(This is a technical article for musicians. Level: intermediate to advanced.)
When we think about harmony, we generally think about tertian structures, that is to say about chords built by stacking thirds. By stacking two thirds we get the so-called “triads,” and adding one more third we get the “seventh chords,” For centuries, these 3- and 4-note chords have been used so extensively in both classical and popular music that harmony has become synonymous with tertian harmony. But tertian harmony is actually just one of five possible harmonic species. In this article, I will introduce the non-tertian species. First of all let’s review the basic principles of tertian harmony.
Using the C major scale as a reference, let’s build the relative 3-note diatonic tertian chords. The principle is simple: tertian chords are obtained by stacking thirds “diatonically,” i.e., including only scale tones. So, we consider each scale degree as a root and add two thirds on top of it to form the common tertian triad.
On the first, fourth and fifth degree, we have major triads (CEG, FAC, GBD), symbolized by a capital Roman numeral; on the second, third and sixth degree we have minor triads (DFA, EGB, ACE), symbolized by small Roman numerals; and on the seventh degree we have a diminished triad (BDF), symbolized by a small Roman numeral with a little circle added to it.
Seventh chords are obtained by adding one more third on top of each triad. So, CEGB on the first degree, DFAC on the second degree, etc. If we add another third, for a total of four thirds, we get ninth chords (e.g., CEGBD). Five thirds create eleventh chords, and six thirds thirteenth chords. If we add a diatonic third above the thirteenth, we’ve added an additional root. Therefore, the thirteenth chord is the most extended type of diatonic tertian chord: it’s a 7-note chord including all the notes of the major scale.
So that’s tertian diatonic harmony. (You can easily verify that these considerations are also valid for other common 7-note scales, such as the melodic minor scale and the harmonic minor scale.) Tertian harmony is a “homogeneous” kind of harmony, because chords are built by stacking intervals of the same type, in this case thirds. It’s easy to see that stacking sixths does not produce different chords. For example, if you stack two diatonic sixths on top of C, you get CAF, which is just the F major triad in second inversion. Similarly, if you stack three, four, five or six sixths, then you still get seventh, ninth, eleventh and thirteenth chords, in a different inversion. This happens because the sixth is the inversion of the third. The result is that the sound of chords by sixths is still tertian.
Thirds and sixths apart, there are four more diatonic intervals: seconds, fourths, fifths, and sevenths. However, again, we must observe that the seventh is the inversion of the second, and the fifth is the inversion of the fourth. So, chords built by stacking sevenths have the same basic sound (the same harmonic texture) as chords built by stacking seconds. These types of chords form “secundal harmony.” And I will call 3-note chords by seconds “secundal triads.” Here is the C major scale harmonized by secundal triads.
Similarly, chords built by stacking fifths have the same basic sound (the same harmonic texture) as chords built by stacking fourths. This is called “quartal harmony.” I will call 3-note chords by fourths “quartal triads.” Here is the C major scale harmonized by quartal triads.
1) Tertian harmony: stacking thirds or sixths.
2) Secundal harmony: stacking seconds or sevenths.
3) Quartal harmony: stacking fourths or fifths.
Of course, just as in the case of tertian 7-note chords (thirteenth chords), a 7-note chord by seconds or fourths includes all the notes of the scale. The difference between a particular 7-note tertian chord and a 7-note secundal or quartal diatonic chord built on the same root is exactly the texture, secundal or quartal, respectively. The more you alter the basic texture, the more this difference is blurred. For example, if you rearrange the notes of a 7-note quartal chord so that in the voicing thirds prevail over fourths, then the quartal sound gives way to a tertian sound.
We have therefore derived two more harmonic species, in addition to tertian harmony: secundal harmony and quartal harmony.
Exercise: build the 4-note diatonic secundal and quartal chords and analyze them.
But, as I said, there exist five possible harmonic species. So, there are two more species to consider. These are not “homogeneous” species, because their chords are not formed by stacking the same type of interval. In one case, we build chords by alternating seconds and thirds. I will call it “mixed secundal+tertian” harmony. In the other case, we alternate thirds and seconds. I will call it “mixed tertian+secundal” harmony. Notice the effect of the stacking order: “root, stack a second on top, stack a third on top” (getting CDF) is of course different from “root, stack a third on top, stack a second on top” (CEF). Again, I will call “triads” the 3-note chords within these two harmonic species. Here, for example, is the C major scale harmonized by mixed secundal+tertian triads.
And here is the C major scale harmonized by mixed tertian+secundal triads.
A striking characteristic of these two non-homogeneous (i.e., heterogeneous) harmonies is that a 7-note chord cannot be produced without repeating at least one note of the scale. In the case of mixed secundal+tertian harmony, the alternation of seconds and thirds can produce at most 5-note chords without repetitions (e.g., CDFGB). That is because, if we add a second on top of B, we land on C, which is already included in the chord. In the case of mixed tertian+secundal harmony, the alternation of thirds and seconds can give at most 6-note chords without repetitions (e.g., CEFABD). If we add a second on top of D, we land on E, which is already included in the chord.
If we allow repetitions, then we can build 7-note chords even within these two harmonic species. Notice, however, that, because of the repetitions, these structures have more than seven tones (you need nine tones to have all the scale notes in a mixed secundal+tertian chord; and eight tones in a mixed tertian+secundal chord).
Another, practical way to see that we have five different diatonic harmonic species is the following.
Number each tone of the C major scale with a number from 1 to 7. Then take the first degree as root and build 3-note chords on it by considering all the possible options.
1) 1 2 3 (e.g., CDE) = secundal triad
2) 1 2 4 (e.g., CDF) = mixed secundal+tertian triad
3) 1 2 5 (e.g., CDG) *
4) 1 2 6 (e.g., CDA) *
5) 1 2 7 (e.g., CDB) *
6) 1 3 4 (e.g., CEF) = mixed tertian+secundal triad
7) 1 3 5 (e.g., CEG) = tertian triad
8) 1 3 6 (e.g., CEA) *
9) 1 3 7 (e.g., CEB) *
10) 1 4 5 (e.g., CFG) *
11) 1 4 6 (e.g., CFA) *
12) 1 4 7 (e.g., CFB) = quartal triad
13) 1 5 6 (e.g., CGA) *
14) 1 5 7 (e.g., CGB) *
15) 1 6 7 (e.g., CAB) *
So, you can form 15 different 3-note chords. However, it is perhaps not too difficult to see that most of them are inversions of corresponding fundamental positions in a certain species (these inversions are indicated by asterisks). Actually, there are only five species of root-position 3-note diatonic chords, one for each harmonic species. This confirms the previous description of the harmonic species. Of course, the number of 4-note chords is larger.
Exercise: try to find as many diatonic 4-note chords as possible.
Optional: for each inversion, try to find its corresponding fundamental position. Remember: the inversion can be in any of the five harmonic species.
You can consider each scale degree as root, and get different kinds of 3-note or 4-note chords by adding different selections of diatonic tones. As we have seen, getting tertian triads or seventh chords is just one of five options.
Exercise: using the table above as a guide, build all the possible 3-note chords on each degree of the C major scale and take note of the characteristics of each.
I think this material can be valuable for composing, performing, or improvising. When you go beyond tertian harmony, you will find many options. But you don’t have to memorize all of them immediately. The best approach is to incorporate these chords slowly. First as substitutions in a mostly tertian context. Then by exploring each alternative harmonic species alone. Quartal harmony is especially common in much modern music. Mixing two or more harmonic species is also a way to give a special freshness to music. By comparing each alternative chord with common tertian chords one will find differences and similarities, which can help to select the most appropriate chords given the context. I suggest to concentrate on 3-note chords and take the time to internalize their sound.
Exercise: take any tertian major triad, on any root, and find the four possible 3-note diatonic triads from each of the other harmonic species, built on that same root. See if you can use them as substitutions for the tertian triad.
I hope this presentation is clear enough. Good work and have fun!