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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.

A pine cone showing off its Fibonacci sequence, related to the golden ratio

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.

Cross sections of a nautilus shell. Some sections show stronger golden ratio proportions than others.

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.

Golden ratio line

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.

Inharmonic Strings and the Hyperpiano

Vibrating strings produce (more or less) harmonic overtones. If two strings are tuned in some simple frequency ratio such as 3/2, 4/3 or 5/3, then those harmonic overtones match up nicely and avoid roughness. But if the two strings are tuned in some haphazard fashion then the overtones of each string won’t match up, causing the overtones to clash with each other.

We can actually plot out a graph which shows the interval between two strings and the corresponding dissonance. This is called a dissonance curve, and for a normal string it looks something like this:

So what.

Well, imagine a weird kind of string that produces inharmonic overtones, such that the dissonance curve looks different to the one above. Because the dissonance curve is different, you couldn’t play Air on the G String and expect it to sound good. You could however write new music that would fit with the novel dissonance curve.

Today, such a string is more than just a mathematical curiosity. It exists in the physical world.

Inharmonic Strings and the Hyperpiano” (by Kevin Hobby and William Sethares) is a paper published in Applied Acoustics. The strings in their hyperpiano have a stretched out dissonance curve where the double-octave sounds most consonant and the octave becomes dissonant. Okay so maybe it’s not going to be used on every new pop record, but this kind of freaky instrument can produce game-changing new tonalities.

Since the dissonance curve is stretched out to the double-octave or “hyperoctave”, Kevin Hobby suggests we might try tuning a hyperpiano instrument to 12 equal divisions of the hyperoctave. Wait, isn’t that just 6-EDO – a whole tone scale? Actually, it isn’t! They may be identical tunings, but the octave is considered a dissonant interval on the hyperpiano, analogous to the tritone on a normal piano. So it makes a lot more sense to describe this tuning as 12 equal divisions of the hyperoctave. Really.

The ringing of the strange hyperpiano sounds like a death bell for the unwavering cult-like belief in pure ratios and true frequencies. Tuning and timbre are deeply linked. If we’re willing to experiment with new timbres then we can uncover new musical vocabulary for the future to come.

The next step is to explore all this for yourself – download the sampled hyperpiano and give it a play.

What’s the difference between ET and EDO?

What is the meaning of ET and EDO, and are they interchangeable?

ET: Equal Temperament
EDO: Equal Divisions of the Octave

In practice, yes they are interchangeable. For example, 12-ET and 12-EDO both refer to the exact same tuning which has 12 equal notes per octave. But there is a slight difference in their meaning.

12-ET suggests that the tuning is a temperament, i.e. it tempers some other interval, usually a just interval. 12-ET tempers 81/80, the syntonic comma, and other intervals.

12-EDO suggests that an octave has been divided into 12 equal parts, but otherwise doesn’t imply that tempering is of importance.

Some people will even say ET for 12-ET, 19-ET and 31-ET, while using EDO for 8-EDO, 13-EDO and others. Perhaps because 8-EDO and 13-EDO are not thought of as temperaments, whereas 12-ET, 19-ET and 31-ET are all useful meantone temperaments.

Personally, I always use EDO in my own thinking and private communication with other microtonalists, but will use TET or ET when I need to be understood by a larger, mixed audience.

To complicate things further, some folks use ED2 or ED2/1 synonymously with EDO, because the octave is equal to the ratio 2/1. The good thing about this format is that we can generalise it for other scales that divide some interval into equal parts (e.g. EDphi, ED3/2, ED4). I welcome the move to this kind of generalised terminology that helps us describe more tunings with less words.

The world of xenharmonic jargon is often difficult to navigate. Once you get your head around it, you can forget about the tuning theory politics and remember that the important part is to make inspiring and enjoyable music!

Microtonalists vs 432 Hz advocates

Microtonalists vs 432 Hz advocates

And don’t get me started on those solfeggio frequency guys…