Harmonic analysis employs an ancient notational system called figured bass. While commonly required in Bach’s time, few contemporary musicians are even aware of it, and so instruction in figured bass is usually part of the preparatory training that I give to students who are new to harmony.
While Gustave could easily read a jazz chord chart, he required an introduction to the workings of figured bass in order to make use of the illustrations and exercises in his harmony textbooks. This email was the first in a short series of lessons through which he quickly mastered this very old art.
My comments are in black or grey Verdana , while Gustave’s are in Times New Roman.
I hope this finds you well, and in great spirits. This lesson is going to address one of the questions you brought up earlier:
. . . you can try some of Piston’s exercises. . .
I would love to, but I don’t understand how Piston’s figured bass works. I can easily and spontaneously create chord voicings (open and closed) from jazz chord charts, but I must be missing something here — I might have to go back to some basics.
Aldwell has some information on figured bass, and there are some decent explanations from other texts, as well. I'll scan and paste them here, and go over them with you. Before referring to them, though, I would like to make a few important points:
“Figured bass” — also known as “thoroughbass” and "basso continuo" — was
a practice used for certain accompanying, chordal instruments (mainly key-
boards and various precursors to the guitar) during the Baroque period (ca. 1600 - 1750). (In fact, some figured bass was also occasionally used by Mozart and his contemporaries.)
It was a specific kind of musical shorthand — analogous, in some ways, to today's jazz fake sheets — that made life a little easier for the composer, and allowed for some creative improvisation on the part of the performer. (Improvisation was far more prevalent in Baroque practices — especially in melodic and rhythmic ornamentation — than was the case in classical music from that time until the 1960s). In brief — the texts will explain it in more detail — a bass line was given, with accompanying figures — ie, numbers — written below each note. These numbers stood for intervals — specifically, the intervals to be found above the written bass note.
Usually, the intervals given would correspond to those found in a typical triad or 7th chord. Since the intervals in a “root position” triad (one in which the root is in the bass) are a 3rd and a 5th — both measured from the bass (this is always the practice) — the figures “3” and “5” would be written below the bass note — which, in this case, would be the root of the triad — like this:
. . . and would mean this:
. . . or any other voicing of this chord — eg, with open spacing, doubling, etc. — so long as it has its root as bass note.
The standard figured bass notation has the ‘figures’ stacked, one above the other, as shown. Because we’re using word-processing programs, and typing these lessons, however, we can ease up on tradition, and write our figures as — for example — “5/3”.
In figured bass (as well as in harmonic analysis), the interval indicated by a given figure can be either a simple interval — ie, any interval up to an octave in size — or its compound equivalent. For example, the figured bass “3” can mean a 3rd, or a 10th or a 17th, etc. — the choice is up to the player. The same is true in reverse: if you wish to write a figured bass for a given chord, a 3rd, 10th, or 17th within that chord are all written as “3”.
Since any chord can be inverted one, two or three times (first inversion means that the 3rd of the chord is in the bass; second inversion means that the 5th of the chord is in the bass; third inversion means that the 7th of the chord is in the bass), the corresponding intervals change accordingly. For example, the intervals in a first inversion chord — measured from the bass, again — are a 3rd and a 6th. The figured bass is therefore written:
. . . which means this:
The intervals in a second inversion triad are a 4th and a 6th:
. . . which means this:
The figured bass for 7th chords and their inversions is a little more complicated, but is addressed in Aldwell (if not in the material that I'll paste here).
Since figured bass was used all the time, during the Baroque period, composers and musicians developed a further simplification for it: since everyone knew that the intervals in a root position triad are always a 3rd and a 5th, the simplification was to write no figure at all under a root position bass note. For example, in figured bass, this:
. . . means:
So, a bass note with no figure beneath it means that that bass note is the root of a chord.
Along the same lines, the first inversion chord was figured as “6” — instead of “6/3”. Therefore, a bass note with a “6” beneath it automatically means that that bass note represents the third of a chord, and that there will be two notes — one at an interval of a 3rd, and one at an interval of a 6th — above it.
(Please note that these numbers refer to intervals, and not to chord members. In the case of a first inversion chord, we see a bass note — which represents the third of the chord. The first figure — the “6” — means that the next note is an interval of a 6th above that bass note — it doesn't mean that it's some kind of “6 chord”!. It turns out that that note will be the root of the chord — since the root, in a first inversion chord, is found at an interval of a 6th above the bass.)
The figures for a second inversion chord are always written in full:
Figures can also refer to notes that do not belong to a triad or 7th chord — they can also indicate suspensions, anticipations, passing notes, etc. For example: let's say you want to write a Dsus, followed by an ordinary D major chord. In a non-simplified figured bass, this would be written as:
The dotted line simply means the 5th is held while the 4th resolves to the 3rd:
In simplified form (which is the usual form), the above would be written
“4 3” , since the 5th is understood.
In a way, a performer could read and play a figured bass without knowing the name or inversion of the chord being played — he could simply read the figures — a kind of “paint-by-numbers”. Of course, we also want to know everything about the chord -- not just its figures!
In both your harmony books and in the material that I'll paste here, you will have to distinguish between two different things: “figured bass” — as used during the Baroque — and the combination of figured bass and roman numerals that is used in typical “harmonic analysis”. True figured bass does not involve the use of roman numerals — it simply uses the figures, as described above. However, almost all harmony text books utilize roman numerals along with figured bass, to show not only the scale degree upon which the chord is built (eg, ii, IV, etc), but also the inversion that is being used.
Okay, on to the different texts. First, from Basic Harmony, by David Cameron:
Here is another perspective on the same thing, from our second text — Harmonic Practice in Tonal Music, by Robert Gauldin:
Why not try what the author next suggests — and then email me your results:
Now we get to Aldwell — page 53 in my version — from chapter 4 (Triads and Seventh Chords). Here Aldwell goes over specific procedures pertaining to writing figured bass. Note that in illustration 4-9, Aldwell intends that you understand that he means the performer would be playing both clefs:
All of this is, of course, a brief summary.
Here are the steps I would suggest you take, in order to begin to assimilate figured bass:
- Memorize both the full and simplified figures used for root position, first inversion and second inversion chords. If you would like to make up some exercises — eg, write a chord progression in full, then re-write it as figured bass — please go ahead, and then email me the results, so that I can check it.
- Study and memorize the full and simplified figures used for seventh chords — go to Aldwell (same chapter), paragraph 14: “Inversions of seventh chords”. If you have any questions about any of it, please ask me!
- When you think you've assimilated the basics, try the following exercises:
- Exercises “A” and “B”, by Gauldin (above).
- Try to ‘realize’ the following figured bass, by Handel.
- the circled notes in the figured bass are not meant to be ‘realized’ — they are non-harmonic tones — passing notes, appoggiaturas, etc.
- I’ve entered a few ‘figures’ — in parentheses — that would not have been necessary during Handel’s day, but that may make your task a little easier. (I’ll explain the thinking behind the original practice, in the future.)
Send me your work when you've completed it. And please feel free to ask me any questions as you go along.
Last, here's a figured bass from one of the most beautiful pieces of music in existence — it's from the opening of Bach's St. Matthew Passion. The cello and organ play the figured bass — it’s on the lowest staff, with figures highlighted in yellow — but all the other instruments (and voices) realize it on their own, too. You can try to realize the figured bass, and then see if you're correct, by examining what everyone else is playing.
Note: the entire passage involves the use of a “tonic pedal” — read about it in Piston and Aldwell, and ask me about it if necessary.
Please don’t hesitate to email any questions you may have. . .
. . . and have fun!
All the best,
Michael Leibson is a composer, music consultant, and music educator, who specializes in jazz and classical harmony. To learn more about him, click here; for information about studying with him, click here; and for copyright information, click here.