Tutor profile: Michael G.
Subject: Music Theory
Leonard Cohen's classic song Hallelujah is a great example of functional harmony in music. The song is set in C major which includes the notes c, d, e, f, g, a, and b with no sharps or flats. This gives us our simple diatonic triads, C Major, D minor, E minor, F Major, G Major, A minor, and B diminished. The most obvious function to spot is the dominant function chords whose job is to cadence to the tonic which helps establish the key center. The first chord progression shows this well as it goes C, Am, C, Am, F, G, C. The G chord resolves to C. However later on in the chorus we see this progression. C, F G, Am, F, G, E, Am, Am. The chord E major has the note G# in it which is not part of our key. What is the purpose of the E chord going outside of our diatonic notes?
This is another example of dominant function in the song but it's being used to move to a new key, A minor, which is the following chord. The relationship between the E chord and the A chord is the same as the relationship between the G chord and the C chord; a dominant chord resolving to a tonic chord.
Let's say a bank is offering a very generous deal on interest. If you put in $1 for one year, it will receive 100% interest. So at the end of the year you will have $2. There's another deal going on at the same time where you can get 50% interest over a 6 month period (and then again after another 6 months). In fact you can pick your interest deal, 33% over a third of a year, 25% over a quarter of a year, and so on. Which deal is best? If we keep dividing up the year, what's the most money you could have?
For the 50% deal, you will have $1.50 after the first 6 months, then add another 50% to your $1.50 which comes out to $2.25. Already this is a better deal! If we make a model we can say that our total at the end of the year equals (1 + 1/n)^n where n is the number of times a year we are adding the interest. If we look at the last example, four times a year, our total comes to $2.44. And now we see we're not getting as much growth as we saw before. This tells us there might be a limit to how much we can make this year and it happens to be around $2.72. That limit as n goes off to infinity in our model is equal to "e", a convenient number that has many properties that make it helpful in describing growth.
Imagine you can travel to the center of the earth. What would the gravity feel like there? How would it be different if the earth had the same mass but was only as large as a basketball? And most importantly- which scenario would you rather be in?
The force of gravity between two objects is proportional to the mass of those two objects divided by the distance between their centers squared. Our two objects here are you and the earth and your masses aren't changing so we can ignore that for now. On the surface of earth, all the earth's mass is pulling you down. As you go deeper you notice some of the mass of the earth is now above you and therefore pulling you up. Once you finally reach the center, the mass of the earth is pulling you in every direction equally so you feel like there's no gravity. If the earth were instead as small as a basketball, it would be much easier to get near the center. Unfortunately in this case, we need to remember that the force of attraction grows as the two objects come closer together. Now the earth is very massive and you are now very close! The force of gravity would be immense and uncomfortable to say the least!
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