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I am working through a figured bass exercise in the "Ninth, Eleventh and Thirteenth Chords" chapter of Piston's Harmony.

The first bar of the figured bass is given as: enter image description here What is the chord represented by the G# in this figured bass? My usual approach is to assume that if I don't recognise something, it means it is likely to be one of the new types of chord introduced by the chapter I'm working through. This would mean a 9th, 11th or 13th chord of some sort. However, in this instance, none of those seem to fit.

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Check out Piston's Example 24-12 on page 380. Note that the second through fourth chords progress from a root-position tonic triad through this 6 5 4 harmony and into a first-inversion tonic triad. I've recreated those three chords in the example below, before the first double barline:

enter image description here

So his intent seems to be that this 6 5 4 chord is a second-inversion V9 chord. I've transposed this into F# minor after the first double barline to show one possible realization of this.

Note that the first example has parallel perfect fifths between the first two chords, but in minor this motion is a P5 to a d5 (which some, including myself, have no problem with). This is really impossible to fix when you start off with a doubled chordal fifth, so the third excerpt in the example is a better voice-leading realization. (Even then some people will get fussy with "similar fifths" or "hidden fifths." Ignore them.)

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The notes I derive from this figured bass, from the bottom up, are G#, C#, D, E#. Stacking these in thirds, we get C#, E#, G#, D. This creates a C# major with a 9. I would label this a C#b9, however, I'm coming from more of a Jazz tradition in this explanation and clearly figured bass is not a Jazz thing but studied in a Classical setting. I believe in the Classical world, you wouldn't specify the 9 being flat, as it is a diatonic note, however, the frequent use of melodic minor in the Classical world might call for some distinction as to the quality of the 9.

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The numbers in figured bass represent the interval distance away from the given bass note. To find the notes in the chord, simply use the numbers in your figured bass to count upwards, stacking those notes in the same order as the numbers of your figured bass.

Thus:

6# = E#

5 = D#

4 = C#

As for what it represents harmonically, that's an interesting question. The figured bass you presented doesn't reflect any of the normally-seen inversions (6/5/3, 6/4/3, 6/4/2). To me, I wonder if we're not getting the full story here - that one of those numbers is perhaps setting up a suspension through anticipation? (The Ex and the D# could be both suspensions for a C# triad, while the C# could in fact be a 4-3 suspension for G#, creating a secondary dominant).

It is difficult to say with any certainty the chord's function without context. Perhaps a picture will help. (I'll edit my answer accordingly).

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    I would agree with Wibbs on this. Since the E is not # in the key signature, adding a sharp would make it E#. This also makes sense in the context of the key, where E# is the leading tone. Additionally, it would be highly unlikely to find an Ex in F# minor, as its enharmonic equivalent is F#. If this were some sort of suspension, I would expect it to be spelled with an F#, which would resolve to an E# or, less likely, an E. – Basstickler Feb 2 '17 at 14:58

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