28 October 2008
Lavazza
I was too harsh on Lavazza earlier. I still like the Illy medium roast a little better, but the Lavazza Caffe Espresso at our neighborhood store is nearly as good, and a bargain by comparison. Last weekend, we were shopping away from home. That grocery store didn't have Lavazza or Illy so I thought I would try another brand, which shall remain nameless. The roast is so dark I just find it unenjoyable. I think I will get a couple cans of the Lavazza and mix it in with the other stuff until it's gone.
27 October 2008
Perfect Fifths
There are some pretty weird conversations in our house sometimes. A topic that has long interested both my wife (a piano technician) and me (amateur musician and former engineering student) is: Why can’t you have true perfect fifths in an equal temperament?
To answer that question, you also have to address a related topic: Why are there twelve semi-tones per octave? Most people never wonder about this, but if you stop to think about it, having twelve notes per octave is entirely arbitrary. There could as easily be three notes in an octave, or 42. There are instruments and compositions for other divisions of the octave. Go listen to Blackwood’s microtonal compositions sometime. While they are fascinating, you can quickly get an intuitive understanding of our preference for twelve notes per octave. Most other divisions just don’t sound very good, at least to our western ears.
The harmonic series is a way of describing any tone in terms of a fundamental pitch and its overtones. The frequency of each overtone is some whole integer multiple of the frequency of the fundamental. Say you play a note on the piano whose fundamental tone is 100 Hz. That means the string is vibrating back and forth 100 times per second. The overtones are then at 200 Hz, 300 Hz, 400 Hz, etc. The string is also vibrating at these frequencies, albeit less intensely than at the fundamental frequency.
There’s good reason to believe that this harmonic series has had an important influence on the evolution of human hearing and on our subjective experience of sounds. Further, it has had a huge influence on our subjective experience of music and harmony – of what sounds go well together.
The first overtone above the fundamental, having twice the frequency, is an octave above the fundamental pitch. Think of a guitar string. Play an open string, then put a finger on the twelfth fret and play it again. You’ve made the string half as long, so you’ve doubled the frequency. And you played a note that’s an octave higher. "Double the frequency" equals "octave." The special relationship of two notes an octave apart is the most basic, most important interval in music.
What musical interval comes next in importance? Well, what’s the next overtone in the harmonic series? It has a frequency three times that of the fundamental pitch. Musicians will recognize it as a "perfect fifth" above the previous overtone. Go back to your guitar. Play the open string again, then put a finger on the seventh fret and play it again. The string is two-thirds as long now, so the frequency increases by three-over-two. That’s a perfect fifth.
So even if we don’t go any further up the harmonic series, it’s safe to say that we would subjectively like a musical scale that incorporates these two most basic intervals, the octave and the perfect fifth. What does this tell us about how to chop up an octave? One approach we could take is this – if we start at a given note and go up a number of perfect fifths, it would be nice if we returned to some octave above our starting note. (Musicians know this as the "circle of fifths".) Can we do that?
It turns out that if you start at some note, and go up a perfect fifth twelve times, you reach a note that is really close to seven octaves above the starting note. Really close, but not exactly! How close? Let’s start with our 100 Hz tone to make the math a little easier. A note seven octaves higher would be:
100 Hz x (2 ^ 7) = 12,800 Hz
And twelve perfect fifths higher would be:
100 Hz x (1.5 ^ 12) = 12,975 Hz
A difference of 175 Hz may seem like a lot, considering we started at 100 Hz, and 200 Hz would be a whole octave higher. But the higher you go, the bigger the intervals get, so it’s really not that bad. The relative difference is:
(12,975 – 12,800) / 12,800 = 0.0137
It’s roughly 1.4 %, over seven octaves. That’s pretty close, but it’s enough to drive piano tuners crazy (and keep them employed). It means that every perfect fifth they tune isn’t quite perfect, but a tiny bit smaller instead.
Just a little more math – A perfect fifth implies a factor of three, and an octave implies a factor of two. For some number of perfect fifths to line up exactly with some number of octaves, you would need some power of three to equal some power of two. That just won’t happen. No number that factors into all threes could ever equal some other number that factors into twos. It’s just a happy coincidence for musicians that they come as close as they do.
So if we chop an octave into twelve equal pieces, we can have a scale with perfect octaves and nearly, but not quite, perfect fifths. And it is because twelve perfect fifths happen to be so close to seven octaves that we generally agree on twelve notes per octave, in case you ever wondered.
I’m certain that this has been explained by others, and much better than I have done here, but I wanted to give it a shot myself.
To answer that question, you also have to address a related topic: Why are there twelve semi-tones per octave? Most people never wonder about this, but if you stop to think about it, having twelve notes per octave is entirely arbitrary. There could as easily be three notes in an octave, or 42. There are instruments and compositions for other divisions of the octave. Go listen to Blackwood’s microtonal compositions sometime. While they are fascinating, you can quickly get an intuitive understanding of our preference for twelve notes per octave. Most other divisions just don’t sound very good, at least to our western ears.
The harmonic series is a way of describing any tone in terms of a fundamental pitch and its overtones. The frequency of each overtone is some whole integer multiple of the frequency of the fundamental. Say you play a note on the piano whose fundamental tone is 100 Hz. That means the string is vibrating back and forth 100 times per second. The overtones are then at 200 Hz, 300 Hz, 400 Hz, etc. The string is also vibrating at these frequencies, albeit less intensely than at the fundamental frequency.
There’s good reason to believe that this harmonic series has had an important influence on the evolution of human hearing and on our subjective experience of sounds. Further, it has had a huge influence on our subjective experience of music and harmony – of what sounds go well together.
The first overtone above the fundamental, having twice the frequency, is an octave above the fundamental pitch. Think of a guitar string. Play an open string, then put a finger on the twelfth fret and play it again. You’ve made the string half as long, so you’ve doubled the frequency. And you played a note that’s an octave higher. "Double the frequency" equals "octave." The special relationship of two notes an octave apart is the most basic, most important interval in music.
What musical interval comes next in importance? Well, what’s the next overtone in the harmonic series? It has a frequency three times that of the fundamental pitch. Musicians will recognize it as a "perfect fifth" above the previous overtone. Go back to your guitar. Play the open string again, then put a finger on the seventh fret and play it again. The string is two-thirds as long now, so the frequency increases by three-over-two. That’s a perfect fifth.
So even if we don’t go any further up the harmonic series, it’s safe to say that we would subjectively like a musical scale that incorporates these two most basic intervals, the octave and the perfect fifth. What does this tell us about how to chop up an octave? One approach we could take is this – if we start at a given note and go up a number of perfect fifths, it would be nice if we returned to some octave above our starting note. (Musicians know this as the "circle of fifths".) Can we do that?
It turns out that if you start at some note, and go up a perfect fifth twelve times, you reach a note that is really close to seven octaves above the starting note. Really close, but not exactly! How close? Let’s start with our 100 Hz tone to make the math a little easier. A note seven octaves higher would be:
100 Hz x (2 ^ 7) = 12,800 Hz
And twelve perfect fifths higher would be:
100 Hz x (1.5 ^ 12) = 12,975 Hz
A difference of 175 Hz may seem like a lot, considering we started at 100 Hz, and 200 Hz would be a whole octave higher. But the higher you go, the bigger the intervals get, so it’s really not that bad. The relative difference is:
(12,975 – 12,800) / 12,800 = 0.0137
It’s roughly 1.4 %, over seven octaves. That’s pretty close, but it’s enough to drive piano tuners crazy (and keep them employed). It means that every perfect fifth they tune isn’t quite perfect, but a tiny bit smaller instead.
Just a little more math – A perfect fifth implies a factor of three, and an octave implies a factor of two. For some number of perfect fifths to line up exactly with some number of octaves, you would need some power of three to equal some power of two. That just won’t happen. No number that factors into all threes could ever equal some other number that factors into twos. It’s just a happy coincidence for musicians that they come as close as they do.
So if we chop an octave into twelve equal pieces, we can have a scale with perfect octaves and nearly, but not quite, perfect fifths. And it is because twelve perfect fifths happen to be so close to seven octaves that we generally agree on twelve notes per octave, in case you ever wondered.
I’m certain that this has been explained by others, and much better than I have done here, but I wanted to give it a shot myself.
22 October 2008
Electronic Voting
Security researcher Ed Felten spoke at the University of Illinois at Chicago last week. I took an early lunch break and went to listen. Felten described research that he and his students have performed on electronic voting machines. He is really a proponent of electronic voting, but only if it’s done openly and properly. He described vulnerabilities in existing machines, steps to improve the machines, and new and efficient ways to audit the outcome of elections to ensure that the results reflect the intent of the voters. It’s very important work, and there’s much more to be done.
A Very Expensive Haircut
It’s very difficult to believe, especially for my daughter, but I once had long hair. It was long enough I could almost pull it into a pony tail. In the front it reached the tip of my nose. This was during my last year at the university, almost 20 years ago. The long hair symbolized nothing. It was the product of frugality and laziness, mostly laziness.
My mom hated it.
That year for Christmas, my parents gave me a very special present. They arranged a trip for me to visit our family in Guatemala . My aunt married a man from Guatemala , and his family has always treated our family as theirs, and vice versa. I was to go in early January for over a week, and spend time with my Tia Carmen and my cousins in the capital city and in various towns around the country.
One catch: the long hair had to go. So I got the biggest haircut of my life, and - Hasta La Vista! An amazing trip, worthy of several posts sometime.
Did my mom arrange the trip just so I would get my hair cut? Perhaps not, but it’s fun to speculate.
Ira Sullivan
The week of the Chicago Jazz festival, Maria and I went to the new Jazz Showcase (hooray!) to see Ira Sullivan. The new Showcase location in Dearborn Station is very nice, and better suited to its purpose than the previous location. The acoustics are good, there are no pillars obstructing the view, and there are even some easy chairs and a sofa along the side. It was a weeknight so the audience was sparse. We could only stay for the first set.
Sullivan played mostly alto but switched to trumpet and bass flute as the set got going. There was an older gentleman on piano, who played one song on vibes, and a couple “kids” on bass and drums. They were all terrific. Apparently, the bass player (wish I got his name) had rehearsed little if at all with the group, so things were a little loose, but he really came through. It was an enriching musical experience.
Sullivan played mostly alto but switched to trumpet and bass flute as the set got going. There was an older gentleman on piano, who played one song on vibes, and a couple “kids” on bass and drums. They were all terrific. Apparently, the bass player (wish I got his name) had rehearsed little if at all with the group, so things were a little loose, but he really came through. It was an enriching musical experience.
Tony Bennett
Late this summer, Maria and I finally went to see Tony Bennett perform at Ravinia. Bennett has appeared at Ravinia for many years, and we’ve been meaning to go for the longest time. We had seats in the pavilion, towards the back. He had a small group of excellent musicians – guitar, piano, bass and drums.
We were disappointed right away by the sound. It sounded like we were all listening to the stage monitors. I don’t know if I remembered the sound at the Ravinia pavilion as being better than it is, or if we’ve been spoiled by the wonderful sound at the Pritzker Pavilion at Millennium Park.
Bennett’s singing was fantastic. The arrangements were “crowd-pleasing” – each song was short and each ended with a magnificent crescendo. But as a result, each song sounded very much like the previous one. It was a very good show in any case, tailored to the masses, and acoustics notwithstanding.
We were disappointed right away by the sound. It sounded like we were all listening to the stage monitors. I don’t know if I remembered the sound at the Ravinia pavilion as being better than it is, or if we’ve been spoiled by the wonderful sound at the Pritzker Pavilion at Millennium Park.
Bennett’s singing was fantastic. The arrangements were “crowd-pleasing” – each song was short and each ended with a magnificent crescendo. But as a result, each song sounded very much like the previous one. It was a very good show in any case, tailored to the masses, and acoustics notwithstanding.
Giving Directions
A couple weeks ago, I was walking home after work in our neighborhood. Crossing a busy intersection, I was summoned by a distraught looking woman driver waiting for the stoplight. “Is this River Forest?” she asked. I heard her fine, but not believing someone would mistake the northwest side of Chicago for River Forest , I had her repeat the question. “Not even close,” I said. She was visibly upset. I asked where she wanted to go. “Dominican University.”
Given that the light was about to change, I did a pretty good job, I think, of giving her directions quickly. I kept the turns to a minimum, and steered her clear of the west side of Chicago. I even sent her off with some words of encouragement. But since then, I thought of a few things I could have done better. In the interest of self improvement I will list those here:
- Keep the sarcasm to yourself. “Not even close” was really uncalled for. “Sorry, no,” would have been just fine.
- Estimate the distance between turns. It would be three miles to her next turn, and four miles to the one after that. She obviously would not have known that. I could have spared her a lot of second guessing along the way.
- Better yet, draw a map! There’s a gas station at that corner where she could have pulled over. A map would have made a lot of things much clearer.
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