3-Part Counterpoint
First Species 3-part counterpoint and the creation of the Harmonic Triad
This final chapter on counterpoint serves as a transition between two-art modal counterpoint and our work with triadic harmony, the central concept on Western classical music.
Discovering the Harmonic triad
If you think back to the Harmonic Series, you will notice a curious phenomenon. The fourth, fifth and sixth harmonics in the series form a lovely sonority: a perfect fifth "filled in" with a major third above the lower tone. This is the "harmonic triad" that Fux refers to at the beginning of Part Two in his Gradus ad Parnassum, and forms the basis for all of Western harmony.
It is important to point out that the use of this triad in musical composition pre-dates the actual scientific discovery of the harmonic series. So it is not true, as you will sometimes read elsewhere, that the harmonic triad is employed because it is part of the harmonic series. But it is true that is was employed in music because of these two properies:
- The Harmonic Triad can be dervived from Just intonation, the 5-limit system that allows integer ratios up to and including 6 (one less than the next prime number of 7). When we create musical ratios based on the 5-limit system, the major triad becomes a fundamental harmonic building block. As we'll see in the next chapter, the minor triad can also be dervied from the same series, taken as the reciprocal.
- The Harmonic Triad is also aesthetically pleasing to the ear. Since the early Middle Ages, European composers of tuned their ears to the beauty of the major and minor third and how it provides a certain "softening" to the open quality of the perfect fourth and fifth. Indeed, this interval was first employed outside of strictly composed music, finding its first expression in the songs of the ancient bards of England and northern Europe. This pleasing sound, perceived to be more earthy, and earthly, in contrast to the heavenward and "heavenly" music of the perfect consonances, was used to express the more human (and simple) emotions of love, empathy, joy, celebration, indeed, inebriation! These sounds found their way from the street and the ballad into the more formal work of the trained composers (more formal only in that composers were skilled at musical notation whereas the Celtic bards were not -- not so different from today, where rock and folk songwriters write from the heart directly to chords and melody, often not able or not interested in writing them down in formal notation.)
Here, since in this course we are learning the more formal art of notation, we will see how the harmonic triad grows out of our work with modal counterpoint. In the next chapter, with its multiple sections, we will see how this harmonic triad is used to create Functional Harmony.
The Harmonic Triad in Modal Counterpoint
In 3-part modal counterpoint, we seek to create as many harmonic triads as possible through the creation of a third and a fifth above the lowest tone. However, as a constant string of 1-3-5 triads does not lend itself to good voice leading (there would be too many parallel fifths or parallel octaves), one can often substitute a sixth or an octave in place of the fifth, which will still result in consonant harmonies.
Thus, there are two possible harmonic triads: that of 1-3-5, and that of 1-3-6. In this style, the '1' is always in the bass, while the third and the fifth or sixth can appear in either of the other two voices.
In Fux's first example, he presents a simple rising bassline, which he harmonizes in two different ways (the second one suggested by the student)
You see here how in the first version, the middle voice moves in smooth contrary motion to the bass, creating some full harmonic triads, but other triads have a doubled octave. In the second version the middle voice jumps down to the low A, creating an additional full harmonic triad.
Notice how the upper notes of the full harmonic triads can be placed in either voice — the third can appear in the middle or upper voice, and the fifth/sixth is then placed in the remaining voice above the bass note.
(In this and all subsequent examples, the intervals are written in as they occur above the bass. Thus in the first measure, the 10 refers to the E an octave and a third above the bass C, while the 12 refers to the G and octave and a fifth above the bass C. This carries on for all other measures.)
Creating 3-part first species counterpoint using a cantus firmus
The process of creating 3-part counterpoint with harmonic triads, as outlined in Fux, is as follows. This will use the same Dorian c.f. by way of example.
Here are his examples, with the cantus firmus above, with the cantus firmus in the middle voice, with the cantus firmus in the lowest voice. The player below will play these straight through. Listen to the sound quality of these simple harmonic structures, and try to carefully hear the voice leading in each case.
What can we observe here? In each case, harmonic triads are created, sometimes using 1-3-5 above the bass, and sometimes 1-3-6. As before, in order to make the voice leading smooth, somtimes the octave is used instead of the fifth or sixth, creating a triad with two roots and a third.
This is the basic process that you will use in completing your 3-part first species counterpoint exercises.
3-part Counterpoint, second species
As you can probably guess, second species 3-part counterpoint is a simple combination of second species 2-part counterpoint combined with the triadic harmony of third species.
In this style, one voice is the cantus firmus, one voice remains in 1:1 counterpoint with the c.f., and the third voice is allowed to move in a 2:1 rhythm, sometimes creating dissonances on the off beats. A few examples will suffice to demonstrate this style, and you can then infer enough to create a few of your own examples.
This concludes our study of modal counterpoint. If this were
a class dedicated to the modal Renaissance style (something you may take
if you are a music major and go on to graduate school), this study of counterpoint
is enough to provide a good foundation for the study of Functional
Harmony, the subject of Chapter Five.