Why is there no full triad as the last chord in 4 part counterpoint?

Question

I read that it is impossible to have a full cadence using all three notes of the final triad in a 4 part counterpoint piece. Here is the exact bit of info:

"we can’t easily arrive at a full triad in the final measure. The bass and alto (or whichever voice has the CF) must both go to ^1, the tenor is free to go to ^3 or ^5, but the soprano usually has ^7 in the penultimate V chord, so the leading tone must resolve to ^1. It is very normal to end on an incomplete I chord for these reasons."

Can someone please clarify if this is true and why? Most music I have listened to always has a full triad at the end of a cadence. And when I studied harmony it has always been a V > I progression, meaning that the whole I chord is used. Where am I getting confused?

Answer 1

Although it's not impossible for both chords to be complete, it does require specific circumstances. The quotation offers a broad explanation for why this is.

First, the quotation implicitly assumes the bass voice is not the cantus firmus. The "rules" state that the bass must move to 1 and so must the CF. Thus, for a complete I chord, the other two voices have to land on 3 and 5.

The quotation makes a second assumption, which is that the soprano voice most often moves from 7 to 1. Thus, if the soprano moves from 7 to 1 and is not the CF, we now have a tripled root (because bass, CF, and soprano all proceed to 1), so a complete I chord is impossible.

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As an example, I went through every possible combination of V - I cadences in which the bass and tenor start off doubling the root of the V chord. Of the 16 possible ways to move from there to the I chord, only 4 result in complete I chords. The remainder are either incomplete or disallowed by various voice-leading rules.

X: 1
T: V - I cadences, both triads complete
T: Tenor and Bass double the root of the V chord
M: 4/4
L: 1/1
K: C
%%score {(V1 V2) | (V3 V4)}
V:V1 
V:V2
V:V3 clef=bass
V:V4 clef=bass
[V:V1] D | E |] D | E |] D | E |] B | c |]
[V:V2] B,| C |] B,| G,|] B,| G,|] D | E |]
[V:V3] G,| G,|] G,| G,|] G,| E,|] G,| G,|]
[V:V4] G,| C,|] G,,|C,|] G,, C,|] G,|C, |]

(Note that bass and tenor share the same pitch in the first and last cadences. This is to avoid a hidden octave between the bass and upper voice moving from B to C.)

Answer 2

"we can’t easily arrive at a full triad in the final measure. The bass and alto (or whichever voice has the CF) must both go to ^1, the tenor is free to go to ^3 or ^5, but the soprano usually has ^7 in the penultimate V chord, so the leading tone must resolve to ^1. It is very normal to end on an incomplete I chord for these reasons."

Somehow that quote seems to be about a specific example. It's so indirect about what exactly the penultimate chord is I don't see why you would formulate a rule from it. At the very least normally ending on an incomplete tonic does not necessarily mean it is impossible to end on a complete tonic chord.

If you really want to study species counterpoint, you need to get Fux's Gradus ad Parnassum. It's the original source for species counterpoint.

Fux's four part examples end with chords voiced both with roots and fifth only and also with several examples of root, third, and fifth, full triads.

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Here are Fux's four part 1:1 example endings with full triads. All transposed to C to compare the variety of voicings and melodic moves. All melodic moves are by step except the occasional ^7 to ^5 move which happens in the tenor, alto, and soprano.

Notice how the scale tone movements in Fux don't match up to the description in your quote? The quote implies ^7 is a leading tone, meaning the tone one half step below the tonic. Based on that, the stuff you're working from is probably species counterpoint applied to major/minor harmony. Fux's harmonic style is modal, and so the examples in phrygian mode have a ^7 a full step below the tonic. At least in these few examples when ^7 is a leading tone it does move up to ^1, but when ^7 is a subtonic in phrygian it either moves up to ^1 or down to ^5.

I think you should talk to your teacher about the quote and try to better understand the context in which it was given. If it's part of species counterpoint method, it should be reconciled with Fux. Fux is the original source for species counterpoint. And Haydn, Mozart, and Beethoven all study counterpoint from Fux!

Answer 3

This doesn't make sense. In a G>C (or G7>C) cadence, you can have G>C in bass, D>E or F>E in tenor, G>G in alto, and B>C in soprano. The D note (V5) does not have a unique resolution, so D>C and D>E can even both be used in the same cadence if you have a step down in soprano. The only unique resolution in a V>I and V>i cadence is V7>I3 (or i3) and V3>I1.

Another option for a D>C soprano is G>C bass, G>E tenor (the "alto cadence"), B>C alto and D>C soprano. It's completely fine to have a 3rd down in the alto cadence (in the tenor voice in this case), going from V1>I3. This is quite cool as you don't have I5 altogether; 5ths in general don't sound well in tune, so you avoid having this out-of-tune feeling in the final chord.

In a lesser number of voices, you need to cover at least one of the sensitive notes (V3 or V7), and then you might miss something. Still, you can have both: F>E (V7>I3) in the lower voice and B>C (V3>I1) in the upper one and it will work just fine.

The idea that alto has to go to I1 is a complete non-sense.