During the Baroque era at the time that functional harmony was a precursor of tonal harmony a system of musical shorthand called continuo, also known as basso continuo, Generalbass in German, or figured bass (thorough bass) in English, was in use by composers as a form of musical shorthand to indicate which notes above the written bass note were to be played by the right hand to fill out the harmony.
In those days this technique was used very often because it save composers time, and it saved them money because paper was so expensive at the time, so they simply wrote out the bass line and put in some numbers under them to indicate the other notes that belonged with it.
So this is an abbreviated system in which there is only a bass line written for the left hand, below which are Arabic numerals; based on those numbers the player then fills in the chords above it with their right hand.
How these chords are filled in depends upon the tempo of the piece; if the tempo is fast, then only chords are sufficient, but in a very slow tempo more can be added, such as thirds and sixths.
These intervals sound nice and are good intervals for functional harmony whereas the perfect pure intervals sound empty and lead to faulty progressions when in parallel motion.
Major and minor 3rds and 6ths however are the most beautiful in tonal harmony and could be used in alternation or parallel motion.
When J.S. Bach would play continuo, it's said that he would add one extra voice, always ... one completely thorough composed voice; if it were a duet, he would add a trio texture; if it was a trio, then a quartet would sound.
Bach was supremely gifted at this and could think very fast horizontally as well as vertically, which is an amazing, very hard thing to do.
Most people, if they had to do it in written form, could probably do this too because they would have time to think about it, but if they had to do it on the spot while sitting at the instrument it would be very difficult for them.
This is because the closed position (of harmony for the right hand) is so perfect for continuo playing.
It's a matter of practice, of course, how fast we can think, and to be able to think as fast as we can play, but for starters it's plenty good enough to be able to add the correct notes to the bass line while given only numbers below it.
Actually, this isn't quite as hard as it sounds because we just have to think about the melody that our other voices are playing.
It's important for a serious student of the organ to become familiar with continuo and its realization.
It's necessary for an understanding of the inversions of chords and for the playing of certain scores from the repertoire, as well as being part of the language with which musicians communicate with one another.
The top staff (photo) shows the realization of the continuo notated in the bottom staff, in this example.
The first bass note is C, unfigured; this refers to a root position of a C major chord, with the root in the bass; if there were a sharp (#) sign, flat (b) sign, or "natural" sign, all by itself above the bass note, it refers to the 3rd of the chord being altered by that accidental.
A root position of a chord always has the root of the chord in the bass and is unfigured.
So an unfigured bass note always calls for the root position of the major or minor chord above it.
This root position chord will be major or minor depending upon what accidentals are, next to the clef, i.e. the key signature.
Sometimes composers would add additional accidentals to the clef by writing a "#" sign, "flat" sign, or a "natural" sign above the bass note into the notation for the numbers.
This would always refer to the 3rd of the chord.
For example, an unfigured bass note with a "#" sign below it would mean that the performer would have to raise the 3rd from the bass a half step.
The second bass note is an E, figured with a 6; this means that the root of the chord is a 6th above the bass note; in this case, a C major chord again, this time with E in the bass, or a 1st inversion.
A 1st inversion of a chord always has the 3rd of the chord in the bass.
So a bass note figured with a 6 always calls for the 1st inversion of that major or minor chord.
The third bass note is a G, figured with a 6-4; this means the root of the chord (C) is a 4th above the bass note, and the other note in the chord (E) lies at a 6th above it; this is a C major chord again, only this time with the 5th in the bass, or a 2nd inversion.
A 2nd inversion of a chord always has the 5th of the chord in the bass.
So a bass note figured with a 6-4 always calls for the 2nd inversion of that major or minor chord.
In the case of 7th chords, which are 4 note chords, the root position dominant 7th chord is figured simply with a 7 to indicate root position.
Technically, this is a 7-5-3 chord, but for the sake of simplicity, it's figured a 7 chord.
Any bass note figured with a 7 indicates a root position dominant 7th chord.
In a 1st inversion 7th chord, the 3rd of the chord is in the bass, so it's figured as a 6-5 chord, meaning that the root is a 6th above the bass note, and the 7th is a 5th above.
Technically, this is a 6-5-3 chord, but since the 3rd of the chord is implied, it remains unfigured.
So any bass note figured with a 6-5 implies a 1st inversion dominant 7th chord.
The 7th chord in 2nd inversion has the 5th in the bass and is figured a 4-3 chord to indicate the root is a 4th above the bass note and the 7th is a 3rd above.
Here again, technically, this is a 6-4-3 chord, but the 3rd of the chord is implied, therefore unfigured.
Therefore any bass note figured 4-3 implies a 2nd inversion dominant 7th chord.
In a 3rd inversion 7th chord, the 7th is in the bass; this is figured as a 4-2 chord, or, more commonly, simply a 2 chord, to indicate that the root is a second above the bass (and the 3rd of the chord is a 4th above).
Technically, this chord is a 6-4-2 chord, but the 3rd, being implied, is unfigured ... and sometimes even the 5th of the chord, being implied, is also left out of the figuring.
So any bass note figured 4-2 or 2 indicates a 3rd inversion dominant 7th chord.
Any bass note figured 5-4 indicates a suspension, which leads to 5-3, so it's always necessary to count from the bass up.
All of this seems like rocket science at first, but it's just a matter of practice and familiarity.
On the organ, the bass note would be played in the pedal usually, with the other notes taken in the right hand.
On an instrument without pedals the bass would of course be taken in the left hand.
In some older music, composers have added a soprano line for the right hand (2 voices, for the soprano and bass) in the continuo part, so the player only has to add the 2 middle voices (alto and tenor) to realize the implied harmony.