Generating a scale through successive divisions of the octave by the Golden Ratio

From Xenharmonic Wiki
Jump to: navigation, search

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.)

Step 1: Divide the octave into two intervals related by the Golden Ratio.

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

0.000

741.641

1200.000

...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.

Step 2: Divide the largest interval, 741.641 cents, into two intervals related by the Golden Ratio.

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

0.000

458.359

741.641

1200.000

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

0.000

283.281

458.359

741.641

1024.922

1200.000

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

0.000

175.077

283.281

458.358

633.435

741.639

916.716

1024.920

1200.000

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

0.000

108.204

175.077

283.281

391.485

458.358

566.562

633.435

741.639

849.843

916.716

1024.92

1133.124

1200.000

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[13] 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.

0.000000

66.873708

175.077641

283.281573

350.155281

458.359214

566.563146

633.436854

741.640786

808.514495

916.718427

1024.922359

1091.796068

1200.000000

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.