Doc Snow has been an online writer for over seven years. He's a lifelong musician who loves to record his own compositions.
This article investigates the patterns governing triads in second inversion. Here, you will learn to navigate the tricks and traps with video examples and interactive practice questions! Let's get to it!
It probably sounds as though second-inversion triads--also known as 'six-four chords'--are a bit different from root position and first-inverrsion triads. If so, then the impression given is entirely correct. Traditionally, chords were classified by the intervals they contained, reckoning things upward from the bass tone.
According to this view, root position triads were the most stable, since above their bass tones one would find a perfect fifth and either a minor or a major third. (Diminished and augmented triads are considered dissonant in this position, precisely because their chordal fifths are not the stable perfect fifth, but the decidedly unstable diminished or augmented fifth.)
Less stable are first-inversion triads; they contain a minor or major third, just like the root position triads, but substitute a sixth for the fifth. Both intervals are consonances, though the sixth is less stable than the fifth. Still, first-inversion triads may be used quite freely, as we have seen.
By contrast, second-inversion triads are characterized by an unstable fourth, as well as the more stable sixth. That fourth above a bass tone had, by the time when chordal theory was coalescing, been treated as a dissonance for many centuries. Therefore, the second inversion triad took on a somewhat dissonant quality too.
That does not mean it is not a useful sonority, however. It just needs a little care to use appropriately. We'll examine in turn the normal ways in which sixth-four chords appear in common-practice music.
Alternating and Arpeggio Six-Four Chords
These two patterns are closely related, and operate on the same psychological principle. In both cases, the root position voicing of the triad occurs first--and very often on a strong beat--and is then and only then followed by the second-inversion voicing. This sequence allows the instability of the six-four to be subsumed in a larger, and stable, context. So the second-inversion triad can freely follow the root position voicing of the same chord--even, in a pinch, the first-inversion voicing. But the reverse does not work.
So what differentiates the alternating and arpeggio six-four chords? In a word, the melodic pattern of the bass. Here is how the alternating six-four works, as shown in a lead sheet for the American folk song, "John Henry."
Note how the bass works in the first measure. Although the entire measure falls within a G major chord--see the chord symbol above the treble staff?--the bass alternates between root (beats one and three) and fifth (beats two and four) of the triad. That is the origin of the term 'alternating.' Note, too, how prevalent this pattern is throughout the excerpt--it is not uncommon for this style of bass to continue for long stretches of music.
By contrast, here is an example of the arpeggio six-four--the setting of "Auld Lang Syne" which we have examined twice before:
Note the many bass arpeggio patterns. (For instance, mm. 1, 3, 5, 9, and 11 all contain arpeggio six-fours.) In each, the arpeggio pattern begins on the chordal root, or sometimes with the third (first inversion)--never with the fifth (second inversion.) It then proceeds through the other chordal members in some variant of a arpeggio pattern. The second-inversion voicing can occur anywhere within the pattern except the beginning; it must be 'prepared' by a root-position or first-inversion voicing.
Again, this is a common accompanimental style, and can continue for considerable stretches.
Let's work an example. If you are new to this series, our standard practice has been to write out questions on paper before looking at the answers given--physically writing out the questions will help you retain the knowledge gained and to build your skill.
Also, while the format of these Hubs lends itself to the voice-by-voice presentation of the answers--and that's how I present them--in reality most composers work on all voices simutaneously, or at least in a rapid-fire back-and-forth process. Working on paper lets you work that way, too.
If you don't have music paper handy, fear not--the sidebar has a link to a site from which you can download and print paper, free.
So let's get to that practice question. First, identify its key (below.)
The key is Bb.
Look at the brackets above the staff; they show where you should write alternating and arpeggio six-four chords. (Hint: One implication of the brackets is that you need to change the chord on beat four of bar two.)
Now write a bass line, using the six-four chords as just explained, starting on second-line "Bb."
Just to get you started, the first two beats should be an alternating six-four chord. The tonic chord is the obvious possibility, since the soprano tone would then be the third.
For any readers who may be unfamiliar with standard Roman numeral chord notation, the Roman numerals given with the bass voice indicate chords (as identified by their roots), with the quality of the chord--major (or minor)--being shown by upper (or lower) case of the numeral.
The vi triad on beat four of bar two is indicated because it differentiates the preceding and following chords: the brackets and their enclosed notes imply that the preceding chord is I, while the succeeding chord is IV. The only other triad containing the soprano's "Bb" is vi.
If your bass line is close to the one shown, differing only in an octave placement or two, don't fret--there are several places where the bass could be an octave higher or lower without creating a part-writing error. If you put a six-four before a root position, or had a chord change in the wrong place, fix those errors. Then go ahead and add alto and tenor.
Let's try one in minor, too. (Not to mention the triple meter!)
Note that this time, the melody includes some passing tones. Ignore them when figuring out the harmonic progression!
Once again, brackets denote the spans--these are often called 'prolongations,' since they extend the harmonic 'influence' of a single chord over a longer time--defined by the arpeggio and alternating six-fours.
Write your bass line.
As you may have noticed, there are two choices for the prolongation beginning bar two--iv and VI.
Both are quite workable, so we'll write the inner parts for both versions, beginning with the submediant version.
Now work out the subdominant version, too. (I'll give the video for the two versions together, just a bit later.)
Which do you prefer?
Neighbor Six-four Chords
In the strict sense of the word, the previous patterns have been linear--that is, they are defined by melodic motion, not by harmonic function. But because of the leaps, some might tend not to think of them by that description; many of us associate "linear" primarily with stepwise motion. But we speak of 'disjunct lines,' too, to describe lines moving by leap.
Those who feel that way may be a bit more comfortable with the 'linear' tag as applied to neighbor six-four chords; they are both linear and conjunct (stepwise.) In the previous Hub we mentioned that 'auxiliary tones' are also called 'neighbor tones.' This term is very much connected with the notion of neighbor six-four chords. Consider this example:
The bass does not change at all, while two of the upper voices execute simultaneous upper neighbor motions. (The upper voice pair may be any of the three possible pairs, though pairing of the upper two is perhaps most common.)
In principle, the neighbor six-four chord could be used to prolong any major or minor triad; but in practice, the chord almost always prolongs either I or V. These are, after all, the main structual chords in common practice style.
So, when looking for opportunities to use a neighbor six-four, one should look for four specific lines--if a neighbor six-four prolongs the tonic, the lines generated will be 5-6-5 and 3-4-3; if the dominant, 7-1-7 and 2-3-2.
Look for such lines in the following melody, then use what you find to write a suitable bass line. There are opportunities (not, I trust, too hard to find) to use two neighbor six-four chords. For good measure, also use a viio6 on beat 2, as indicated by the asterisk.
In this version, the bass sustains through the neighbor six-fours, but this is not necessary. (It is common.) The bass could have repeated the tone instead (either at pitch or displaced by octave), as shown in this variant:
Now write the inner parts--an open voicing will work well to start.
Another question on neighbor six-fours is warranted, but let's be tricky and mix prolongations (with arpeggio or alternating six-fours) in as well.
We'll try the following melody; as in the first question, brackets indicate prolongations using the alternating or arpeggiating six-four patterns. Locations for six-fours are given.
Work out the implied harmonic progression and write the bass.
If the repeated notes at the beginning of the soprano are not harmonized with alternating or arpeggio six-fours, then they make a wonderful chance to use a neighbor six-four, using the repeated note option menioned above and placing the moving voices in the inner parts.
The arpeggio six-four in the second bar could also have be an alternating six-four. The last two beats of bar three, however, must be separate harmonies, since there is no bracket. Scan the question one more time to see if you have taken advantage of all possible opportunities to write a neighbor six-four, then write the inner voices.
The last bar is a perfect chance to echo the neighbor six-four at the beginning.
Let's try one last question to before moving on to Part 2, in which we'll consider two new paradigms: the Passing and Cadential six-four chords. First, identify the key of this melody:
The A# is a prime clue that we are in B minor.
As above, the brackets show places where prolongations using six-four chords should be used. Also as above, let's use a viio6 triad in our harmonization. Write a bass line embodying these ideas.
An arpeggio six-four will fit two-beat prolongations (mm. 1 and 3); neighbor six-fours fit three-beat prolongations (mm. 2 and 4). The bass line shown includes an octave shift for the neighbor six-four in measure two; at the final cadence, by contrast, the bass holds. (You may have chosen differently for these details, and that is fine.)
Of the three remaining chords, only the downbeat of m. 3 can be the viio6, since neither "B" nor "G" fits into that chord, whereas "E" does.
One might wonder why the arpeggio prolongation in beats 3 and 4 of measure 1 uses iv6, not root position iv. But the soprano's "C#-B" would create parallel fifths with an "F#-E" bass.
In writing the inner parts, use a technique Bach was fond of: have your alto parallel the first measure's soprano eighth notes a third lower. (That will also force you to put the root of the iv chord in the tenor voice.)
In measure two, the use of iv6 is again necessary to avoid parallel fifths, which would result from using a VI chord. And in m. 3, I've had the tenor parallel the soprano's eighth notes--a nice 'frill', but not necessary.
And that's it for Second-Inverion Patterns, Part One. Check out Part Two to learn about the very important Passing and Cadential six-four chords!
Doc Snow (author) from Camden, South Carolina on February 05, 2015:
You are more than welcome. I very much appreciate such a great question, and was thankful to have the chance to respond to it. (And I think I got so absorbed that I forgot to say 'thanks' in my first return comment! My bad…)
Moreover, the need you articulate gives a great hint for future possible Hub projects--a writing opportunity for which I'm also grateful.
Unless and until that Hub gets written, I'd suggest looking at keyboard works with an eye to voice leading. Check out that Invention--and indeed, any or all of them. Check out, say, Mozart or Haydn piano sonatas, and look for the things we've been talking about: how do tendency tones resolve? how do you make sense of any apparent parallels you may come across? how are chord progressions stated or implied? (Remember, in all of this, that the harmonic rhythm will not randomly fluctuate--that is, the rate at which chords change will be somewhat, though not completely, uniform and consistent.)
You could even try 'reverse engineering' those keyboard textures to create 'chorales'. It might not always work, but trying would, I suspect, be pretty instructive.
J.T. on February 05, 2015:
Thank you. That really made a lot of sense even though I am still hazy as to how the "Big Picture" is conceived and flushed out in a common practice accompaniment piece (aside from imagination, and general awareness of timbre and range). But as I was looking for insight regarding this uncomfortable middle ground, I found a lot of people who have newly acquired the skill of part writing find themselves at a similar loss, that is being able to apply their hard earned awareness of movement in applications other than voice or being able to reconcile what they have learned with what they see in practice (common practice specific). What would you suggest as the next rung in ones informal education to build upon the training that was developed in absorbing your in depth look into part writing? Thank you again, really appreciate you taking the time to respond to my question. I will mine everything I possibly can from your response.
Doc Snow (author) from Camden, South Carolina on February 05, 2015:
An excellent question, but not easy to answer, especially in a reasonably concise way.
I think my starting point would be relative to the style of the work you were concerned with. As noted, the 'rules' promulgated in my part-writing Hubs are not of universal applicability; you wouldn't expect them to apply uniformly in music by Debussy, Schoenberg, or Paul McCartney.
And it's also true that keyboard music is a bit different as well. For one thing, there's a long-standing tradition in that realm of what the Germans call "freistimmigkeit":
What that means is that often keyboard works (and lute, guitar and similar sorts of string compositions) often exhibit a constantly changing number of chord tones. That obviously creates difficulties in even defining a 'voice'! There can be another difficulty in inter-mapping keyboard parts to 'voices', too, which is the phenomenon of 'implied voices'--J.S. Bach is particularly famous for writing lines which, though apparently monophonic, actually imply multiple 'virtual' voices which are 'time-shared' within the one 'surface' line.
Freistimmigkeit also means that you can have more than 4 parts. When you do, you are going to see more doubled 3rds in primary triads.
(I hope that's somewhat clear; some examples would be any of the 'cello suites or violin partitas. The practice can also be observed in many keyboard works--to give a specific instance, consider the 8th 2-part invention in F. There are many places in the piece where you can see and hear this at work, but perhaps the most obvious is the cadence in mm. 11-12, where the implied ii7-I64-V-I progression could scarcely be any clearer. And note, by the way, how that progression follows the part-writing norms we've been talking about.)
Part of this difference is practical: keyboard music needs to adapt to the abilities of normal handspans! And part is esthetic; keyboard music has its own 'soul' and its own sound. It uses freistimmigkeit partly because it can, partly because it must, and partly because that is a kind of strength.
Still, a common-practice piece will *tend* to follow the norms. You are correct that parallel octaves are common, but that's almost always not an infraction, because reasonably systematic doublings come across, not as 'bad parallels', but as a single voice which is reinforced by the doubling. Similarly when a vocal or instrumental melody is doubled in the piano part--a very common tactic.
Parallel fifths are quite a bit rarer, though they are better tolerated than in choral writing. But it's very unusual, in my experience, to find them between outer voices: the soprano-bass counterpoint is the most critical in common-practice style. (That's not true in contemporary popular styles, where parallelisms are 'a feature, not a bug,' because independence of voices isn't a consideration.)
So, if you needed to write a piano accompaniment for the vocal/string work you mentioned, I'd look (if possible) at the keyboard writing by that same composer, and use that as a model. (If you can't find models by that composer that are sufficiently close, you could try looking at published keyboard realizations of similar works by later editors--especially when the editor is also a good composer! Brahms and Mahler are two whom I know did such work on occasion.)
One last comment: bear in mind that 'the rules' aren't all of equal weight in practice. Doubling, in particular, is 'more of a guideline, really.' Doubling the third of a primary triad may be option C, but it is still an option, even in choral writing. What's relatively rare in keyboard writing is not to have a third at all: it's an important tone in preserving a chordal sound. (A partial exception is dominant seventh voicings omitting the third, which aren't that rare in Classical keyboard textures. But with root, fifth and seventh, you certainly have a 'chordal sound.' Even so, omitting the fifth is more often the choice.)
Resolution of tendency tones is more critical, but with freistimmigkeit the voice in which the resolution 'should' occur may very well 'disappear', allowing the ear to infer it. Consider, for example, m. 18 of the F major Invention, which I referenced above. There's a clearly implied diminished seventh chord, presumably vii43 of D minor. The seventh would then be the Bb heard in the upper tones of the right hand part. Does it resolve? Maybe, but not before the A on beat 3 of the following measure.
Last sub-point: you asked "Should I observe the counterpoint movement of the contrabass and the vocal?" My answer would be "yes, do!" Very, very few exceptions to that, and almost all of them would be because another part is momentarily taking the bass role (say, the cello or viola.)
J.T. on February 04, 2015:
Hi Doc Snow its been very informative to work through these hubs. I am very thankful for your work. I was wondering if you can point me in the right direction as to the correct application of part writing for use other than constructing four part chorale textures. So far all of the examples have been in this style and I find it discouraging when I am unable to adequately observe proper usage in common practice accompaniments that feature instrumentation as well as voice. In my own attempts at textural reduction for accompaniments I have observed plenty of parallel fifths and octaves and the doubling of thirds for primary chords (where the melody may sing a third, while that third is played in an instrument as well). While careful not to make a case for every infraction, I have yet to pinpoint a direct adherence to four part style writing in anything other than chorale writing. If this is the case, what criteria is observed when constructing an accompaniment that includes voice and instrumentation. An example would be in observing a work that features a violin section,viola, contrabass, and vocal. Should I observe the counterpoint movement of the contrabass and the vocal? The (soprano) violin and the contrabass? Put another way, what would take precedence in outlining the general harmony? If it is the voice and the contrabass what observances are then followed in the other instruments that are tasked with completing the chord voicing (other than the obvious make sure each defining chord member is voiced for the length of the harmonic rhythm). It is not clear to me what is allowable and an infraction other than that. How about a piano accompaniment and a single voice? Is my melody line in the soprano(voice), a third for example mimicked in the piano or complimented, say starting with a fifth. The rest of the triad being constructed in the piano with careful observance of the bass movement. Or does it even matter at all? Your "Auld Lang Syne" example does a great job of contextualizing voice leading in an applicable form for voice and/or piano but unfortunately common practice music doesn't do so explicitly. I apologize for the length of the question I wanted to be as clear as I possibly could, even though it still may be a bit confusing. Thank you Doc Snow
Doc Snow (author) from Camden, South Carolina on January 30, 2013:
Done. Luckily, the error was in the body of the text, which is much more convenient to edit than is the graphic.
Again, thanks to thrig for taking a moment to point this out.
Doc Snow (author) from Camden, South Carolina on January 30, 2013:
I love it when people pay attention--but I must admit, I kind of hate it when that turns up mistakes! Thanks, Thrig, I'll fix that as soon as possible.
Jeremy from Seattle on January 30, 2013:
"As you may have noticed, there are two choices for the prolongation beginning bar two--vi and VI." But the graphics indicate iv and VI...
Doc Snow (author) from Camden, South Carolina on January 04, 2013:
Thanks for working through these second inversion patterns! Hope it was a helpful exercise...
What do you think? Suggestions for improvements or new topics are welcome!