I was surprised when I discovered the extent of the visual and musical culture of the circle of fifths. It is like a huge tree with many branches and colourful fruit.

There was a second surprise when I started investigating why the circle of fifths is so useful, and why it exists at all. The tree that we see above ground turns out to have a huge root system which spreads into deep areas of music theory, mathematics and psychology.

**Consonance and
dissonance**

There are many ways into this subject. But the one that works for me is through the concepts of consonance and dissonance. Like, y’know, the perceptual quality of musical sounds when two or more are heard together or in close succession.

Over the centuries there have been at least five distinct ways of thinking about this subject[1]. I won’t attempt to describe them all or choose the best. But one idea has survived the centuries. This is the idea that consonance is high, or dissonance is low, when the notes of an interval or chord have frequencies in a ratio of small whole numbers. Those ratios need not be mathematically perfect, but the margin of permissible error is small.

This assumes harmonic timbres, i.e. repeating waveforms, a fair assumption in European music performed on voices, pipes and strings. The theory is different when bells, gongs and xylophones are the main instruments[2].

Frequencies in the ratio 4:5:6 form a major triad, which sounds highly consonant in harmonic timbres. 2:1 (the octave) is consonant in harmonic timbres. But in other timbres, neither of these things is necessarily so. From here on, I shall assume harmonic timbres. But I shall resist ascribing any special status to the octave.

**Making scales**

Not all genres of music put the same values on consonance and dissonance. But it is never a problem getting all the dissonant intervals you could wish for. If you are creating a scale – a collection of pitches for music-making – your efforts will probably be concentrated on obtaining an adequate supply of consonant intervals and chords.

Fortunately there is a piece of magic that makes this easy. Choose a few small-integer ratios to use as generators and let them breed. Start with 2:1 and 3:1, for example, and by multiplying and dividing them together, or with themselves, you can generate other small-integer ratios such as 4:1, 3:2, 4:3 and 9:8. Throw in 5:1 and you can generate 5:4, 5:3 and 6:5. Most of these are intervals of high consonance, though the effect weakens as the numbers build up. 9:8, for example, is significantly more dissonant than 3:2.

I am deliberately not giving intervals or notes their conventional names, because those names imply a particular scale, and we are still trying to keep our scale options open. But I will now use a real scale, the pentatonic, as an example.

**Generated scales**

We’ll take a base pitch with frequency *f*, and apply generators 2:1 and 3:1.

First use the 3:1 generator to make a set of five pitches: *f*/9,* f*/3, *f*, 3*f *and 9*f*. (Later I will refer to such a sequence as a generator chain.)

These are a bit spread out for a musical scale, with just five notes over a frequency range of 81:1. Not many good tunes in there, certainly not the kind you can hum in the bath.

But now we can use the 2:1 ratio to parachute some notes into the gaps. Introducing factors of 2 generates some pitches much closer to *f*, namely: 3*f*/4, 8*f/*9, (*f*), 9*f*/8 and 4*f*/3. That’s five tones within a range of 16:9 (less than an octave). Distinctly more hummable.

The next five notes, moving upwards, are: 3*f*/2, 16*f*/9, 2*f*, 9*f*/4 and 8*f*/3.

Or downwards: 3*f*/8, 4*f*/9, *f*/2, 9*f*/16 and 2*f*/3.

By continuing to multiply or divide by factors of two, we can extend the scale over the full range of human hearing (and if necessary, beyond)

And – this is where the magic of generated scales [3] does its stuff – the interval between any two notes of this scale will have a ratio within the restricted universe of numbers made by multiplying and dividing twos and threes. By using these small numbers, we have minimised the cost (in dissonance) of adding new notes to the scale.

Continuing the process will generate some bigger numbers, so the promise of limitless consonant intervals eventually runs out. For example, the frequency ratio between* f*‘s two immediate neighbours, namely 8*f*/9 and 9*f*/8, is 81:64, which does not look promising [4].

Allowing unlimited factors of one generator, while setting limits on all other factors, has the effect of creating a repeating pattern. The interval between repeats, which in this instance is an octave (ratio 2:1), is known as the period of the scale.

**Pitch classes**

In the scale we have made, each repeat of the pattern contains five pitches. There are five distinct roles or positions in the pattern. If we give each role a symbol, we get something like this.

The interval between one mushroom and the next is the period. Likewise for any other symbol.

What is the generator doing here? Well, each position or role in the scale is associated with one point on the generator chain. Here is the generator chain, with symbols added.

All pitches with the same symbol are said to belong to the same pitch class. Each pitch class is associated with a position on the generator chain. (This could be generalised to any repeating scale.)

Pitch classes are sometimes defined in acoustic terms, with specific reference to the octave. By adopting a structural definition instead, we’ll be able to apply the concept of pitch class to scales with non-octave periods. You don’t meet those every day, but they do exist, musicians have used them, and they are of considerable theoretical interest.

**The Tenney lattice**

The notes of the pentatonic scale, or any scale with two generators, can be spread over two dimensions to form a lattice. Here we have drawn it so that one step upwards represents a 2:1 interval (octave), and a step to the right represents a 3:1 interval (a ‘tritave’, or perfect twelfth).

This kind of lattice, where the intervals are drawn to scale, is known as a Tenney lattice, after the 20th-century American composer James Tenney. The music theory literature includes countless varieties of lattice, but most are like the London tube map – they don’t show true distance. Tenney’s insight was that if tuning is important, our map of pitch space should show not just connectivity but distance.

Our example lattice extends without limit in the vertical dimension. Its width is set by the generator chain, (shown in green). The shading, and the red pitch height axis arrow, indicate that high pitches are at top right, and low pitches at bottom left. The purple ‘unison’ line separates pitches higher than the base frequency f from those below it. Differences in pitch height correspond to distances in the SE-NW direction.

Here is the same lattice, with five pitch classes identified. The strip of lighter colour includes all pitches within half an octave of the base pitch. Each octave of the scale (or in general, each period) includes one note from each pitch class. This provides the repeating unit of the scale.

Let’s now extend the generator chain to seven iterations, so we have this as raw material for a scale:

Breed those pitches with unlimited factors of 2, and the empty spaces fill up. We need to zoom in a bit, to see what is happening.

This is an important type of pattern known as a diatonic collection. The white keys on the piano are a familiar example. This is easily seen if we match *f* to D on the piano. But *f* could be any note. It’s the pattern of intervals that makes a scale diatonic.

Enough mushrooms. We can give the notes of this scale their conventional names. Here is its generator chain, with seven pitch classes.

###
**Tonality, modulation and the chain of tonalities**

Something interesting happens if we take a diatonic scale, add a pitch class at one end of the generator chain and remove one at the other end.

Effectively we have shifted everything one step to the right. This has the effect of transposing the scale by the generator interval (here, 3:1). The new scale shares six out of seven pitch classes with the original one, though their roles in the scale have changed. For example, G now occupies the role vacated by C.

Any 2D generated scale can be transposed in this way, with the loss of one pitch class and the gain of another. It’s not hard to see what has happened: we have changed key. More precisely, we have moved from one diatonic collection into another. It was an easy move, since the two collections have six out of seven pitch classes in common.

These close key changes aren’t the only ones ever used; but they are common in many musical styles.

After G replaces C, at the next modulation its role is taken over by D; then A, E and so on, in the generator-chain sequence. In the white-keys collection, C or A might be a tonal centre, depending on whether the musical context is felt to be ‘major’ or ‘minor’. So there is a chain of major keys that follows the generator chain, and a chain of minor keys which does the same.

I will call this the chain of tonalities. But it is really a chain of collections. Similar chains exist for other 2D generated scales, including those which don’t have the major-minor thing or a recognised way of establishing tonal centres.

###
**The
F/F# property**

The pentatonic and diatonic scales have a property that their 6- and 8-note close relatives lack: the outgoing and incoming pitch classes are particularly close, and the net effect is simply to sharpen or flatten one scale tone. In terms of voice-leading, this could not be better. It’s musical synchromesh.

Gerald Balzano, one of the pioneers of modern scale theory, called this the F-F# property. It is a structural property, not dependent on consonant generators or exact tunings.

It underlies our conventional system of letter-names, accidentals, staves and key signatures.

When we get to the circle of fifths (soon), this will mean that though every 2D generated scale has a chain or circle of tonalities, the diagrams for scales with the F/F# property would look familiar, because they can show staves with key signatures (albeit slightly weird key signatures).

Here is a circle of tonalities which looks simultaneously familiar and weird. It is from a preprint by Brandon Wu. We’ve got 19 keys here. Sharps accumulate in one direction, flats accumulate in the other, and at one point flats flip into sharps. The generator is 8/19 of an octave, and each key uses 12 of the 19 available pitch classes. Those numbers are not random. They are chosen because the resulting scales have the F/F# property.

### **Harmonic distance**

Structural properties like F-F#, which are robust within a certain tuning range, influence the generator chain’s role as a modulation path, and the sense of closeness or distance between keys.

The generator chain also creates a sense of harmonic distance between pitch classes. This, however, depends on the scale’s acoustic properties. It is sensitive to tuning.

We have already seen that pitch height can be measured from the lattice diagram, as a straight-line distance in a particular direction.

James Tenney suggested that we can also read another kind of distance off this chart. He called it harmonic distance. It is not a straight-line distance. It is what mathematicians sometimes call a taxicab distance or Manhattan distance. It is often simply called Tenney distance or TD.

If an interval has ratio p:q, where p and q are whole numbers with no common factor, the interval’s Tenney distance is log_{2}(p)+log_{2}(q) = log_{2}(p.q). (Adding the logarithms of numbers is equivalent to multiplying the numbers.)

The basic idea has been around in one form or another since the sixteenth century, when Giovanni Battista Benedetti suggested that if two tones have frequencies in the ratio p:q, their degree of dissonance can be predicted by multiplying p and q together. The ‘Benedetti distance’ of the octave is 2*1=2. For the perfect fifth it is 3*2=6, and for the perfect fourth, 4*3=12.

Tenney used the logarithm of Benedetti’s formula. This stops the numbers growing so rapidly as intervals become more dissonant. And it means Tenney’s numbers can be usefully interpreted as geometrical distances.

I am not sure whether Tenney knew about Benedetti. Perhaps there is a reference somewhere in Tenney’s writings, but in my limited reading I have not found it.

**Pitch
class Tenney distance and the chain of pitch classes**

Provided the interval between two pitches can be stated as an exact frequency ratio, Tenney’s formula measures the harmonic distance between them. It would be useful to have a similar measure for the harmonic distance between pitch classes, if only because musicians and theorists so often refer to pitch classes rather than specific pitches.

One approach is to measure the Tenney distance between representative members of pitch classes – usually pairs which are close in pitch height. But it is difficult to choose representative pitches in a systematic way.

A look at the lattice diagram suggests a simpler approach. Pitch classes appear as vertical columns, so why not simply measure the horizontal distance between the relevant columns? Pitch-class Tenney distance (PCTD) would then simply correspond to distance along the generator chain. Tenney called this “generalised distance” but I want to emphasise that it is a distance between pitch classes, and categorically different from TD, which is a distance between pitches.

In 2D systems, PCTD between any two pitch classes can be measured along the generator chain.

It turns out that generator-chain distance is a surprisingly good proxy for the actual Tenney distance of typical musical intervals.

I have put the mathematics of this in a separate document. The gist of it is that if a scale has two generators, each generator is responsible for roughly half the TD of any small interval. PCTD measures the share of dissonance due to the generator, but excludes the share due to the period. Simply double the PCTD figure of a typical small interval, and you will have a reasonable estimate of its Tenney distance.

If it is only approximate, why bother? It means we can stop pretending the note names on the circle of fifths refer to specific pitches. We can avoid the inelegance of “add a fifth, and reduce the pitch within the octave” – which begs the question, “which octave?”. There is no right answer, because whichever octave you choose, you end up with a circle of some fifths and some not-fifths.

The PCTD concept can in principle be applied to any 2D generated system, provided the period and generator are both consonant, simple-ratio intervals. Sufficiently close approximations can be treated as if they were exact ratios, though the definition of ‘sufficiently’ is open to debate.

Subject to these conditions, we can form a chain of pitch classes ordered by harmonic distance. The chain of pitch classes has the same sequence as the chain of tonalities based on easy modulation. Despite their different rationales, both sequences are manifestations of the generator chain.

All this applies to 2D generated systems. There is no direct equivalent in systems with more than two generators.

###
**Chains
into circles**

A generator chain can be endless, or looped into a circle. The size of the circle is a choice, exercised by fine tuning of the generating intervals. Beyond a certain size, it really does not matter how big the circle is. Mathematics can distinguish between an infinite chain and a long cycle, but the ear cannot.

Having said that, there are good reasons for embedding the diatonic and pentatonic scales in a cycle of 12 pitch classes or tonalities. It is one of the smallest cycles offering acceptable intonation, and from the 17th century or earlier, it became popular with instrument makers. From the 19th century onwards, in genres ranging from opera to jazz, composers increasingly exploited the harmonic possibilities of the full cycle of 12, and the sub-cycles of lengths 6, 4 and 3 that we get by taking two, three or four generator steps at a time.

A couple of examples. Richard Wagner apparently made a complete circuit of the 12 keys in his *Meistersinger *overture.

And John Coltrane famously used a sub-cycle of three tonalities, B, G and Eb, in *Giant Steps*.

###
**Temperament**

Cycles or circles are a consequence of temperament – a subject already touched on in this series.

If the period and generator(s) involve different prime numbers, pushing into new keys will create new pitch classes for ever, slicing the audio spectrum into ever smaller steps. This is obviously a problem for fixed-pitch instruments. Modulation within a piece is clearly an issue. But more than that, it’s the simple wish to play pieces in a variety of keys.

Temperament is the practice of adjusting the generating intervals, so as to limit this endless spawning of pitch classes. There are many temperaments, invented by everyone from Renaissance organ-builders to 20th-century microtonalists. Each temperament causes some small interval to vanish.

Best known is 12-tone equal temperament (12-TET), which depends on the fact that 2^{-19}.3^{12}=1.0136… This implies that 12 perfect fifths exceed 7 octaves by a small interval of approximately 1/4 semitone. That small interval (the Pythagorean comma) disappears if we replace the generator 3:1 by 2.9966..:1.

Every temperament will stretch, squeeze or ruck the relevant Tenney lattice by a small amount. We’ll concentrate on the 2D case. A small interval can be made to vanish by finding its lattice point – which must already be close to the unison line – and tweaking the lattice so that the point lies exactly on the line. For 12-TET, it’s the point with coordinates (-19, 12) that has to be nudged into unison.

In the space between the origin and that point, there are 12 pitch classes. Beyond that, once the lattice has been adjusted, the pattern repeats. No new pitches appear.

We can roll the lattice into a cylinder so that (-19,12) coincides with the origin (0,0), and everything else will join up perfectly. Importantly, pitch height and pitch class are consistent across the join.

A cross section of the cylinder reveals what we usually call a circle of fifths. More precisely, it is a circle of pitch classes – though it is only a short step from this to a matching circle of tonalities. The circle of pitch classes is easily overlooked, but it is put to creative use in the tuning diagrams of Yoshio Okamoto (English language site) and in Tessitura Pro, an ingenious app from MDecks Music.

It doesn’t matter where we slice the cylinder, provided we cut along a unison line. We may hit no lattice points at all, but that doesn’t matter. Lattice points are scale tones, and we are looking for pitch classes, not scale tones. The circle reveals how pitch classes are configured in a ‘sideways’ dimension that is independent of pitch height.

Pitch class F is like a wire linking all the Fs. It’s there, even in places where there is no actual F note. Running parallel to it are wires for 11 other pitch classes.

We can now bring Tenney distance into line with this new geometry of pitch space. It makes sense to measure pitch-class Tenney distances along the shortest arc on the circle, so B to Db would be two steps, not 10. Similarly, Tenney distances between tones should now be measured by the shortest city-block route on the surface of the cylinder.

(Theoretically all 12-TET pitches could be generated from a single generator, the semitone. But the diatonic scale has the structural and acoustic properties of a 2D generated scale. Temperament – closing the generator loop – leaves most of those properties unaffected, and we want to keep sight of them. It is the dimensionality of the embedded scale, not the background, that matters. That is why I have modelled 12-TET as a lattice on a cylinder, rather than a simple ruler graduated in semitones; in other words, as a temperament rather than an an equal division of the octave or ‘EDO’.)

**Thank
you for choosing 12 pitch classes. We know you have a choice**

12 pitch classes may be some kind of sweet spot: it’s a way to get reasonably consonant intervals in up to 12 keys, without driving instrument makers to despair. But 12 is a choice, not a law of musical nature. If we want other ways of tempering the 2:1, 3:1 generators, we need only explore the lattice a little further, to find other small intervals ripe for tempering. For example it is possible to close the generator chain at 41 or 53 pitch classes, in each case with only the tiniest of tuning tweaks.

The possibilities of temperament expand if other primes such as 5 and 7 are drawn in as generators. If N primes are involved, you can arrange for up to (N-1) small intervals to vanish. (You’re now working in a mind-bending N-dimensional Tenney lattice.)

In the 20th century this led to the discovery of many new temperaments. Things resembling the circle of fifths exist in this wonderland, but in some temperaments it is impossible to find a period and a generator, both of which are simple, consonant intervals. So you may have all the structural features of generated scales, including a circle of tonalities. But the acoustically-sensitive circle of pitch classes may lose its rationale.

I don’t have psychoacoustic research to settle this, but it is hard to see what psychological reality a circle of pitch classes could have, if there were no sense of harmonic closeness between neighbouring pitch classes.

So, as musicians explore away from scales generated by the simplest ratios, 2:1, 3:1 and 3:2, I suspect only the ‘circle of tonalities’ will remain with us. It will survive because it depends on the structural properties of generated scales, rather than the harmonic properties of the generators.

But how much musical relevance can the circle of tonalities retain, without the harmonic underpinning of a circle of pitch classes? Can you feel a modulation, without those bonds of consonance between the tones involved? I really don’t know. This is not a simple question of physics or physiology, though both are relevant. It is also a cultural question about how we make music, how we listen to music and what we expect from music, in a particular place and time.

**Summing
up**

- 2D generated scales were adopted for acoustic reasons, but they turned out to have structural properties that musicians could exploit.
- European music relies on the harmonic timbres of voices, pipes and strings; not so much on bells, gongs or xylophones.
- In harmonic timbres, frequencies in ratios of small whole numbers form consonant intervals.
- Scales rich in consonance can be generated from as few as two simple, consonant intervals.
- Restricting all generators except one (the period) produces repeating scales, leading to the idea of pitch classes.
- Using the simplest generators (2 and 3) and short generator chains, the pentatonic and diatonic are among the first scales to emerge. They have some nice properties.
- Keys or tonalities arise when a scale is transposed. Keys with many pitch classes in common are regarded as close.
- The generator chain is a yardstick of distance between tonalities, and ease of modulation. This is a structural property, not requiring consonant generators. When the chain is closed into a loop, the result is a circle of tonalities.
- The generator chain is also a chain of harmonically-close pitch classes. This is an acoustic property, dependent on consonant generators. Closing the loop results in a circle of pitch classes.
- What closes the generator chain into a loop? Temperament.
- What makes a circle of fifths (or other interval)? Temperament applied to a 2D generated system.
- There is a world of scales out there, which have their own “circles of fifths”. Along with many that don’t.
- Diatonic scales embed neatly in a cycle of 12 pitch classes. Once this happened, composers exploited it. But 12 is a practical choice, not a law of nature.

###
**And
finally: why, out of a million ways to approach this subject, I chose
this one**

I chose to approach the subject through acoustics and the Tenney lattice, because the lattice depicts a prehistoric musical universe, as it was after simple scales were created, but before the emergence of the circle of fifths. We find ourselves in a space which extends infinitely in all directions. There is, as yet, no indication that the generator chain, or anything else, will close into a circle.

In this loosely historical approach, the circle of fifths emerges as a practical response to a musical problem. How do we make a harpsichord which can play in more than a few keys? Almost accidentally, the solution makes the world safe for Wagner and Coltrane.

I’m sure one could equally well find a justification for the circle of fifths in structural scale theory. But much of the scale theory literature begins with the assumption that scales are embedded in a cyclic universe of 12 pitch classes – or at least, a fixed number* c*. Many wonderful properties of the diatonic scale and 12-TET have been discovered on this basis. It is possible to look back and say “hey, didn’t that idea work well?” which is logically fine, but somehow disquieting, like retrospective planning permission.

## Free circle of fifths worksheet

Six circle-of-fifths images per sheet. For teaching, study or just getting your head around chords and scales. Good printed out on paper. Even better dropped into electronic-ink apps like Microsoft Onenote.

**Download **(20KB PDF)

**References/Further
reading**

This article has taken many ideas from James Tenney. The collection* From Scratch: Writings in Music Theory* includes *The Structure of Harmonic Series Aggregates* in which Tenney introduces his measures of harmonic distance and generalised harmonic distance. It is available online from Google Books. *A History of ‘Consonance’ and ‘Dissonance’ *(1988) is downloadable from http://www.plainsound.org/JTwork.html.

Paul Erlich’s concept of harmonic entropy addresses some of the deficiencies of Tenney’s harmonic distance. Erlich argues the case for 2D generated systems – the ‘middle path’ – in his unfinished paper *A Middle Path Between Just Intonation and the Equal Temperaments* (http://sethares.engr.wisc.edu/papers/erlich.html). This is a manifesto for temperament as a positive force rather than a regrettable necessity. Remarkable for its vision, it nevertheless leaves much unexplained and – how shall I put this? – may be unrewarding for the casual reader.

The best description of Erlich’s harmonic entropy is in an appendix to *Tuning, Timbre, Spectrum, Scale* (2nd ed., 2004) by William Sethares. The main text is a deep inquiry into the question ‘where do scales come from?’ The answers are fascinating, and anyone who knows this book will see its influence in what I have written.

One of the best ways to understand 2D generated systems is to play with them on a computer.

It was the HEX software (https://dynamictonality.com/hex.htm), and a 2012 article about it in Computer Music Journal (http://oro.open.ac.uk/30157/1/Hex_-_submission.pdf) that helped me to understand these systems and grasp concepts like the valid tuning range.

Hex is very good for observing how scale tones slither and slide in response to tuning changes. I noticed that each generated scale has a character which survives substantial changes in tuning. The generator chain is the constant. It can be sometimes be closed into a circle, but the size of the circle is totally a choice.

In Hex, all scales are by default embedded in an infinite lattice, related to the Tenney lattice but more abstract and fluid. No circles are explicitly shown. We are in pre-circular Eden. But with a bit of thought you can see how endless opportunities for temperament and circle-formation arise.

### Footnotes

1. James Tenney, *A History of ‘Consonance’ and ‘Dissonance’ *(1988). There is a good summary in *Tuning, Timbre, Spectrum, Scale* by William Sethares.

2. This is the central idea in the book by Sethares. His approach is based on sensory dissonance – a quality of complex sounds experienced without musical context, implied tonality or harmonic function.

3. By ‘generated’ I mean a scale made from two or more generators. Some authors restrict the term to two-generator or ‘two-dimensional’ (2D) scales, but if I mean 2D I will say so.

4. In fact it doesn’t sound bad, probably because it is close to 5:4, a consonant interval which comes from a larger universe which includes a 5:1 generator.