Here’s a look at the complete set of possibilities for anhemitonic pentatonic scales–that is, five-note scales that don’t contain semitones. Some of the most widely used scales in various musical traditions belong to this collection, including the major pentatonic (1, 2, 3, 5, 6) and the minor pentatonic (1, b3, 4, 5, b7). While you can build a dizzying array of pentatonic scales with semitones, it turns out there are only a few you can build without them–the possibilities boil down to fifteen scales belonging to only three modal families. It’s not hard to see why the options are so limited. If we’re building a pentatonic scale, we have to include five notes from the twelve available notes in the octave–let’s represent these “included” notes as white dots on the vertices of a pentagon:
If our scale has no semitones, then there must be an excluded note between any two included ones. We’ll represent these as red dots on the pentagon’s edges:
Without making any choices to “customize” our scale, we’ve already accounted for ten notes: there must be five excluded notes somewhere between the five included notes. We only have two notes in the octave left to play with, and all we can do is decide which edge or edges of the pentagon to put them on. Since we know that each edge has at least one excluded note, we might as well remove the corresponding red dots from our diagram and start with blank edges. (All we’ll be doing from now on is adding more red dots, and the order in which we place multiple red dots on the same edge isn’t significant, so the first five red dots have no bearing on our upcoming choices.)
So how many different ways are there of positioning our two remaining dots on blank edges of a pentagon? We’re looking for arrangements that are distinct in the sense that they can’t be rotated to match each other (in musical terms, this is the same as requiring that two scales can’t be transposed so their notes coincide). There are only three distinct arrangements:
For each of these arrangements, we can create five modes, depending on which white dot we choose as the root. An edge with one red dot means there will be two excluded notes between the white endpoints (remember that we dropped one of those excluded notes to simplify the diagram), which means the endpoints will be separated by three semitones or a minor third (m3). Similarly, an edge with two red dots represents a major third (M3). An edge with no red dots represents a whole tone (M2). Walking around the pentagon clockwise, you can extract the interval pattern for the scale as a sequence of whole tones, minor thirds, and major thirds.
Our first template (two red dots together on one edge) represents the interval pattern M2-M2-M3-M2-M2 and generates all of the five-note subsets of the whole-tone scale. This is fascinating material, but not what comes to everyone’s mind they hear the word “pentatonic.” Our second template (red dots on adjacent edges) represents the interval pattern m3-M2-M2-M2-m3 and also creates some uncommon pentatonics, although one of these (1, b3, 4, 5, 6) can be seen as a subset of the Dorian mode of the major scale, or as a subset of the melodic minor scale. The third template (red dots separated by one edge on one side, and two on another) represents the interval pattern M2-M2-m3-M2-m3 and generates the familiar major and minor pentatonics.
The diagram below includes all fifteen possibilities. The five modes of each of the three template are arranged around the points of three large pentagons. As you move clockwise around one of the large pentagons, you’ll see that the root note in the small pentagon (the white dot labeled 1) shifts around, while the red dots remains in place. The remaining vertices in each small pentagon are labeled as scale degrees relative to the given root.
[Note: I’ve proofread this diagram once, but please let me know if you spot any errors in the scale-degree labeling.]
This is a Venn diagram showing the relationship between any three adjacent keys in the circle of fifths, with special attention to the pentatonic subsets of each key.
This diagram is not meant as a replacement for the standard circle of fifths, which is usually presented with note names and key signatures for the twelve major (and corresponding relative minor) keys in the diatonic system, arranged clockwise in ascending perfect fifths. Rather, the diagram is a way of visualizing the overlap between keys that occur close to each other in the standard circle.
All notes are labeled with respect to an arbitrary major key that we’ll treat as the home key or I. Its notes are 1, 2, 3, 4, 5, 6, 7. Stepping counterclockwise in the Circle of Fifths would take us to IV and stepping clockwise would take us to V. To label the notes of those keys with respect to the home key, we also need b7 (flat seven) and #4. You’ll see that the notes of all three keys appear in the diagram, left to right, in ascending fifths: b7, 4, 1, 5, 2, 6, 3, 7, #4.
All the notes of the home key appear inside the large blue oval (it contains everything but b7 and #4). Within that oval, there are three circles representing the three major pentatonic scales based on IV, I and V. The yellow circle on the left, containing {4, 1, 5, 2, 6}, represents IV-pentatonic. (To play these notes as a major pentatonic scale, you’d order them as 4, 5, 6, 1, 2.) The red circle on the right, containing {5, 2, 6, 3, 7}, represents V-pentatonic (which you’d order as 5, 6, 7, 2, 3). The circle wedged between the yellow and blue circles contains {1, 5, 2, 6, 3}: this is I-pentatonic, which you’d play as 1, 2, 3, 5, 6.
The diagram shows that if you take C major as your home key, you can find the notes of F-pentatonic and G-pentatonic within it. In fact, C major can be seen as the union of F- and G-pentatonic. (Another way to say this is that if you take the pentatonics based on any two keys a whole tone apart, their union gives you the major key that falls right in between them on the Circle of Fifths.) Why is that interesting? Well, for one thing, it shows that if you stick to pentatonic scales, you can modulate between F, C, and G, all while staying within the seven notes of C major. When I hear “modulation” I usually think either “accidentals” or “change of key signature,” but if we’re modulating between closely related pentatonics we can do it all inside the confines of one seven-note major scale. How do you modulate between I-pentatonic and V-pentatonic? Look at the diagram: you’re moving from the middle circle to the red circle on the right. Just stop playing 1 and start playing 7. Another way to say this is that shifting the root of a major pentatonic scale down a semitone gives you new pentatonic rooted a fifth above the original root. (In the seven note world, you modulate up a fifth by raising 4 to #4, which becomes the leading tone of the new key. In the pentatonic world, there are no leading tones! You modulate up a fifth by lowering the root!)
The full seven-note major keys based on IV and V are represented by the two large gray circles in the diagram. The gray circle on the left represents IV and contains the yellow circle as its pentatonic subset — its two additional notes are the tritone b7-3. The large gray circle on the right represents V and contains the red circle as its pentatonic subset — its two additional notes are the tritone 1-#4. Notice that the small middle circle encloses all the notes in the intersection of the two large gray circles: {1, 5, 2, 6, 3}. What this means is that I-pentatonic consists of the notes in common between the two heptatonic major scales at IV and V.
It’s also interesting to observe that the notes 5, 2, and 6 (which we could interpret as a quartal chord on 6, or a sus2 chord on 5, or a sus4 chord on 2) are the most extensively shared notes in the diagram, being the intersection of I-pentatonic, IV-pentatonic, and V-pentatonic.
Here’s a look at some of the pieces you’ll find in the full diagram. You can think of this as F major on the left, with its pentatonic in yellow; G major on the right, with its pentatonic in red, and C major in the middle, shown as the union of F-pentatonic and G-pentatonic — all labeled from the perspective of C.
Here’s the main diagram again, with each circle labeled as Heptatonic or Pentatonic:
About the Diagram — Update 5/15/2014
After seeing some discussion of this diagram on Reddit I wanted to explain why I structured it this way — in particular, why are the 5 and 6 out of line with the other numbers? My reason for doing that was visual, not conceptual. Indeed, you could make this diagram by first writing out b7, 4, 1, 5, 2, 6, 3, 7, #4 in one straight line and then drawing ovals around various subsets of notes. What you might find is that all these overlapping ovals create a visual jumble. My goal was to take that jumble and refine it so the eye could easily pick out any of the six sets we’re discussing and quickly see how that set relates to the others. To help my own eye navigate the diagram, I tried to use contrasting colors, size, and shapes. Putting the 5 and 6 where they are allowed me to draw perfect circles around the IV and V pentatonics and heptatonics, contrasting with the ovals around the I pentatonic and heptatonic. Doing this also helped keep the diagram somewhat compact.
Whenever I come across an unfamiliar musical scale, I wonder how it might be related to scales I already know. In particular, is it part of a modal family that I recognize, like the seven modes of the major scale, or the seven modes of the jazz melodic minor scale?
Grouping scales into modal families might seem like a largely theoretical project. After all, these families are motley crews, with members that have totally different musical personalities. Locrian and Ionian belong to the same family, yet one is eccentric and unstable while the other is the terra firma of Western music. If they behave so differently, what can we learn from the fact that Locrian and Ionian have a “family connection”?
First, there’s a harmonic implication. Any two members of the same modal family generate the same chord types. (A harmonized Locrian scale has 3 major triads, 3 minor triads, and 1 diminished triad, just like a harmonized major scale.) Second, there’s a melodic implication. You can shift from one mode to another while staying in the same pool of notes. (An interesting improvisational exercise would be to play phrases in C Major and see how quickly you can evoke B Locrian and then return to C Major.) Third, there may be a performance implication. Depending on your instrument and approach to fingering, you may be able to reuse the same fingering patterns for all members of the same modal family (see my Note Neighborhood approach to the guitar fretboard, for example).
It would be convenient if all “interesting” scales belonged to a small collection of modal families, but this is simply not the case. For example, the 72 ragas of the South Indian Melakarta system belong to a total of 36 modal families — still a large number to get to know. (I’ll write more about the Melakarta system in a separate post). However, if we place a certain “natural” restriction on the kinds of scales we consider, we end up with fewer scales belonging to dramatically fewer families.
In this post I’d like to consider seven-note scales that do not contain two semitones in a row. The major scale satisfies this constraint: in C major, for example, the semitones B-C and E-F are not consecutive. But if we flatten the second degree of C major, the resulting scale would violate our constraint because B-C and C-Db would form consecutive semitones. It’s worth noting that there are many beautiful scales that violate the consecutive semitone constraint. It seems that consecutive semitones work well when they are centered around the first or the fifth degree of the scale. For example, there are many ragas in North and South Indian classical music that use natural 7 in conjunction with flat 2 (forming a cluster around the root as we just considered); or sharp 4 in conjunction with flat 6 (forming a cluster around the fifth). The North Indian Rag Poorvi or Purvi has both of those clusters: its pattern is 1 b2 3 #4 5 b6 7. Another famous scale with consecutive semitones is the Blues scale: 1 2 b3 4 b5 5 7. (Here, the semitones are centered around b5 instead of 5.) On the other hand, much Western music is built from scales that do obey the consecutive semitone constraint, and the theorist Dmitri Tymoczko has written extensively about this constraint as a point of connection between classical Impressionism and Jazz.
In this post I’d like to show that all scales meeting the consecutive semitone constraint belong to one of only six modal families. (I’m assuming a twelve-tone equal-tempered octave and am only considering scales with matching ascending and descending forms.) Writing in 1997, Tymoczko presented essentially the same conclusion, although his list of modal families is slightly different because he frames the constraint a bit differently. (Instead of focusing on seven-note scales, he allows scales of any size but disallows intervals larger than a major second between any two consecutive notes; later, he allows augmented seconds but disallows scales that are subsets of any larger scale meeting the criteria.) In any case, the claim is not hard to verify with a software analysis of all seven-note scale possibilities, and it’s easy to obtain from combinatorial methods of scale construction such as those discussed in From Polychords to Polya by Michael Keith. What I hope to offer here is a simple, visually-oriented explanation. This style of presentation might help build intuition into the dynamics of scale construction, and I also hope it might be enjoyable as a bit of equationless recreational math.
So how can we find all of the modal families that don’t have consecutive semitones? To get started, let’s observe that if we’re building a seven-note scale, we need to pick five notes from the octave to leave out. We can think of these five omitted notes as dividing the octave into five sections. All of the notes we include in our scale will fall into one of those five sections, and to obey the consecutive semitone constraint, we can’t have any three notes in one section.
In a general sense, if we’re placing any seven “items” into five sections, and if we can’t put three in a section, then there are only two possibilities:
Two sections will have two items each, and three sections will have one item each
Three sections will have two items each, one section will have one item, and there will be one empty section
Translating this into musical terms, if we have a seven note scale without consecutive semitones, then it must have:
Two semitones and three singleton notes, or
Three semitones, one singleton note, and one augmented second
I’m using the term “singleton” to refer to a note that does not have any immediate neighbors in the scale that would form a semitone with it. And I’m observing that an “empty section” in our terminology means that the two omitted notes forming the section would occur right next to each other in the scale, creating the interval of three semitones or an augmented second between the notes on either side of the gap (see Ab and B on the right side of the diagram above).
In each of these cases, we’re free to move the section contents around as we like (for example, deciding whether our empty section in case 2 falls between two full sections or next to a singleton). Different arrangements of the section contents might yield different sequences of intervals representing different modal families. Our task is to figure out how many distinct arrangements are possible. To explore this, we’ll represent the section types as beads on a necklace. Instead of including one bead for each of the twelve notes of the octave, we’ll work with simpler necklaces that have only five beads. We’ll use a red bead to represent a section containing two notes that form a semitone. We’ll use a blue bead to represent a section containing a singleton note. And we’ll use a yellow bead to represent an empty section that creates an augmented second between adjacent notes. (It’s important to mention that this analogy between scales and necklaces only works if we keep the necklace on a flat surface and never flip it over; what we’re interested in is the clockwise order of the beads, and that order might change if we flip the necklace!)
Now that we’ve reduced the scale problem to a necklace problem, let’s ask how many necklaces we can make with two red beads and three blue beads (or two semitones and three singletons — our first case above). We’ll consider two necklaces as distinct if we can’t rotate one to match the other while keeping them on a flat surface. In fact there are only two possibilities: the red beads can be separated by blue or they can sit right next to each other. (If they’re separated, there will be one blue bead between them on one side, and two blue beads on the other side; and if they’re together, all the blue beads will fall together as well.)
Before moving to case 2, let’s look at a variant of case 1. What happens if we take away one of the red beads above and replace it with a yellow bead? How many necklaces can we make with three blue beads, one red, and one yellow? Well, now we have four possibilities: each of our original two-color necklaces gives us two new necklaces since we can choose one of two red beads to replace with yellow. Each choice yields a necklace that’s distinct by our definition.
Of course, to address case 2 of our scale problem, we need to know how many necklaces we can create with three red beads, one blue, and one yellow (three semitones, one singleton, and one augmented second). But this is basically the same question we just answered, since what matters in a mathematical sense is not the specific colors of the beads, but how many of each type of bead we have to work with. To get our answer for three red beads, we can just take the 3-blue/1-red/1-yellow necklaces above and swap blue with red. Here then, are all six necklaces satisfying our scale constraints, with the two-color possibilities on top, and the three-color possibilities below:
In fact, these necklaces represent the six modal families that include all possible seven-note scales without consecutive semitones. If we unpack the meaning of these necklaces we can see what scales they actually generate. In the diagram below, I’ve marked scale degrees in a clockwise order on each necklace, reflecting the most common choice of a root note in each case. The scale degrees are natural, flat, or sharp according to the pattern of intervals dictated by the necklace. Of course, if we position the root differently, we get a different interval sequence relative to the root — in other words, a different mode in the same underlying family — which would have a different pattern of natural, sharp, and flat degrees. Each necklace gives us seven possible modes, corresponding to seven positions of the root, for a total of 42 possible modes that satisfy the consecutive semitone constraint. (And since none of the necklaces is rotationally symmetric, we can gather that all of those 42 possibilities are distinct.)
In this set of six families, you’ll notice three items that are likely familiar. There is the major scale, the melodic minor scale (which we take here to have the same descending form as the ascending form, as it is used in jazz), and the harmonic minor scale (which we also take in the version with matching ascending/descending patterns). The so-called “harmonic major” scale is a fourth item that might be familiar as it comes up in jazz theory — it’s basically a major scale with a flat 6.
The two less familiar items in the lower right corner can be seen as subsets of the eight-note diminished scale. Remember, the diminished scale is a sequence of half and whole steps. If we take the mode of the diminished scale that starts with a half step, we could label the scale degrees as 1 b2 b3 3 #4 5 6 b7. If we omit the b3 above our chosen root (leaving natural 3 in place), we get the raga known as Ramapriya in the South Indian Melakarta system (1 b2 3 #4 5 6 b7). If we omit the natural 3 instead, we get the raga known as Shadvidamargini (1 b2 b3 #4 5 6 b7). We can look at these ragas as two of the most musically workable or “coherent” seven-note subsets of the diminished scale, because they have a perfect fifth above the root and all the other degrees occur in easily recognizable varieties (there is no #2 which sounds like b3, no #5 which sounds like b6 etc.). It’s interesting to note that Ramapriya has no modal relatives that satisfy the constraints of the Melakarta system; Shadvidamargini has one, called Nasikabhushani, but it contains a #2.
Next time you come across an unfamiliar seven-note scale, check to see whether it has consecutive semitones. If it doesn’t, I’ll be happy to wager you anything that it’s a mode of major, melodic minor, harmonic minor, harmonic major, Ramapriya, or Shadvidamargini. And if you’re looking for some interesting scales to explore, start with the members of these six families.
Appendix I: How does it sound?
One of the interesting things that happened for me in working through this analysis was “discovering” Ramapriya and Shadvidamargini. Of course, they’re both present in the full list of Melakarta ragas, and they probably turn up under different names elsewhere, but it was interesting to encounter them by exploring a limited space of possibilities, suddenly finding my attention focused on two uncommon gems. Here’s a sample of my own first exploration of Shadvidamargini. (I’m playing in my personal improvisational style and am not attempting to treat this in Carnatic style. And throughout the post, although I’ve been using the Melakarta name for “Shadvidamargini,” I’m really referring to the scale as an abstract construct outside the context of any specific musical tradition.) On a technical level, this improvisation travels through a series of modulations around a fully diminished seventh chord.
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.
Rudi Seitz 