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This question stems from some thoughts on List of possible trichord/triad names..

When you stack chords in 3rds, you get all the standard chord names of Major, minor, 7 etc, but what about when stacking other intervals, like in the case of 20th century music.

What is a basic stacked 2nd chord called, if anything! I'm open to the possibility that a naming system doesn't exist in the same way that we have major,minor etc.

I read that Stacked 4ths are called a Viennese trichord, but I imagine that's because it's come up enough times to merit it's own name.

for example (say in C major)

  • stacked 2nds making a chord on scale degrees 1 2 3
  • stacked 4ths making the chord 1 4 7
  • Stacked 6ths making the chord 1 6 11

How do you name chords in 20th century music?

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2 Answers

up vote 9 down vote accepted

Actually the way you describe them in your question is one of the ways, called pitch class sets (http://composertools.com/Theory/PCSets/).

EDIT: Except you use C major as a starting point, which I didn't notice at first. With pitch class sets you use the chromatic scale, so your chords would be (putting C at zero) [0,2,4], [0,5,10], [0,9,17]=[0,9,5].

EDIT2: To classify pitch class sets one can use prime forms. With traditonal third-based chords one can usually reorder the notes in a chord so it forms a tower of thirds, and then use that to decide the quality of the chord. For example, given the notes G,Bb,E,C, one reorders them to C,E,G,Bb, and decides that it's a dominant 7th chord. Obviously one cannot build a tower of thirds (or any other fixed kind of intervals) from an arbitrary pitch class set, so something else must be done.

The idea is to rotate the pitch class set until it is as small as possible. We also consider the inversion (the "write it upside down" kind of inversion) of a set as the "same", since it has the same interval structure and therefore sounds very similar. In particular, every major and minor triad has the same prime form. The basic algorithm to calculate the prime form is (from the link earlier):

  1. Keep rotating your chord until it is as small as possible.
  2. If there are ties, then use the rotation that has the notes most packed towards the bottom.
  3. Check to see if the inversion is better packed.

Let's do this for [0,1,7,9]. The rotations are: [0,1,7,9], [1,7,9,0], [7,9,0,1], and [9,0,1,7]. Of these, [7,9,0,1] is the smallest; no ties. The inversion is [7,8,11,1], which is better packed (more tight on the left). Transposing this to zero, we get the prime form [0,1,4,6].

The prime forms can be ordered and then given a so called Forte number. This answer explains the numbering and Wikipedia has a list of them all. The Forte number of [0,1,4,6] is 4-Z15. Now, just like the quality of the chord G,Bb,E,C is dominant 7th, the quality of C,C#,G,A ([0,1,7,9]) is 4-Z15. Of course, some special chords have more descriptive names. For example, 4-Z15 is one of the two all-interval chords. The dominant 7th chord is 4-27; prime form [0,2,5,8] (which, as you can see, is actually the inversion of a dominant 7th chord).

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I thought that using chromatic scale might be the case, Does that then mean that the chords don't necessarily have a name, and are rather Identified by their construction mathematically? –  Alexander Troup Oct 23 '13 at 10:51
1  
@AlexanderTroup: Pretty much, yes. I'm not aware of any system of assigning "real" names to all chords. However, there's the concept of a prime form of a pitch class set (explained in the link in my answer), and these can be ordered systematically and named by Forte numbers. The C-major triad would then be 3-11, which is actually the name of every major and minor triad. Of course, this is probably not what you mean by a name. –  nonpop Oct 23 '13 at 12:04
    
sweet, can you expand on this in your answer, then I'll upvote :D –  Alexander Troup Oct 23 '13 at 13:11
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This will calculate prime for you and give you the Forte name for any set of pitches.

http://www.mta.ca/faculty/arts-letters/music/pc-set_project/calculator/pc_calculate.html

I enjoyed learning atonal theory from Strauss Introduction to Post-Tonal Theory.

I find it helpful to relate atonal pitch collections to something from Jazz or Traditional harmony in my mind to get my bearings. For your example, 123, I might think of that as a Major chord with the 9 and no 5. But this is just a personal mnemonic device. The "official" name for the collection would be (024) or 3-6 as the website calculator will confirm. They are numbered rather than named because there are so many possible pitch collections.

Another important property of any pitch collection is its interval content, which is essentially an accounting of all the intervals that exist between each possible combination of two pitches in the collection. The interval content has a lot to do with how the collection sounds. Two different collections with the same interval content are "Z related". Read the book, it's fascinating.

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