I've done some jazz rehearsals and while other members were discussing chords I overheard them talking about an 'Italian sixth', 'German sixth' and a 'French sixth.' I have no idea what they are referring to, it could just be the language (lingo jingo) of the musicians. So if anybody knows what they were referring to, what is an Italian sixth, German sixth and a French sixth? Is the term a broader thing? Example: There could be Italian sevenths too?
These are tone combinations often used in classical music. They consist of a bass tone (usually a major third below the tonic), a major third above the bass, and an augmented sixth above the bass and some other tones. They are the same chord in major and minor. I'll use the key of C for examples.
The Italian Sixth is Ab-C-F#. It's written this way because of the usual resolution in classical music. This would be to G-B-D-G (The C resolves both directions here.) Note that the Ab and F# each move by half steps which makes is a strong melodic and harmonic motion. It's possible to resolve this chord to G-C-E-G before proceeding to the Dominant. These three chord have (in classical practice) a strongly Subdominant function (sometimes called a pre-dominant function).
The French Sixth has an added fourth above the bass: in C this gives Ab-C-D-F#. This collect can resolve the same way to either the Dominant in root position or the Tonic 6-4 thence to the Dominant. (In Jazz theory, this chord is a triton substitute for the Dominant and would resolve to the Tonic. In this case, the bass would be Db rather than Ab in the bass in the key of C. Classical music can do this but it isn't common.) This chord has two tritones which makes it strongly dissonant.
The German Sixth has an added perfect fifth above the bass. Again in C, this chord is Ab-C-Eb-F#. It generally resolves through the Tonic 64 to avoid parallels. The German Sixth is enharmonically identical to a Dominant seventh on Ab and could be resolved to Db (non-standard to not too uncommon.) This chord is also the Dominant of the Neapolitan Sixth (F-Ab-Db-F in C) and can be used as such.
In a cadence (like ii-V7-I or ii06-V7-I for Major or Minor respectively), these chords sound good between the ii chord and the V7 chord (or V chord) and one can eve have strings like ii6-It6-I64-V7-I which extends the cadence. (The chords are often notated as It6 Fr6 and Ge6.)
They are often approached from the Tonic, the minor Tonic, or the Minor Subdominant.
This chord pattern can be used on other scale degrees. The chords are "rootless" in classical theory as they are perceived (by the theorists anyway) as arising from melodic rather than formal harmonic processes. They are not uncommon on a flattened second degree where they often proceed directly to the Tonic somewhat as in Jazz harmony.