Tuesday, December 1, 2009

Lesson 2: Pitch relations

Allright, so now that we have the basic vocabulary down, it's time to start dealing with the nuts and bolts of what makes music music.

If you ask a theory professor, someone who will probably have glasses, a sweater vest, and lots of tweed, what a scale is, they'll probably give an answer like "A scale is a collection of unique pitch classes". Awesome. Of course, this tells us absolutely nothing. That could describe, like, everything in music.

What we're mostly concerned about, what you'll deal with most often, are "Diatonic scales" Wiki says a Diatonic scale "is a seven note musical scale comprising five whole steps and two half steps, in which the half steps are maximally separated" Awesome. Well that's correct, but what the fuck? What does that even mean? Maximally separated? Allright, so what it means is that there are 2 or 3 whole steps between each half step, depending on which diatonic scale you're using as well as when you ascend through octaves. Basically, a major scale is (W=Whole step, H=Half Step) W-W-H-W-W-W-H, and a natural minor scale is W-H-W-W-H-W-W.
This is a little easier when seeing and hearing it, so here's a C major scale
C Major
To make it a natural minor scale, we would flat the third, sixth, and seventh scale degrees, or E, A, and B for C minor.

Here's a C major scale being played, both ascending and descending.  You'll notice it sounds "Happy" 

And here's a c natural minor scale ascending and descending.  It's "Sad"

There are two other scales commonly used in common practice tonal theory, both of which are variations on the minor scale.

The first is the harmonic minor scale, which raises the 7th scale degree(from minor.  So it is in fact the normal, unaltered scale degree in major), both ascending and descending.  It sounds something like this:

The reason this is known as a harmonic minor comes a little later, when we start talking about the basic chords in a key.

And then finally we have the "Melodic minor", which is confusing because it changes depending on whether you're going up or down.  If you're ascending, you raise the Sixth and Seventh, so that the only note that is different from the major scale is the lowered third.  However, descending, the melodic minor is exactly the same as the natural minor.  It sounds like this.

Now, when we talk about "Scale degrees", we count in ascending order, and while in chord theory later on we'll be talking about 9ths and 11ths and 13ths, for the most part, after the 7th scale degree, we loop back to the 1st scale degree, but in a different octave.  So the third scale degree Now, n C Major is E.  The fifth is G, the seventh is B. 

And speaking of octaves, I've used that term a lot, what exactly is that? In order to lay the groundwork for some later theory stuff, each specific frequency, as well as any frequency which is derived by x^2/f or f/(x^2), where x is any whole number, is a "pitch class".  That is to say, a pitch class is any frequency as well as its double, its quadruple, eight times it, its half, its quarter, it over eight, and so on(Any math people, feel free to correct me if that formula doesn't say that). in other words, all "C"s are of the same pitch class. But they are not all the same note, because there are really high and really low Cs. What an octave is is the interval between two neighboring notes of the same pitch class. It looks something like this

Both of those notes are C, and the interval between them is an Octave(So names because they are the 8th scale degree apart from each other). Since an octave has the same pitch class for both notes, it's also an easy to discuss boundary for ranges of notes. So when I say that something is "In a different octave" or "crossing octaves", I'm normally referring to it being between a different octave of the 1st scale degree.

  So, for example, the scales I posted are all within the same octave, in that none of the notes extends above the 3rd space C or below Middle C the ledger line below the staff, with C being the reference because we're in the key of C

Which gives me a nice segue into keys.  When we're talking about key signatures and what key something is in, we're essentially talking about what notes are modified to fit into a specific diatonic scale.  So if we're in the key of A Major, as we looked at last update, we see that F, C, and G are sharp.  That's because in order to build a major scale based on A(meaning that A is the first scale degree, also known as the "Tonic"), we need to have those notes be sharp.  If we had no key signature, we would be in a minor, as the a natural minor scale needs no modified notes.  We name the keys based of the the Tonic, or the first scale degree.  So the key with modifications to build a major scale with the tonic of D is D Major.  The key with modifications to build a minor scale with the tonic of B is b minor.

For reference, here's the key signature for D major:

D Major 

And here's b minor:

b minor 

HEY WAIT!  I messed up and posted the same key signature for both!  D Major and b minor are examples of "Relative" keys.  Relative keys are keys that share the same key signature, but a different tonic.  The relative minor of any major key is based on the 6th scale degree, and the relative major of any minor key is based on the 3rd scale degree.  So C and a are relatives, D and b, E and c#, etc.  The other common relationship you'll see is what is referred to as "Parallel" keys.  A parallel key is one with the same tonic, but a different key signature.  So the parallel minor of C Major is c minor.

There's a trick here to telling what key you're in, too.  If the key has sharps, you go one step above the last(right-most) sharp in the key signature, and that's your major tonic.  So the last sharp in D Major/b minor's key signature up there is the C#.  One step above? D.  The sixth scale degree?  B, so the relative minor is b minor.  Specifically you go one half step above the last sharp, but you only go there when in really awkwardly written keys like B# or E#.

If the key has flats, then the second to last flat is the Major tonic.  So if you have two flats, Bb and Eb in the key signature, then you're in Bb Major/g minor.  If you have Bb, Eb, Ab, Db, Gb, you're in Db major/bb minor.

Also, there is a strict order to how you add sharps and flats that makes that work out correctly.  With sharps the order is: F C G D A E B.  With flats it's B E A D G C F.  There's a trick to this that won't make sense until the next part, which is that Sharps you add by going up by perfect fifths, and flats you add by going up by perfect fourths.  Which is also how the keys loop around.  The keys in order of increasing number of sharps are C(no sharps) G(one) D A E B F# C# G# D# E# B#.  With flats it's C F Bb Eb Ab Db Gb Cb Fb.  Again, that's a little more advanced, so I'll return to it later.

In fact, let's look at intervals now, so that makes a little more sense.

Intervals are the distance between notes.  They're also the term used for the notes when played together or sequentially.  So a "Third" is a combination of notes a third apart.

Lableing them is pretty easy, we name them based on how far apart they are.  If the notes are one step apart, they are a "Second".  If the notes are three scale degrees apart, they're a third.  Here are... well all the natural major intervals.  For the audio, they're all in C major, and they'll play the notes C - interval above C - notes together.

Major second

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Click here to download Major second.mp3

Major Third

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Click here to download Major third.mp3

Perfect Fourth

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Click here to download Perfect fourth.mp3

Perfect Fifth

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Click here to download Perfect fifth.mp3

Major Sixth

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Click here to download Major Sixth.mp3

Major Seventh

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Click here to download Major 7th.mp3

Ok, so quick thing, why are some intervals "Major", and some "Perfect"?  And for that matter, you may noticed I didn't mention "Minor", "Augmented", or "Diminished" yet.  Well, there's a historical explanation for the specific term "Perfect", but ultimately, There are 3-4 ways of defining any specific interval.  The interval remains the same in terms of diatonic scalar distance, but the pitch class distance isn't necessarily the same.  So a minor third is between, for instance C and Eb, while a Major third is between C and E.  They're both between a C and an E of some sort, but the chromatic modifier is different.  For fourths and fifths, there is no such thing as a "Major" or "Minor" interval.  If you look at Major v. minor scales this will start to make some sense, in that the fourth and fifth scale degree aren't modified.  Now I know what you're saying "But Khavall!  The second scale degree also isn't modified!"  That's true, but again, that's specifically the historical and mathematical reason.

For intervals with Major and Minor sonorities, the Major is the one that would occur in a scale based on the lower note in major.  It's a little confusing with that saying, but for instance, a C Major Third is C and E, and a c minor third is C and Eb.  An A major third is A and C#, an a minor third is A and C.  For diminished intervals, you shrink the interval one step chromatically from the minor, so a diminished third would be, for instance, D and Fb(Yeah, I know), where D and F is a minor third.  For augmented intervals, you expand the interval one step chromatically from Major, so C and E#(Yeah, I know), while C and E are the Major interval.  For perfect intervals, Diminished is one chromatic step smaller than Perfect, and Augmented is one Chromatic step larger than perfect.

Ok, so... some of you may have noticed something here.  Like, if you play a diminished triad on your instrument of choice it sounds exactly the same as a major second.  For a visual version of this idea, take a look at these two intervals:

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Image and video hosting by TinyPic

They look different.  They sound the same.  If you go one half step up from G, or one half step down from A, you arrive on the same key/fret.  So are they different?  Are they the same?  Why do we have both?

Well here's an important concept in theory: Behavior, and context.  You'll notice that I've been trying to separate the concepts of "Pitch class" and "note".  G# and Ab are the same pitch class, but they are different notes.  The difference between them are one of two things.  One is purely notation, the other is function.

     An Augmented fifth is probably going to be followed by the interval expanding, where a minor sixth is probably going to be followed by the interval collapsing.  Even without the interval, a G# will probably be followed by ascension, where an Ab will probably be followed by descent, if they aren't simply functions of the key signature.  They sound the same, but the specific note used is an indication of behavior.

Now, I mentioned key signature there... that's the other function of different notes for a specific pitch class.  In keys with flats in the key signature, a flat or even a double-flat is simply easier to read than a sharp, because if a performers mind is thinking in flats, suddenly reversing that to add a sharp in can be very confusing.  Sometimes a double-flat or double-sharp are even easier to read than Natural signs.  Essentially, if certain notes are in a chord are modified in one way, modifying other notes in a similar way means that the thought process is only "Down by X or Y" instead of "X is down by Y but Z is up by A and B is up by C from being down by D".  

Either reason can be why a certain pitch class is portrayed with a certain note, and like many things in theory, there isn't really a hard and fast rule to a lot of it.  Note that, for instance, I mentioned that Augmented fifths normally expand.  I'm sure with even cursory searching someone could find an Augmented fifth that collapsed, but the general use of an Augmented fifth is that it will expand.

And now that we know intervals, let's briefly return to key relations.  Once again, the order of keys with sharps is: C G D A E B F# C# G# D# A# E#, and flats is C F Bb Eb Ab Db Gb Cb Fb.  So what's the relation between them?  Sharp keys are what are known as the "Circle of fifths", while Flat keys are the "Circle of fourths".  That's because to get from one sharp key to the next, you ascend by a perfect fifth.  To get from one flat key to the next, you ascend by a perfect fourth.  

So today we've covered Scales, Keys, and Intervals.  Next update we'll cover chord construction, and then we pretty much will have covered the basics of construction of music, and can move on to what to do with all of it.

As always, if you have any questions, please don't hesitate to leave a comment and I'll try to get back to you.

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