Generating a scale through successive divisions of the octave by the Golden Ratio
Jake Freivald here.
The Golden Ratio, represented by phi or ϕ, has interesting properties. I'm going to act as if you know enough about phi already and are just interested in building a scale.
The basic idea I'll follow is to keep dividing intervals such that they are in the proportions of the Golden Ratio. Since the ear hears logarithmically, I will use cents rather than ratios to do so.
I use the term "MOS" and things like "LsLLs" below. An MOS is a scale made up of only two step sizes, and the Ls notation tells the order of large and small steps. (There's more to MOS than that, but that's enough for now.)
Start by dividing the octave, 1200 cents, by the Golden Ratio, 1.6180339887... to get 741.641 cents.
(Equivalently, you can multiply by the Golden Ratio Conjugate, which is 0.6180339887... Yes, weird as that seems, the conjugate of phi is phi-1. Phi is special that way.)
So now my scale is
...for scale steps of 741.641 and 458.359. (That's a trivial MOS of LS.)
(Let's agree, as a convention, that every time we divide an interval into two steps, we'll put the larger interval on the bottom and the smaller on top.)
Note that 1200/741.641 = 741.641/458.359 = phi. That's the way phi works. If you did this with e or pi, you'd wouldn't get that result.
The largest interval in this really simple scale is 741.641 cents. Divide that by the Golden Ratio, just like we did with the octave before: 741.641 / 1.6180339887 = 458.359 cents.
This is really important: Note that this value, 458.359 cents, is the same as the remainder from the first split! This will result in us creating an MOS throughout this process.
So this splits the 741.641-cent interval into two intervals of 458.359 cents and 283.281 cents.
The cool phi relationship still holds, too: 741.641/458.359 = 458.359/283.281 = phi.
In our original scale, my scale steps were 741.641 and 458.359, and when I replace the 741.641-cent step above with 458.359 and 283.281, I get the step sequence 458.359, 283.281, and 458.359. That's an MOS of LsL, and my scale is
Step 3: Lather, rinse, repeat.
The largest step I have in my scale now is 458.359 cents. I have two of them.
Divide each of them into Golden Ratios: 458.359 / 1.6180339887 = 283.281 cents.
This splits the ~458-cent intervals into two intervals each of 283.281 cents and 175.077 cents. (Cool phi relationships: 458.359/283.281 = 283.281/175.077 = phi.)
In my last scale, my steps were 458.359, 283.281, and 458.359, so when I replace 458.359 with 283.281 and 175.077, I get scale steps of 283.281, 175.077, 283.281, 283.281, 175.077, which gives an MOS of LsLLs and a scale of
Do it again: The largest steps we have in the scale are 283.281 cents. Break them into two parts of 175.077 and 108.204: Scale steps of 175.077, 108.204, 175.077, 175.077, 108.204, 175.077, 108.204, 175.077. The scale is an LsLLsLsL MOS with values
Do it again: Break the 175.077 interval up and replace 175.077 steps with 108.204 and 66.873. Scale steps are now 108.204, 66.873, 108.204, 108.204, 66.873, 108.204, 66.873, 108.204, 108.204, 66.873, 108.204, 108.204, 66.873, which is an LsLLsLsLLsLLs MOS. The scale is
Again, because we're doing this with phi, this process will always generate an MOS.
Points of Interest
1. If you do the math with only this level of precision, Scala will tell you that there are small, medium, and large steps. In reality, the "medium" step is a rounding error.
2. As we've seen, there are MOSes at 5, 8, 13, and 21 notes, and of course ultimately an infinite number. The final MOS I gave -- call it phi or something -- is LsLLsLsLLsLLs. Because it's an MOS, you can generate it (without the rounding error) by using 741.64078644 as a generator and choosing the LsLLsLsLLsLLs mode of the resulting MOS.
3. The scale above was made by dividing intervals into Golden Ratios with the larger section going on the bottom and the smaller section going on top, e.g., the octave was split into 0-741-1200 for scale steps of 0-741-458. If you want the mode where the smaller section goes on the bottom and the larger section on top -- 0-458-1200 for scale steps of 0-458-741 -- use the sLLsLLsLsLLsL mode. It's similar, but not identical.
4. This wasn't built using traditional notions of harmony, and has no basis in the way we think the ear works; therefore, it's possible that, if it works at all, the Golden Ratio scale may be most interesting to use in melody. Do melodies that define a boundary between notes, and then sections them off in Golden Ratio segments, sound interesting? I don't know -- I've only played with the scale a little bit.
5. However, despite the fact that the scale wasn't built around traditional harmonic ideas, it has some tones that are close to familiar notes, which gives it familiarity and some harmonic cohesion anyway. And besides many people have shown that they can make all kinds of interesting-sounding music out of "dissonant" or non-harmonic scales. so there's no reason not to try with this one, too.