SPEAKER 1: Let's dive into some music.
Before we get working on the actual problems
you'll need to solve as part of this problem set, let's take a moment
and look at some background about music, notes, and how we might represent them.
What we're looking at here is a portion of a piano keyboard.
And you'll notice we have seven different white notes and five
black notes.
And so as you may know, each note is assigned a letter
where we have C on the left side, then D, E, F, G, A, and B,
where the white keys on the piano are each assigned to a different letter.
A semi-tone, as we'll refer to it, is just the interval from one
note to the very next note, including both the white and the black keys
on the keyboard.
So between C and D, that's an interval of two semitones,
because we go from C, first to the black note in between C and D,
and then to D. D and E are also two semitones apart,
because there's a black note in between them.
But E and F are one semi-tone apart, because right next to E,
then immediately following it, is the note F.
So semi-tones are just how we're going to refer to a step of exactly one
note on the piano keyboard.
But how, then, are we going to represent the black keys on the piano keyboard
if the white keys are all assigned to letters?
Well, in order to do that, we'll need to use accidentals.
Accidentals adjust notes up or down by exactly one semi-tone.
So the sharp accidental, represented here by the pound sign,
means we're going to move up one semi-tone.
So these are how the black keys would be represented in terms of sharps.
Up one semi-tone from C is C sharp.
And up one semi-tone from D is D sharp, for example.
Likewise, we can also represent notes as flats, where a flat just
means move it down one semi-tone.
So here, we're looking at D flat, which is one note down from D,
and E flat, which is one semi-tone down from E.
And notice here that there are multiple different ways to name the same note.
That note in between C and D can be represented as either C sharp or D
flat.
And they both mean the same thing.
In fact, it's not just the black keys that
can be referred to as sharps and flats.
Since F is one semitone above E, and likewise E is one semi-tone below F,
we can call F, E sharp, and we can call E, F flat.
And those are totally valid ways of representing those notes.
But as we know, a piano does not just consist of these keys.
It consists of far more.
And what we say is that the piano is divided
into octaves, where an octave is just going to be a sequence of these 12
notes that gets repeated over and over across from the low range of the piano
to the high range of the piano.
And so, generally, the middle of the piano
we'll call octave four or the middle octave.
And you'll see that we have a C there, a D there, an E there,
and all the other nodes that are present in octave four.
But that same pattern of notes repeats again in octave five,
to the right of octave four, where all the notes sound similar,
but are actually at a higher frequency.
More on that later.
We can actually refer to notes not just by what letter they are
and what accidental they have, but also by what octave they're in.
For example, we might call the C that is located in octave four, C4.
And we might call the D sharp that's located in octave five as D sharp 5.
And so what you're looking at here is a representation
of a bunch of notes on the piano and how we would represent them
in terms of what note they are, what accidental they have, if any,
and finally, what octave they're in.
Finally, let's take a look at how we would actually
represent these notes if we were to try to write them down
in sheet music, a conventional way for musicians to write down notes
and how long they last, and what they mean.
This is an example of the treble clef in sheet music, which is just part
of the way that notes are represented.
And each one of these lines or spaces between a line
is a space for an individual note.
And so right below that bottom line, if there were a note there,
that would be the note D4.
And after that, each time you go up one line or space between a line,
you just go up by one letter.
So after D4, this note, which is located on the bottom line, is E4.
And the note immediately above that, located in the space between those two
lines, is F4.
And next is G4.
And remember, after G4, we've run out of letters.
And so we need to go to A4 and loop back around.
And then to B4.
Next would be C4, but if you remember from our picture from earlier,
you'll notice that the octaves change numbers once we get to C. Notice
that we have B3, but the note immediately after that is C4,
because C is generally where we'll call the beginning of the next octave.
So if we take a look at this sheet music again,
we have D, E, F, G, A, and B all in octave four.
But the note immediately after that is actually going to be C5.
And now we're in the fifth octave.
So following that is going to be D5.
And following that is going to be E5.
So every line and every space between a line
gets its own letter and octave number in order to represent it.
So this, for example, is A4.
But how would we represent, for example, A sharp 4, A sharp
in the fourth octave?
Well, if you remember that we represented sharps
as kind of just that pound sign, we can just
draw that symbol to the left of the note, and now this note is A sharp 4.
Likewise, if we wanted to represent a note as flat,
where flat just means down one semi-tone,
we could likewise to the left of the note add the flat symbol.
And now this note that we're looking at here is A flat 4.
And so using that representation, we have the capacity
to represent any note using any accidentals,
just by drawing the notes on the correct line
or space between the line that corresponds
to the right note in the right octave.
Now, not all notes are the same length, though.
So how, in sheet music, would we tell apart
a note that is a short versus a note that is a longer note?
Well, there are a bunch of different lengths of notes,
and the notes look different depending upon their length.
So on your left, we're looking at an eighth note.
And an eighth note, we'll say, lasts the duration of one eighth,
and generally looks like a note with a little curvy tail at the top of it.
Next to that we have a quarter note, which
last twice as long as an eighth note, for a duration of one fourth.
And it looks just like an eighth note but doesn't
have that curvy tail at the top.
Next to that, we have a dotted quarter note,
which is a quarter note with a dot to the right of it.
And that lasts three times as long as an eighth note,
or a duration of three eighths, we might say.
And finally, on the right hand side, we have a half note,
which looks like a quarter note except the center of the note is not filled in
and is hollow.
And a half note lasts the length of four eighth notes,
or a duration of one half.
There are more notes than just these, but these are the ones
that we're going to care about for this particular problem set.
And so using these different symbols for notes that determine their length
and placing them in the right place, we have the ability to represent,
in sheet music, a note, which note it is, what accidental it has,
what octave it is in, and also how long the note lasts.
And so one other thing that you might find
when it comes to the lengths of notes is that sometimes when two eighth notes
are next to each other.
They'll be represented in sheet music as just two
notes that have a little bar that connects the two of them.
So if you ever take a look at that and are confused by what that means,
generally, that symbol on the left just means two eighth notes.
And that's all the background we'll need in order
to be able to dive into the problems here.
So with that in mind, even if music notation
might seem confusing or complicated at first, I'm sure that in no time at all
you'll be a natural.