Rarely asked questions: tritones and modes

At the end of the two-voice part of the book, I had two theory questions. I think I now have educated guesses at the answers.

Q: Why is there no mode corresponding to B?

A: Because this would be a mode without a perfect 5th. (B-F is an augmented 4th. B-G is an augmented 5th. The perfect 5th above B requires an accidental-- that's the only white key for which this is true.) Aloysius seems adamant that the perfect 5th is the interval that establishes the mode in the listener's ear, and so there can be no mode associated with B.

Q: What's the difference between an augmented 4th, a diminished 5th, and a tritone?

Augmented 4th and diminished 5th refer to the same degree in every mode. They are synonyms. "Tritone" refers to this same degree, and is, in an equi-tempered world, a third synonym. Near as I can tell, it has its own name because, in a world of different temperings, there were some keys in which the augmented 4th/diminished 5th was especially hard on the ears. It was sometimes called "the wolf," after the terrifying howl it made. I assume "tritone" or "devil's interval" was reserved for these particularly heinous instances of the augmented 4th (which is not, in equi-temperament, especially hard on the ears).

This answer is extrapolated from an excellent and fascinating article on tuning by Jan Swafford: The Wolf at Our Heels.

Quality of tools and standards of success

I was assigned about a dozen exercises from Fux as a Sophomore at St. John's. I remember them going very quickly, maybe ten minutes each, and so thought I could tear through the sixty two-voice exercises in a week. It's actually taken me about six weeks. Why?

I think there's an interesting interplay between my workflow and my standards of success. When I did these as a 19-year-old, I worked on paper in a dorm room. I'd usually show up for class a little early so I could play my exercises on the piano. (I've never been able really to read music, so the only way I can know what they sound like is to play them.) Every time I played one, I was impressed that some part of it sounded neat. I was following these rules, marking down notes that I couldn't hear, making nice-looking patterns on the page, and when I played them, some part always sounded nice.

Now, as soon as I finish a first pass, I do it up in my score editor and have the computer play it back for me. It's still the case that some parts sound nice, but it's always the case that some parts could be nicer. So I fiddle with it, which is easy to do. I move some notes around on the screen, then hit "play" again, then fiddle some more. So, instead of being happy that some bit of it sounds nice, I keep working on it until I'm basically satisfied with the whole thing. This can take a long time-- I spent three hours on one of the fifth species exercises. Because tweaking my work is much easier now, I'm much less willing to be satisfied with mixed results.

The new way is probably the better way to learn, but it's definitely made the project more daunting than I thought it would be. I hope I have the stamina to make it through all 90 of the three-voice exercises. I'd love to do that. But I don't expect I'll do more than a handful of the four-voice exercises. Unless I somehow get much, much faster, there's no way I'm going to do all 120 of those.

Recap: two voices

Last night I finished the 60 exercises that make up Part I of the Fux book. Musical lessons learned:

1. Occasional skips greater than a third help change the character of a line from an accompaniment to a melody. This is counterintuitive to me-- my instinct is to use steps whenever possible.

2. Parallel motion really does serve to undermine the momentum of the counterpoint and the interest of the harmony. Fux isn't kidding when he says that in florid counterpoint, parallel motion should be generally avoided.

3. However, I find parallel motion can be a nice effect near the end. It's a kind of pre-resolution calming. Like the piece has found its stride before it resolves.

4. Skips in the cantus firmus-- especially skips of more than a third-- introduce the possibility of collapsing the distinction with the counterpoint. This problem is acute when the skip narrows the distance between the two lines. One way to deal with this in florid counterpoint is to "cover" the skip with quarter notes. A run of quarter notes keeps the ear focused on the counterpoint, and helps keep it distinct from the cantus firmus.

5. In florid counterpoint, Fux allows eighth notes on the second and fourth beats. I tried using them in a bunch of exercises and always hated the way they sounded. I didn't end up keeping any eights.

6. I don't yet know if it's the nature of the modes, or the particulars of the cantus firmi Fux supplies for them, but there's no doubt that some modes are easier to write and pleasanter sounding than others. I've found that Dorian, Phrygian, and Ionian modes usually come together pretty easily. Lydian and Mixolydian are tough slogs that often sound bland. Aeloian is sometimes hard to get right, but I usually like sound of the result.

7. A lesson I learned from Wire's "Pink Flag" that is recapitulated by counterpoint exercises: with music, familiarity fosters enjoyment. If I spend two hours getting an exercise into a form I like a lot, then let it sit for a few days, I might dislike it when I listen to it with fresh ears. But if I listen to it four or five times in a row, I start to like it again. This helps explain why so much old classical music is formulaic, and so much popular music is repetitious. When you aren't dealing with a 30-second snippet that can be heard six times in short order, you've got to find some other way to secure the familiarity that fosters enjoyment.

Marking my own work

It's getting harder to be confident in marking my own work. I've adopted a multiple pass technique that seems reliable so far. I'll edit this post if I end up changing my approach.

First pass: check for fifths, eighths, and unisons on the first beat, and make sure they're entered via contrary motion. (I'm especially prone to error when entering the penultimate bar.)

Second pass: scan through for bs and fs. They're always dissonant against each other, so double check any b-f harmonies. Also check for exposed tritones in the counterpoint any time a b or f appears at the peak or trough of a melodic run. If the counterpoint uses any accidentals (for example, in the closing formula) run the same set of tritone leap/exposed tritone checks on the accidental.

Third pass: locate all dissonances, and double check that they are entered and left stepwise, or are part of a cambiata formula.

Fourth pass: scan through the counterpoint and note every jump larger than a fifth. Double check that none of those leaps are forbidden. Also double check that any leap larger than a fifth is "recovered."

5th Species Dorian

Fifth species combines all the rules of the previous species: whole notes, halves, quarters, and ligatures. Moving from fourth species, in which there's next to no flexibility, to fifth, in which there's a huge amount of flexibility, is a time-consuming relief. With so many possibilities, I can mess around indefinitely with bars I don't like until I find an approach that's better. This first exercise, in Dorian mode, took more than an hour to write 11 bars, but I like the result:

4th Species, Dorian Mode

This species is also called "ligature," or "syncopation." Two half-notes are set against each whole-note, but the half-note on every downbeat is tied to the half-note that came before it. So, to the ear, we've got whole notes against whole notes, with the counterpoint offset by half a bar.

I've only done the Dorian exercises, so far. My impression is that the rules governing 4th species are much more confining that the previous species. Several times I found myself in a position where I had only one option in three or four consecutive bars. The impression of confinement is supported by the fact that my solutions are identical to Josephus's for big stretches.

I think 4th Species will sound wonderful when it's put together with a third voice, but with only two voices, it's a little bit dull:

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.