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I have been studying Pat Martino lessons on the parental forms. It absolutely blew my mind that if you lower any note a halfstep in a augmented triad you get a major chord. And if you raise a note you get the relative minor of the lowered note.

I have been playing around with this for months and today I made a discovery. If I lower any note two halfsteps (one wholestep) in the augmented, I get 3 notes out of a 7b5 chord. If I raise a wholestep in the augmented I get a 7th.

In the Caugmented cluster you have these 7b5 chords if you lower any note a wholestep.

F#7b5(no3) A#7b5(no3) D7b5(no3)

I also note that if I raise any note of the augmented I get a 7chord. These 3 7th chords are offsprings of the Caug:

C7 (raising the Ab) E7 (raising the C) Ab7 (raising the E)

My question now is how do all of these relate? If you lower or raise the notes in a augmented triad a halfstep they relate because they are relative minors and majors. Is there a similar thing happening if you raise and lower a wholestep? Is there a mindblowing sacred geometry happening here or is it just that the chords are chromatic neighbors so it kinda works to play them with the Caug? I know that you can play any diminished chord next to any major chord as a passing chord, to modulate to a different key. And since 7b5 KIND OF reminds me of diminished chords because they sound a little bit “odd”, is that why you always get a nice 7b5 if you lower the augmented?

Here is a chord progression I made and the first four chords are offsprings of the Caug. Lowering to get the F#b5.

Amin9->Caug->C->F#7b5(no3)->Fmaj7->Emin->Ebdim->D9->Emaj

You can throw in a note to make the F#7b5(no3) into a F#min7b5 it sounds just as good. Also throw in 9ths and do any kind of inversion whatsoever the F#7b5 still sounds good in the context.

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  • Sounds exciting, have you tried brute forcing this by computing earth mover's distance between chords to see if there are more examples?
    – Emil
    Commented Jan 10, 2022 at 7:00
  • Who is earthmover? The only bruteforcing i done is just to move chromatically. I also drew the thing to see if i cant find some interesting relationships. I found that two tones in the F#7b5 contain two tritones in a diminished chord. I simply drew out the geometrin shape of a diminished, and then the shape of a 7b5, and two notes of the 7b5 cut right through the diminished symmetrically. So possibly, this could mean that one wholestep away from a augmented - there will always be a tritone relationship to a diminisjed chord. I am not 100% sure tho i just sketched it fast.
    – j a
    Commented Jan 10, 2022 at 18:54
  • I meant computing the minimal number of half steps between a lot of types of chords, it's a pretty general way to measure distance between a bunch of lists of numbers used in various sciences. It should be possible to let the computer cluster them is what I mean.
    – Emil
    Commented Jan 10, 2022 at 19:18
  • Ah i see. Unfortunately i am no programmer so i wouldnt know how.
    – j a
    Commented Jan 11, 2022 at 0:13
  • A critical factor here is the symmetry of the augmented triad, which comes about because it comprises only intervals of four half steps, and 12 is an integral multiple of 4. The same symmetry is found in the diminished seventh chord, but the effects are somewhat more complicated because it has more pitches: if you raise any two adjacent components by a half step, you get a minor seventh chord. If you lower any of its components by a half step, you get a dominant seventh chord. If you move any two components toward each other, each by a half step, you get a major seventh chord.
    – phoog
    Commented Jan 12, 2022 at 14:03

3 Answers 3

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I don't think I have enough of a handle on this to create an proper answer, so take this as an extended comment, but I have heard from many older jazz musicians about this whole alternative way of viewing jazz theory, or some would say ORIGINAL jazz theory, that centre's heavily around the augmented chord.

I've had it laid out to me at some point by a trumpet player called Neil Yates who's a lecturer at Leeds College of Music in the UK, I made some notes but I can't find them annoyingly. But I do remember him using augmented triads to suggest many things, almost everything, and using them to sidestep around in the way you are talking about. He coached me through a slow improvisation (now, play this, now that, use the leading tone on THIS note to suggest THIS) and I remember laughing out loud as the lines I created almost effortlessly had a hint of Bird and certainly felt more real, original, Bebop than anything I have been able to muster up otherwise. We were both teachers at a very drunken weird workshop for a very high flying, rich, company out in Austria. When I got back I remember trying to collate the notes I made but couldn't really work out exactly how to do it. He was often talking about a sort of lost theory that was the main thing in the 40's, and alluded to several times musicians like Herbie Hancock etc. had mentioned 'the other way' of looking at things in interviews etc. In truth it seemed almost conspiratorial, but I certainly hadn't came across that way of thinking in my learning and it's true that the Bebop musicians weren't thinking scales, however chord tones, leading tones and encapsulations doesn't quite seem to be the full story of what they did.

I remember him talking of a 3d system of related augmented triads that suggest related 7ths etc, and then treating leading tones or notes 2 semitones down to shape chords in parallel keys, all linked by diminished scales that you could set-up, then jump on top of and ride out to the end when you resolve with a 'be-bop' (he described it as thats what the name means, anecdotally at-least). Barry Harris seems to touch on this idea sometimes with his lessons, though it's not quite the same thing, but it has a similar feel of 'another way to view it'.

Anyway, not much of an answer, but wanted to do a full comment as I think there may be some truth when you talk about a 'whole other' sacred geometry in this stuff. Coltrane had his own geometric theories, though that's not the same thing, and related more to his sheets of sound period. But Coltrane's explorations do suggest there are many ways to view basic jazz theory with very interesting connotations for the improviser/jazz composer. With any luck the workshop me and Neil Yates taught on will happen again this year or the next and I can ask him again, I will report back if it does! It may not be 100% the same thing but it does sound like you have found the start of something along the same lines, it definitely involved using augmented as pivots to get to 7's or 7b5's via leading tones and then the relation between THOSE chords in different keys formed corners of a shape that you could navigate by H/W and W/H diminished, I remember when he explained it and drew it out on a piece of paper I was pretty blown away by how complex yet simple it appeared.

Please forgive the rambling non-answer, but very interesting question and I'll save it in case I'm one day able to add more!

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  • Very interesting, and very intriguing. By "a 3d system of related augmented triads" , do you mean a "third" or a "three-dimensional" system?
    – Jos
    Commented Jan 10, 2022 at 10:50
  • 3 dimensional, it was like a cube (or similar) and each corner was a 'home' augmented triad and you could tell the relationships between it and the other augmented triads by how many steps you had to make to get to each.. a little similar to the relationship between the 4 diminished scales and their modes. I wish I could remember more, it was literally on the back of a napkin at 1am standing at a 'free bar', I'm busy this week but I'll try get my head around it a bit more clearly and answer again, or maybe I should just call the guy who explained it to me!
    – OwenM
    Commented Jan 10, 2022 at 12:30
  • Pat martino also talks about 3D. I havent understood it yet tho. I’ll try to find the segment and i will come back and link it. Appreciate your comment, please get back if you hear from that guy!
    – j a
    Commented Jan 10, 2022 at 16:28
  • I found that the 7b5 has two notes that are tritones in a F#-diminished yesterday.. Idk if i am on to something with that but it doesnt hurt to just throw that out there.
    – j a
    Commented Jan 10, 2022 at 18:56
  • Following! Would love to learn whatever that concept was. Commented Feb 2, 2022 at 3:02
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It feels a lot less mystical if you think about it linearly instead of drawing lines on a circle. Like on a piano keyboard, or in terms of modulo arithmetic. Modulo arithmetic means dividing all results by a number and taking the remainder, for example 4+3+3+4 = 2 (modulo 12). Which means that the 9th in a 9 chord, which is 14 semitones above root, is the same as the 2nd which is 2 semitones above root. Modulo 12 returns all pitches to the base octave. (You can see the same thing on a piano keyboard or music stave without any arithmetic mumbo jumbo though.)

Let's start with the notes of an augmented triad, and we'll play its notes in all octaves. It's completely symmetrical, all intervals are the same. It looks like this on a semitone grid, showing three octaves but you can imagine more:

augmented triad in several octaves

Let's arbitrarily select one of the notes as the one to move, shown in red:

augmented triad 2

We'll call that the zero note, being at offset 0, and other notes are higher than that by +4 and +8 semitones. Modulo 12, because when you move by an octave, you get the same thing.

augmented triad 3

It should be intuitively clear that whatever you do with the red note, the same thing relative to the red note would have happened regardless of what was selected as the red note. Let's move the note UP by a semitone. The other notes become +3 and +7 relative to the red note. That's a MINOR triad:

red note moved UP by a semitone

If we move the red note DOWN by a semitone, we get 0, +5 and +9, which is a major triad. Well, second inversion if the red note is the lowest, but this was an infinite pitch grid. You could also see this as a rootless m7 chord.

red note moved DOWN by a semitone

If we move the note down by TWO semitones, we get 0, +6 and +10, which can be looked at as a rootless m7-5 chord like you said. Or the character tones of a 9 chord where 0 is the chord's third. Or moving the 9 chord's root by a tritone, the 7th, 9th and 13th of a 13 chord.

red note moved down by TWO semitones

If we move the note down by THREE semitones, we get 0, +7 and +11, which could be a maj7 chord without a third. Or m maj7 without a third. ;)

red note moved down by TWO semitones

I'm sure you could get more mystical feelings from this by drawing everything in a circle instead of a straight line. But very few actual instruments have their pitch dimension curled up in a circle, so a circular layout seems less practical.

To get a similar view into a dim7 chord, you use every third instead of every fourth semitone.

dim7 grid

Select one of the notes as the red one, move it somewhere and see what you get.

dim7 grid 2

My own intuition about things like this is to emphasize practice. How did I figure out that the rootless m7-5 is the same as the character tones of a 9 chord? I didn't calculate it, and I didn't draw mystical geometry pictures. I imagined a piano keyboard in my mind, and the notes looked like what I would typically play as a 9 chord with my right hand. The bass plays the root, and the fifth can be left out anyway. If you want to emphasize the mystical geometry side, that's completely ok. People have been able to come up with very interesting new expression by looking at music from a different perspective.

Here's another question about Pat Martino's augmented shapes Pat Martino chord concept - Augmented Forms - Guitar Fretboard shapes

My summary of the concept is, Pat Martino is utilizing the augmented shapes as a tool to locate note patterns on the guitar fretboard in some kind of a unified and general way, supposedly requiring to learn fewer special cases. There's the CAGED system and others which, AFAIK, try to achieve the same goal: to be able to command the entire fretboard, all strings and positions. But even with Pat Martino's "mother" shapes, it will take several years. And for a keyboard player, this thinking might not be very meaningful at all.

Whatever the approach, you need lots of practice. Even if you know about the 9 chord containing a m7-5 chord, you need to be able to locate it quickly. Noticing forms and patterns and geometric shapes can provide ideas for practicing, but ultimately you need to practice to make the shapes a part of your practical vocabulary.

If your primary instrument is guitar, it might be useful to learn piano or keyboards a little bit, to explore the note geometry. Or if you're a programmer, you might be able to brute-force search for hidden patterns like "9 chord contains a m7-5" with something like Python and the Music21 library. Good old classical staff notation can help too, even though seeing enharmonic equivalences there is less trivial than on a piano keyboard.

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    Now you said something very interesting. The 7b5 forms the shape of a 9chord? That’s very interesting. This is what I am looking for. It helps me build a perception of the fretboard of the guitar, a cohesive view of it and not just like 20 different scale patterns in different spots. I want to be able to look at the frets and just see one big shape that covers everything. And the geometry, I don’t think the application is mystical. It’s almost embarrassingly simple. But the origins is definitely mystical. It’s almost like nature is communicating with you by showing you these shapes.
    – j a
    Commented Jan 10, 2022 at 20:52
  • There is also the argument that because the guitar isn’t a linear instrument it’s difficult to think in pianoterms. In a guitar you can move vertically and horizontally, but a piano you only move horizontally. So that becomes a problem. The geometry works then on the guitar because when you have both a X and Y axis you literally have a graph. Like, you literally see shapes if you imagine them between notes on different strings. It’s not just a triangle in the chromatic circle, it’s also a triangle literally on the fretboard.
    – j a
    Commented Jan 10, 2022 at 21:02
  • Thanks for sharing btw. I actually found these lessons by pt Martino from searching the web if there is any mathematical sense to the guitar. So when you mention modular arithmetic you give me a hardon. I want to see how far I can go by playing the instrument mathematically. So far it has been extremely fruitful. Not familiar with modulo tho. Gonna have to read on that. It’s a style of multiplication?
    – j a
    Commented Jan 10, 2022 at 21:05
  • @ja I added an explanation of what modulo arithmetic means. In music you'd usually think about modulo 12. Commented Jan 10, 2022 at 21:22
  • @ja But really, learning to play piano might be the easiest way to get an overview. Certainly it requires less calculation, because it's all graphically laid out in a nice linear picture you can look at. You can do the "calculations" with your eyes and fingers. :) Commented Jan 10, 2022 at 21:43
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Within the context of the derived chord progression, "the theory" behind it is the (re-)discovery of a core principle of voice-leading: the voices should move primarily by (whole or half) step. Once this chord progression is achieved, the relationship to the augmented chord becomes secondary, even though that was the starting point for its creation. However, it also could have been created by starting with the A-9 chord and then moving voices one step at a time, with no consideration to a single reference chord (i.e., the augmented chord).

Here are the chords laid out in five voices. They're presented on separate staves to emphasize the (horizontal) voice leading at the expense of (vertical) chordal readability.

Voice leading of OP chord progression

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