The “Spiral” of Fifths – An Alternative to the Circle of Fifths
April 3, 2019
The “Spiral” of Fifths – a “self similar fractal”:
If you start a new major scale on the fifth step of any current major scale, the new scale will have one more sharp than the previous scale. Sometimes this new sharp is in the form of a flat from the previous scale that has been “sharped” into a natural in the new scale.* Or a sharp from the previous scale that has been ‘sharped’ into a double sharp. The new sharp always appears om the seventh step of the new scale.
If you start a new major scale on the fourth step of a current major scale, the new scale will have on more flat than the previous scale. Sometimes this new flat is in the form of a natural in the previous scale which was been “flatted” into a flat in the new scale. Or a flat from the previous scale has been ‘flatted’ into a double flat.
These two processes (adding one sharp at a time or one flat at a time) can go on forever. If one keeps adding a sharp, the new scales will begin to contain double sharps, then triple sharps, etc.. If one keeps adding a flat, the new scales begin to have double flats, then triplet flats, etc..
The correct drawing of these two interlocked, and inverse procedures is not a circle of fifths (or fourths) but a spiral of fifths. Just as interestingly, there is no beginning or ending to this spiral. There is always another layer of the spiral outside the last layer currently displayed in any representation of this spiral, and so on indefinitely. Similarly, there is always another layer of the spiral compressed inside the smallest layer of the spiral currently displayed, and so on.
Most of us learned about key signatures using the diagram of the circle of fifths. Yet, a lot is left unanswered by this image. It does not explain, for instance why, when you got about half way through the circle sharps somewhat arbitrarily turn into flats. Why just then and not sooner or later in career of the circle?
Just as a C# means something different than a Db, and just as Cx means something different than D, theoretically there is nothing to prevent us from having a C-triplet-sharp which is different than a D# or Eb. We may never see a triple sharp throughout our playing lives but it exists as sure as ultraviolet and infrared extend the boundaries of the visible portion of the electromagnetic.**
In the spiral of fifths, if we are located at C Major, then the next seven positions (or “o’clocks”) on the spiral will bear the addition of one more sharp, until at 8 o’clock double sharps start to appear in the scale. And seven hours later than that “triple sharps” will start to appear, and so on every seven hours.
Similarly with flat. If we go in the other direction from C Major, the next seven keys bear the addition of one more flat, until at “4 o’clock’ double flats begin to appear, and so on every seven o’clocks further inwards around the spiral.
Advantages to the spiral shaped diagram:
We no longer have to treat as arbitrary the point when we switch from flats to sharps, or vice versa. We can also see the dynamic relationship of sharps and flats to each other, not as having two separate, essential identities, but both as being an expression of the same, single reciprocal principle of flatting and sharping.
If you draw a straight line from any key in the spiral inwards towards the center of the spiral, all the keys that lie along that line are all enharmonic equivalents of each other, moreover the letter of the alphabet in the name of the tonic progresses one letter at a time through the musical alphabet.
The letter “C” is always special:
C major has the distinction of being the only key consisting of only naturals and no sharps. C sharp major has the distinction of consisting of only sharps. C double sharp major has the distinction of consisting of only double sharps. C-triple sharp major consists only of triple sharps. Etc..
Going in the other direction: C flat major has the distinction of being the only key consisting of only flats. C double flat major has the distinction of being the only key consisting of only double flats. Etc..
* For instance, the key of F major stands to the key of C major as having “one more flat” than the later, the note B-Natural in the C Major Scale being flatted into a B-Flat in the F Major Scale. Going in the other direction, the “new” sharp in the C Major scale is the B-Natural on the seventh step, which had been, in the F Major scale, a B-Flat.
** A curious thing happens when we have ‘sharped’ C-natural twelve times, so that we now have C-dodectuple-sharp. We find that C is its own dodectuple sharp one perfect octave higher. In the same way we can ‘flat’ C-natural twelve times, and find that C is the dodectuple-flat of itself. And then the process could continue until we find ourselves beyond the range of frequencies covered by the piano keyboard, going past vintuple sharps or flats (or would it be vigintuple), then centuple, and so on indefinitely. For there is no lowest or highest frequency that a sound can have in theory. A frequency of a billion vibrations per second is just as possible theoretically as a frequency of one billionth of one vibration per second. Whether they are audible is another question. And at some point when we get down to the size of molecules of air, perhaps there is no higher frequencies physically possible (got to think that one through).