This post is a followup to Counting the 19 Trichords, where I looked at how we can list all the possible types of three-note chords that are distinct under transposition (shifting all notes up or down by the same interval) and inversion (displacing individual notes by an octave). Here, I’d like to give a diagram for each of the nineteen trichords and say a little bit about its musical significance.
I’m using the numbering scheme from my previous post, where I organized the trichords in terms of their smallest interval, starting with trichords that contain two notes separated by a semitone, and progressing to the augmented triad (#19), where all notes are separated by major thirds. See that post for a diagram explaining the numbering. Here, I provide a clock diagram as the main illustration of each trichord, but you can also click on the View Star links in the comments to see an alternate “Star of David” diagram of each chord.
Nickname: Semitone Cluster
Comments: [View Star]
Interval pattern: Reading clockwise from 12 o’clock, you’ll see that the first and second notes (in yellow) are separated by 1 semitone, the second and third are separated by 1 semitone, and the third and first are separated by 10 semitones. So I list the interval pattern as [1 1 10] in semitones, or [m2 m2 m7] in interval names (i.e. m2 = minor second, etc.)
Comments: Reading clockwise from 12 o’clock you could think of this as the first three notes of the Phyrgian mode. Or, if you take the root at 1 o’clock, you have the outline of a Major 9th chord, including the Major 7th and 9th, but excluding the 3rd and 5th. [View Star]
Interval pattern: [1 2 9] or [m2 M2 M6]
Nickname: Minor/Major 7th
Comments: If you take the root at 1 o’clock, the other notes form a minor 3rd and major 7th above it, giving the signature of a “minor/major 7th chord,” missing a fifth. [View Star]
Interval pattern: [1 3 8] or [m2 m3 m6]
Nickname: Major 7th
Comments: If you take the root at 1 o’clock you have a major third and a major seventh above it, giving a major seventh chord without a fifth. [View Star]
Interval pattern: [1 4 7] or [m2 M3 P5]
Nickname: “Lydian Sus4”
Comments: I call this a “Lydian Sus4” because if you take the root at 6 o’clock, you have something that could be thought of as a Isus4 chord from the Lydian scale (i.e. where the 4 is sharp). Alternatively, if you take the root at 12 o’clock, you have a b5 and b9 above it, so this could be seen as an incomplete 7b5b9 chord. [View Star]
Interval pattern: [1 5 6] or [m2 P4 T]
Nickname: “Phrygian Sus2”
Comments: With the root at 12 o’clock this could be thought of as a Isus2 from the Phrygian mode. Or, if the root is at 1 o’clock, the chord could be a maj7b5 missing a third. [View Star]
Interval pattern: [1 6 5] or [m2 T P4]
Nickname: Major 7th Shell
Comments: With the root at 1 o’clock you have a Major seventh chord missing a third. When the chord is missing a third I refer to it as a “shell” in the nickname, since we can’t tell what quality it has. Compare with Trichord #4. [View Star]
Interval pattern: [1 7 4] or [m2 P5 M3]
Nickname: Augmented Major 7th
Comments: If the root is at 1 o’clock, this could be an Augmented Major 7th chord missing a third. [View Star]
Interval pattern: [1 8 3] or [m2 m6 m3]
Comments: Starting at 10 o’clock, you can think of this as the first three notes of a minor scale, which can occur together in a minor ninth chord. If we think of this as a minor ninth chord, it’s missing a fifth and seventh. [View Star]
Interval pattern: [1 9 2] or [m2 M6 M2]
Nickname: Whole Tone Cluster
Comments: Starting at 12 o’clock, this can be seen as the first three notes of a major scale. If we take the root at 2 o’clock, this could be seen as a minor or dominant ninth chord missing its third and fifth. [View Star]
Interval pattern: [2 2 8] or [M2 M2 m6]
Nickname: Minor 7th
Comments: Take the root at 2 o’clock and you have a minor seventh chord without its fifth.
Interval pattern: [2 3 7] or [M2 m3 P5]
Nickname: Dominant 7th
Comments: Take the root at 2 o’clock and you have a major third and a minor seventh above it, creating a dominant seventh chord (missing its fifth). [View Star]
Interval pattern: [2 4 6] or [M2 M3 T]
Comments: Depending on which note you treat as the root, this could be a Sus2 chord, a Sus4 chord, or a Quartal chord (stack of two perfect fourths). [View Star]
Interval pattern: [2 5 5] or [M2 P4 P4]
Comments: If the root is at 2 o’clock we have a half-diminished seventh chord missing its third. If the root as at 8 o’clock we have majb5 chord. [View Star]
Interval pattern: [2 6 4] or [M2 T M3]
Nickname: Minor 7th Shell
Comments: If the root is at 2 o’clock this is a minor seventh chord missing its third. Of course if we added a major third instead of a minor third, it would become a dominant seventh chord. [View Star]
Interval pattern: [2 7 3] or [M2 P5 m3]
Comments: If the root is at 12 o’clock this is a diminished triad. If the root is at 3 o’clock this could be thought of as a minor sixth chord (minor triad with added major sixth), without a fifth. [View Star]
Interval pattern: [3 3 6] or [m3 m3 T]
Comments: If the root is at 12 o’clock this is a minor triad. With the root at 3 o’clock this could be thought of as a major sixth chord (major triad with added major sixth) missing a fifth. [View Star]
Interval pattern: [3 4 5] or [m3 M3 P4]
Comments: With the root at 8 o’clock this is a major triad. [View Star]
Interval pattern: [3 5 4] or [m3 P4 M3]
Comments: The only trichord with perfect symmetry! Also, the only trichord that has fewer than 12 distinct instances as we transpose it (or rotate the template around the clock face). [View Star]
Interval pattern: [4 4 4] or [M3 M3 M3]
7 thoughts on “The Nineteen Trichords”
I’ve really been enjoying your recent posts, Rudi. I’m particularly intrigued by this one because of the way it connects “horizontal” melodic thinking with “vertical” harmonic thinking, and clean mathematical abstraction (combination of twelve tones) with intuitive phenomenology (I can summon the sound and feeling of many, if not all, of the trichords).
In an earlier post you mentioned your frustration at harmonic systems based on triads. It seems to me that the tough (interesting) job is to elaborate a harmonic system based on these trichords. What relationships to they bear to one another; when do we hear them as stable and unstable; how do they want to resolve into one another? I look forward to that part!
Thanks for the comment, Peter! Glad you enjoyed the post and do let me know if/how you use any of this in your own music.
The frustration I mentioned in a previous post wasn’t so much with harmony based on thirds as with the limitations of “interval structure” as way of organizing our understanding of sound experience. Notes and the intervals they form are the “stuff” of music, in the same way you could say words are the building blocks of sentences… and yet even an analysis of the words in a sentence and a diagram of their relationships may not explain how a sentence comes to impact us as it does; ditto with chord charts and Roman numeral analysis. That said, the theory and categorization of interval relationships is an undeniably useful tool, and this post is part of my own effort to get a better grasp of it. I agree with your assessment of where the really tough/interesting work is, but I wonder how much can be stated (about stability/instability and resolutions patterns) in the abstract, outside the context of a specific piece or style. It always amazes me how something that’s dissonant and/or unstable in one context can be stable in another — even a “grating” minor second can be a resting place if used in a certain way — and that malleability is one of the things that makes music so interesting. So I would take your questions as a starting point for thinking about how to use the trichords in a specific composition (which could be thought of a harmonic system of its own). Thanks again — I’ll post more in this area and I look forward to continuing the discussion!
Thanks for this post. Going around the clock within the twelve tones is a useful perspective on this. I actually used this as an exercise to explore how different chords lie when using Major 3rds tuning (on bass). Overall, the trichords lie pretty well within this tuning. And there are also some “mechanical” advantages compared to standard: stacking thirds, anything involving whole tone, grabbing a half step up or down from a neighboring string. Very cool.
Thanks for comment, Matt — glad to hear you found it useful, particularly as a way of exploring the potential of M3 tuning! That’s a nice application and I’ll give it a try as well…
There seems to be so little on the internet regarding this fundamental question. I suspect that everyone thinks there are so many chords that it is hardly worth bothering about. Considering there are only 19 it amazes me that these haven’t all been named in a sensible way and exploited to death by now.
Anyway, I just wanted to say I did some similar work around the same time as you and also back in the 90s. This question came back up, as a result of asking a question on a Facebook group. Your page was quoted in the answers. My diagrams came out a little different, but are along similar lines. If you’d be interested in seeing a copy, let me know.
Thanks for confirming the number, I was concerned I had missed something.
Hi Joe, Thanks for writing! I agree, it’s surprising that this question doesn’t get more attention. Arriving at 19 certainly took some time for me, as I imagine it did for you too, but we’re not the first to get that number: it had been established by the theorist Allen Forte in the 1970s and possibly by others earlier. Forte came up with a numbering system for all possible chords or “pitch class sets.” In his system, the 19 trichords are numbered as 3-1, 3-2A, 3-2B, …, 3-12 as you can see here: https://en.wikipedia.org/wiki/List_of_pitch-class_sets That said, I don’t find his system particularly intuitive for music making, so when I wrote about this back in 2013 I was looking for an approach that made sense to me. Yes, I’d certainly be interested to see how you went about this. Feel free to share a link to your diagrams here if you have them online, or email me (format: firstnamelastname at gmail). Best, Rudi