Let's ask a question, is there a difference between a major 3rd and a diminished 4th? Well seeing how diminished means to lower a perfect interval by one semitone, that would mean they are the same interval right? Yes and no. Although they are exactly the same thing physically (at least in 12tet), the difference is in function. If you play a major scale and play C and E, you get that bright, lame, major third sound haha. But now try playing the scale C Db Eb Fb and do this alternating playing the C and Fb. Fb is the same note but now it feels different, more dark and dissonant, not like before. This is what I call functional difference. There is physical and contextual difference. Although there are only 12 intervals physically in our musical tuning system, there are far more functional intervals. For practicality, we don't consider every one of these intervals though, some we use more than others, or is it that we're just lazy.
Now let's talk about Hyper Extensions or HSE's. What are they? It's not some shoulder exercise. They are hypothetical extensions that exist beyond the 13th of the major scale. I'm going to make up a term that we will call the scale's limit. This is not like the harmonic limit of a tuning so bear with me. When I say scale limit, I'm talking about the note at which the scale stops going up in fifths. If you read the beginner theory course, you should know that a major scale is created by building a cycle of 6 fifths and making the 2nd note of the cycle the tonic, the collapsing it into a single octave.
Like this: F C G D A E B into C G D A E B F into C D E F G A B.
However, notice how B is where it stops? This is what we call the scale's limit. The major scale has 6 fifths as a limit. But if we wanted to go higher, can we? Of course, there are no rules in music after all.
We can extend it to add a fifth between B and F# which changes alot.
F C G D A E B F# So what did we do here? We added a note logically to the major scale. Though it's not exactly a major scale anymore, or it is but it's got a new note now. This is what we call a Hyper Scale Extension. Hyper meaning beyond the scale.
Putting Hyper Extensions into extended harmony
The easiest way to do something with Hyper Extension is to put them into an extended chord.
This is where things get a bit complicated though because we now have to think beyond two octaves.
Look at this F C G D A E B, in this we learned that half steps are five notes above the lower note in the half step. That means that F# is a half step from F. That means that it's F# not Gb! Why does it matter? Because if we called it Gb, it would get confusing. So just call it F# for now and you'll see in a minute why it's not Gb.
Let's build a cycle of thirds on C. C E G B D F A, this is as high as we've gone, but now let's go another third higher. If you remember how the cycle of thirds moves Major third to minor third forever. That means the next highest note is C. So we have C E G B D F A C right? Ok so what does this mean? C is just the octave right? Yes but not exactly, because this C is Two octaves, we'll treat it like extended harmony which mean we will call this a 15th not an octave. Ok so, there isn't a difference between the 15 and the 8 but there's a huge difference in the #15 from the 8. So what we're doing is adding an aug15th to our chord which gives us C E G B D F A C# which is called a Maj13#15 chord. Now that's a huge chord so we can omit certain notes. The fifth isn't needed so we can leave that out. We can also leave out the 11th because that doesn't sound good with the #15. Can we leave out the ninth? Nope, without context, leaving out the ninth, the C# becomes a b9 which is a Db. It's the sounds of the major seventh from D to Db that gives it it's flavor so the D is very much needed. The A can also be left out but then we would just have a Maj9#15 chord. One other thing, every interval higher than an octave will be altered either diminished or augmented and you just have to figure them out.
So that's it right? Nope, we can go higher if you want all the way until you return back to the Root. I will warn you though, the colors higher up than the #15 are more exotic sounding which is fun but not useful in every aspect. They are cool though, but the world of using the last five notes in extended harmony is uncharted by 99 percent of musicans. If you want the relative minor to the Maj9#15 chord, all you have to do is create a minor scale from the 7 fifths and throw in the 7th fifth.
If we take C E G B D F A and make it A B C D E F G A then add the HSE #15, A B C D E F F# G.
Now just throw in the F# up high. Now you get a Min13#20. Gah, that's a high number.
Have fun!
