# ## The Golden Ratio as a musical interval

The golden ratio, also known as φ (phi) or approximately 1.618, is a number with some trippy properties. It’s no wonder that many people treat the golden ratio with a great deal of mysticism, because (here’s the cliche part) it appears repeatedly in nature and also crops up in many fields of mathematics. And we all know that mathematics is the language of the universe. Duuude.

So I want to talk a little about the golden ratio and how it might sound if we used it as a musical interval. You may have seen or heard some youtube videos which exclaim “this is what the golden ratio sounds like!!” — These are almost always based on the decimal expansion of phi, where each digit is assigned to a note of the major scale. I won’t give an example here because it’s such an unimaginative Western-centric interpretation of nature’s finest ratio. Just imagine the background music for a shitty Google commercial, children and mothers playing, young trendy people smiling while looking into their laptops, and you get the idea.

Let’s have none of this. The truth is that the golden ratio, as a musical interval, is gritty, dirty, dissonant, inharmonic, and not remotely like you’d expect. And it’s explicitly microtonal.

Take for example John Chowning’s Stria, an important electronic work from 1977, using his then-new discovery of FM synthesis. The golden ratio is used as the interval between carrier and modulator, such that the resulting timbre is an inharmonic cloud of golden-ratio-related partials. To get a sense of what the golden ratio may sound like as a musical interval, start from here and let the sounds slowly work their way into your brain. ### Which golden ratio is the real golden ratio?

Here is something that causes confusion time and time again. There are two musical intervals which both claim to be the golden ratio! How is this possible?

Consider the octave in terms of cents. Cents are a musical measurement of pitch which divide the octave into 1200 logarithmically equal parts. This means that 200 cents is equivalent to an equal-tempered whole tone, 100 cents is equivalent to a semitone, 50 cents is equivalent to a quartertone, etc. So our first version of phi will simply divide 1200 by phi:

`1200 / φ = 741.6407865 cents`

~741 cents sounds like a horribly sharp fifth. Well, we already know that the golden ratio sounds dissonant, so this could be the golden ratio that we’re looking for. But first let’s look at the other contender:

`1200 * log2(φ) = 833.090296357`

While both of these intervals could be used to generate scales that you could play interesting music with, only one of them gives me that one-with-the-universe vibe. To find out why, we need to look at a psychoacoustic effect known as combination tones. ### Combination tones

It is a simple fact of psychoacoustics that any two tones that you play will produce additional combination tones. There are two types of combination tone: difference tone and sum tone. The sum tone is calculated by summing the frequencies of the two tones. The difference tone is calculated by subtracting the frequency of one tone from the other.

Let’s play A (440Hz) and E (660Hz) on the keyboard and work out these combination tones for ourselves.

```Difference tone = 660Hz - 440Hz = 220Hz

Sum tone = 660Hz + 440Hz = 1100Hz```

The difference tone sits an octave below the root note of A. This is the reason why you hear the missing fundamental being “filled in” by your brain when you play a power chord or a just major triad.

As for the sum tone, it is tuned to the 5th harmonic above that missing fundamental of 220Hz, because 220Hz*5=1100Hz. The 5th harmonic is a just major third plus two octaves. The sum tone is perhaps one of the reasons why the just major triad is such a stable and pleasing sonority.

Just to recap: If I play TWO tones (440Hz and 660Hz), your brain hears FOUR tones, (220Hz, 440Hz, 660Hz, 1100Hz). The effect is subtle, and the combination tones are heard a lot more softly than the real tones, but the effect can be perceived. Bearing that in mind, let’s work out the combination tones that appear when using the golden ratio interval.

We shall play a 1kHz tone and a ~1.618kHz tone. The interval between these two tones is the golden ratio of ~833 cents.

```Difference tone = ~1.618kHz - 1kHz = ~0.618kHz

Sum tone = 1kHz + ~1.618kHz = ~2.618kHz```

What’s interesting about these combination tones is that they are themselves related to the original tones by the golden ratio. This is easy to demonstrate:

```1kHz / ~0.618kHz = φ

~2.618kHz / ~1.618kHz = φ```

All four of these tones are related by the golden ratio! If we calculate the 2nd order combination tones, 3rd order, and so on, we’ll find the same thing again and again. Every combination tone is connected to some other tone by the golden ratio. This is exactly the recursion effect that we expect to find when we use the golden ratio properly.

However this landscape of recursive inharmonic partials can best be described as a chaotic and complex mess. This is an extreme contrast to the use of the golden ratio in visual proportions, paintings, architecture, flowers and nautilus shells, which most people would agree appear harmonious and pleasing. But that’s just the truth of it. The ear works in weird ways. Not all visual forms have an analogue as sound. ### Generating a scale based on the golden ratio

By now it should be clear that golden ratio music has some interesting properties that the Western equal-tempered scale could scarcely hope to reproduce. If you have some highly accurate micro-tunable instruments and perhaps some scale-designing software such as Scala, alt-tuner or LMSO, then your next step is to design a scale that features the golden ratio.

There are many possibilities when making your own phi-based scale, but today I will only give you one. Developing your own golden ratio scales is left to you as a rainy day exercise.

Almost all scales contain an interval of equivalence, and for most scales that interval is the octave. In octave-based systems, a note such as C appears many times, and each C is separated by one or more octaves.

By the way, an octave is a ratio of 2. If you play an A at 440Hz and then play the A’ one octave higher, the higher pitched note would be 880Hz.

For our scale, we want to pack in as many golden ratios as possible. There’s no room for any boring numbers such as 2. Indeed, our interval of equivalence will be the golden ratio itself. This means the scale will repeat after approximately 833 cents. This scale would fit snugly into the frequency space between 1kHz and ~1.618kHz.

Our next step is to divide this interval into smaller intervals (a & b). Each time we divide an interval into two intervals, those intervals should have the golden proportions. Then take the biggest interval that remains in the scale and break that down into 2 intervals with the golden proportion. Repeat this process until you have enough pitch material to write music with.

I’ve done all the calculations, and the resulting scale (taken to 8 steps) is presented below in Scala format so that you may try it out for yourself:

```! sevish_golden.scl
!
Scale based on the golden ratio
8
!
121.546236174916
196.665941335636
318.212177510552
439.758413685468
514.878118846189
636.424355021105
711.544060181825
833.090296356741```

Some interesting notes about the above scale… When generating the values, I noticed that the scale was a Moment of Symmetry (MOS) after 3, 5, 8 steps. Those are Fibonacci numbers. I stopped after 8 steps because it will map very nicely to a piano keyboard with linear mapping. With this mapping, you will always hear the golden ratio when you play a minor sixth on the keyboard. Coincidentally, this tuning has some fair approximations to the fifth, fourth, supermajor third, minor third, and major second.

If you want to find more tunings with the golden ratio, you might look at Heinz Bohlen’s 833 cent scale for inspiration.

### The golden ratio as a spectral microtuning

When using scales that are based on the golden ratio, you will find that timbre makes a huge difference to the result. If you use an instrument that produces bright and clear harmonic overtones such as the violin or a sawtooth wave, you will hear an unruly clash of a harmonic timbre against an inharmonic scale.

Instead, you could take a similar approach to John Chowning and bake the golden ratio right into the timbre itself by using inharmonic overtones based on the golden ratio. Then suddenly your timbres and your scale will align, resulting in a much smoother sound. It’s just one more demonstration of the fact that tuning and timbre are deeply intertwined.

If you’re a computer musician, download a free copy of the Xen-Arts synth plugins. These instruments can produce inharmonic partials which can be controlled by the user. By default, they include a setting where each partial is tuned to some multiple of the golden ratio. These instruments could be a quick way to explore the topics I’ve been discussing in this post.

### Exercises

• Jam and improvise in the above golden ratio tuning until your ears tune into it.
• Apply the above tuning method to sequence your own golden ratio rhythms.
• Write a piece of music that combines both of the above.
• Make sarcastic YouTube comments saying that John Chowning’s music heals DNA.
• Change your desktop wallpaper to a picture of a spiral galaxy.
• Get your head around microtonal tunings by listening to my new album Harmony Hacker.

### 11 thoughts on “The Golden Ratio as a musical interval”

1. Chris Vaisvil

Hi Sean, with some help from Jake Freivald I took a different approach – taking the 17th root of Phi. To my ears the tuning is useful and fitted my DIY 17 note controller well (the reason to take the 17th root)

! E:\cakewalk\scales\17th_root_of_phi.scl
!
17th_root_of_phi
17
!
49.00531
98.01062
147.01594
196.02125
245.02656
294.03187
343.03718
392.04249
441.04781
490.05312
539.05843
588.06374
637.06905
686.07436
735.07968
784.08499
833.09030

1. Sevish Post author

What at first glance looks like a stretched 12-EDO is actually way out there, and tunes the golden ratio perfectly. :) Thanks for sharing this Chris, I’ll have to jam in this one as soon as I start my next creative cycle.

2. Other Paul

How about, instead of generating a polytonic scale by stacking Pythagorean fifths (i.e. keep multiplying frequencies by 3/2 as in 1, 3/2, 9/4, 27/8, 81/16 etc and then dividing these by 2s to get these rationals back into the octave range lying between 1 and 2 – i.e. 1, 3/2, 9/8, 27/16, 81/64 etc), generate one by stacking phiths (in much the same way as the western chromatic scale may be generated by successive multiples of 27/12).

Thus the root frequency is 1, the phith is φ, the phith of the phith is φ2/2, the phith of the phith of the phith is φ3/4 etc. Numbering from 0, the n-th ratio is thus φn/2m where m = floor(n log2(φ)) = integer part of (0.69424191363 n)

If you generate (say) 7 such pitches and re-order them, then the 7 ascending frequency ratios are [1, φ3/4, φ6/16, φ2/2, φ5/8, φ, φ4/4] which is approx [1, 1.059, 1.122. 1.309, 1.386, 1.618, 1.714]. If 1 represents C, then these pitches are closest to [C, C#, D, F, F#, A, A#] but with noticeably flat F#, A and A# (each about a quarter tone down).

Of course there’s no need to stop at 7, but it’s only after you’ve made the decision where to stop that you get to sort your sequence, not before.

1. Other Paul

Ah. I see your HTML doesn’t allow for superscripts and subscripts. I hope your readers can guess at the intention!

3. Jake Freivald

Interesting that you choose the 833-cent phi instead of the 742-cent phi. I’ve done a little work (very little work, none of it particularly fruitful) with a phi-based tuning, and I accepted the octave as an interval of equivalence to do it. Octave equivalence is natural, after all — although some people say they can get, e.g., the tritave to be an octave of equivalence, I can’t hear an octave and not perceive it as “the same not only higher” over the root. Phi-based tuning is odd enough on its own, so I didn’t feel the need to add the additional complexity of using a non-octave scale.

So I thought of the 742-cent phi dividing the octave up into proportional sections, and then repeating at the octave. Three octaves together might then be analogous to three panels on a wall, each one equivalent to the other, but the divisions inside each panel being related using the golden section.

One challenge with this way of thinking is in creating music where this “golden section of the octave” is obvious. I think five or eight notes is better than 13 or 21, in that case — using too many notes means we fall back on the ratios that the phi-based tuning approximates, like 5/4, 11/9, 13/11, and so on.

And by “we” I mean “I”, of course. Your mileage may vary. :)

4. Randy S

One such consideration could be to view the graphs of the constituent parts of the golden ratio [phi^1=0.5*(1+sqrt(5)) and phi^-2=0.5*(3-1*sqrt(5)); so for x=1 plot points at y=+1 and y=+sqrt(5), and for x=-2 plot points at y=+3 and y=-sqrt(5), continuing on with powers of phi being the independent variable] and see that the two sets of points produce two separate functions, y=(phi^x+phi^-x) and y=(phi^x-phi^-x), with each point set having alternating (even-odd) points lying on these two graphs.
This is very similar to how y=e^x can be drawn as the addition of y=cosh(x) and y=sinh(x) (respectively).
This shows that, because of the fact that negative powers of phi contain negative constituent parts [such as phi^-3 = 0.5*(2*sqrt(5)-4) and phi^-4 = 0.5*(7-3*sqrt(5)] similar to e^x, the midpoint of these two graphs produces the graph of y=phi^x. Noticeably, the graphs of y = cosh(x) and y = sinh(x) are described as y=0.5*(e^x+e^-x) and y=0.5*(e^x-e^-x) (respectively). This indicates that, similar to e^x, phi^x would actually be the sum of the two graphs y=0.5*(phi^x+phi^-x) and y=0.5*(phi^x-phi^-x).
If that hasn’t dissuaded anyone from continuing to see the point of all this yet, I’ll continue by just adding in that I had taken the time to look at the lines connecting the two different point sets [one set being whole number y-values, the other set being y-values that are whole multiples of sqrt(5)] to notice how they interact with one another and “cancel out” each other to “create” the tangible graph of phi^x.
The whole purpose of this, is to view the graph(s) as the creation principle (since all values of each of the powers of phi are sums of the two values of the powers before them, if that makes any sense to those who don’t know what phi is).
My own concept was to see where on the graph of y=phi^x the value would be 2, since the entire premise of creation is reaching powers of 2. This turns out (not surprisingly) to be equal to log(2)/log(phi)=1.4404200904…, so that would be the time(t) at which one cycle was complete. This means that, for every whole number multiple “n” of x=1.4404200904.. on the graph of y=phi^x, y=2^n.
This could be seen as similar to the division of cells over time, or the reproduction of a species with no limiting factors as proposed by Fibonacci when he derived his sequence from studying nature. However, I provide a disclaimer as to having no knowledge of IF cells divide every 1.4404200904.. seconds, that claim is not part of my entire claim.
What, then, would one do with this number (1 cycle/1.4404200904.. seconds = 0.6942419136306.. Hz)? What I did, was use it in an equation to produce a function which gives frequencies similar to (but sharper than) standard concert pitch. The resulting A4 note ended up being about 447.84128…Hz, which provides an uplifting feeling when playing harmonies.

5. Stephen Boyes

How about golden meantone! Where the ratio between a tone and a semitone equals phi (ie 1.618/1 instead of 2/1)
This produces a tempered diatonic scale, that continues in a spiral of fifths, that fill in the gaps with smaller and smaller intervals, in the proportion of phi.

This is related to the sequence
7 EDO
12 EDO
19 EDO
31 EDO
50 EDO

6. Thomas

What does he mean when he says second or third order combination tones?

7. Douglas Blumeyer

Your scale is intriguing. It’s a hybrid of acoustic phi and logarithmic phi. I’d like to know if this was your intention when designing it. Let me try to clarify my perspective:

Bohlen’s 833 cent scale is purely acoustic, functioning like a JI scale except using phi instead of rational ratios. It focuses the recursive power of phi acoustically, i.e in the realm of frequency, on a cloud of combination tones.

Golden meantone (like all other golden moment of symmetry scales) is purely logarithmic, iterating a generator in order to divide a period up into step sizes which are in the golden proportion to each other. It focuses the recursive power of phi logarithmically, i.e. in the realm of pitch, on a scale for which every general step size is in the same proportion with other general step sizes.

But your scale includes a bit of both effects. Because you repeat the scale at acoustic phi instead of the octave, you get the the acoustic effect outside the scale, between notes of the same pitch class (up or down repetitions of the scale). But then you get the logarithmic effect within the scale, between notes of different pitch classes.

Furthermore, yours is also the only phi-based scale I’ve seen which entirely eschews the octave. There’s nothing binding them to the octave, but other logarithmic-based phi scales typically choose the octave as the period because they are not after combination tones, but rather spicing up the typical tempering objective to approximate low-limit JI intervals like 2/1 and 3/2 with a little fractally interval spice. And even Bohlen’s 833 cent scale involves the octave; while it repeats at phi instead of the octave, strangely the octave is still involved because the steps are created by stacking 833 cent intervals and octave-reducing them.

I have one more question: since you only discovered after the fact that your scale satisfied the conditions of a moment of symmetry scale, it seems you created your scale by manually breaking down intervals. Or did you use rabbit sequences to achieve this?

8. Charlie

The Xen-Arts website linked is down, and I can’t seem to find the new site or any mirrors of their instruments :(

1. Stewart Miller

Xen-Arts has been shut down for a little while now. If you’re looking for a synth that can manipulate harmonics in a similar fashion you can try ZynAddSubFx.