This is part of the Toronto visit series.

So I inadvertently took a picture of the running propeller on my side of the plane, and it came out weird. And I mean, really weird (1):

The propeller does not look like this. It has like six blades spaced out evenly and all of them straight. So of course this is due to aliasing of the damned camera. But wait now, I just said not too long ago that this is photography, so indeed, I only took this and

this (2)

and this (3)

along with this (4)

because I was strangely mesmerized. Not only is this pseudo-reality only observable through a cheap phone camera like this, but I’m measuring the camera as much as I’m measuring the propeller.

See, the effect is only possible because the phone camera has no shutter (so DSLR’s shouldn’t exhibit this). The image is literally read out line by line, top to bottom. By the looks of it, all pixels on one horizontal scanline are read out or replicated to memory in parallel, but different scanlines belong to different times, with a sampling time gap between lines. Meanwhile, the propeller moved on. Since the angle between attachment points of the blades is reduced in (1) and expanded in (4), the propeller is spinning clockwise.

However, what appears to be the blades in these photos, are not the blades! Otherwise how could some of them hang around in mid-air and also have two ends. Well, first note that the entire region where there are fin-like objects is circumscribed by the elliptical region that the real propeller traces out (from this viewpoint). This clue suggests that each discrete piece is a portion of an image of a single blade sweeping through a certain angular range. It’s only a portion, because the blade in question is captured only when it is within the current scanning region. Once it’s out, the camera doesn’t “see” it. There are gaps between these discrete pieces because there is an angular gap between two adjacent blades on the propeller.

If the rotational speed of the propeller is known then we can even figure out the rate at which image lines are scanned. As is, there are about a dozen or so discrete pieces in a single image, so given that there are only six blades, the propeller makes nearly two turns during the shot.

This is part of the Toronto visit series.

I sit in this small airport in the middle of Downtown Toronto, wondering why airports can’t all be like this. YTZ (Billy Bishop) is small, with a single terminal (if it can be called that). There is just one commercial airline, also a small company. And its planes are small, four-seat-across propeller aircrafts that I’ve sworn not to take again after one particularly unpleasant ride years ago, but am taking anyway. More on that later.

The inbound wasn’t all that impressive. You go through Canadian customs, then cross a small canal on what must be the world’s shortest ferry ride — I mean you wait for the ferry longer than the ride, which is like 20 seconds — to the parking lot which also belongs to the airport. The airline provides two shuttles, one to the York Hotel, one to Union Station. To go anywhere else you’re kind of on your own, but the nearest public trolley buses are a five-minute walk away — if you know where they are, because there is no information desk. This is the downside of small.

But downside is replaced by upside on the outbound. For such a small airport, US customs is not here, but back on the US side. This is great for saving time. YYZ (Pearson International) is not like this; it has both the Canadian and US customs checkpoints outside the security perimeter. Somehow this also translates into a friendlier security checkpoint at YTZ. You don’t have to take off your shoes unless they tell you to, you can put your laptop in the same bin as other stuff. Mostly they are not in a hurry because there is nobody in the airport this day — almost empty. Also note the following:

It makes air travel bearable again. A tradeoff exists with small planes, too. They don’t do too well in turbulence, but in exchange for this possibility, they are quiet and have large seats. Also they seem so… primitive, and their mechanism so… accessible, that I couldn’t help but watch the propellers go around.

1:2 (8)
2:3 (5)
3:4 (4), 3:5 (major 6)
4:5 (major 3), 4:7 (harmonic 7)
5:6 (minor 3), 5:7 (augmented 4), 5:8 (minor 6), 5:9 (minor 7)
6:7 (septimal minor 3), 6:11
7:8 (*augmented 2), 7:9 (*augmented 3), 7:10 (diminished 5), 7:11, 7:12, 7:13
8:9 (major 2), 8:11, 8:13, 8:15

The dyads with stars are tuning dependent, and any dyad with an integer larger than 10 basically do not have names since they don’t naturally occur in low order tunings.

Next, we classify consonant triads, by combining intervals

1:2:3 (P8+P5)
2:3:4 (P5+P4)
3:4:5 (P4+M3, major)
4:5:6 (M3+m3, major), 4:5:7 (M3+A4, partial 7), 4:5:8 (M3+m6), 4:6:7 (P5+sm3, partial 7)
5:6:7 (m3+sm3, diminished), 5:6:8 (m3+P4, partial major), 5:6:9 (m3+P5, partial dim7), 5:7:8 (A4+A2, partial 7), 5:7:9 (A4+A3, partial dim7), 5:8:9 (m6+M2, partial 9)
6:7:8 (sm3+A2, partial m7), 6:7:9 (sm3+A3, harmonic minor), 6:7:10 (sm3+d5, diminished), 6:8:9 (4+2, quartal triad)
7:8:9 (A2+M2, getting weird…), 7:8:10 (A2+M3, partial 7), 7:9:10 (A3+M2, partial m7)
8:9:10 (still stranger)
10:12:15 (m3+M3, “standard” minor)

Though there are countless more of these, we stop here. From the above, we begin to realize that most chords are just partial suggestions of the natural harmonic series. If we only take integers no larger than 10, we will get the harmonic series with the indicated intervals between harmonics:

1:2:3:4:5:6:7:8:9:10
(P8+P5+P4+M3+m3+sm3+A2+M2+M2)

The simplest chords just build from the base up. The more harmonics taken, the higher order the chord would appear to be. In practice, modifications are made by omitting (and implying) the lower harmonics, especially the zeroth, which is almost always omitted.

To generalize even further, omitting harmonics is a special case of applying weighting (or a spectral filter) to all the harmonics. This is perhaps what people aptly call “color” of the sound. Thus all chords and chord colors can actually be identified entirely by the weight vector on the harmonic series. (This may not be as trivial as it sounds, because it implies additional degrees of freedom in musical activity, by varying weights dynamically and continuously, for example, and generalizing discrete progression of chords.)

Depending on the psychoacoustic circuitry, these weight vectors somehow can be partitioned into equivalence classes, so there is more complication here. I haven’t thought this one through, but for instance, not only are 3:4:5 and 6:8:10 equivalent, but 3:4:5 and 4:5:6 are also somewhat equivalent as they are inversions of each other. The latter case is possibly due to psychoacoustic masking of 6 in the implied chord 3:4:5:6 in 4:5:6. Not sure.

In any case, choices not taken from the series below 10 will tend to be dissonant, because the higher harmonics, if they are not clearly resolvable in frequency psychoacoustically, will tend to combine as colored noise in the limit. In particular, the (non-harmonic) minor triad is exceptionally dissonant, but I suppose we have learned to accept it, if only for contrast.

The first paragraph of this old paper is intriguing. Don’t agree with the rest of the conclusion.

But I understand… playing ads too many times in error is of no concern to NBC, of course.

So what’s on NBC’s site? Most shows have only the latest episodes on a time delay (to not preempt live broadcast and DVD sales, I suppose), but a few have all past episodes available. One of these I’ve been watching is Friday Night Lights. Actually never saw any ads for this, but all of Season 1 was surprisingly good. Unfortunately, Season 2 was crap and total BS. Sadly, there is only so much good material for writers to crank out.