes at either the A (because it's the integer frequency) or at C (because that's where a lot of people think of the traditional scale/octave beginning). In this episode, Sandy uses the C-based subscripts (which is also the notation used in the Equal-tempered frequency chart mentioned in note 1).

Note 1: I knew the basic concepts behind the above difference, but had forgotten a lot of the details. A web-search took me to the Michigan Tech Physics of Music site, primarily the "Scales: Just vs. Equal Temperment" page. They've also got a page with the frequencies on the Equal-tempered scale. Another good site to look at is at http://tyala.freeyellow.com/4scales.htm.