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In many hours of doodling, I found that nested Stars of David make an interesting way of visualizing the musical octave. In particular, they’re a good way of visualizing the even divisions of the octave: as two sets of six notes (the two whole tone scales), as three sets of four notes (the three fully diminished seventh chords), and as four sets of three notes (the four augmented triads).

In mathematical discussions of music it’s common to represent the octave as a clock face (see my previous post on that), and it’s common to visualize note sets as polygons inside the clock face.  But when you try to represent the whole tone scales, diminished seventh chords, and augmented triads all at once, you get two overlapping hexagons combined with three overlapping squares, combined with four overlapping equilateral triangles: a tangled mess!  It’s an interesting information design problem to try to represent all of those relationships clearly, in the same diagram.  I think nested Stars of David do that in a very beautiful way. The idea is to place a note at each of the outer points of the Star of David, progressing clockwise, with C at the top; then draw another star inside the main star.  Each line segment on the outside of the larger star represents a half step between two notes, and each triangle edge represents a major third.

Here you can see the two whole tone scales represented in yellow (B, C#, D#, F, G, A) and light blue (C, D, E, F#, G#, A#).  You can “ascend” each scale by moving clockwise from one point to the next point of the same color:

Musical Star of David: The Two Whole Tone Scales

Here you can see the four augmented triads as triangles in light blue (C, E, G#), fuschia (D, F#, A#), light green (A, C#, F), and yellow (B, D#, G).  Comparing this diagram with the first one, you’ll see that each whole tone scale contains two of the augmented triads:

Musical Star of David: The Four Augmented Triads

And here you can see the three fully diminished seventh chords as rhombuses, represented in light blue (C, D#, F#, A), fuschia (C#, E, G, A#) and yellow (B, D, F, G#).  Each one has a combination of notes from all four augmented triads:

Musical Star of David: The Three Fully Diminished Seventh Chords

Note that the coloring scheme is somewhat arbitrary–if you’re one of those people who has strong synaesthetic associations between notes and colors, these diagrams might be painful to look at (sorry!).  For everyone else, hopefully they’ll be fun to check out.

From these diagrams you can deduce some interesting musical facts.  For example, did you know that if you look at any augmented triad and any fully diminished seventh chord, they’ll have exactly one note in common?  In fact, every one of the twelve tones can be identified as the common tone in one specific augmented/diminished pairing.  The way to see this in the star diagram is to notice that each of the four main triangles (the two that make the outer star and the two that make the inner star) share a vertex with exactly one of the three rhombuses.  For example, here is A aug in green and A# dim7 in fuschia.  The common tone (shared vertex) is C#:

Musical Star of David: Augmented/Diminished Intersection

Most texts on harmony follow a similar path from triads to seventh chords: first you learn the three-note chords you can build by stacking thirds, and then you look at the four-note chords you can build by adding a seventh. In this jump from three to four notes, there’s a fundamental question we pass over: how many types of three-note chords could we build if we weren’t limited to thirds as the building block? There’s so much complexity in seventh chords and their extensions that it’s easy to get lost and never return to the three-note universe to fully map it out.

A list of all the possible three-note chord types (or trichords) crops up in a number of places, but as far as I know it hasn’t become part of “mainstream” music education. You can find it in atonal theory (which has various ways of labeling all possible “pitch class set classes”), in Hindemith’s Craft of Musical Composition, in various explorations of math and music (in particular Michael Keith’s From Polychords To Polya), and in some online resources.  What I’m hoping to do here is provide a simple way of examining this question, since I think it deserves more attention that it gets.

So how do we find all the three-note chord types?  In my post on Musical Clocks I reviewed the idea of representing a three-note chord as a division of the clock face into three slices, made by the three hands of the clock. A chord type can be seen as a pattern or template for sizing and arranging those slices: for example, any major triad divides the clock into a slice of size 4, followed by a slice of size 3, followed by a slice of size 5, reading clockwise. This is the same as saying that a major triad divides the octave into a major third (4 semitones), a minor third (3 semitones), and a perfect fourth (5 semitones)–with two of those intervals audible at any given time, depending on what inversion is played.

By spinning the major template (4/3/5) around the clock in increments of one hour, we can create twelve chord instances (C maj, C# maj, D maj…), one for each landing position.  Remember, what we’re trying to count in this post are templates or chord types, not instances or actual chords.  In fact, not every chord type generates twelve distinct instances (hint: augmented), so we can’t simply divide the number of possible three-note combinations by 12 to get the number of chord types.  Go ahead and try it though: (12 choose 3) = 220.  That’s the number of ways we can pick three notes from the octave to create a chord instance.  Divide 220 by 12 and you have 18.33333…. which isn’t an integer!

Now, one template for slicing up the clock is “equivalent” to another (it represents the same chord type) if we can rotate them so they line up.  For example, you might make your slices at 12 o’clock, 4 o’clock, and 7 o’clock, and I might make mine at 12, 5, and 9. Starting from 12 and reading clockwise, your slices span 4 hours, 3 hours, and 5 hours; while my slices span 5 hours, 4 hours, and 3 hours.  If I spin all of my slices as a unit around the center of the clock face they’ll eventually line up with yours, because my slices are sized like yours and occur in the same clockwise order.  In musical terms, we’ve both built major chords and I can make the notes of my chord match yours by transposing them all by the same interval. But if my slices were of size 3, 4, and 5 (reading clockwise) I’d have built a minor chord: there would be no way to get my slices to line up with yours on the clock face, and no way to transpose the notes of my chord so they match yours.

So the question, “How many types of chords can we create with three notes?” is the same as asking, “How many distinct templates are there for separating a clock into three slices?” Again, “distinct” means that we can’t spin one template around to match another.  Now it turns out that the generalized version of this question–where you have an arbitrary number of hours (or possible slicing points) on the clock, and you’re making an arbitrary number of slices–is difficult and requires a tool called the Polya Enumeration Theorem.  We don’t need anything so powerful, however, since we’ve only got 12 hours on our clock and we’re only making three slices.  We can count all the possibilities by hand without too much trouble.  What we do need is a systematic way of counting, so we won’t include duplicate templates in our count.

Let’s organize our count around the smallest slice size that occurs in each template.  In a minor triad, for example, the smallest size interval is 3 semitones (the minor third). In a diminished triad, it’s also 3: in fact we’ve got two slices of that size. In an augmented triad, the smallest size is 4: the chord is built entirely of major thirds.

Since all the slices on our clock face have to add to 12, the smallest slice size in any template must be 1, 2, 3, or 4. (It can’t be 5, because if we had three slices of size 5 or greater, we’d need a larger clock to fit them!) The diagram below looks at each of these four cases separately. The yellow slice in each clock is the one we designate as the smallest.  Imagine you’ve already cut that smallest slice and you’re looking for the possible “cut points” where you can make a final incision to create the two remaining sections. The blue dots represent the viable options, the places where you can cut without creating a slice that’s smaller than the yellow slice.  (I’ve given each blue dot a number to represent the trichord it gives rise to.)  The gray dots represent “cut points” that are off limits, either because they’d yield a slice that’s too small, or because they’d lead to a repeat of the pattern created by an earlier cut point.  [For example, in the first clock where the yellow slice is size 1, there's a blue dot labeled "1" which gives slices of size 1, 1, and 10 (reading clockwise from 12 o'clock). The gray dot is off limits because it would give slices of size 1, 10, and 1, which can be rotated clockwise to match 1, 1, 10.]

Counting the 19 Trichords

I find it intriguing that the markings on these four clocks represent all of the harmonic possibilities in the three-note universe (within 12-tone equal temperament, that is).  The standard triads are on the bottom two clocks: 16 is diminished, 17 is minor, 18 is major, 19 is augmented. In my followup post, I take a closer look at all 19 trichord types shown here and examine the musical significance of each one.  Think about it, any complex chord you’ll ever play will be made of some combination of these 19 trichords–they are worth getting to know!

(Note to self: don’t get too excited.)

The idea of chord type is what I’d call a devious concept. It’s devious because it appears so close to the concept I actually want that it makes me assume it is what I want. As a musician, what I want is a way of understanding and categorizing the different sounds (or sound experiences) I can create by playing groups of notes in a connected way. I want a way of thinking about compound sounds. What does the notion of “chord type” actually do? It helps us identify the intervallic structures that arise when we combine notes in certain ways–it’s a technical, essentially mathematical notion that relates to sound but doesn’t describe sound.

For example, a major triad in root position is a major third with a minor third stacked on top it: four semitones plus three semitones. And the major chord type extends to any transposition or inversion of that interval structure. We get so used to thinking in terms of chord type that we equate it with sound: a major chord has a major sound; if you want a major sound you should use a major chord. To analyze a piece of music often means “figure out what chords (i.e. interval structures) are being played and how they relate to each other functionally,” and once we’ve done that, we think we understand what’s going on. And yet it’s easy to forget that interval structure is not the be-all end-all of sound experience — in fact, it’s far from a natural way of conceptualizing sound experience. As a student, you don’t always know what concept you want so you take what’s there and forget there’s something missing.

Now the holy grail of sound classification is probably unattainable, not only because sound experience is subjective, but because it is influenced by so many factors including context (musical, psychological, and cultural), register, articulation, tuning, and so on. So we’re left with this idea of interval structures and the names we’ve given them — major seventh, minor seventh flat five, Lydian dominant, etc. We go on assuming that (harmonic) music is made of chords and chords are made of interval relationships, and anything you want to know is contained therein.

This post is a “rant” — I needed to get it off my mind because I’m about to publish some material on chord types. And yes, I realize that most musicians know there’s a lot more to sound than interval structure.  My real reason for writing this post is to stamp out a reductionist bias that I find myself falling into from time to time.

This post is a follow-up to my article about Note Neighborhoods, a technique I developed for navigating the guitar fretboard.  In this post I’d like to provide charts that can be used for practicing this approach — these are the charts I use myself.

First, a recap of the Note Neighborhood concept.  In guitar playing it’s common to practice ascending and descending scale patterns that start from the root of a scale or mode and stretch across all six strings (for example, see my post on Guitar Modes Unified).  In the Note Neighborhood approach, you dispense with the idea of ascent and descent from a starting point, and you focus on smaller patterns that extend outward from a central note.  You practice one pattern for each scale degree, taking that scale degree as the center and learning to find all of the other notes that surround it physically on the fretboard (within a three-string/three-note-per string circumference).   If you’re playing in major, melodic minor, harmonic minor, or any of the their modes, each scale degree will have a unique neighborhood pattern, a unique “geography” including eight notes around the center.  Once you’ve memorized all seven neighborhoods, you can play freely in your chosen scale anywhere on the fretboard, as long as you recognize what degree you’re playing at a given time and can recall its neighborhood pattern.  This approach works particularly well in All Fourths tuning because you don’t need to memorize variants of each neighborhood pattern for different string sets; however, it can be applied to Standard Tuning as well.  I find it particularly useful when working in Melodic Minor and its modes, since the six-string patterns for Melodic Minor lack the regularity of the Major patterns and (for me at least) are harder to internalize.  This approach also frees you from always searching for the root as the anchor of your fingering because you learn to be equally comfortable taking any scale degree as your “anchor” — that’s helpful if you’re improvising music that’s not “root-centric” but moves back between different points of modal focus within an available pool of notes.

I don’t see the Neighborhood concept as competing with other approaches for understanding the fretboard; it can be used to complement them.  In fact, if you know the three-note-per-string fingering patterns for the major and melodic minor modes, you’ll see that the Neighborhoods are just fragments of those patterns, labelled in a different way (focusing on the center as opposed to the lowest note).  If you’re an experienced player, you might think of the Neighborhood concept as a different way of organizing fretboard knowledge that you already have, so you can apply it differently and see new possibilities.  Or if you’re just starting out with scale patterns, you might find that Neighborhoods cut down on the amount of rote memorization you have to do the first time around.

Here are the seven neighborhood patterns for Major.  (If you’re playing in standard tuning, these will work on the bottom four strings as shown, but you’ll have to shift things up by a half-step when you cross from the G string to the B string.)

Major Note Neighborhoods (guitar)

And here are the neighborhood patterns for Jazz Melodic Minor:

Melodic Minor Note Neighborhoods (Guitar)

To get started with these, try playing the notes of C major up the A string, focusing on C, then D, then E, etc., and using the patterns for 1, 2, 3, etc. to give you the neighboring notes.  Then do the same for C melodic minor.  As for fingering, try placing your middle finger on the central note of the neighborhood you’re playing–keep it there and explore all the intervals you can play between that central note and the other notes in the neighborhood.

Addendum I: Comparing The Neighborhoods

One way to get to know the neighborhoods is to compare them–to notice what they have in common and how they differ.  A great way to do this for the Major neighborhoods is to lay them out side-by-side in Circle of Fifths order: 4, 1, 2, 5, 6, 3, 7.  When you do this you’ll notice that each neighborhood differs from the previous one by only one relationship, as shown in the diagram below.  For example, look at the 5 and 2 neighborhoods: 5 has a major third above it, while 2 has a minor third above it.  This means the finger position in the top right region of those two neighborhoods is different; otherwise, the fingering patterns are identical.  Now compare the 2 and 6 neighborhoods, and you’ll spot only one difference, this time in the bottom left corner.  Note that between the 4 and 1 neighborhoods, and also between the 3 and 7 neighborhoods, you’ll spot two finger positions that are different.  That’s because each neighborhood contains some doubled notes, and when you go from the 4 to the 1 neighborhood, or the 3 to the 7 neighborhood, it’s one of these doubled relationships that changes.  Take a closer look and you’ll see that the two positions that shift are an octave apart.  (As always, click on the image for the full size version.)

Major Note Neighborhoods in Circle Of Fifths Order

Addendum 2: Theory Behind The Comparison

What’s the theory behind the diagram above?  Well, anyone who’s looked at the circle of fifths knows that two major keys a fifth apart differ by one accidental.   A slightly less common observation is that the formulas for modes of the major scale based on degrees a fifth apart differ by one sharp or flat as well.  As we progress from Lydian to Ionian to Mixolydian, all the way to Locrian, we pick up natural 4, then flat 7, then flat 3, then flat 6, then flat 2, and finally flat 5:

  • Lydian: 1 2 3 #4 5 6 7
  • Ionian: 1 2 3 4 5 6 7
  • Mixolydian: 1 2 3 4 5 6 b7
  • Dorian: 1 2 b3 4 5 6 b7
  • Aeolian: 1 2 b3 4 5 b6 b7
  • Phrygian: 1 b2 b3 4 5 b6 b7
  • Locrian: 1 b2 b3 4 b5 b6 b7

The note neighborhoods in the diagram are just different ways of expressing the information in these modal formulas.  A modal formula looks at the notes in an ascending sequence from the root, whereas a neighborhood pattern considers how the notes are situated around the given center, but the underlying relationships are the same.  It’s a remarkable property of the major scale that the modes can be arranged in such a clean progression, where one modal formula differs from the next by only one sharp or flat.  That’s to say, it’s remarkable that when you look at the interval relationships that one note forms with other notes in the major scale, and then you do the same analysis for its fifth, there’s only one difference.  I call this “remarkable” because it’s not true for many other scales, namely melodic minor.  You can’t arrange the modes of melodic minor so that successive formulas differ by one accidental, and you can’t arrange the neighborhood patterns in a clean progression like what’s shown in the diagram above for Major.  For example, try comparing 1 and 5 in melodic minor and you’ll see two relationships that are different: 1 has a minor third and a major sixth above it, while 5 has a major third and a minor sixth above it!  You can try other comparisons too–1 with 4, 1 with 7, etc.–and in each case you’ll find at least two differences in the neighborhoods.

There’s a very natural mapping between the equal-tempered musical octave and the face of a clock.  There are twelve notes in an octave and there are twelve hours on a clock. If we let twelve o’clock represent C, then the notes of a C major triad (C, E, G) would fall at twelve o’clock, four o’clock, and seven o’clock.  We can represent the triad by aligning the hour, minute, and second hands of the clock with the corresponding marks.  The spaces between any two hands of the clock show us the size of the interval between those two chord tones.

The idea of representing the octave as a clock is sometimes attributed to the composer Ernst Krenek and these octave-clock diagrams are sometimes called Krenek diagrams, or pitch constellations.  They’ve become very popular among people who study the so-called “geometry of music” and other connections between math and music.  I use them all the time in my own music practice when I’m trying to understand the structure of a chord or scale.  In fact, I’m so accustomed to drawing these diagrams that I sometimes assume most musicians are familiar with them, and I’m surprised when people say “Clock?  Huh?”  I’d like to use clock diagrams in some upcoming posts on music (particularly an upcoming investigation of the Nineteen Trichords), so I wanted to create some examples that show how these diagrams work.

Here you can see the notes of a C major root-position triad mapped out on the clock.  When looking at staff notation it’s common to read the notes from the bottom up: C, E, G.  On the clock, it’s a bit different: you take the lowest note of chord as the starting point and read clockwise till you’ve hit all the other notes.  You can see that the major third (M3) between C and E becomes a slice of 4 hours (corresponding to 4 semitones) on the clock face.  The minor third (m3) between E and G becomes a slice of 3 hours/semitones.  Finally, there’s a gray slice representing the non-sounding interval between the top note of the chord (G), and the bottom note transposed up an octave (click on the diagram for higher resolution):

C Major (Root Position) Clock Diagram

You can get an inversion of the chord by picking a different starting note on the clock face and reading clockwise from that note.  Here we start at E and read off the notes E, G, C, giving the triad in first inversion.  In this case we traverse a minor third (m3) and then a perfect fourth (P4):

C Major (First Inversion) Clock Diagram

Finally, you can transpose a chord by taking all the hands of the clock and rotating them as one unit  (so the spacing between the hands stays the same).  Here we transpose the C major triad up a semitone to C# major by moving all the hands clockwise by 1 hour.  Compare this diagram with the first diagram above:

C Sharp Major (Root Position) Clock DiagramIf you haven’t come across Krenek diagrams before it’s worth becoming familiar with them, because they make a lot of things about music a lot clearer.

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