Will it go round in circles?

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People love circles, especially 12-sector circles. So although the significance of music’s circle of fifths can take a while to appreciate, the diagram itself seems disarmingly familiar. With a bit of imagination it can become a clock face, a zodiac, a colour wheel or a calendar of months.

Whoa, are all those things really connected? Well, only in the vaguest of ways. It’s possible to get really mystical about this stuff. Which I don’t plan to do. Sacred geometry and the music of the spheres are beyond the scope of this blog.

Here is a typical circle of fifths chart, from Wikipedia. It shows key signatures for 12 acoustically-distinct keys, in major and minor modes, plus three enharmonic equivalents.

By Just plain Bill (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)%5D, via Wikimedia Commons. Original file is an SVG.
Most students first encounter the circle of fifths as a way of learning key signatures. However, key signatures are a notational wrinkle resulting from our use of the five-line stave. The diagram represents relationships between pitches and keys which would exist even if we replaced the five-line stave with some completely different notation.
The basic diagram has spawned thousands of variants. There are versions showing key signatures in bass clef as well as the more common treble. Tenor and alto clefs too, I should not be surprised. Never underestimate the power of viola players.
Search “free printable circle of fifths” and you will be offered dozens of charts and worksheets, including partly blank charts for student exercises. Then there are the gadgets, the merchandise, and the apps.
There are discs with rotating parts. The Dialanote disc has a 13th sector inserted between F and C,  making room for a window where masses of additional information can be brought into view.


The Chord Wheel is another spinning disc, invented by guitarist Jim Fleser and sold by the Hal Leonard organisation. By using three concentric rings, the Chord Wheel manages to show all seven diatonic chords for each key, plus a couple of secondary dominants, in their correct qualities. The chords are framed by the bold outline on the transparent oversheet.
This works fine for triads, but a slight disappointment for jazz musicians is that it doesn’t really work for seventh chords.
Possibly the most complex rotary gadget of this kind is the Guitar Wheel. As far as I can tell, it is based on a chromatic circle rather than a circle of fifths, so we shall merely mention it and move on.
The Guitar Wheel is not just for guitarists. On the back, there is a whole section for pianists and other musicians.
Rod Smith’s rotating Music Mandala™ combines the chromatic and fifths circles. As you can see on this June 2011 blog post, it is as much a work of art as a practical device for musicians. I am not sure of the current status of this project.
The internet’s gift shops, with their nose for human obsessions, can offer you posters, decals and framed prints of multiple circle-of-fifths designs. Nothing says you are serious about music like a circle-of-fifths framed print.
The colourful 7-pointed star, second from left, is an example of the ‘in-key’ circle of fifths, which includes a tritone. This version won’t take you through the twelve keys, but it shows some of the things that can be done with the seven notes of the C major scale.

Here is the circle of fifths in T-shirt form, from The Jazz Shop. Chord pairs suitable for tritone substitutions are helpfully picked out in matching colours.CoFTshirt.jpg

Search eBay and at any time you will probably find a choice of circle-of-fifths clocks. Ideal if your tutor has told you to practise for twelve hours a day, one hour in each key (cue Whiplash movie theme).
The Chromatic Watch Company, which we fear may have ticked its last, offered circle of fifths watches (below, centre) as well as the classic chromatic version (left) and the widdershins ‘circle of fourths’ watch (right) which we’d have liked even more if the hands went around anticlockwise.
chromatic watch
Want more CoF randomalia? There is a collection of guitar and CoF-based images at a website which looks a bit like Pinterest but is inexplicably named after a painful medical condition.
And then there are all the apps. Some of these may be brilliant. (I will cover mDecks’ Tessitura Pro in a future post.) Some, I suspect, provide no more information than you would get from a printed chart. Though I guess most of them make noises, which printed charts don’t.
Apps aplenty at Google’s Play Store. Apple iOS users also have a big choice.
Many versions of the diagram show equilateral triangles and squares inscribed in the circle. Four equilateral triangles can be drawn, representing augmented triads, and three squares representing diminished 7th chords. This happens because the 12 chromatic notes can be subdivided as 4×3 or 3×4.
Guitarist Pat Martino applied these ideas in his unique approach to the fretboard, and illustrated them with a beautiful diagram. The chart is diamond-shaped, and shows some extra stuff – four cardinal points (NSEW) and (in some versions) four seasons – which can only be relevant to music in a metaphoric way. But it is still basically a circle of fifths.
Now, hold onto your hats, for we are about to digress, bigtime. Pat Martino has placed the points of the compass on a 12-point circle, and that may seem unusual. But in medieval Ireland, there were twelve winds.
And each of them was ascribed a colour.
Who would have thought a mere 30 degree veer could change the wind from yellow to red? What is a red wind, even? My source for this lore is the wonderful Twitter feed of the art historian Anne Louise Avery. It was retweeted to a wider audience by the nature writer Robert Macfarlane.
The science fiction writer Ursula Le Guin, who sadly died in January, was tapping into the same tradition when she named her 1975 short story collection The Wind’s Twelve Quarters, which itself is a quote from an A.E. Housman poem that Ralph Vaughan Williams set to music. Phew. Music. We are back on topic, more or less. I did warn you that 12-point circles crop up all over the place.
Here’s some real research data, on harmony in the work of six composers, beautifully presented by Kelly Schroer in infographics based on the circle of fifths. Schroer, a user experience designer based in Chicago, is also a classically-trained pianist.
And finally, did you know that the notes on a tenor steel pan – the one which plays the melody – are usually arranged in a circle of fifths?
tenor steel pan
Tenor drum pattern, from 21stcenturysteelband.com

The Trinidadian steel pan pioneer Anthony Williams introduced this arrangement in 1953.

It is actually a bit surprising that this arrangement should work. Most musical instruments generate a fundamental tone plus a series of ‘partials’ with frequencies in exact or near-exact multiples of the fundamental. The note we call A (440Hz) would have partials at 880Hz, 1320Hz, 1760Hz and so on, which are known as harmonics.
Notes separated by a perfect fifth have frequencies in the ratio 3/2, and many of their harmonics therefore coincide. This is thought to be why, on most Western instruments, a perfect fifth sounds consonant.
The physics of many percussion instruments is different. The partials of a bell or drum are generally not harmonics. On a steel pan, you would not expect fifths to work their usual harmony-creating magic. Why then does a circle-of-fifths design work so well?
Physicist and steel pan player Ulf Kronman has investigated this. Against expectations, steel pans do have harmonic partials. But these are not the natural vibration modes described by the standard physical theory. They result from a phenomenon known as non-linearity, which distorts the fundamental vibration and converts some of its energy into harmonics. The same thing happens – less pleasingly – in cheap, tinny loudspeakers. Non-linearity helps to explain why a steel band sounds so harmonious when performing music that was written for instruments like flutes, cellos and horns.
I hope this whirlwind tour of circle-of-fifths culture has aroused your curiosity. My next post will investigate the origins of the circle of fifths, and the acoustic and musical reasons for its usefulness. This iconic diagram was not simply handed down to us on a stone tablet. It was invented, probably at least twice. What are the secrets of its success?


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