The "Game" of Counterpoint: A Tutorial on CANTUS FIRMUS

The following material is repurposed and largely extracted from a scripted slide show I researched, wrote and presented in video form, approximately five years ago. A comprehension of how to create a short cantus firmus line is basic to mastery of the twin arts of counterpoint and harmony. So, I deem this material to be of sufficient value to not leave sitting idly on a computer storage drive, and have modified and adapted that material to this pictorial essay form. Unfortunately, I have not as yet worked out an efficient way to provide the brief audio examples on this site, so the reader is advised to also personally play the notated examples on a piano-type keyboard. Please feel free to comment and share suggestions below.

Before you play the game of chess (or any other game), you must first learn the Rules, then apply those Rules every time you play the game. From then on, mastery is a matter of forethought, application, practice, mentoring perhaps, and time.

"Come, Watson, come! The game is afoot,” exclaimed the great fictional detective, Sherlock Holmes. Our investigation of musical harmony leads us to the topic of counterpoint, which requires our learning a set of game Rules. So, “Come,” let’s begin!

W hat is the result of playing or singing two or more musical tones simultaneously? Harmony. Chords! Like those written above. Try playing them on your piano-keyboard.

Next, what happens when we play or sing two or more melodies at the same time?

Hopefully, harmony again! Above, we have divided up the same previous chords into four separate voices. Now, try playing them again! We today may not think of a chord progression as being a synthesis of separate musical lines – or melodies – but centuries ago, such a horizontal, or “linear” view was the original concept of harmony and the generator of many of the Rules we meet in music theory class.

Our “Rule Quest” begins with a serious look at linear things: MELODY, and VOICE LEADING. The ancient concept of harmony was NOT the modern view – i.e., not vertical stacks of notes – but horizontal, linear. Therefore, WHAT IS this “voice leading” all about?

As you see above, Voice Leading, a.k.a. counterpoint, is a scientific approach to the vertical relationships between multiple horizontal lines. We may say that a bass line ACCOMPANIES the melody of a song. That means there are TWO independent lines – melody, and bass – and when properly synchronized, they imply chords. To experience this phenomenon, play and listen to the above example on your piano.

Like an architectural structure, each musical line, whether melody, or bass, has a design. We can analyze structures in theoretical terms. First, we’ll study guidelines for making a simple, coherent, no-frills, musical line, called a CANTUS FIRMUS, which means “fixed song.”

For learning purposes, we’ll treat cantus firmus as a “mini-game” within the larger context of the game of counterpoint. A cantus functions as a scaffolding for synchronizing other voice lines in counterpoint. All writers of melody will benefit from understanding the basic principles of cantus firmus construction.

A cantus firmus (CF) is a meaningful series of tones with no accompanying chords, and no distinctive rhythm. Two modern music theorists and authors, Felix Salzer and Carl Schachter, have said that CF is an “abstraction from melody in which the linear element is separated from the other aspects of melodic design.” – p. 3; italics mine. – See references below.

Ground Rules

First and most important, the individual tones of a CF are all the same duration; there is no metric emphasis – no downbeats or upbeats – and no memorable rhythm. As above, our first cantus firmi (pronounced “fear + my”), will only have notes of equal rhythmic duration, e.g. all whole notes, a.k.a. semibreves.

1. NUMBER OF NOTES – A proper CF is like a little story, having three arbitrary segments: “Once upon a time,” “this happened,” and “The End.” According to Salzer and Schachter, a practical CF of between eight and 16 tones will suit our work here just fine.

1. VOICE RANGE – So, do we just “toss together” any old series of eight to 16 whole notes? Well, no. We must think in terms of being able to SING the CF within the range and capacity of someone’s voice, either our own or that of another singing being.

1. SPAN – Most examples of actual cantus firmi lie within the span of one octave. Regardless of voice range, our studies will limit the span of each CF to between a Fifth (smallest), and a Tenth (octave + third), from lowest pitch to highest pitch.

Intervals - Acceptable and Unacceptable

The human voice is not a mechanical musical instrument. Instruments are machines and usually have no trouble playing large leaps, dissonant intervals, and chromatic intervals. These are not easy for voices to sing, and must NOT be included in our cantus firmi. So, …

1. There (above) are THE ONLY ACCEPTABLE INTERVALS, according to the Rules of our “Cantus Firmus Mini-Game”: Perfect Octaves, Fifths and Fourths, Major and Minor Sixths and Thirds, Diatonic Whole- and Semitones (i.e., Major and Minor Seconds). Abbreviated: Diatonic P8, P5, P4, M6, m6, M3, m3, M2, m2.

1. UNACCEPTABLE INTERVALS – Unacceptable are Compound Intervals (those larger than an octave), Any Seventh, Any Diminished or Augmented Interval – including the Augmented Second – and the Chromatic Semitone (i.e., Augmented Prime).

Analysis of Melodic Intervals in a Cantus Firmus

Now let’s analyze the short CF above. After a leap of a P4 at the beginning, the line descends a m2, then a P5. Climbing a M2, a m2, another M2, it skips down a m3 to the LT “B,” then rises a m2 to cadence on the final, the tonic.

For your information, the example above is written with a “C-Clef” and the first note is Middle C. So to sing or play it, you begin with the middle C, or “DO,” and play or sing the correct intervals from that note on. We'll save a discussion of the use and usefulness of C-Clefs in another essay.

Incidentally, that funny-looking note in the last bar is called a “breve.”

The same cantus is shown above. Are there any “Unacceptable Intervals” in it? Any 7ths, diminished, augmented, or chromatic half-step intervals? Not at all! By the way, the breve is the same duration as two whole notes, or two semibreves.

In another article, we’ll provide more information on the science behind cantus firmus and this game of counterpoint. If you have any questions, by all means post a comment! If you like this material, be sure to say so and also share the link with a friend.

Food for Thought – Though we have more Rules to learn, you should begin experimenting now. You may write in whatever mode, key and clef are appropriate.

Here is a summary of things to do, as covered in this essay:

• Choose a tonic key; start and end with the tonic pitch.

• Create one or more brief cantus firmi, based on the Game Rules regarding Number of notes, Range, Span, and Acceptable Intervals.

• Practice playing and singing your original CFs for friends, family, and other students. If you are curious about how to add words to your cantus, consider using the one-syllable-per-note approach, and have fun with it!

Tap/click here to Read Part 2

REFERENCES CONSULTED

• Jeppesen, Knud. Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century. New York: Dover, 1992.

• Mann, Alfred. The Study of Counterpoint, from Johann Joseph Fux’s Gradus ad Parnassum. New York: Norton, 1965.

• Salzer, Felix and Carl Schachter. Counterpoint in Composition: The Study of Voice Leading, pp. 3-12. New York: McGraw-Hill, 1969.

• Schubert, Peter. Modal Counterpoint, Renaissance Style. New York: Oxford, 1999.

FLICKR IMAGE CREDITS, all commercial use and mods allowed, in order of appearance:

• Glass Chess Pieces 8-26-09, by Steven Depolo: https://flic.kr/p/6Tbyvm

• Sherlock Holmes, by Mark Hillary: https://flic.kr/p/76VNKG

• Cougar Looking Serious, by Eric Kilby: https://flic.kr/p/q6SZSC

• Mackinaw Bridge, by Srinivas Anupoju: https://flic.kr/p/8eeSGP

• Cantus Firmus (screenshot), p. 2 of Manual of Counterpoint, by Salomon Jadassohn: https://archive.org/details/manualofsingledo00jadaiala

• Cantus Firmus (screenshot), p. 45 of Gradus ad Parnassum, by Johannes Fux (reprint of original, in Latin):

• Wiener Sangerknaben (Vienna Boys Choir), by My Past: https://flic.kr/p/wJWEy