How to Harvest Scales
Erv Wilson's musical phyllotaxy
Ever stop to really look up close at all those pianos and synthesizers growing in your garden? I do. Fascinating plants. I’m awe-struck every time, and every time I wonder: why are their leaves so often in groups of five, seven, and twelve?
Did someone design… even invent these patterns? How did we land on twelve, for example (and skip ten and eleven)?
I want to believe that the best musical structures are inevitable, not invented. I imagine them revealing their patterns in parallel with the evolution of human perception and culture. I dream of learning mathematical truths that allow me to logically predict where they will take us next.
I've recently come across one explanation of our pull toward those five, seven, and twelve-note scales that finally satisfied me. It’s simple, mathematically elegant, and universal. It's also observable in the natural world.
For anyone with a basic music theory knowledge the explanation is actually quite simple and my version requires 501 words so if you want a tl;dr I’m putting in in this footnote ➡️ 1. I hope it will be 🤯 for my musician friends.
But I’m choosing to focus this post on the intuitive side of this explanation, and its direct connection to mathematics observable in nature, because it’s confirmed for me that the resources humans draw from when we make music aren't always invented. Much like plant life on earth, they often self-generate according to universal mathematical properties. They grow, and we harvest.
Erv
Last week I briefly mentioned the work of Erv Wilson (1928-2016) and shared some new music created through experimentation with his ideas. Wilson wasn't a composer, he was a musical botanist. In his greenhouse he observed, studied, mathematically analyzed and painstakingly documented a vast array of musical structures, searching for every possible varietal — as if they were plants.
He spent his life in this pursuit not for his own sake, or for the cold, silent libraries of academia, but directly in service of musical gardeners, florists, chefs, landscapers… the creative people you and I might call composers.
Moments of Symmetry
Wilson’s work has applications all over music, but particularly with regard to scale and keyboard design. You might say he was searching for every mathematically possible tuning, a system for organizing them, and a generalized keyboard structure to play them.
He also came up with the idea for blue corn chips. This made hopi growers and some chip manufacturers very wealthy.
Central to his musical discoveries is a particular set of scale structures he deemed “moments of symmetry,” which he saw as the basis of his keyboard designs.2 This concept (MOS for short) is gorgeous and archetypal as a framework for composing elegant melodic material in a theoretically infinite array of tunings.
I believe the MOS are the closest any music theory concept gets to a plausible explanation of those familiar five, seven, and twelve note keyboard and notation systems. I also believe they are the key to eventually escaping from them.
And they aren’t specific to western music — the evidence of a magnetic pull toward moments of symmetry scales can be found across musical cultures all over the globe.
They can also be observed in plant life.
Tuning Spirals
Phyllotaxy is the science of how leaves are arranged on the stems of a plants. The most common arrangement is a spiral3, where each consecutive leaf grows at a constant angle. Call that angle the generator.
When viewed from above, the generating angle will create a spiral of leaf growth that either repeats every period, or doesn’t:
Left: the ninety-degree generating angle results in a single size of gap between adjacent leaves, no matter how many new leaves it sprouts — call this a “one gap” pattern.4
Right: With this generating angle there appear to be several different size angles between adjacent leaves. How many sizes of gaps are there here? How many could there be, theoretically?
In the answer to that question is a crucial mathematical truth: as long as the generating angle stays constant, there will never be more than three different sizes of gaps in the spiral. This is known as the "three-gap theorem" (or the Steinhaus Conjecture, after the mathematician who first proposed it).
We can now explain Wilson’s MOS:
Moments of Symmetry are scales with exactly two different sizes of interval: large and small.
The basic structure of the “moments-of-symmetry” is almost embarrassingly simple. Unfortunately what is simple is not always obvious, or visa-versa [sic]. And we use certain devices, repeatedly, without identifying them. Or we neglect to use them, when we might well have chosen to do so, had we only recognized them. - Erv Wilson, 19755
Put another way — musicians have gravitated toward moments of symmetry scales without knowing why for the entire history of music. The appeal of their robust structures, rich in elegant properties, is as intuitive as it is mathematically inevitable.
To begin, select a seed
In the three-gap theorem, we have a mathematical truth that connects the growth of plants to the growth of musical structures. To observe this in action, we can select a generating angle (interval), and see what pattern (scale) results with each additional leaf (note) that grows around the spiral of the stem.
It follows that finding moments of symmetry means simply looking for the scales with exactly two sizes of interval.
Generating angles are like seeds that come in all sorts of mathematical varietals: ratios, irrational numbers, harmonic partials, even tempos. With this single variable,6 a universe of possible resulting patterns opens up.
Diatonic, Pentatonic, Madonnatonic
Here’s a walkthrough of how planting a perfect fourth generates twelve scales. You’ll see that the 2, 3, 5, 7, and 12-note versions turn out to be the moments of symmetry. Those are scales that will be each be deeply familiar to some culture somewhere on earth, from Balinesian gamelan to Bach to our very best early 80s pop.
Wow, that video really went in a different direction than I expected!
Symmetric Gardens
The composition I posted last week used two simple moments of symmetry scales. The first was the (justly tuned) pentatonic generated with that same perfect fourth as discussed in this post.
For brief moments I also dipped a toe into some unfamiliar territory — if you use a 7th partial relationship (7/4) as a generator, the effect is beautiful and alien. It’s also oddly consonant and so satisfying. Here’s what that scale sounds (and looks) like:
You can explore infinite scales created by different generating intervals intuitively using the Wilsonic iPad app developed by Marcus Hobbs. I highly recommend it!!!
To say I’ve only just scratched the surface of the surface of Erv Wilson’s work would be a massive understatement. I am a noob and there are others who have been exploring for decades. But I hope I’ve been able to make it feel somewhat accessible, especially if you are a composer looking for new directions. See below for some tools and resources that can help you get started.
Warm thanks to Kraig Grady for his correspondence on this topic and for all he does as a steward of Erv Wilson’s work.
👋🔊 Subscribe for free:
Tools for exploration
Wilsonic — Start here. These are apps for exploring Wilsonic scales intuitively, developed by Marcus Hobbs. The iPad app in particular could be fascinating for anyone, even non-musicians, as it requires no musical or mathematical knowledge for experimentation. Even babies love it.
Oddsound’s MTS-ESP — Retune any DAW globally. The gold standard for limitless tuning work in the software realm. (Aphex Twin endorsed!)
Resources
The Wilson Archives — Kraig Grady’s online library
Microtonality and the Tuning Systems of Erv Wilson — A deep and thorough dissertation of Wilson’s theoretical work. Terumi Narushima, 2018
The Xenharmonic Wiki
Articles about Erv Wilson
Erv Wilson Remembered - Gary David, 2018
Erv Wilson: In Words - Compiled by Jose Hales-Garcia
Ervin’s Life - Younger Years - Grant Wilson, 2009
Grateful for Erv Wilson - Marcus Hobbs, 2019
Here we go.
The first three musical intervals generally discovered all over the world are the frequency ratios 2:1 (octave), 3:2 (pure perfect fifth), and 4:3 (pure perfect fourth). They are universal perceived as musical “special relationships” by all humans. This is physiological fact, not “theory.”
They are also the relationships between the first four partials of the natural harmonic series, which makes them the simplest possible mathematical ratios as well.
One natural impulse in exploring intervals is to stack them. When stacking octaves, they have the sameness of a repeating period. When stacking 3:2 or 4:3 intervals, they generate a “scale” within that period. As each new note is added, a new scale is born.
What nobody (and I mean nobody, until the 20th century) consciously realized is one simple mathematical fact: there can only ever be a maximum of three different sizes of intervals between consecutive notes in a scale built by stacking intervals in this manner. This was conjectured by Hugo Steinhaus and proven by various mathematicians in the 20th century. It’s known as the “three-gap theorem,” and it’s not specific to music — it can be observed in nature, too.
The commonly used metaphor for the three-gap theorem is that of taking equal sized steps around a circle and observing the space between the footprints. There can only ever be a total of three gaps or less between footprints no matter what step size, where you stop, and how many circles you make.
In the three-gap theorem we have a plausible explanation for a completely intuitive process that has been followed unconsciously by musical cultures all over the world. As it turns out, the two-gap scales have robust structural qualities that simply make them feel more proportional for melodic material than the three-gap scales.
Irv Wilson deemed the two-gap scales “moments of symmetry.”
And when that pure perfect fifth or fourth is the generating interval, the two-gap scales happen to be the ones with 2, 3, 5, 7, or 12 notes. The development of music all over the world in perfect fifths and fourths, as well as Madonnatonic, pentatonic, diatonic, and chromatic structures naturally follows.
Over time, musicians tweaked this scale. For example: using a 5:4 interval for the fifth note in the chain of pure perfect fifths (a dissonant major third, interval 81:64) brought it closer to the tonic and created a major third that is more consonant, vertically. Further tweaking that to an equal-tempered major third followed, for practicality. Eventually we landed on a twelve note equal tempered scale — but the pattern of pentatonic (five black keys), added to diatonic (seven white keys), to make a chromatic scale (twelve keys total) remained, as this reflects the inevitable origin scale that twelve tone equal temperament is forever approximating.
The next moment of symmetry in that chain is at seventeen notes. The five new added notes create a literal five-note pentatonic scale (!!!), added to the twelve-note chromatic. Western music searched high and low for where to go next in the 20th century… might I suggest…
Musicians: does this track? For me it was revelatory and like… a weight off my shoulders?
Source: Kraig Grady
I’ve seen the Three Gap Theorem explained as “at most three, and at least two, different lengths,” (van Ravenstein 1988, p. 360 - emphasis mine) which would imply that there cannot be a “one gap” scenario as I have presented above.
I’ve come to the conclusion that this was an error, or else I’m misunderstanding the quote: if the generating angle is a rational multiple of pi, there will only ever be one size gap, so there can clearly be “one-gap” scales. This is an excellent mathematical parallel to octave-repeating equal-division scales (for example the exhaustively documented “circle of fifths” of twelve tone equal temperament).
To me, the idea that all equal tempered scales are “one-gap,” and all linearly-generated rational scales are either two-gap moments of symmetry or more complex three-gap structures is too delicious an organizational idea for me to let go of. But I’m open to arguments!
Source: Microtonality and the Tuning Systems of Erv Wilson (Narushima, 2018).
The circumference of the spiral itself, also known as the “period” could also be seen as an independent variable, but a ratio of 2:1 (aka “octave”) is so natural, and offers so many possibilities, that it has hardly occurred throughout human history to anyone but the most theoretical of musical thinkers to grow anything other than “octave-repeating” scales.
That said, the trio of Bohlen, Prooijen, and Pierce explored them. So did Wendy Carlos, and you can too, especially with modern software like Wilsonic or MTS-ESP. But I’m leaving them out of the explanation to keep things simpler for now.
Thanks again, Chris.
It's too bad I don't have time to really dig in or engage. Someday soon I hope to plow through everything you've posted that I haven't read. Every single word, I promise you.
Mind blown.