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.

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.

A Mode For Every Day Of The Year

I just got back from two weeks at the Jamey Aebersold Summer Jazz Workshop. It was amazing, as always, and especially important since this is the final session before Jamey retires after running the “camps” for over fifty years. I sat in on Pat Harbison‘s advanced music theory class, and I got some great ideas out of that.

One idea in particular got me thinking about digging deeper into the sound of specific modes. Pat mentioned something along the lines of “twenty-eight modes ought to be enough.” Since there are about twenty-eight days in a month, and twelve months, I saw how you could practice a different mode on a different root note every day of the year. Take the month as your root, and the day as your mode number. The base scales I chose were Major, Melodic Minor, Harmonic Minor, Harmonic Major. For the extra days in the month, up to three in a month with thirty-one days, I added Diminished (mode 1), Diminished (mode 2), and Whole Tone. Today is July 17, so that means my mode of the day is the second mode of Harmonic Minor, with a root note of F# or Gb. The whole scheme is summarized in the charts below.

How to practice a different mode every day of the year.
Figure 1. How to practice a different mode on every day of the year.

Key Takeaways

If you are stuck in a rut practicing modes, try this method. You don’t need to keep it up for a year, but if you try it for a while, you will learn that some of the modes you never practice are really beautiful.