Reimagining the Fretboard: A New Approach to Harmonic Development

[jordanklemons]

Jordan,

besides from my own thesis works I have ghostwritten -- at least in parts -- quite a few thesis papers and two doctoral dissertations when I was young. Also I have done a lot of refurbishing work (content and layout/typesetting) on thesis papers, all that for some additional income. My interest in this kind of published texts is still there, so took a short look into your paper. There are two things that immedeately struck me:

1. Your thesis is a scientific work, therefore it is expected to be written in an appropriate, academic language. If you write passages like: "Look through almost any guitar method book, and you will find chords and scales being expressed as visual patterns, intended merely to be memorized. This way of thinking is not wrong, and I would even argue that it is a necessary part of guitar education due to the physical nature of the instrument. But if we [...]" or "really quite bizarre", then certainly this expectation is not satisfied. In this text you're not casually talking to an abstract reader. This kind of style would be too informal...

2. With regard to the next point, like Don Quijote I am fighting against windmills half of my life--nonetheless it should be said: your text is supposed to be read by human beings. It is a functional text with long passages of continuous sentences. Western culture provides you a 2,000 year old treasure that perfectly meets the ergonomic demands of such a kind of text: serifed typography. Helvetica is ok for display parts and headlines, but for sure it is not appropriate for reading passages. Essential reading for that: Robert Bringhurst "The Elements of Typographic Style".

Of course there might be claims of the house style of your university that would contradict my point of view (then please kick your Dean's tibia with greetings from me). One example could be the the immense leading of your text, which makes it almost unusable. Many universities demand this kind of typographical nonsense for proofreading purposes...

Wish you good success with your thesis!

Robert

Thanks Robert.

The casual style I chose to write my paper in was a conscious decision and one I'd spoken to my advisor about. My original thought was to write a 'method book'. But due to the limitations placed on me, I had to leave a lot of information out, and I chose to make it more of a documentation of exploration... which required a fairly quick sum up (and generalization) of the standard approaches found in the current guitar community. If I went into the kind of detail I needed to, I wouldn't have been able to fit in all the harmonic ideas I wanted to. It wasn't ideal... but it worked well enough.

What font would you recommend? That issue wasn't talked about at all during my advising sessions or colloquy class. And it's too late to change it now, but I'm curious. I just used whatever my writing program defaulted to.

[jordanklemons]

its a big deal putting something like this up on a forum - thanks a lot J.

there is no internet forum in the world on which i would post my phd thesis on perception (philosophy)

Uh oh... now you have me worried.
Should I not have?

i start b in g with a gm6 chord in 1st inversion which has the 6th in the top voice. i love that chord. i suppose its like the C7 with a seventh in the bass - but its also like a Bbmaj#11

How do you voice this out?

how do you stand in general to 'barry harris-type harmony' (i.e. his maj and min6 dim chord scales and their multiple applications)?

I think Barry is brilliant. I went to a BH masterclass very early on in my jazz playing. I understood the theory of his ideas, but I wasn't ready to really work on those things yet. I've come back to it a few times over the years little bits. I imagine that the ideas I'm currently working on will take me a good 5-10 years to really dig through, and probably even longer to really start to hit a solid footing. If I ever get to the point where I feel like I've excavated everything I'm looking for and still want some sort of systematic way of exploring, I've thought about spending some time on BH.

[jordanklemons]

J
Some questions for you:

On page 5 the point is made that the upper structure triads in the paper will be limited to major triads for the sake of simplicity. Can you discuss some guidelines for when a non-major triad would be used for the upper structure triad? Is it arbitrary based on the sounds you are going for or what you can conceive of on the fly, is the type of triad guided by the underlying/left hand harmony, does Stefon focus just on major, or?

My main goal is that nothing should ever be arbitrary. Whether I'm choosing because of what my ear wants, or because it matches the theory more properly, or some other reason. I know a lot of people don't like for chord names to be overly specific. They might not want to see D-7 vs D-9 vs D-11 vs D-11(9) vs D- vs D-(add2). I can respect that mentality and understand why. But all of these chords sound and behave very differently. So for me, I like to try all of them and get familiar with how they all sound (using triads). The more I get to know them, the easier it is to make choices about which triads to use.

But a better starting point is to base it on the 'melodic progression'. I have a couple of threads talking about this and showing examples. The melody of tunes are pretty much based on triads, and often, they deviate from the harmonic root triads. So if I see the melody hinting at a minor triad, then that's where I'm going. But if you're new at this and you're interested in checking it out... I'd stick with major triads for a while. I spent about 6 months just on major triads before adding anything else. You can squeeze a lot of juice out of them.

J
Can you elaborate on how one generates the tension note that was shown on page 16 for each triad in the grid for Blue in Green to form the 'Quadrad' in Stefon Harris' system? Apologies if I am being dense but the pattern isn't jumping out at me and I am not at an instrument to play around with the sounds right now.

Yeah this is a bit of a tricky question to answer typed out. There's a sort of method to it, but there are so many exceptions and sticky issues that can come up. The basic idea is to choose from the lower structure notes. The idea is not only to create tension, but also to glue the triad to the lower structure harmony, so that it doesn't feel disconnected. But there are rules and issues that come up.

Sorry, but that's about the best answer I can give in a typed situation. It's something that I need to go over multiple chords with someone for a while until things start to click for them.

[jordanklemons]

Jordan - A question for you.

I watched and played along with your video of Body and Soul. Did you apply your research approach to this tune in your performance? To be honest I expected to see something "radically" different in terms of fingering or bass and inner voices.

What I heard was excellent melodic predominantly playing with tasteful harmonic additions and good time. Given that you did improvise on the melody if I recall and that will be certainly often the highest pitched note, can we dis-count that note from the equation as a given? In other words the melody note or improvised melody note will be there no matter what. That leaves the rest of any harmony voices and the bass voice. I am assuming that a bass voice of some sort is also a 'given', though one can naturally employ other notes besides the root and fifth including leading voices or extensions.

These presumptions on my part re bass and melody voices leave you 3rds, 7ths, 9ths, and other extensions and alterations as the inner harmony voices. Did this conception regarding the upper triads or use of extensions in a 'top down' harmonic approach (you choose the proper designation) contribute to your recording of Body and Soul? And if so, how?

This is not a trick question or dismissal of your approach. I am trying to hear it in the recording. I would add that given that the harmony was well done but relatively sparse in a melody dominant "arrangement" for lack of a better word, this song might not be the vehicle for applying your approach, as it is fundamentally a blues.

Thanks for the compliments Jay. I was using the approach some. I recorded that video early on in my experimenting with the system. The majority of my single-note lines, if not maybe all of them, were utilizing the quadrads. The chords were mixed. Some standard, some from the top-down approach.

[diminix]

The casual style I chose to write my paper in was a conscious decision and one I'd spoken to my advisor about.

Ah, ok! Then it is appropriate.

What font would you recommend?

If it should be made as good as possible, I would choose a full-fledged typeface (with everything you need for such a task, like small caps, proportional and tabular figures in lining and oldstyle, condensed cut for footnotes) made for book setting. And it should be a beauty too! Air gets thin here, my choice would be FF Scala (the new German edition of Hitler's Mein Kampf is set with that typeface, it's a real work of art--the book setting, not the content). Modern and classic at once, very appropriate for your approach to Jazz. If it shouldn't cost money, then Adobe Minion, very well equipped and almost omnipresent because it is rolled out with Adobe CS products. If not available, any good Garamond-derived typface, but under no circumstances Times New Roman (too long a story to tell, it's just a no-go).

Pair that with a good looking, non-geometric sans serif, not Helvetica, which is your default right now. You Americans have a great tradition in that respect, like Trade/News Gothic or Franklin Gothic. And of course you would need some good tools for music setting. The problem would be, that all these typgraphic capabilities could only be used in a real type setting software like Adobe InDesign. A normal wordprocessor like Word is a pain in the a**, but it is the standard tool.

All these considerations are unusual for a thesis work, but they would make life much easier than the standard procedure if determined before the writing starts (this is not aimed at you but at academic writing in gerneral). And if the content of the work has a real value (I bet your work has it), the form should be appropriate.

Robert

[studiotan]

...So if I see the melody hinting at a minor triad, then that's where I'm going. But if you're new at this and you're interested in checking it out... I'd stick with major triads for a while. I spent about 6 months just on major triads before adding anything else. You can squeeze a lot of juice out of them.

Yes makes perfect sense.

...The basic idea is to choose from the lower structure notes. The idea is not only to create tension, but also to glue the triad to the lower structure harmony, so that it doesn't feel disconnected. But there are rules and issues that come up.

Sorry, but that's about the best answer I can give in a typed situation. It's something that I need to go over multiple chords with someone for a while until things start to click for them.

Thanks that helps, and avoiding a disconnected feeling between the structures is a recurring theme in what you wrote so that tracks.

[jordanklemons]

Ah, ok! Then it is appropriate.

If it should be made as good as possible, I would choose a full-fledged typeface (with everything you need for such a task, like small caps, proportional and tabular figures in lining and oldstyle, condensed cut for footnotes) made for book setting. And it should be a beauty too! Air gets thin here, my choice would be FF Scala (the new German edition of Hitler's Mein Kampf is set with that typeface, it's a real work of art--the book setting, not the content). Modern and classic at once, very appropriate for your approach to Jazz. If it shouldn't cost money, then Adobe Minion, very well equipped and almost omnipresent because it is rolled out with Adobe CS products. If not available, any good Garamond-derived typface, but under no circumstances Times New Roman (too long a story to tell, it's just a no-go).

Pair that with a good looking, non-geometric sans serif, not Helvetica, which is your default right now. You Americans have a great tradition in that respect, like Trade/News Gothic or Franklin Gothic. And of course you would need some good tools for music setting. The problem would be, that all these typgraphic capabilities could only be used in a real type setting software like Adobe InDesign. A normal wordprocessor like Word is a pain in the a**, but it is the standard tool.

All these considerations are unusual for a thesis work, but they would make life much easier than the standard procedure if determined before the writing starts (this is not aimed at you but at academic writing in gerneral). And if the content of the work has a real value (I bet your work has it), the form should be appropriate.

Robert

Thanks for the advice. I'll check these out.

[Nate Miller]

" Can you discuss some guidelines for when a non-major triad would be used for the upper structure triad? "


there are a couple of ways to go about it...If you were thinking in terms of playing in two keys, then it would be the diatonic triad in the key that voice was harmonizing from. If you were doing this like Jordan is doing it, you really just need to consider whether the major or the minor third (to decide between major and minor) is going to be in the home key and use that triad

For example, consider the first chord in Blue and Green where Jordan has a C major triad over a G minor. Clearly you want C major, right? The E natural is in the melody...an Eb would clash horribly

that's how you decide...does the major 3rd clash? yes? use the minor....that's the process. Its simple enough you can calculate that on the fly

[nick1994]

" Can you discuss some guidelines for when a non-major triad would be used for the upper structure triad? "


there are a couple of ways to go about it...If you were thinking in terms of playing in two keys, then it would be the diatonic triad in the key that voice was harmonizing from. If you were doing this like Jordan is doing it, you really just need to consider whether the major or the minor third (to decide between major and minor) is going to be in the home key and use that triad

For example, consider the first chord in Blue and Green where Jordan has a C major triad over a G minor. Clearly you want C major, right? The E natural is in the melody...an Eb would clash horribly

that's how you decide...does the major 3rd clash? yes? use the minor....that's the process. Its simple enough you can calculate that on the fly

but there are other alternatives, if jordan were using minor triads he could've A minor

[Nate Miller]

but there are other alternatives, if jordan were using minor triads he could've A minor

well spotted! And while A minor is a substitute for C, look what happens if we try and substitute an E minor for the C...B natural against a B flat!

I wonder if there is a more general pattern for substitution of upper structure triads. I never thought about that

good stuff, though Nick

how about a C# minor over the G minor? (thinking of more minor triads with an E natural)

[jordanklemons]

but there are other alternatives, if jordan were using minor triads he could've A minor

Absolutely Nick! And the Amin triad would sound fantastic over the G-7 chord! It gets rid of the root note in the upper structure triad and subs in the 9.

And if you wanted to get REALLY fancy, you could even use an AMaj triad. That would yield a G-13#11(9)... suuuuuch a cool sounding chord. If you have a piano, try playing G-Bb-D-F in you left hand, and then an AMaj triad in your right.

If not, try playing this on a guitar

0.10.12.11.11.9

This is an E-13#11(9). It's got the F#Maj triad on the top three strings, and the R-b3-b7 on the bottom three. I use a diagonal barre with my pointer finger to play both the 5th and 1st strings. If anybody has a better way to finger this chord, I'd love to know it. It's the easiest way I've found so far.

[jordanklemons]

how about a C# minor over the G minor? (thinking of more minor triads with an E natural)

You definitely could use the C#min triad if you liked it. My ear doesn't like the sound of the b9 over a minor chord... it just doesn't sound harmonious to me. Obviously that's a big part of the phrygian tonality, and I'm currently playing with how best to approach that sound using this system. It's used so rarely compared to other types of minor chords in standards. But it is such a beautiful and important tonality, so I've been thinking about how to best accomplish it.

But for the general min7 chords, I avoid the b9 in the upper structure triads.

But...

That's the cool thing about this whole system... is that it's not necessarily dictated by scales or theory... it's based on the ear. So the options I come up with might be different from the options someone else might come up with. Which I love and find so great. It can help multiple people find some very particular and personal sounds.

[nick1994]

I like how jordan takes the melody note and shows all the chords that have that note and then he just eliminates it. the way I think is that there are only two diatonic minor triads for G Dorian (apart from G minor) D minor and A minor and only A has the E in it. I relate triad stuff and chord extensions to scales, it just solidifies the sound for me.

I think it's good to do both scale derived triads and finding triads that contain the meldoy note. Like with C# minor it's a diminished derived triad but it's easier to go through jordan's process to get there

[AllFifthsTuning]

Thanks Jordan,


Great reading.
This gives some good food for thought (and work to do!).
ahh that good old mighty triad!

Hans

[Nate Miller]

"But for the general min7 chords, I avoid the b9 in the upper structure triads." -- yes, There's that G# in C# minor. Probably not a good choice after all, really. That's why I don't play keyboards. I type the wrong stuff all the time


nice to hear guitar players talking about triads, though...even wrong ones

[nick1994]

Absolutely Nick! And the Amin triad would sound fantastic over the G-7 chord! It gets rid of the root note in the upper structure triad and subs in the root.

And if you wanted to get REALLY fancy, you could even use an AMaj triad. That would yield a G-13#11(9)... suuuuuch a cool sounding chord. If you have a piano, try playing G-Bb-D-F in you left hand, and then an AMaj triad in your right.

If not, try playing this on a guitar

0.10.12.11.11.9

This is an E-13#11(9). It's got the F#Maj triad on the top three strings, and the R-b3-b7 on the bottom three. I use a diagonal barre with my pointer finger to play both the 5th and 1st strings. If anybody has a better way to finger this chord, I'd love to know it. It's the easiest way I've found so far.

yeah, the A major over G minor is a really cool sound but for me I go more of a scale based sort of thing only because I need a connection with the sound that I'm using. I'm pretty much trying to justify any triad over any chord with scales. but it's much more ear based than what I'm describing.

[pingu]

i start b in g with a gm6 chord in 1st inversion which has the 6th in the top voice. i love that chord. i suppose its like the C7 with a seventh in the bass - but its also like a Bbmaj#11

---

could you write how you voice that on the
guitar ?
maybe
6x875x

[jordanklemons]

yeah, the A major over G minor is a really cool sound but for me I go more of a scale based sort of thing only because I need a connection with the sound that I'm using. I'm pretty much trying to justify any triad over any chord with scales. but it's much more ear based than what I'm describing.

Yeah approaching it diatonically with the sounds you're already familiar with isn't a bad way. It will just be missing some of the fancier options. But you can still get a lot of mileage out of things. You just have to be careful and trust your ear more than anything else. It's possible to find diatonic triads that might not work.

All of my triad options come from sitting at the piano and just testing individual notes against a particular chord. Once I find all the notes I like, then I sift through those and find all of the triads. So it ends up with triads from basic major scales as well as triads that might be found in harmonic or melodic minor, or any number of other places. But I'm looking for them by first deciding what notes my ear likes.

So one funny thing that happened that I wasn't expecting was that I didnn't find the root note of a maj7 chord to sound harmonious when played melodically over the 1-3-7. So I don't use the root triad to develop chords or melodies over a maj7 chord. Things like that are where I have to follow my ear more than the theory. There are theoretical reasons I could come up with as to why that's an avoid note for me. But the theory says that the diatonic scale is just a basic major scale which grows out from the root. But I actually don't want the root.

Funny little issues like that are why I dig the idea of following my ear and not relying on theory or scales for these types of explorations at the beginning. I bring it in later, but it starts with the ear. Because for 20+ years I just assumed the root note not only worked against ANY type of major chord (including maj7) but that it was the strongest resolution point and the most stable note... because that's what the theory told me. Then I sat down and listened to it. And it just didn't function that way in my ear.

[jordanklemons]

"But for the general min7 chords, I avoid the b9 in the upper structure triads." -- yes, There's that G# in C# minor. Probably not a good choice after all, really. That's why I don't play keyboards. I type the wrong stuff all the time


nice to hear guitar players talking about triads, though...even wrong ones

I wouldn't say wrong. Some guys might like it. It's just wasn't a note that I wanted to allow into my minor chords... at least not the minor chords that aren't meant to imply phrygian. But again... I'm still trying to wrap my head and ears around some tonal categories and chords and how to approach them within this system... and phrygian is one of them.

[Robertkoa]

Robert, I'm not sure how android software works. The file is just a basic pdf... so if you're able to open pdfs in general, it should still work. If not, maybe it might be possible for you to download it to a computer?

If that doesn't work, maybe one of the moderators has an idea? Or if that doesn't work, you can pm me and we'll talk.

Thank you very much ...
I was finally able to easily download it by using Chrome rather than the default Adobe App on my phone .




I am going to absorb your Concepts and the Voicings I like for my Expanded R&B Stuff
( essentially more Rhythmic Jazz staying in each Region longer and less frequent 2-5-1s)
( other Modulating Cadences though.. easier to Solo on..lol).

My pet voicing for a Big Minor 6 7 9 11 :

on A minor for example is 5 - 3 - 5 - 7 - 7 -7
the top 3 Notes are a mini barrè with pinky.
Beautiful Chord .
[ One of the few Minor Voicings that can support a i - bvii ( lowered seventh degree wants to be Major ) ]

And the first inversion available also- which would free the pinky for melodies on top right ?

And as a A Minor 7 6 9:

5 - 3 - 5 -5 - 7- 7 doubling the b3 and has more tension than the Minor 6 7 9 11 above.


Also - for improvising your Concept ties in well with what some suggest as another approach ( targeting one Triad or Another in the Chord instead of just one big chord- I would never have thought of that).

I am going to read through your PDF because all the Jazz Theory ( especially Improv) applies to my Harmonically Expanded R&B stuff-

I am used to often playing all the Voices on Guitar because...

I don't have horn sections to play the other Voices ( and Strippers Dancing ) like you Jazz Pros ..☺

[55bar]

Hey guys,

So I've mentioned my thesis a bunch of times on here, and on the topic I was writing about...

I've been planning on, and talking about, posting my thesis here on the forum for anyone interested. However I've been holding off until I could pick up the final, graded copy from my advisor. I wanted to see what notes he left for me so I could make any necessary changes before putting this thing out publicly.

But it's proving harder than I expected to meet up with him. So whatever. I may make a couple changes to this at some point after talking with him. But this was the final draft that I turned in.

It's only about 24 pages, and a lot of that is made up of examples, chord charts, chord melodies, and comping etudes. Here's the thesis abstract to give you a quick summary of what to expect. Feel free to download a copy and check it out. And please hit me up with any questions or comments you might have. I'd love to hear them.


Abstract
This study will examine an advanced view of harmony, both in theory and in application to the physical realities of the guitar. Guitarists generally approach chords with a bottom-up view, building them up from the root. This creates an harmonic environment where voicings are weighted towards the lower end of the chords’ structures. Upper extensions are treated as separate entities added on if possible. I will present a top-down approach to harmony where the emphasis and priority of the chord will be placed in the upper register of its structure. I will use the jazz standard Blue in Green, an analysis of its melody, upper structure triads, and ideas developed by Stefon Harris to create a new harmonic vocabulary for the guitar. For each chord I explore I will chart out a new set of voicings on the fretboard. These will include full chords (5-6notes), mini chords (3-4 notes), and dyads. I will provide examples of how these voicings can be used in Blue in Green both to create chord-melody arrangements and to apply to comping situations. I hope to create a new chordal vocabulary for myself that is more deeply connected to the melodies I am harmonizing and which offers a richer and more colorful palette of tonalities. I am documenting the exploration in this thesis to help other guitarists interested in expanding their own approach to harmony.

Very excited to read this!

[nick1994]

Yeah approaching it diatonically with the sounds you're already familiar with isn't a bad way. It will just be missing some of the fancier options. But you can still get a lot of mileage out of things. You just have to be careful and trust your ear more than anything else. It's possible to find diatonic triads that might not work.

All of my triad options come from sitting at the piano and just testing individual notes against a particular chord. Once I find all the notes I like, then I sift through those and find all of the triads. So it ends up with triads from basic major scales as well as triads that might be found in harmonic or melodic minor, or any number of other places. But I'm looking for them by first deciding what notes my ear likes.

So one funny thing that happened that I wasn't expecting was that I didnn't find the root note of a maj7 chord to sound harmonious when played melodically over the 1-3-7. So I don't use the root triad to develop chords or melodies over a maj7 chord. Things like that are where I have to follow my ear more than the theory. There are theoretical reasons I could come up with as to why that's an avoid note for me. But the theory says that the diatonic scale is just a basic major scale which grows out from the root. But I actually don't want the root.

Funny little issues like that are why I dig the idea of following my ear and not relying on theory or scales for these types of explorations at the beginning. I bring it in later, but it starts with the ear. Because for 20+ years I just assumed the root note not only worked against ANY type of major chord (including maj7) but that it was the strongest resolution point and the most stable note... because that's what the theory told me. Then I sat down and listened to it. And it just didn't function that way in my ear.

see, I don't know if diatonic eliminates the fancier options, it probably makes them a little harder to find. like with A triad over G minor it just comes from harmonic minor or D over E half dim comes from melodic minor. I usually try to use scales to justify the sounds after I find them.

As for diatonic triads that don't work, I was working through the goodrick/miller book (someone mentioned it here not so long ago) and one of the first things is he goes through a scale in triads and its inversions over the changes to stella. He starts with F to G- triads over an E halfdim, now for a long time I jut kept going through this hating the sound of those two triads over that chord (the F more than the G) but then I combined the 2 into a G-11 sound and now I really dig those 2 triads over the half dim (it's a bland sort of sound to my ear but I think it works depending on how you voice it). so I think those initial triads that don't work can be persuaded to work well(still going to be a crunchier sound over an F# triad though).

I follow my ear as well but theory says we can play a b9 over a minor chord and I want to figure out a way to do that. I think what the ear says and what theory says should equal the same thing.

now I'm going to say something that's probably controversial but I think the b9 over works over a ma7 chord play a C lydian #5 but then when you get to the A play A major , it sounds better than going back to the tonic to me.

[AllFifthsTuning]

Jordan,

I was wondering, what software did you use to create the text and diagrams with?

Hans

[Vladan]

Won't go into analysis of your thesis, as I'm no expert, by far, just to tell you one , or two things even before I've read it in ful, after glancing into abstract and first 10 pages, or so and after reading couple of your posts (Maybe not with full attention. You write long posts).

When I read your thesis, as much as I have read it, I get a pretty good idea about what it's all about, what will be shown, how it's done and so on. I see it systematize a part of our usual guitarist struggle. Good that someone put an effort into explaining it in coherent manner.

On the contrary, when I read your posts about it on the forum , as much as I have read them, and your further explanations of it, I seem not to understand that well. Like if you are talking about something different, much more complicated and on different level of abstraction, then what I read in your thesis.
Seem to be analog and related to the nature of music theory where it's often more complicated to explain what was done, than it was to do what's being explained.

Now I don't know which version to trust. The official one, which I think I understand better, or forum one, which is in some space far away from my own reality and usually I have no idea what you are trying to say.

I mean, good work!
Will try to read it in full when I grab some larger chunk of spare time. Hopefully I won't find my self in the "Forum Zone".

[Nate Miller]

I wouldn't say wrong. Some guys might like it. It's just wasn't a note that I wanted to allow into my minor chords... at least not the minor chords that aren't meant to imply phrygian. But again... I'm still trying to wrap my head and ears around some tonal categories and chords and how to approach them within this system... and phrygian is one of them.

you're on the right track thinking in triads. That's how the guys did it. Don't screw it up thinking about freakin Phrygian mode.

Think about key relationships, not modes, that's how they did it.

Keys a half step apart and a minor third apart. And here I thought I was taking that secret to my grave. Look at the triads in those key relationships. Top down secondary key in triads over a bass line in the main key and the other way around.