18 Triadic Permutations

I use the term permutations loosely here. But for any given chord inversion, there are 6 variations of the tones in the pitch you can play in order to play each tone once. What makes this an impure use of the term permutations is that the second and third notes of the sequence can go both above the starting note in one variation, and below it in another.

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Decisions when playing Chromatic Triadic patterns

George Garzone is the Sax players sax player. He is a teacher that has taught the best of the crop that is out there right now. I had the privilege of studying with George back in high school. I can honestly say that no subject I studied before or since taught me how to think better than Jazz improvisation.

Back in the time of DVDs, George published his method for building a solo that he called the triadic-chromatic approach. A recent video from Moon Hooch’s Patreon reminded me of this approach.

The simple take of the approach is play a triad, move chromatically, play another triad, move chromatically, and continue. Simple, but not easy. Like the game Go, you can learn the rules in 5 minutes, and then spend the rest of your life trying to master it.

What I would like to do is lay out the set of decisions I need to make when playing this approach so that I can build a structure to practice it better. One idea I had was a dice version, where you roll a die for each decision, and then build from there. The first decision would be to chose a key. Since there are 12 Keys, we’ll roll a 12 sided die and use this mapping:

1=A 2=b c=3 4=d…7=G 8=A# 9=C# 10=D# 11=F# 12=G#

That only has to be done once.

OK, now you have your starting note. That does not mean you have your starting chord. First, lets determine what kind of chord it is. We could limit our selves to major and minor, which we would get by flipping a coin….but since there are other decisions to make here, we’ll hold off on that.

We need to figure out the inversion. This is a number 0-2:

0 = root, 1 = first inversion, 2 = second inversion

These 6 combinations can be generated with a 6 sided die.

1=root maj 2 = first major 3 = 2nd maj 4=root min 5=first minor 6=second minor

We also need to come up with the order of the notes. This where the permutations start to add up. If we are playing Root position, we could play 1-3-5, 1-5-3, 5-3-1, 5-1-3.

We also have to decide if we are going up or down between each note. That gives the following options: Up Up , Down Up Up, Down, Down Down. However, this is going to lead to some pretty big jumps. 5 up to 3 is actually jumping up a sixth. 3 up to one is also up a sixth. We can work that in, but it is a lot less common in playing, and we might want to weight things such that those show up less often. The combinations get complex enough quickly enough that we would want a Dungeon Masters Guide (first edition) lookup table to fine the right combination. For now, we’ll leave it equally spaced. 16 options…easier to keep them separate and roll 2 @ 4 sided dice, one for each decision.

For the chromatic runs afterwards, we are going to keep things simple. They are one, two, or three note runs. We need to determine the length of the run and the direction, up or down. 6 combinations. 1-3 is Up, 4-6 is down. divide the roll by three and take the remainder as the length of the run.

Ignoring our first 12 sided die roll, we have a repeated pattern of

6 * 4 * 4 * 6 = 576

options in each sequence. That is a pretty big option space.

What is going to limit you? Here are some ideas:

Habit: You are going to play the notes under your fingers that are most familiar. We are used to going from G down to E down to C more than from C down to E down to G. But it does sound cool.

Chords: You are going to gravitate to the chords of the tune you are playing.

Ease of fingerings: For the most part, it is easier on the Sax to play a natural than an accidental. Certain fingerings are just more comfortable and easy to hit. It is easier to play a C# than a G#.

Size of the Saxophone: You can only play so high and only play so low. have to turn around at some point or you run out of notes

Humans are not random: We are not going to make all of those decisions all at once every time. We are going to fall back on heuristics like : mostly play 135 up or 531 down, but after a while realized we have not played 1 5 3 down and throw that in.

Making a George Garzone simulator would be a fun task.

So I did: https://github.com/admiyo/chooch

The Chooch is one of George’s early tunes. It seemed appropriate.

chooch requires abcMIDI. On Fedora:

sudo yum install abcMIDI

Here’s some sample output.

I suspect there are bugs. It seems to get stuck at the bottom of the staff, althoug I tried to reset it to the middle if it gets too low.

The Bird Run

The Summer Youth Music School (SYMS) at the University of New Hampshire runs each summer. I attended it a couple years back in High School. The second year, I prepared a piece called Au Privave (No. 1) from the Bird Omnibook. This is a transcription of solo by Charlie Parker.

Charlie Parker played on alto, but I played tenor sax, and prepared the version from the B flat book, which was transposed down a fifth. It turns out that this radically alters the fingerings. Parker played insanely fast, and in doing so, he naturally sought out the fingerings that flowed smoothly and naturally on the Saxophone. And then he played them at superhuman speed. It turns out that transposing it up a fifth radically changes some of the fingerings, and also puts many of the notes out of the range of the Saxophone

I bombed the audition.

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Harmony and the Harmonic Series

Why does the G# work so well when the rest of the song is in C major? Why, if the song is in A minor (the relative minor to C) does the G# become the note that turns it into a Harmonic minor? The answer would be based on the Harmonic series.

Here is the Harmonic series as explained in Wikipedia’s page: https://en.wikipedia.org/wiki/Harmonic_series_(music)

The G# we are discussing is based on the 13th element of the series. In a non-tempered instrument, this would be the +41 Cents indicated, meaning that it would be a fairly sharp note.

Let’s Ignore Octave for a moment and just write out the notes of the Series.

C C G C E G Bb C D E Gb G Ab Bb B C C# D D# E


You can think about how much color a given note has by how far down the series you need to go to get to that note. The 1, 3, 5 and Dom 7 all are in the Early part of the series. That Dom 7 is fairly flat (-31 cents) from the tempered tuning. I think it is fascinating that the Minor Third we hear so often in a minor chord is so far down the Harmonic series. A C minor uses the Eb, which is the 19th note of the series. However, if we think of the A minor, then the Third is the base note of the series, while the root of the chord is based on the 27th note of the series. There must be a simpler explanation for that.

The E minor chord is much earlier in the series…the E is the 5th note of the series, and the G is the 3rd. But the B (the V of the E minor) is the 27th note of the series.I suspect that what we actually hear is not based on a single harmonic series, but on multiples. If we were to translate the C scale shown above to the E scale, then the V of the chord is, again, the 3rd note of the series. Same with the A minor. So You would have to transpose around the harmonic series to see where the G# fits relative to the other notes of the chords.

I suspect that what you are going to find is that the Chords are based on multiple Harmonic series. For example, the B that makes up the Major 7th in the C maj chord is very closely related to the G in the Harmonic series based on the G root. It does not even show up in the first 20 notes of the Harmonic series based on C.

Looking again at our series of notes, we can see the earliest Minor chord is the G minor. The G is the 3rd and 6th entry of the Series, so very strong root. The Bb is the 7th entry and the D is 9th entry.

The sequence also shows why a Dom7 with a Ninth added to it sounds so strong. The Chord Ninth is the D, coincidentally it is also the 9th element in the series.

I also find it quite interesting that the E to Bb tritone shows up so early in the series: 5th to 7th entries. This sound is so strong and jarring, but it is right up front.

8 Tone scale for that strange chord in Take The A-Train

You must Take the A Train…if you want to improvise over a standard. But this standard tune has a non-standard chord in Bars 3 and 4. If you are playing the “Real Book” version in C, the song starts with two measures of C Major 7, and then goes up a whole step to D. If we stayed in the Key of C, this would be a Dminor chord. Billy Strayhorn was much more creative than that, and he put in a chord rarely seen anywhere else: D7 b5.

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