Species 3, Dorian Mode

These are getting harder fast. In third species counterpoint, four quarters are set against each whole note. My big lesson from the exercise I've posted below: 'f's in the cantus firmus make it difficult to write a counterpoint below. That's because you've got two adjacent dissonances: c-f is a 4th, and b-f is an augmented 4th. In the sixth bar, here, the only way I could see to get out of that bind was the cambiata formula.



It takes forever to mark, then tinker. It's hard to believe things are going to get still more complicated. (On the other hand, it's also hard to believe how much improvement I can observe in working through an exercise set. These Dorian mode exercises I've been posting are always the first ones in a set. It usually takes me about 1/3 as much time to complete the Ionian mode-- the last one in every set-- as it did the Dorian.)

Melodic Rules

These rules governing the melodic flow of each counterpoint are common to every species. I'll add to the list, or clarify the wording, as I come to understand them better.

1. Forbidden skips: augmented fourth, major sixth (up or down), minor sixth (down only), seventh, any interval greater than an octave.

2. "Exposed" tritones are forbidden. That is, no run of notes in a single direction should be an augmented 4th from end to end.

[NOTE: Since all of these exercises are written in their natural modes, the special notes to watch out for are F and B-- an augmented fourth when taken together. There can be no jumps between F and B, and no runs that begin on one can end on the other. Beware of F and B!]

3. Leaps of an ascending minor sixth or octave, or a descending octave must be "recovered." That is, such a large skip must be followed immediately by a step back into the range covered by the leap. (Fux often "recovers" using a skip of a third back into the big leap.)

4. Do not repeat notes. (Though Fux makes many exceptions in Species One, repeated notes are rare in the other species.)

5. Avoid multiple skips in the same direction. (This is a guideline-- Fux contradicts this advice all the time. See, e.g., page 46, in the 2nd species phrygian example, he skips 4th, 5th, and 4th in the same direction: c-f-c'-f'.)

Species 2, Dorian Mode

Second species is two half-notes set against each whole note in the cantus firmus. It's much harder to get right than whole against whole...



Also, I found a pretty amazing online counterpoint tutor. It abstracts Fux's rules, much like I've been doing for myself. The best part, though, is a Java applet that detects errors in your compositions. I've been using it to make sure my counterpoints follow the melodic rules-- those are the ones I have the most trouble remembering, so far.

Typesetting music: lilypond and latex

I made it through the first four exercises before I flipped ahead and thought, "wow, there a lot of exercises. It would be nice if someone had mad a workbook to accompany the text." Then I thought, "I should make a workbook to accompany the text!" Then I went off the deep-end figuring out how to typeset music in lilypond and LaTeX.

I've now typeset the first 60 exercises-- the two-voices section of the book. Check out a sample page in PDF-- it looks cool! I'm going to stop, now, for fear of burning out. I'll go back to working on the music, and typeset three-voice counterpoint after I finish two-voice. Here are a few of the things I've learned:

To integrate lilypond scores into a LaTeX document:

First, in your LaTeX document, reference the file with your score, e.g:

\lilypondfile{WB.1.2.E.a.ly}

Then, when you want to compile the typeset file (mine is called "WORKBOOK.tex") do these steps:

$ mkdir out
$ lilypond-book --pdf --output=out WORKBOOK.tex
$ cd out
$ pdflatex WORKBOOK.tex

This is pretty self-explanatory: lilypond-book creates a new TeX document, converting the score sources into images and LaTeX code, then dumps it into the "out" directory.

Naming conventions:

I've settled on this naming convention, to help me keep track of proliferating lilypond files:

<species>.<voices>.<mode>.<above/below>

For example: 1.2.E.b.ly

Meaning first species, two voices, phrygian mode, counterpoint below the cantus firmus.

Because Fux re-uses the same cantus firmi throughout the book, I can re-use the same scores five times in each section of the workbook, once for each species. Including the species number helps me keep track of completed exercises, though.

Landscape formatting in LaTeX:

Including "landscape" as an option under \documentclass will format landscape. If you use pdflatex to compile the source, it handles everything. If you're using latex to compile the source, you've got to tell latex to respect the landscape option. And then you have to tell dvips to respect the landscape option. Pdflatex is the way to go.

Empty bars in lilypond:

Since I'm making a workbook, I need blank staves that I'll fill in by hand. Lilypond freaks out about bars with no notes in them, though. The way around is to fill in every bar with a whole note, and then use \hideNotes and \unHideNotes around the set of bars I want empty. A similar trick can make roomy, regularly-sized empty bars for workbooks.

Formatting single-line scores:

All the Fux exercises are one-line scores. I want to use all the available page space, to make them easier to write on. That means reducing the default indent and turning off the ragged right edge. (Lilypond indents the first system of any score, and doesn't justify full the last system in a score.) To accomplish this, I've included this bit at the top of every score:

\paper {
indent = 10\mm
ragged-right = ##f
}

Also at the top of every score is this: \version "2.12.3"

Apparently lilypond somewhat regularly breaks backwards compatibility. When they do, they release a script that converts old scores to the new version, and that script looks for a version string at the top of the file it's crunching.

Notation software

I haven't decided about the best way to do Fux's counterpoint exercises. I copied the first few down on music paper. (There's lots of good blank staff paper available online. For two-voice counterpoint, I've been using the "instrumental duet" sheet from this extensive collection of blanks. As things get more complicated, there will be increasing returns from a software score editor, though.

I spent a while today running down some score editors for GNU/Linux. I want something with a reasonably intuitive graphical interface that can also output lilypond format. (I wonder if it might be fun to make a blank workbook of Fux's exercises as I do them. If so, I'll want lilypond markup for that, and I hate the idea of duplicating work.) It looks like the best option is probably NtEd, so I think I'll start with that.

I also stumbled on a web program called noteflight that looks like it will be great for sharing work on this blog. I don't think I can use it as my primary editor, but I'll probably do up some occasional samples. Like this, my favorite of the four exercises I've done so far:



That's in the dorian mode, with the counterpoint below the cantus firmus. It sounds nice!

What modes are

I've never understood what modes are. When I see people writing about them, it seems like sometimes they're the equivalent of a key, and sometimes they're the equivalent of a scale pattern. (Like major, minor, blues, etc.) Well, I think I finally get it.

Modes are scale patterns, but they're patterns associated with particular tonic notes. Consider just the white keys on a piano. If you play any eight contiguous white keys, you'll have played a scale in one of the ecclesiastical modes. (The exception is B-- for some reason there's no mode associated with B.) For example, if you choose C, you'd play C-D-E-F-G-A-B-C. The interval pattern of that series of notes is W-W-H-W-W-W-H. That's the scale of the Ionian mode, and it's familiar to us as a major scale. If you choose A, you'd play A-B-C-D-E-F-G-A. That interval pattern is W-H-W-W-H-W-W. It's the Aeolian mode, and familiar to us as a natural minor scale.

The other modes each have their own interval pattern that you can discover by starting on a different white key as the tonic. Those other modes haven't survived in popular music, but are all over the place in medieval music.

Mode	Interval Pattern	Tonic
Aeolian	W H W W H W W	A
Ionian	W W H W W W H	C
Dorian	W H W W W H W	D
Phyrgian	H W W W H W W	E
Lydian	W W W H W W H	F
Mixolydian	W W H W W H W	G

The upshot: when Aloysius says to Josephus, "find a counterpoint to the cantus firmus which I am writing down for you in E," he doesn't mean the key of E. Instead, he means the Phrygian mode. In the Phrygian mode, if you choose E as the tonic, you don't need to use any accidentals-- you can confine yourself to the white keys of the piano.

Definitions, and the only rule of counterpoint...

This is the only background information Fux expects.

Intervals

Perfect consonances: Unison, fifth, and octave.

Imperfect consonances: Six and third.

Dissonances: Second, fourth, diminished fifth, tritone, and seventh.

I have a question: What's a tritone, if neither a fourth nor a diminished fifth?

Motion

"Direct motion results when two or more parts ascend or descend in the same direction by step or skip."

"Contrary motion results when one part ascends by step or skip and the other descends-- or vice versa."

"Oblique motion results when one part moves by step or skip while the other remains stationary."

The four rules of counterpoint

1. From one perfect consonance to another perfect consonance one must proceed in contrary or oblique motion.
2. From a perfect consonance to an imperfect consonance one may proceed in any of the three motions.
3. From an imperfect consonance to a perfect consonance one must proceed in contrary or oblique motion.
4. From one imperfect consonance to another imperfect consonance one may proceed in any of the three motions.

It seems to me the four rules are a roundabout way of saying one thing: never use direct motion to enter a perfect consonance.

It is necessary for you to know that in earlier times, instead of our modern notes, dots or points were used. Thus one used to call a composition in which point was set against or counter to point, counterpoint; this usage is still followed today, even though the form of the notes has been changed. By the term counterpoint therefore is understood a composition which is written strictly according to technical rules.

Fux: lacking means and a teacher

All Sophomores at St. John's College take a semester of music. Part of that semester is a face-meltingly fast flip through Fux's The Study of Counterpoint. As with most of the St. John's program, I was too slow to get much out of it at the time, but could recognize it as something worth closer attention later.

Well, it's fifteen years later. (Sweet merciful crap.)

Near as I can tell, The Study of Counterpoint is the first effective textbook for the study of music composition. Fux set out to write a book that approached advanced concepts through a series of manageable steps. Thus the title of the full work, Gradus ad Parnassum, roughly "steps up the mountain of inspiration."

My object is to help young persons who want to learn. I knew and still know many who have fine talents and are most anxious to study; however, lacking means and a teacher, they cannot realize their ambition, but remain, as it were, forever desperately athirst.

Seeking a solution to this problem, I began, therefore, many years ago, to work out a method similar to that by which children learn first letters, then syllables, then combinations of syllables, and finally how to read and write. And it has not been in vain. When I used this method in teaching I observed that the pupils made amazing progress within a short time. So I thought I might render a service to the art if I published it for the benefit of young students, and shared with the musical world the experience of nearly thirty years, during which I served three emperors.

Now that I've been teaching for a few years, I can better appreciate how difficult it is to do this-- to imagine what a subject looks like from the perspective of a beginner, and to understand what content and what approach will be both accessible and useful to students at each moment along the way. Fux seems to have nailed it, though. The book was used by Bach, Mozart, Haydn, and Beethoven. They all learned from it, and many of them used it as a textbook for their own students.

My plan is to work through as much of the book as I can without a teacher. I'll abstract and post the rules, and post the exercises along with my solutions. Presumably, if I make it to the end, I'll be the equal of Bach, Mozart, Haydn, and Beethoven.

Knowledge Ratchet

Five years ago I bought an accordion on ebay. I worked through the first few volumes of the Palmer-Hughes accordion method, which entailed learning how accordions work and how to read music. After a while I tried writing my own songs, which entailed learning scattered bits of music theory. And once I had a couple of songs I liked OK, I started messing around with recording them on my Ubuntu box, which entailed learning a hodgepodge of things about hardware, software, and sound.

Then the philosophy department assigned me three new courses in a row and all of that stopped. I spent 18 months developing new syllabuses and learning a lot about political philosophy and applied ethics, while forgetting most of what I'd learned about the accordion, songwriting, and home recording.

There's little I find more discouraging than forgetting things I worked hard to learn. So now, as I try to get back into the music stuff, I hope to use this blog as a knowledge-ratchet. As I recover the things I knew, and learn new things, I'll post them here. Then, the next time I'm distracted by other subjects for a year or two, I'll have a music notebook to fall back on.

That's the overall goal, anyway. My short-term focus is narrower: to work through Fux's The Study of Counterpoint.