Circle of fifths: reflections

Again, the particular way the author of that site chose to do his inversions does indeed introduce all sorts of notes that come from a different key. But that doesn't mean all inversions do.

For one thing, you can do a diatonic inversion - keeping the basic size of the interval (third, fifth, etc) but not worrying about whether it is a major or minor third, perfect or diminished fifth, etc. That can keep a melody in the same key. Also, while the site referenced performed its inversions around D and Ab because that happened to yield a nice visual symmetry on the piano, you can perform inversions on whatever axis you want.

For instance, if you're in the key of Bb and perform your inversion around C (or Gb), then all diatonic notes remain diatonic, and all non-diatonic notes remain diatonic but shifted in a musically useful way. The key of Bb has two flats: Bb and Eb. The two most useful chromatic alterations you'd add would be E (backing out the most recently added flat) and Ab (adding the next flat in line) - and these two notes end up being inversions of each other if you invert around C or Gb.

Basically, any time you invert around the second scale degree of a major key, you get this same property: diatonic notes remain diatonic, and closely related non-diatonic notes remain closely related, but on the other side, as it were. The second degree works because because a major scale is symmetric around that note. And that's the real reason why D works as the mirror point in the site you referenced: it's the second scale degree of C, and of course the physical layout of the piano is designed around that key. Had the piano been designed so that the white keys formed a Bb major scale, he'd have written that same article but with C as the mirror point.

BTW, the circle of fifths can apply to all sorts of things - keys, key signatures, chords, individual notes, etc. It has lots of applications. The specific one being discussed had to do with keys, yes. But of course, keys consist of notes, and notes can form chords, so it's all related.

Yes, there is a rule governing what keys end up getting implied if yinertiaert non-diatonic notes around, but that's part of what I was describing previously that may have seemed unnecessarily complicated. No, it is not as simple as what you just said. Please read my responses again. It is not a case of every time you invert a prime in Eb you get one in A. That's only true for the specific case he was discussing - the case where the piece as a whole is in C and you happen to choose to nevertheless around D and then temporarily modulate to Eb. If you do that, and keep inverting around D, then yes, that particular portion of your piece - which otherwise remains in C - will move to A. What I was explaining is that if you want a general rule, don't always invert around D, but instead, invert around the second scale degree of the key your piece is actually in. It is complicated, so if you wish to understand this, you have to start thinking in terms of scale degree and temporary modulations, beause the answer does not have to do with absolutes like Eb always becomes A.

Assuming you do invert around the second scale degree, then anything that started in that key remains in that key when inverted. And if you temporarily modulate to a different key, what happens is that a key "n" degrees flatter than the original on the circle of fifths maps to a key "n" degrees shapers, and vice versa. Using the example I already gave, use of E while otherwise in Bb suggests the key of F - one degree sharper than Bb. The E maps to Ab, which suggests the key of Eb - one degree flatter. Two degrees sharper than Bb is C, which comes from also adding a B. The B maps to Db, which implies the key of Ab - two degrees flatter. I haven't worked it through the rest of the way, but I see no reason to doubt the results shown In the web site you reference, which demonstrate it working that way I the specific case of inverting around D, the second scale degree of C.

How important this is depends if you plan to write music this way. I can assure you this is *not* something very often (if ever) done, so it isnl't important in general to an understanding of music and how to compose music that sounds good to the average listener or is readable to the average musician (those are important skills for *all*). That's what I meant when I called it an "idle curiosity" before - the vast majority of musicians would never care to think about this. It wasn't clear if you understood this. But if it helps you create music unlike what others have done, then indeed, it's important to you specifically.

I am not assuming anything, and for the record, I have a degree in mathematics, so I understand perfeclty what you have been asking. It's just the answer is not as simple as you want it to be. I am sorry you are having trouble understanding my explanations, but I *am* answering your questions quite literally, but I will try to do so again.

*If* you invert around D or Ab (which are the specific notes chosen by the author of that web site), then yes, the key of Eb turns into the key of A, each and every time. Try it yourself and you'll see; you shouldn't need me to explain that much. If you limit yourself to inversion around D or Ab, then everything that author wrote is always true, just as you can see for yourself very easily by working it through what each note of an Eb major scale maps into when inverted around D or Ab and seeing what set of notes you end up with. You'll find that if you start with the notes of Eb major, you end up with the notes of A major. Eb becomes C#, F becomes B, G becomes A, Ab because G#, Bb becomes F#, C becomes E, D becomes B. The original Eb major scales has now turned into the A scale, and there is no way to make that not be true. That much is quite simple.

What I am trying to explain is the "general rule" you are asking about - what happens if you invert around notes *other* than D or Ab. In other words, what happens if you perform inversions the way they are normally done, which is *not* always around D or Ab, but around other notes chosen by the composer. Certainly Bach, Mozart, and other composers performed inversions - but they did *not* limit themselves to inversions around D and Ab. As I keep saying, *if* you choose to limit yourself to inversion around D or Ab, then yes, Eb maps onto A, and you can stop asking about general rules because you aren't dealing with the general case - only the specific case of inversions around D and Ab.

But if you invert around some note other than D - as the composers you mention all did - then Eb will map onto something other than A. So if there is a "general rule", then it needs to be able to predict what Eb will map onto given the point of inversion. Whether or not you plan to ever invert around notes other than D or Ab, a *general rule* needs to handle cases other than those two. And there can be no denying: neither Bach nor Mozart nor anyone else would normally limit themselves to inversions around D and Ab. Yes, inversions are common, but no , inversions that limit themselves to inversions around those two axes are *not*.

And as it turns out, there *is* a general rule that allows you to predict what key Eb will map on from any given axis of inversion. This is precisely the rule I have already tried to explain. I'm sorry, but it just can't be easily summarized in one sentence. That's what you will have to read and work through what i wrote above if you wish to understand. But I gave you the literal answer the your question - the general rule that allows to predict with perfect reliability what key any given key will map into - *generally*, which is to say, given any point of inversion.

If you *always* use D as your point of inversion,l the Eb will always map onto A. On the other hand, if you want the key of Eb to remain Eb (as it appears is your goal), then you need to invert around the second scale degree of Eb, which is F - just as the general rule I have already explained predicts. If you invert around F, then the key of Eb will *always* remain the key of Eb. The rule works: exact inversion around the second scale degree preserves the diatonicity of any melody.

For some other examples, if you invert around G, then the key Eb maps to the key G (for the reasons i previously explained). If you invert you invert around Bb, then the key of Eb maps to the key of Db. If you invert around C, then the key of Eb maps to the key of F. The rule that explains this is as I explained in my previous posts. It's not a simple rule, but if you understand it, you can apply it.

As for the importance of this, I didn't you were the *only* one concerned with this. Just that it isn't at all common - and the people you mention did not actually use inversions in this way very often. Mostly, they used inversions in much more ordinary ways that don't require these sort of calculations (eg, using diatonic as opposed to exact inversion). But in any case, you asked previously why there aren't other web sites explaining this. And like it or not, the reason is as I already suggested: there aren't many other web sites explaining it because most people don't consider it important. So I did *literally* answer your question.

Now, the *reason* most people don't consider this topic important is another matter. I would guess that a) only a fairly small number of people perform inversions, and b) most of the ones who do perform inversions are either doing diatonic inversions - in which case none of this applies - or else they are doing exact inversions but are writing atonal music, in which case - again - none of this applies. The set of people performing exact (as opposed to diatonic) inversions in tonal music is just tiny. Again, that doesn't mean it isn't important to those people, but the fact that it is such a small set explains why most would consider it not important, which in turn explains why you won't find many web sites on the topic.