Friday, February 5, 2010

Lesson 10: Roman Numberal Chordal Theory I: Foundation

Ok, so.  Finally, right?

It took us long enough to get here, but now we're at roman numeral theory.   Roman Numeral theory is great for about a billion reasons, and if you're conversant in it and understand it, it is easy mode for music in so many ways.

You see, we talked before about how specifically to talk about chords, by talking about Co7/Eb or whatever, and that's great, and tells us a bunch, but it is also like, completely in a vacuum in terms of how it relates to anything around it as an intellectual object.  While that's how you'll see specific things written, and you'll be using written out sheet music for, you know, reading sheet music, when discussing music in a theoretical way, we normally don't get that specific, because we're dealing with relationships, not specific things.

You know what, that's sort of a confusing thing to say without explaining first the basics of Roman Numeral theory.

Ok, so we've all heard people saying things like "Yeah, it's just a I-vi-IV-V-I thing" or shit like that, I'm sure, and maybe we even have a basic idea of what all that means, but what does that all mean?

Well, let's look at a basic major scale.  We have 7 unique pitch classes in a line.  To start, let's assign each of those pitch classes a number.  We'll take C major for fun, and so we start saying C is the 1st one, D is the 2nd, etc.  These are scale degrees, and are what we use to talk about notes in any nonspecific scale.  That is to say, the 3rd scale degree in C major is E, but the 3rd scale degree in Gb major is Bb.  So the advantage is that now when we talk about the 7th scale degree, we can know how that relates to the 1st scale degree in every scale we talk about, not just, for instance, how B relates to C

We also have names for scale degrees, because apparently numbers were just a little too simple.  Actually, the names are really useful, because they give us some terminology that is pretty descriptive of the job the notes do instead of remembering, for instance, that scale degree 7 naturally leads to scale degree 1.

These names are:
(Scale degree - - Name)
1 - Tonic
2 - Supertonic
3 - Mediant
4 - Subdominant
5 - Dominant
6 - Submediant
7 - Leading tone/Subtonic

Ok, so the most confusing one there is the Submediant being above the mediant.  Other than that, the supertonic is above the tonic(and sounds awesome), and the subdominant is below the dominant.  So what's the deal with it being the "submediant"?  Well, measure a third up from the tonic and you get the mediant.  Measure a third below and... submediant!  The other way to think about it is that the Mediant is the middle note in a triad with the root of the tonic, and the submediant is the middle note of a triad with the root of the subdominant. 

Now, I'm willing to bet that some of you also know the scale degrees by other names, which would be Do(or Ut), Re, Mi, Fa, Sol, La, and Si(or Ti).  I will not be using these, for two reasons.  One reason is that this has no advantage over knowing the scale degree numbers or names in terms of theroy, so it's really just another thing to learn. Another is that Solfege(the system where we call the notes those names) comes in two varieties.  One is "Fixed do" and one is "Movable do".  In fixed do, Do is always C, regardless of where it is in the scale.  In moveable Do, Do is always the tonic, regardless of where it is absolutely.  So essentially, solfege is either an absolute or relative measurement.  There are reasons for using both, but really, I still don't think they really have any advantage over the other absolute or relative measurements we have for Theory.  Now, I will probably be using the fixed do system while hoping to also phrase things in movable do if I cover sight-singing, because solfege is a great tool for that, for a variety of reasons.  But for theoretical purposes?  I really don't see a reason to use them.

Anyways, those are the scale degrees.  Now, let's not alter any notes from the key signature and build chords on each scale degree, and we get 7 chords.  We refer to these by the scale degree they're built on in root position.
 So basically that.  We use roman numerals here.... well, because we do.  This makes things easier than using normal numbers actually, since you'll notice that some of the roman numerals are capitol and some aren't.  Well, look at the chord makeups and you might see why.  You know how instead of writing "CMaj" or whatever, you can just write "C" and assume major?  Well it's the same thing with the roman numerals.  I is a Major chord based on the Tonic(Scale degree 1), and i is a minor chord based on the Tonic.

In Major, the chords are as we see in that image.  I is naturally Major, ii is minor, iii - minor, IV - Major, V - Major, vi - minor, viio - diminished.

In minor, things get a little wonky.  In its natural position, also known as the natural minor scale, we have the following:
i - minor
iio - diminished
III - Major
iv - minor
v - minor
VI - Major
VII - Major

Now, the reason things get wonky is because as we'll learn, the V chord is a really, really strong move to I(or i).  Like it is the way to lead back to I(or i).  Even more so if we make it a Dominant 7 chord(I'm sure you've noticed that the Dominant tone of the scale is based on V.  And the Dominant 7 chord is... well, it's called the Dominant 7 chord.  These two are related.  V leads to I, and Dom 7s have the V-I motion).  There's a really nice explanation for this involving the tritone collapsing and all that, and I'll get to chord relations and voice leading with that, but basically all you need to know is that V goes to I(or i).  A reasonable substitute for V or V7 is viio, again, for a reason we'll go over(Actually, this one is a little simpler.  Take a V7 chord in any key... for now let's say C Major to keep things simple, and then take a viio chord.  Look at the notes there.  G7: G, B, D, F.  boB, D, F. Hey look).  Anyways, so they lead super-well to the tonic chord.  Well, in minor, you'll notice that v is naturally minor and VII is naturally Major.  So to compensate for this, in minor it's very common in tonal practice to raise the leading tone to make V major and viio diminished.  Also, having the leading tone raised to give it only a half step away from the tonic makes it really... well, lead as a tone.

Now, this also brings up a slightly confusing thing.  We will find we reuse terminology a lot in this section.  For instance, the Dominant can refer to the scale degree, the Chord built on the scale degree, or the type of 7th chord.  The only real way to figure out which is in context, unfortunately.  It's safe to say that, for instance, if I say "x based on the subdominant", I'm probably talking about the note.  If I say something like "The Dominant-Tonic motion", I'm probably talking about the chords, for reasons we'll get to, and if I say "Dom7" I'm talking about the type of the 7th chord.  That one's easier.

Anyways, I know this is a super-short post, but I just came back from a long-as-fuck night, and I need to go to sleep.  Also, the next thing I'm going to cover is chordal tendencies and holy shit you guys is that post going to be a doozy.  I don't want to mix them up, so for now, just take these basics and make sure you understand them before we move on.

Tuesday, February 2, 2010

Lesson 9: Counterpoint II

Quick note: In the last blog post I accidentally referred to P4s as consonances.  While they are considered consonant intervals by any measure outside of counterpoint, and really are just inversions of P5s, in strict species counterpoint they are considered vertical dissonances.  Sorry for the confusion.  Also then the no parallel 4ths thing is unnecessary because... well no 4ths in general is the rule. 

Ok, so if First Species counterpoint is 1:1, and Second species is 2:1, third species is 3:1, right?

Unfortunately, no.  While first and second are really nice and helpful with their names, third is stupid.

Third species is 4 notes in the added line per one note in the C.F.

Also, to bring back the whole historical aspect, I'd like to note the way this is fitting together.  Remember how in early Organum duplum(which I brought up last post too), we just made harmony by copying and pasting the Chant, and then after that we started having more stuff over a pedal line?  Well that pedal line was always the actual Chant.  I think I forgot to mention that during the history, but the pedal line that gave us sort of rudimentary chords was the chant, and what went on above it wasn't based on the chant.

Anyways, the reason I bring this up is because the Cantus Firmus being the basis for the added lines in counterpoint is pretty close to the same idea, where the moving line is the added line, and the C.F. is slower and in something like third species, sounds almost like a bass line

So onto Third Species.  For the pedagogical purposes of dealing with rhythms, similar to how we normally have first species as whole notes and second as half notes over whole notes, for third, think quarter notes over whole notes.

Similar to second species, we can either begin third species with a downbeat or have a rest on the downbeat and begin after that.  In third, that rest can only be a quarter rest, and we start on beat 2.  Also similar to second species, the first note in every measure is bound by the same rules as first species.

Now third gets some confusing rules too.

1:  The first note of any measure must be consonant.  The second and fourth may be dissonant, and the third.... well the third's tricky.  The Third quarter can be dissonant under only two circumstances.  The first is if the second and fourth(and obviously 1st) are consonant.  The other is if it's in a "double passing" gesture.  That is to say, both the second and third can be dissonant if the 1st and 4th are consonant and the motion between them is all passing motion.
2:  We can use unisons now!  But not on the first quarter note of any measure.  We can also now have dissonant neighbors.  And we have a new nonharmonic figure: The double neighbor.  In the double neighbor, we have both a lower and upper neighbor. Also, we must continue the motion from the second neighbor to exit the figure.  So for instance, if we start on a G against a C, the double neighbor would be G-A(Upper neighbor)-F(Lower neighbor)-G-A(continuation).  The continuation must also be consonant and work in all the other rules.
3: P5/P8s are to be separated by at least 2 notes if they occur in different measures.  If they are against the same C.F. note, there is no such restriction.

So those are really the new rules.  There aren't too many, they're just super-fragmented and confusing.  Also, Third species is really difficult to write in any way that doesn't suck.  You'll all notice, I'm sure, that I haven't talked about writing counterpoint that doesn't, you know... suck.  Part of that's because technically the rules of counterpoint, again, are really mainly used as a pedagogical tool that can sort of overlap with writing modern stuff.  However, when we write music of any kind, even just as an exercise to follow some rules, in general we try to make it not terrible.  All this stuff gets a lot harder when you're trying to make it feel like it has a purpose and isn't just a collection of random notes.  And it becomes a billion times harder when you have to fill every measure with four notes.  Third species is the hardest in terms of rules because they are very specific and just keep piling on each other, but it's even more so the hardest in terms of not sucking while writing, because keeping a line going with that many notes while still following the rules is really hard to do.

And now we go onto my favorite species of counterpoint, 4th species.  4th species is really easy to make sound not stupid, and we scale back a ton on the rules.

4th species counterpoint is basically 1st species, but with the added line offset by a half note.  So the added line starts with a half rest, then is two half notes tied together over bar line breaks.  This has the effect of making pretty much everything a suspension.  Also, this gives us the magical ability to have dissonances on the strong part of measures(holy shit!), assuming we prepare and resolve it correctly.  The way we do that is by having the suspension be consonant when the to-be-suspended note is hit, and have the resolution, which is usually a downwards resolution, be consonant.  The only change to this is that the final note occurs with the C.F. Also, there's a specific rule for approaching it I'll get to.

In fact, in 4th species counterpoint, the only suspensions we're allowed to have are 4-3, 9-8, 5-6, 6-5, and 7-6 in the upper voice.  In the lower voice, we just invert all of those(So for instance, 2-3 is the same in the lower voice as 7-6 in the upper).  Note that those are given as "suspended note-resolved note".  And preparation interval that's consonant will work.

If the held over note isn't a dissonance, then we don't have to worry about the suspension rules, and can leap out of it or really do whatever we want out of it as long as we obey the other species rules and we're not setting up an illegal suspension.

After that, 4th species is basically 1st, but with the second note in the measure of the added line against the first note in the measure of the C.F.  So the measure can start with a P4, but then the second half must go to a Third, following both the rules of the suspensions and satisfying First species rules.  We can't, however, have, for instance, more than three 4-3 suspensions in a row, as while the suspensions are all dealt with just fine, we have four parallel thirds which is forbidden by the 1st species rule.

Basically, for 4th species, you have to think both in terms of 1st species were the offset removed and both lines just played at the same time, as well as within the rules governing suspensions.

Also, in 4th species, we must end with a 7-6 suspension that then resolves at the beginning of the next measure to an appropriate first species final.  So basically, a suspended C to B against a G in the C.F. resolving back to a C over a C is acceptable.

And finally, in extreme cases, we can do something special in 4th species and break species, returning to 2nd species if we're caught in an otherwise inescapable line of suspensions.  This is only acceptable if it's the only way out of the suspension string though.

Anyways, why is 4th species my favorite?  Because it always is super easy to make sound really pretty with all the suspensions, and it's just sort of a douchebag way of writing counterpoint, because it's like applying a liberal amount of hamfistery to the entire exercise.  In almost any other circumstance, chaining together a ton of suspensions would(in a perfect world) be punishable by death.  Like, it's really awful writing.  Except in 4th species you get to do it the entire way through.  It's like doing standup and only telling one joke over and over again, but you're allowed to now.

And finally, Florid counterpoint, a.k.a. 5th species.

Florid is a giant mess of stuff, because it's pretty much... every other species combined.  Plus eighth notes.  Kind of.  I'll be using links from to show examples here, their examples are nice and clear and clean.  They also have a big thing on counterpoint which is pretty spot on, if a little confusing to read.

First consideration for 5th species is that we try to start and end basically with 4th species, with a suspension to the tonic to end, and starting with a slower suspension figure too.

Also, eighth notes.  We can use them under very specific circumstances.  We can use eighth notes in a double passing tone figure, or as a single neighbor, resolved with the second eighth note.  Also, we can only use them max once in a measure, on either beat 2 or 4, and we try to avoid overusing them.  The reason is that technically counterpoint is still considered a vocal technique writing instead of an instrumental technique, since originally it really was vocal stuff, with chants and counterpoint over.  In vocal lines, slower lines are easier to sing, so we still go by that convention.

Now, another thing we see in fifth is a change to the way we can handle suspensions.  While we still need to approach them correctly, we can play with the resolution a bit, by using a quarter note on beat 2 to anticipate the resolution(note: This is different than the nonharmonic anticipation, since we're going to a harmonic tone.  I know, it's a little confusing) or by using an eighth note figure with a neighbor to do the same thing as the quarter note resolution but... well, with the added neighbor.  We can also use an escape tone from it, or a figure that's pretty close to a double neighbor thing, all by using a quarter note on beat 2.  Escape tone we handle just like the normal escape tone nonharmonic, and the double-neighbor-like figure we leap from the suspension by a consonant interval, to a consonant interval, before going to the expected resolution.

Also, as you'll notice in all those linked examples, you'll notice that we still obey the rules of 4th species in terms of the third beat always being consonant.  Even though we're breaking species in regards to 4th for these embellishments, we still obey the rules in the strong beats.

Other than that, basically we just combine all other forms of counterpoint in fifth.  On the one hand, this makes things a little easier on us, since, for instance, we don't have to deal with the incredible ease of writing really meandering parts in 3rd species or really boring, drawn out parts in 1st.  On the other hand, with more freedom comes more choices, and that means we have to think more about how we want to do rhythms.  In early species of counterpoint we really don't have to think about rhythm, it's there.  Now we have a whole dimension added.

Anyways, that's counterpoint.  Next we'll start on Roman numeral theory, which was the initial impetus for this blog, I just had to get all of this out there so we made sure we were on the same page in terms of these basics.