Fun With Triad Cycles

Ex. 4 Triads Pivoting Within a Major 10th

My teacher, Mark Sherman, gave me a simple assignment at our last lesson: practice cycling through 12 keys using only major triads, mostly as 4-note chords, all while restricting yourself to a narrow range. I think this one is worth dedicating some serious time.  

OK, so let’s think about this algorithmically. The basic rules are simple: 

  • The only notes allowed are from the major triad 
  • Play 4 notes from each triad
  • Keep going in the same direction until you hit an upper or lower boundary, at which point, change direction 
  • No repeated notes 
  • After 4 notes, cycle to the next triad in the descending cycle of 5ths 

First question: what is the most narrow range into which you can fit the entire cycle? Answer: ok, maybe a bit of a trick question? Obviously, you need all 12 notes, so the smallest range would be 12 semitones, or a major 7th. Another obvious observation (obvservation?): there are no octaves anywhere, so every triad will have one note played twice in the 4-note grouping.  

Ex. 1 Triads Within a Major 7th
Ex. 1 Triads Within a Major 7th

Well, that’s a good exercise, but it is rather limited! What if we expand the range by one semitone? Now we have a full octave, C to C. That one note actually opens thing up quite a bit! Now we have an octave on the root of C major, on the 3rd of Ab major, and on the 5th of F major. Cool!  

Ex. 2 Triads Within an Octave
Ex. 2 Triads Within an Octave

What if we wanted to find the minimum range that would allow each of the 12 triads to have at least one octave? Well, on the one hand, every note is part of 3 different triads, it’s either the root, the 3rd, or the 5th. So every note we add will add 3 more triads, right? Well, not quite, because notes that are a 3rd apart will have a triad in common. 

So let’s see how this plays out. Let’s add a B below middle C. That gives us three more triads: B major, G major, and E major. That’s a total of six triads, we are half-way there! Let’s keep going down, adding Bb gives us Bb major, Gb major, and Eb major. Wow, 9 out of 12 covered! 

This time, let’s add a higher note; we’ll add C#. That gives us C# major, A major, F# major. Oops! Not quite 3 more, because F# and Gb are the same. It makes sense, because we said that notes a 3rd apart will share one triad, and Bb and C# (or Db) are a 3rd apart. 

OK, we are at 11 out of 12. Let’s add one more higher note, D. That gives us D major, Bb major (which we already had) and G major (which we already had). Woohoo, 12 out of 12. Why did D add only one new triad? Well, it is a 3rd away from Bb, so shares a triad there, and it is a 3rd away from B, so shares a triad there too!  

Now we have all 12 triads with at least one octave. By the way, you know those triads we said we ‘already had?’ F#/Gb major, Bb major, and G major? Well, those ones have two different notes with an octave.   

Ex. 3 Triads Within a Major 10th
Ex. 3 Triads Within a Major 10th

So, to summarize, the smallest interval that will allow you to cycle all 12 triads with at least one octave per triad is a major 10th

OK, this is all good so far, but the motion rule is a little bit dull. Let’s try throwing in what Barry Harris calls pivoting. Pivoting breaks up the motion of an arpeggio, or broken chord, by adding an octave displacement, either up or down, and then continuing. In this example, we will play a note, pivot down to the next chord note, and continue upward for 3 notes. We’ll then chose the closest note in the next chord and repeat the process. This way all of the 4-note groups will have a similar countour.  

Ex. 4 Triads Pivoting Within a Major 10th
Ex. 4 Triads Pivoting Within a Major 10th

That’s all for now!

The next note is never more than a whole-tone away

I first learned of the exercise of running different scales into each other from Barry Harris. You start near the top of your instrument, and as the chords change, you change your scale but keep going in the same direction. Relatively easy on a tune like Autumn Leaves, pretty difficult on a tune like Giant Steps. My teacher Mark Sherman also emphasizes this drill, and has in fact taken it to a whole new level.

In discussing this drill a few weeks ago, Mark told me, the next note is always either a semitone or a whole-tone away. I thought of a simple way to prove this.

First off, we need to review the concept of locally diatonic scales. In short, they are scale in which all groups of three consecutive notes can be mapped onto three consecutive notes of the diatonic scale. There are only four scales that qualify: diatonic, melodic minor, whole-tone, and diminished. None of these scales contains an interval greater than a whole-tone. Both the harmonic minor, and its inversion, the harmonic major, contain an augmented second, that’s why they are not included.

Ok, so moving between any of our four scales, at any transposition, the next note will always be a semitone of a whole-tone away, and here’s the proof.

  • The current note will either be a member of the next scale or not
  • If the current note is a member of the next scale, the next note will either be a semitone or a whole-tone away
  • If the current note is not a member of the next scale, then the next note has to be a semitone away, by definition
  • Therefore, the next note is always a semitone or a whole-tone away

By the way, when running locally diatonic scales into each other, the aggregate result is definitely not guaranteed to be locally diatonic. It should be easy to think of examples where running two scales together results in two consecutive semitones, which would break the rules for locally diatonic.

A fun way to derive the melodic minor scale

This thought just occurred to me as I was listening to the audiobook of Frank Wilczek‘s excellent Fundamentals: Ten Keys to Reality.

  • The melodic minor is a 7 note scale consisting of only semitones or whole tones, but with no 2 consecutive semitones (put another way, it is “locally diatonic“)
  • The whole tone scale is a 6 note scale consisting only of consecutive whole tones
  • The melodic minor contains 5 continuous notes of the whole tone scale
  • To create a 7 note scale from the 5 continuous notes we need to add 2 more notes
  • These 2 notes can be either whole tones or semitones
  • Neither of them can be a whole tone, because that would result in a full whole tone scale
  • Therefore, both added notes must be semitones
  • Therefore, the melodic minor consists of 5 notes from the whole tone scale, bounded at both ends by a semitone
  • Corollary: since the whole tone scale is symmetrical, the melodic minor scale is also symmetrical

The “Minor 6/9 Shape” Harmonized

During our weekly jam session this week, my friend Tony and I got into a discussion about a harmonic topic we’ve both been looking at. It’s an intervallic shape that’s pretty useful harmonically and melodically, and has a good ‘modern jazz’ feel to it. You can see the shape in example [A] below. You might call this a “minor 6/9” shape, since it has the 3rd, 5th, 6th, and 9th of a minor chord.

But what makes this shape so useful is its ability to outline different chords. Example [B] lists five of these. If we think of the Eb as the root of a chord, we have an EbMaj7#4. If we think of it as a minor 3rd, we have a Cm6/9. If we think of the Eb as a D# serving as a major 3rd, we have a B7#9b13, which is an ‘altered dominant’ sound. If we think of the Eb as the b5th of a chord, we have an Am11b5. And if we think of the Eb as the 7th of a chord, we have an F13 chord.  

As I’ve been exploring this sound, it dawned on me that since the shape is a subset of the melodic minor, then it can be harmonized with all of the notes of the harmonic minor, expanding the number of chords from five to seven. These are all listed in example [C].

The shape also lends itself to quartal voicing, which is great since quartal harmony is another key component of modern jazz, and, as a guitarist, quartal voicings fit on the neck beautifully. Looking at example [D], we can see that the shape contains the “Viennese trichord” with a perfect 4th on top. If we label the chord according to its constituent 4ths, we can call it an APP.

I did a post on quartal tetrachord harmonizations a couple of years ago, and if we refer to that, we see that the quartal tetrachord APP appears (get it, APPears?) in not only the melodic minor, but also in the major, the harmonic minor, and the harmonic major. That means we can also harmonize the shape with all the notes of each of those scales.

If we look at the major scale, we see that the APP is on the 4th degree of the scale. Keeping Eb as the bass note, that would mean our new scale is Bb major. Bb major has only one note different from C melodic minor: the Bb itself. That means the chord with Bb as the root is a new one, the other six notes are the same as chords we have already discovered. This new chord is shown in example [E]. I didn’t label this chord, but you could think of it as EbMaj7#4/Bb or Bb13sus. We are now up to eight chords.

Looking at harmonic minor and harmonic major, the APP appears on the 6th degree of the scale. Eliminating chords we have already discovered, this gives us one new chord, with an F# as the root. I will leave it to you to label that one.

So now we have a total of nine discrete chords across these four scales. A good next step would be to see how we could use parallelisms across these chords in different progressions, but I will leave that topic for another day.

A few tidbits on the melodic minor

I’ve been thinking about the melodic minor a lot this week. It occurs to me that in jazz pedagogy, it’s common to refer to the 7th mode of the mm as a “diminished / whole-tone” scale. (In addition to the other common nomenclature, which is “altered scale.”) This makes sense, but why limit it to the 7th mode? Why not just think of the entire melodic minor as a diminished /whole tone? I just spent some time practicing the scale this way on the piano, and it has really opened up my thinking.

Earlier in the week, it had occurred to me that, instead of thinking of a “locrian #2” on a IIm7b5, it makes much more musical sense to think of it as a minor IV, using the melodic minor. That sets you up for some lovely parallelism going from mm on IV to mm on the b9 of V–or in other words, up a minor 3rd.

And then, once you’re thinking that way, it’s easy to drop in a mm on the 5th of V, which gives you the so-called “lydian dominant.” A nice way to practice that is to play a mm scale in the rh, start with the root in the lh, and then drop a 5th to turn it into a dominant. Play that, and I dare you to try and stop from playing Debussy-esque riffs.

Tidbit: Locally Diatonic Scales

I was reading an article by Dmitri Tymoczko this morning, Stravinsky and the Octatonic: A Reconsideration, and came across a useful term: locally diatonic. This refers to a scale whose seconds are all minor or major, and whose thirds are all minor or major. This includes the following scales: major, ascending melodic minor, whole-tone, and diminished (octatonic). Any three consecutive notes from any of these scales can be mapped onto a segment of the major scale.

Tritone Fun Facts, Part One

This post kicks off a miniseries on tritones. I’ll use the terms tritone and diminished 5th somewhat interchangeably. This post covers some basics, the next one will explore an idea about key signatures, then I’ll look at diminished 7th chords. 

Natural Fifths
The seven natural 5ths

Let’s start by considering the seven natural notes. If we place a natural note a 5th  above each of the seven natural notes, we see we see that we now have six perfect 5ths and one diminished 5th.  

  • FF1 There is only one diminished 5th made of natural notes: B-F 
  • FF2 There are six perfect 5ths made of natural notes 
  • FF3 Any 5th made of natural notes, other than B-F, is perfect 

OK, let’s look at each of these natural perfect 5ths. There are two ways we can turn a perfect 5th into a diminished 5th. We can raise the bottom note with a sharp. Or we can lower the top note with a flat. 

Sharp Tritones
The six “sharp” tritones
Flat Tritones
The six “flat” tritones
  • FF4 Other than B-F, any 5th with a sharp on the bottom and a natural on the top is diminished; there are six of these 
  • FF5 Other than B-F, any 5th with a natural on the bottom and a flat on the top is diminished; there are six of these 

So, there are six tritones with one sharp and one natural, and six tritones with one natural and one flat. But six intervals adds up to twelve notes, so the six “sharp” tritones add up to the entire chromatic scale. And then, so must the six “flat” tritones also add up to the entire chromatic scale.  

  • FF6 The six tritones with a sharp on the bottom and a natural on the top make up the entire chromatic scale 
  • FF7 The six tritones with a natural on the bottom and a flat on the top make up the entire chromatic scale 
  • FF8 For every tritone with a sharp on the bottom and a natural on the top, there is an enharmonically-equivalent tritone with a natural on the bottom and a flat on the top 

Hey, if the tritones with only one sharp cover the chromatic scale, and so do the ones with one flat, then what about B-F? Is that an extra tritone, since it has no sharps or flats? Oh good question, but B-F is covered enharmonically in both cases: E#-B, and F-Cb. 

  • FF9 The tritone with no sharps or flats, B-F, has an enharmonic equivalent with one sharp, and another equivalent with one flat 
Enharmonically-related Tritones
All six enharmonically-related tritones

The diagram above summarizes all six enharmonically-related tritones that can be spelled with either two naturals, a natural and a sharp, or a natural and a flat. (There are actually two more that can be made with two sharps or two flats, but they are also enharmonically equivalent. I’ll cover those in the next post.) Oh, let’s go ahead and formalize this with its own fun fact!

  • FF10 There are only six tritones and their enharmonic equivalents

Actually, this makes sense from another perspective. On the piano, B-F is made of white notes only. We said earlier that there are six tritones with one natural and one sharp. But wait, there are only five black keys. So one of those sharps has to be a white key. Same goes for tritones with flats. 

  • FF11 Every tritone consists of one white key and one black key, except for B-F and its two enharmonic equivalents 

So, B-F is kind of weird, right? What’s also interesting is that to turn B-F into a perfect 5th, you have to do the opposite operation: raise the top note with a sharp, or lower the bottom note with a flat. 

  • FF12 To turn a diminished 5th into a Perfect 5th, you can either raise the top note with a sharp, or lower the bottom note with a flat 

OK, so this is mildly interesting (or wildly, in which case, you may have a career as a music theorist ahead of you!), but is there any practical application? Why, yes! This should help with automatic recognition of intervals. Remember, other than B-F, two naturals always means perfect; sharp on bottom and natural on top always means diminished; natural on bottom, flat on top always means diminished. This should also help with instant recognition of 5ths with double flats and double sharps.  

That’s enough for today. Next time, we are going to talk about key signatures.

Quartal Harmony: Tetrachords

In today’s post we will continue our series on quartal harmony with a quick look at quartal tetrachords. When we began with dyads, we saw there are three quartal dyad types: P, A, and D. With quartal trichords, there are nine types (3 * 3). With quartal tetrachords, we now have twenty-seven types (3 * 3 * 3). These are summarized in Figure 2. Of the twenty-seven, only eleven occur in our four scale harmonizations.

As before, the most common tetrachord type is based on stacked perfect fourths, PPP. This type occurs nine times in total across the harmonizations, making it once again the most harmonically ambiguous chord type. PPP occurs four times in the Major harmonization, twice in the Melodic Minor, and once in each of the other harmonizations.

After PPP, the next most common tetrachords are PPA and APP, both of which occur four times, once in each of the harmonizations. Both of these are very useful voicings. APP can easily serve as a Maj7#11 and PPA can serve as a dominant, as we shall see next.

When looking across the four harmonizations, we see an interesting detail on the both the supertonic and dominant degrees: they each share the same tetrachord across the harmonizations. In all cases, the supertonic is a PPP, and the dominant is a PPA. In other words, the II-V quartal tetrachords are the same for Major, Melodic Minor, Harmonic Minor, and Harmonic Major.

We mentioned the Viennese trichord in our last post. Any tetrachord that contains either PA or AP is a superset of the Viennese trichord. Our next most common tetrachord is PAP, which occurs three times in the harmonizations. Once can see PAP as an interlocked PA and AP, or an interlocked Viennese trichord. A Viennese tetrachord perhaps? This voicing also functions well as Maj7 11 chord.

Quartal tetrachords harmonized with the major, melodic minor, harmonic minor, harmonic major scales.
Figure 1. Major, Melodic Minor, Harmonic Minor, Harmonic Major scales harmonized with Quartal Tetrachords.
Quartal Trichords in scale harmonizations.
Firgure 2. How many times does each tetrachord appear in the four scale harmonizations?

Key Takeaways

From a practical standpoint, here are a few things to keep in mind using quartal tetrachords for comping or soloing:

  • The texture of a quartal tetrachord is getting fairly thick and might be a practical upper limit for comping, especially on guitar
  • The quartal tetrachord material for II and V is the same for each of the four scales, so work on those quartal II-Vs and you will get a lot out of them!

Quartal Harmony: Trichords

Today we continue the discussion on quartal harmony with trichords. Yesterday’s topic on quartal dyads was a bit of a warm-up. Things are getting more interesting now. There are three varieties of fourth: perfect, augmented, and diminished, or using our labels, P, A, and D. In order to construct a trichord, we need two intervals. That gives us a total of nine different types of quartal trichord. These are shown in the Fig. 2 chart below. When looking at the harmonizations of our four scale types, Major, Melodic Minor, Harmonic Minor, and Harmonic Major, we can make some observations about how often the trichords appear.

Two of the trichords, AA and DD, do not appear in any of the harmonizations. In AA, the outer voices form an augmented seventh, which is enharmonically equivalent to an octave. In DD, the trichord is enharmonically equivalent to an augmented triad.

We saw yesterday that P was the most common dyad, and see now that PP is the most common trichord. It occurs five times in the Major harmonization, three times in the Melodic Minor, and two times in each of the Harmonic Minor and Harmonic Major. So then, just like the P dyad, the PP trichord is the most harmonically ambiguous of the quartal trichords.

The next most common trichords are two of my favorites. They combine an outer major seventh, and inner perfect and augmented fourths, or PA and AP. I love both of these sonorities and used them often in my early composing. (Berg and Webern loved them too, so much so that they are sometimes referred to as the Viennese trichord.)

Two of the trichords occur in only one harmonization. PD occurs in only the Harmonic Major harmonization, and DP occurs in only the Harmonic Minor. As a result, these two trichords are the most harmonically specific.

I find it interesting to look at the similarities and differences among the four harmonizations. One thing that leaps out is that all four harmonizations share the same trichords on modes 1, 2, and 5, or on the tonic, supertonic, and dominant. So, for example, a riff on the tonic and supertonic trichords would be completely harmonically ambiguous; it would fit with any one of the four scales. Somewhat surprisingly, the dominant is the same for all four.

Not surprising is where the most variation occurs between the harmonizations: on the mediant and leading tone. Both of these degrees (or modes) contain both the third and the sixth, which is where all of the variation between these four scales takes place. Contrast this with the subdominant and submediant. The subdominant trichords contain the third, so the two with the lowered third are the same, and the two with the natural third are the same. For the submediant, the two with the lowered sixth are the same, and the two with the natural sixth are the same.

Quartal trichords harmonized with the major, melodic minor, harmonic minor, harmonic major scales.
Figure 1. Major, Melodic Minor, Harmonic Minor, Harmonic Major scales harmonized with Quartal Trichords.
Quartal Trichords in scale harmonizations.
Firgure 2. How many times does each trichord appear in the four scale harmonizations?

Key Takeaways

From a practical standpoint, here are a few things to keep in mind using quartal tichords for comping or soloing:

  • The texture of a trichord is very useful for comping, especially on the guitar
  • Any quartal trichord material built on I, II, and V will work for each of the four different scales
  • The quartal trichord material on II, VI, and VII is where all the harmonic differences are

That’s it for today’s installment. In later posts, I will look at quartal tetrachords, as well as usage of inversions for the trichords we looked at today.

Quartal Harmony: Dyads

More follow-ups to my recent visit to the Aebersold Summer Jazz Workshop. I attended master classes with four great jazz guitarists: Corey Christiansen, Dave Stryker, Mike Di Liddo, and Craig Wagner. All four of these musicians gave me things to work on. One of the discussions that we got into with Corey was about quartal harmony, something I had worked with him on previous visits. I decided that it would be good to do an in-depth study on this material, so I plan on doing a series of blog posts.

My ultimate goal will be to incorporate this material into my jazz comping and soloing. But I am going to start out with a high-level survey of the territory. To do this, I’ll look at quartal chords of various size, dyads, trichords, tetrachords, pentachords. For each size of chord, I’ll look at all of the different quartal chord types, that is, all of the different combinations of the interval of a fourth. I’ll also look at where these different types of chord appear in harmonizations of four different scale types: Major, Melodic Minor, Harmonic Minor, and Harmonic Major. (These four scales correspond to the twenty-eight different modes I made reference to in my previous post.)

Let’s start with dyads. There are three different qualities of fourth that I will look at: Perfect Fourth, Augmented Fourth, and Diminished Fourth. Since we are only dealing with fourths, I’ll use a simple label and just call them P, A, and D. The first figure below shows where each of these three different dyads occur in harmonizations of our four different scale types.

Quartal dyads harmonized with the major, melodic minor, harmonic minor, harmonic major scales.
Figure 1. Major, Melodic Minor, Harmonic Minor, Harmonic Major scales harmonized with Quartal Dyads.

The next figure contains some statistics about how often each dyad type occurs in the different harmonizations.

Quartal Dyads in scale harmonizations.
Firgure 2. How many times does each dyad appear in the four scale harmonizations?

There are a few things to note. First, it’s clear that the prefect fourth appears the most often: eighteen times out of twenty-eight in total. In a sense, this makes the perfect fourth the most ambiguous of the three dyads, since it occurs in so many contexts. Another interesting insight is concerning the augmented fourth. Since the augmented fourth only occurs once in the Major scale, it has traditionally been taken that it serves well to establish the scale type and key. In contrast, the augmented fourth appears twice in each of the Melodic Minor, Harmonic Minor, and Harmonic Major scales. This arguably makes each of these scales more harmonically ambiguous than the Major. It also strikes me that the Melodic Minor, Harmonic Minor, and Harmonic Major scales have the same number of occurrences of the different dyads P (four times), A (two times), and D (one time).

That’s it for today. Next time, I’ll do the same treatment for quartal trichords.