When a major 6th (or an octave thereof) is in a chord, when is the chord symbol written as a 6th vs. a 13th?

Question

I'm not asking about obvious cases like regular 6th chords or extended chords with 13ths. For example, I'm clear that C E G A is a C6 chord. I'm also clear that C E G B♭ D F A is a C13 chord. (And yes, I know that, in practice, some of those notes—the 5th, 9th, or the 11th—may be omitted.)

I'm interested in theoretical and edge cases. Let's begin with an example like C E G A B♭. Wikipedia says it's a C7/6 chord. Fine.

The 7/6 notation seems to begin and end with sevenths, however. There is no 9/6 chord that I can find, for example. (That's not to be confused with a 6/9 chord, which does not have a 7th.) So, if we had C E G A B♭ D, what would that be, assuming C is the root and we wanted to leave nothing to ambiguity? Our choices are:

• C9(add13)
• C7/6(add9)
• C9(add6)
• C13(omit11)

Again, I'm sure someone's going to say that you can omit the 11th from a 13th chord, so C13 is appropriate. But what if I wanted to find a chord symbol that was unambiguous about what's in and what's not, irrespective of the voicing?

If we root the chord on G, we have a similar problem:

• Gm6(add9,11)
• Gm(add9,11,13)
• Gm13(omit7)

(I thought of Am11(♭9), which is perhaps the simplest of all, but that's not useful to the question.)

What plays into the decision to call it a 6th or a 13th? Sure, we could say it's the proximity of the 6th to the root, but that doesn't feel totally comfortable to me because chord symbols don't necessarily imply or reflect chord voicings. So what is the rule?

Update

I understand the answers I'm getting. Thank you. I was hoping for answers which don't take context into consideration, though. (Think of an algorithm which generates possible chord names given nothing but a set of pitch classes.)

Answer 1

This is a tricky question, and I think answering it requires digging into various music notation systems and their purpose.

Sometimes as a composer, you want to specify exact notes for people to play. Standard notation is perfect for this. Other times, you want to lay out a structure and let your musicians decide on the exact voicing and other details. Chord symbols and slashes are perfect for this.

When you get into these extremely detailed extended chord symbols, it often seems like an unholy union of these two ideas. If you're composing something and you really want me to play C E G A B♭ D, then just write that (in standard notation), don't try to communicate that to me with a crazy symbol. If you're analyzing a piece of music and need a label for that set of pitches, then again I ask for what purpose? If it's so that you can give it to someone and have them recreate it then I have the same answer as before. If it's to understand the theory of the piece, then are the extreme extensions of the chord actually important? Is the absence of a note (the 11th in your case) actually important? The ambiguity should be a hint that it doesn't really matter.

As far as the validity of the labels you suggest, my opinion is that multiple "add"s are bad. I would probably go with C13(no11) or C9(add6) for a C root.

Answer 2

When naming chords, composers can hardly find them agreeing with each other, especially when they are in different eras, genres or instruments. For example, the C6 chord may refer to C E G A, which is a C major triad plus a major sixth; but it's more common to be C E A in classical music, as the first inversion of the A minor triad, with the 6 notating the A being the major sixth of the inverted root, C; which is more often written as Am^/C in modern music. It all depends on the context.

P.S. I personally like to notate C E G A as C₆ and C E A as C⁶ (if necessarily not written as Am^/C), to avoid confusion. In my system, I use superscript for intrinsic attributes such as major/minor/seventh/extension/bass note etc., and subscript for added/omitted/suspended notes.

Every composer is entitled to invent his/her own way to notate chords, as long as it can be understood by the musicians who read his sheet. C₆ can be more ambiguous than C_add13, but if the chord appears many times in a score, no one would blame you for using the more abbreviated form. It's just you are obligated to make sure no one is confused, most times a small line of text will do the trick - which in the C₆ case is not necessary, because it's so common that everyone can recognize it.

The C7/6 in your question, can be found in publications by minimalism musicians. I myself prefer to write it as C⁷_add13, so it's clearly a dominant 7th chord with an added 13th. The so called "C9/6" chord should be a C⁹_add13. It's technically equal to a C¹³_omit11, but when you say "C9/6" in the first place, you are emphasizing it's functioning as an extended 9th chords rather than a 13th.

As for the 6th vs 13th, I personally always try to avoid using the 6 notation. Some may interchange them, some may use 6 only when there is a note which has to be exactly a 6th above the root. Then again, it all depends on the context and which system the composer/editor uses. I respect them as long as they are self-consistent.

Answer 3

To be exact, one needs to write in all the appropriate dots. Writing merely a chord name just won't do it, in so many cases. When we see 'C' written, there are numerous ways in which it can be played/voiced, all of which would be construed correct. Even seeing C/G only guarantees that the lowest note would be G, under a however-voiced C chord.

Drop chords will to some extent make playing closer to what the writer expects, but those aren't often used. Leaving out 9 and 11 is commonplace in 13th chords, but not as accurate as the OP wishes for. Even seeing a carefully thought out C13,no11 or C9add6 will result in several different versions being playable - although restrictive on guitar - so for me the only concrete way is to prescribe exactly what you want played in the dots. But be careful when writing for guitar - it may not be possible to reach each in the written order!

Answer 4

I think the issue is that there are a few different ways to say a lot of things and none of them are necessarily more correct than another. Theory is a means of describing what is happening in the music, so when we try to describe what's happening and have to choose between 6 and 13, what's the difference? They're the same note and tend to have the same effect on the harmony, so I think the terminology that is most commonly used is going to depend on context. This isn't to say that there is no way to spell out exactly what notes you want to be in your chord, just that the standard notation that you will find people using doesn't provide this. The best notation that I've seen to describe exactly what notes are in a chord is to basically define the chord type, then, in parentheses, write the extensions and alterations that apply. Since 9, 11, and 13 are all enharmonically equivalent to 2, 4, and 6, there shouldn't be too much concern for which of those you choose to use. I think there are some better choices to be made based on context but in an attempt at using notation without context, I would think that 9, 11, and 13 are more commonly accurate representations of those notes.

So for your first example (C E G Bb D A), the notation I am referencing would look like this:

C7 (9, 13)

This allows you to show exactly which notes are in the chord. This can get tricky in a few areas, for instance, the altered chord, which you could potentially spell out this way:

C+7 (b9, #9, #11)

This is difficult because the way the world of jazz talks about an altered chord does not include an augmented fifth, they refer to it as a b13. With that in mind, there wouldn't really be a standard chord symbol to indicate a 7 chord with no 5, so I think writing it as I did would be the easiest way to spell out every note in the chord, even though it goes against the standard terminology.

On the whole, context does need to be considered. You can find instances of the same terminology being used in two different realms of music and find that they don't mean the same thing. For example, the dreaded 13 chord. In Jazz, you will find that a 13 chord is basically a dominant chord that has a 13. In Classical, a 13 chord (which I believe they call a 13th chord), would be based on the scale degree that it was formed on, so in C Major a C13 would be a major chord but an A13 would be a minor chord. On top of that, the 13 of each chord would be different, where C13 would have a Major 13 and the A13 would have a Minor 13. Taking it a step further, if we were in A Major, A13 would be a major chord with a major 13 and C13 would be a major chord with a major 13 (derived from the parallel minor)

We also see plenty of instances of one concept having more than one name. In rock/pop/country/etc you don't really see 13 very commonly; you are much more likely to see 6. Add2 and Add9 are another example. These are effectively the same thing but someone might understand one and not the other.

Answer 5

Your main question is "how to decide whether a chord be called 6th or 13th" and the answer is: well yes the proximity to the root IS a deciding factor. Not 100% (because again context) but a very strong one. What makes it stronger is the fact that there's NO OTHER factor. The only other factor is particular context, that might invalidate the guiding principle.

Now, you say that it's not comfortable, as "chord symbols don't necessarily imply or reflect chord voicings". But in fact they can imply some details about the voicing, even if they don't imply the complete layout. So in this case notation might strongly suggests, the 6th should not be played in root octave.

So if you apply this thinking to your first example, and will check only for that single aspect of the voicing (namely where is the 6th), then you will end up with choice of two labels for each chord, out of each there's always one that is a definitely simpler description of what is going on (C7/6(add9) and C9(add13))

And anyway from this point on the only factor playing into choosing the label is wider musical context, which in case of your algorithm that works without context doesn't exist and makes the choice purely aesthetic.

To elaborate a bit about the 'simplicity' and 'aesthetics' for example: the choice between labels: C9(add13) and C13(omit11) is really a matter to sticking to certain convention. I can't see any harmonic context where choice of one over another would carry any information. I personally like the convention where 'omit' is applied to base voices and 'add' is used to selectively apply extensions, as it reflects how we naturally think about building chords - we add extensions and decide to omit some core voice.

Answer 6

You're asking for a definite ruling on 'theoretical and edge cases'. Sorry, almost by definition one isn't available. Chord symbols cannot be a full description of every possible chord. Sometimes we need notation.

Answer 7

I understand the answers I'm getting. Thank you. I was hoping for answers which don't take context into consideration, though. (Think of an algorithm which generates possible chord names given nothing but a set of pitch classes.) - trw

Step 1: Identify the intervals that best fit the pitch classes specified.

This step would provide you with:

• interval set (p1-m3-p4) when the pitch classes are (0,3,5)
• interval set (p1-m3-d5-dd7) when the pitch classes are (0,3,6,9)
• and so on ..

the algorithm for this is not straightforward and would include parameters as:

• fit the intervals against a set of available qualities (dom7, dim7, min, maj, ..)
• fit added, altered notes against the reference scale (major/mixolydian)
• avoid as many as possible intervals with conflicting degree classes
• prefer certain intervals or combination of intervals over others
• and many more subtleties ..

Step 2: Identify the (compound) degrees that best fit the intervals calculated.

The following context-free algorithm can be used to identify compound degrees for chords that adhere to tertian standards. (chords with mixed sixths are excluded)

• If no sevenths
  • any sixth would be of degree 6 (A)
• otherwise
  • if no ninths and no elevenths
    • if the sixth is major
      • this sixth would be of degree 6 (B)
  • otherwise
    • any sixth would be of degree 13 (C)

Examples:

• (A) Cm6add#11, D#6no3
• (B) C7/6(b5), D#(7/6)
• (C) Cm7b13, D#13b5b9

Answer 8

The most important element of the question is the "not considering context" element. Calling a note the sixth or thirteenth is specifically dependent on context or, once given, implies a context.

Edward Aldwell and Carl Schachter offer a very useful distinction (in the context of classical music): in a chord with a 6, the 6 is a suspension delaying the arrival of 5; in a chord with a 13, the 13 replaces the 5.¹ (They do not address the possibility that both 6/13 and 5 are present.)

Further, by giving the chord a name (Cm6, Cm7add13, etc...) there is also typically an implied meaning (i.e., context) to the chord (Cm6 suggests a tonic chord; Cm7add13 suggests a ii chord).

If you really want to name the chords consistently and accurately without context, then use a contextless notation. To take the first example from the OP, C E G A B♭ D, which otherwise could be

• C9(add13)
• C7/6(add9)
• C9(add6)
• C13(omit11)

would all simply be (02479T) (i.e., a set of "pitch classes").² If note order is important (i.e., some minimal context), then you could call it [0479T2] (ordered pitch classes). And if you really want to get fancy, you could name it according to the intervals <43214> ("interval classes"), which is non-specific in terms of the chord root. Then, to designate the above example, it would be T_0(<43214>) (T_0 here means [T]ranspose to C (= 0)). To transpose the identical chord to C# would be T_1(<43214>).

---

¹ Edward Aldwell and Carl Schachter, "Harmony and Voice Leading", 2nd ed. (Harcourt Brace Jovanovich, 1989), pp. 449-53.

² The notation here is called "pitch-class notation", which assigns 0 = C; 1 = C#; 2 = D; ...; T = 10 = A#/Bb; E = 11 = B. There is a closely related interval-class notation which assigns intervals based on the number of half-steps (mod 12): 0 = unison/octave; 1 = one half-step; 2 = two half-steps; T = ten half-steps; E = eleven half-steps. Given a set of pitches (or intervals) they can be transposed by X half-steps by the function T_X: thus, T₁(CC#E) = T₁(014) = (125) = C#DF. Searching terms like "pitch class", "interval class", and "musical set theory" will provide a wealth of information. Text books on "post-tonal theory" also will generally discuss these concepts.