Circle of fifths part 3: tunings and temperaments

Last time we covered some of the better-known uses of the circle of fifths. Our subject this time is one that you may not have met, unless you are an early music enthusiast or a member of that endangered profession, a keyboard tuner.

The mismatch between 12 pythagorean fifths and seven octaves, known as the comma, is about a quarter of a semitone. In this temperament devised by Andreas Werckmeister, four of the 12 fifths are flattened by 1/4 comma each. This and other ‘well temperaments’ made keyboard instruments playable in all 24 keys. Diagram from http://www.hpschd.nu

This illustration comes from the website of Carey Beebe Harpsichords of Sydney, Australia. The site is a huge resource of all things harpsichord. It makes me want to acquire one of these beautiful instruments and learn to play it, if only for the meditative pleasure of tuning it.

The example shows a tuning devised in the seventeenth century by Andreas Werckmeister, using eight fifths in the mathematically simple frequency ratio 3:2, and four non-3:2 fifths¹. The fifths C-G, G-D, D-A and F#-B are each shrunk by 1/4 of the small interval known as the pythagorean comma, which itself is roughly 1/4 semitone. The comma arises as the slight mismatch between twelve perfect 3:2 fifths and seven octaves. It’s the reason why you can’t just set all fifths to a 3:2 ratio and be done.

Werckmeister’s objective was to absorb this mismatch gracefully. But that involves tradeoffs which are far from simple. Adjusting the pitch of a single note affects some 11 intervals, and you can be sure they won’t all change in the way you would wish.

The circle of fifths articulates the problem of tuning an instrument which has 12 fixed pitches per octave. Ideally, each of those pitches should be reusable in as many as 24 major and minor keys, and should make consonant intervals with the other pitches in each key. But why use the circle of fifths rather than, say, the chromatic circle? (The chromatic circle is superficially similar, but shows the notes in the order of the chromatic scale, C, Db, D…)

On the circle of fifths we can see that in Werckmeister’s tuning, most fifths have the desirable pythagorean value of 702 cents, while four are flattened by a barely audible 6 cents. The chromatic circle reveals that the semitones vary much more widely, over the range from 90 to 108 cents. This is acceptable, since the semitone is an inherently dissonant interval. For a keyboard tuner, fifths matter and semitones don’t. Major thirds also matter, but they can easily be added to the circle of fifths diagram. (1200 cents = 1 octave.)

The rationale for using the circle of fifths to specify tunings might look something like this:

• There are 66 intervals in the octave. Ideally you would like them all to sound as good as they can
• Any 12-note tuning is fully specified by 11 intervals
• You see the problem? You are trying to get 66 intervals right, but you are only granted 11 ‘wishes’. You need to set priorities, devise a strategy, and make compromises
• No diagram is going to carry out that intricate process for you, but you would, at least, like a diagram which tells you how close you are to achieving your objective
• To specify a tuning you need to set 11 intervals, or provide equivalent information. 11 semitones will work, fairly obviously. So will 11 perfect fifths or fourths (you know this from the circle of fifths). No other interval, by itself, generates the full set of 12 pitch classes
• These two ways of generating the 12 pitch classes can be pictured on  circular diagrams: respectively the chromatic circle and the circle of fifths. (The twelfth interval, the one which closes the circle, is determined by the other 11. So 11 wishes are all you get)
• Fifths (and fourths) are the most important intervals after the octave: the ones you really have to get right, to avoid excessive dissonance. The semitones are going to be dissonant anyway. So it makes sense to specify tunings in terms of fifths rather than semitones. This is why keyboard tuners use the circle of fifths rather than the chromatic circle
• The major third is also important. But you can’t tune the major thirds (or any other interval) independently of the fifths…
• …and crucially, you can’t build the whole scale from major thirds. If you try, you will be back at your starting note after just three steps. There is no circle of major thirds to provide an alternative to the circle of fifths

A historic mean-tone temperament (see below), recreated with Scala and displayed on a circle of fifths. Intervals are shown in cents. Eight of the 12 major thirds achieve the desired 5:4 frequency ratio. They are identified by orange lines.

Scala is a freeware software program for creating, investigating and playing musical scales. Among its many ways of displaying a scale and its properties is this one. It’s a circle of fifths, with the value of each fifth displayed in cents. The tuning shown here has been set up to provide eight major thirds with frequency ratios of exactly 5:4. To achieve this, eleven of the twelve fifths are on the small side (696.6 cents), and consequently the remaining fifth (Ab to Eb, 737.6 cents) is too big by about a third of a semitone. Clearly this tuning will be unsuitable for keys which contain both Ab/G# and Eb/D#. But with that reservation, this will be an attractive tuning for those who regard a 5:4 third as essential to a beautiful major triad.

You can read yards of technical articles on tunings and temperaments, on Wikipedia and elsewhere. Briefly, there have been four main historical stages:

• Pythagorean: clean-sounding ratios for most intervals, but 3rds and 6ths not necessarily to modern tastes. On a keyboard with 12 notes per octave, an ugly ‘wolf’ interval prevents playing in more than six keys.
• Mean-tone: all fifths are slightly detuned, in the interest of better-sounding thirds. Wolf interval is still there.
• Well-tempered: wolf interval eliminated by selective detuning of fifths. You can now play in any key, but the keys don’t all sound the same, because their triads and other intervals vary.  Werckmeister and his contemporaries explored many such temperaments, each with its pros and cons.
• Equal temperament. All intervals built from 12 equal semitones, so there are no exact 3:2 fifths or 5:4 thirds. Complete freedom to play in all keys. No wolf interval.

The idea of dividing the octave into 12 equal steps was widely discussed in 17th century Europe, and may have much older roots in China and Greece. Why was equal temperament not widely used until just over 100 years ago? I think the answer is that it is very difficult to tune an instrument in equal temperament by ear. Most of the historic tunings were devised by the people who actually tuned the instruments. They wanted a good sound, but they also wanted to keep their own job relatively quick and simple.

Here is another, more elaborate diagram of the Werckmeister III temperament. Near the centre, the size of each fifth is shown in cents, together with any discrepancy from the 3:2 value of 702 cents. (A cent is 1/100 of an equally-tempered semitone, so an equally-tempered fifth is 700 cents.)

This design by Yoshio Okamoto is packed with information about a historic keyboard temperament. English-language site: http://well-temperament.us/

The pie-slice between C and G contains all the information about the C major triad. The width of the oval reflects the size of the fifth C to G (696 cents, 6 cents flatter than ideal). The length of the oval indicates the size of the major third, C to E which, at 390 cents, is only 4 cents sharp of the desired frequency ratio of 5:4. The shapes and colours of the ovals show how closely each triad approaches the ‘just’ 4:5:6 frequency ratio.

No triad is perfect here. It is impossible to tune the thirds independently of the fifths, and compromises have been made. C and F are the best triads in this tuning, and Gb, Db and Ab (dark blue ovals) are the worst.

Even this incredible diagram doesn’t show everything. To work out the other intervals – semitones, whole tones, minor thirds and tritones, you might have to get out your calculator. But why bother? Most of those intervals are going to be dissonant, whatever you do. Only the minor thirds might be a significant issue. But if a tuning provides a number of acceptable major triads, then at least that number of minor thirds will also sound OK (unless your fifths are rubbish). There is little to be gained from putting additional effort into the minor thirds.

Yoshio’s research on historic tunings and temperaments is brought together in the chart below, which – even to a non-Japanese reader – is one of the most impressive musical infographics I have ever seen. All, needless to say, based on the circle of fifths.

Why you probably shouldn’t play Coltrane’s Giant Steps on a Baroque organ. In Yoshio Okamoto’s timeline of keyboard tunings and temperaments, enhanced circle-of-fifths diagrams show visually how historic tunings gave each key, and its major triad, a unique character. From http://pianotuning.jp/?attachment_id=2800

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(1) The Pythagoreans observed that stopping a string at 2/3 of its length produced the interval of a fifth. By the time of Galileo it was understood that this corresponds to a 3:2 ratio in vibration frequencies. On keyboards and other fixed-pitch instruments, it is only possible to create a closed cycle of fifths if some or all of them depart slightly from the exact 3:2 ratio.

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