Closing the circle and ending the spiral: of well-temperament and equal temperament

I’ve been struggling for a while with how to present this in an accessible way. Stumbling upon this video below made me want to try: let’s have a go at this.

“Circle of fifths” or “spiral of fifths”?

It’s been well-known since the days of the ancient Greeks that the musical intervals we perceive as “purest” are the ones where the frequencies are simple fractions.
• octave: 2:1
• perfect fifth: 3:2   (and its inversion the perfect fourth, 4:3)
• pure major third: 5:4 (and its inversion the minor sixth, 8:5)
Less consonant are smaller ratios such as:
• pure minor third: 6:5 (and its inversion the major sixth, 5:3)
• greater whole tone, 9:8 (and its inversion the minor seventh, 16:9)
Now if you have a continuous-pitch instrument such as the violin or the human voice, you can just sing/play the intervals perfect (in “just intonation“) and be done with it. But what about fixed-pitch instruments such as guitars/lutes or pianos/harpsichords?
Big deal, you say: just stack up pure fifths, thus build the circle of fifths, and we can span a whole scale. Twelve fifths up and seven octaves down, and you should be where you started.
Sounds good in theory, except: the circle of pure fifths isn’t — it’s really a spiral of fifths. (3/2)^12 divided by 2^7 works out to 1.013643265…, almost a quarter of a semitone too wide. This small interval 531,441/524,288 is known as the Pythagorean comma.
Big deal, you say, if it doesn’t close: at least our fifths are pure. But what about the thirds?
Okay, let’s stack up four fifths, C-G-D-A-E, and drop two octaves down. We get… ((3/2)^4)/4=81/64, a Pythagorean third which is considerably wider than the pure major third. The ratio between them, (81:64)/(5:4)=81:80, which is known as the syntonic comma (or just plain “comma”). It’s just slightly (2% of a semitone) flat of the Pythagorean comma.
Even to an untrained listener, Pythagorean thirds sound unpleasantly sour. There is no way to fix this without either detuning the fifths or adding microtones to the scale.
So what if instead we start stacking up major thirds? Well, let’s see: C-E-G#-B#=C. That’s (5:4)x(5:4)x(5:4)=125:64, or 3:64 short of an octave!
In short: pure octaves, fifths, thirds: pick two.
The octave is the one interval nobody wants to mess with. (Well, there are “xenharmonic” scales, but nobody outside academia has even heard of them.)

Pythagorean intonation: pure fifths at all cost

You may effectively say: I must have the fifths (and hence fourths) pure, and if that means the thirds are sour, I’ll treat them as dissonant.
This is exactly what happened in Western music until the Renaissance.
Since all intervals in Pythagorean tuning are rational fractions that have no prime factors larger than 3, Pythagorean is also known as “3-limit rational tuning”.  [According to the same classification, “just intonation” as practiced by a cappella vocal ensembles is also known as “5-limit rational tuning”, since the largest prime involved is five (e.g., in the major third 5:4, the minor third 6:5).]
String instruments naturally lend themselves to Pythagorean tuning: anybody with musical hearing can tune pure fifths (or their inverses, fourths) by ear, just by tweaking until the “beats” stop. Violins are tuned in fifths; bass guitars, and the lowest four strings of a guitar in standard tuning, are tuned in fourths.

Quarter-comma meantone: pure major thirds at all cost

Alternatively, we can sacrifice the pure fifth in such a way as to restore the pure major third. The simplest way to do this is to narrow all fifths down by one-quarter of a Pythagorean comma, such that four narrowed fifths minus two octaves come out exactly a pure major third, 5:4. Such a fifth would be 5^(1/4)=1.495348781… (This quarter-comma meantone temperament was first proposed by a Spanish monk of Jewish origin named Pietro Aaron.)
Fifths in 1/4CM do “beat” (they are flat by about 1/20th of a semitone), but one can get used to them. The trouble: twelve fifths now stack up to three perfect major thirds (5:4)^3,  which we’ve seen above work out to 125:64, or 3:64 short of an octave. If you like: where Pythagorean tuning creates an expanding spiral of fifths, quarter-comma meantone gives rise to a contracting spiral of fifths.
Thus, you end up somewhere with one last fifth that is really bad, a so-called “wolf fifth” wide by two-fifths of a semitone. Since there are twelve possible places to start tuning, you can pick one such that the wolf fifth does not appear in the most frequently used keys. (Typically, it is put on G#—D#.) There are also several “wolf thirds” in the most remote keys.
Quarter-comma meantone was the prevalent tuning for much of the Renaissance. If one stays in the “safe” keys (with no more than two sharps or flats, say) and does not modulate to the more “remote” ones, it is quite tolerable. But don’t even think of playing in F# or Db on a keyboard tuned in quarter-comma meantone.

A first compromise: sixth-comma meantone

Musicians soon started experimenting with different meantone tunings.
In sixth-comma meantone [Ed.: a.k.a. Silbermann temperament], the wolf fifth can be reduced to about one-sixth of a semitone, at the expense of making the thirds just a little bit wide. This tuning still enjoys some popularity among “authentic Baroque practice” performers. Twelve fifths are now 1/5 semitone short of seven octaves.
Eleventh-comma or twelfth-comma meantone are nearly impossible to tune by ear, but are actually as close as makes no difference to equal temperament (see below).

Well-temperaments: closing the circle

As the Renaissance morphed into the Baroque era, composers started becoming ever more adventurous with modulations, and solutions that retained playability (to a greater or lesser extent) for all twelve major and all twelve minor keys were sought.
This led to the family of so-called “well-temperaments”, in which the Pythagorean comma is spread out over all twelve fifths, (at least) initially in an unequal fashion. Such temperaments are also called “circular”, in that twelve fifths now stack up to exactly seven octaves.
The term “well-tempered” (wohltemperiert) was originally coined in 1691 by the German organist and music theorist Andreas Werckmeister. He himself proposed several well-temperaments, one of which (Werckmeister III) is still in some use today among the HIP (historically informed performance) community.
In Werckmeister III, six of the fifths are tuned a quarter-comma flat (F-C, D-A-E, F#-C#-G#) while the remote G#–D#, to compensate, is made sharp by a quarter-comma and the remainder are tuned pure.
Another example is Vallotti temperament, in which the six diatonic fifths F-C-G-D-A-E-B are all tuned 1/6 of a comma flat and the rest are tuned pure. Young temperament, developed by the physicist and polymath Thomas Young, is based on the same pattern except cycled by one fifth to C-G-D-A-E-B-F#. There are many others: Kirnberger (advocated by a pupil of J. S. Bach), “tempérament ordinaire”,…
Common characteristics of all these well-temperaments include the following:
• all keys are at worst tolerable
• “nearer” keys approach just intonation
• “remote” keys approach Pythagorean intonation with its sharp thirds
Some of these are easier to realize by ear (i.e., without a digital tuner or other assistive device) than others.

Equal temperament: nothing perfect, everything equally imperfect

Mathematicians like Simon Stevin and Marin Mersenne in the West (and independently, Zhu Zaiyu in China) had proposed a more radical solution: to simply divide the octave into twelve equal parts, 2^(1/12)=1.059463094…, which is equivalent to narrowing all twelve fifths by one-twelfth of a Pythagorean comma to 2^(7/2)=1.498307077…
This is known as “equal temperament”, specifically “12-tone equal temperament” (12-TET). It is a special case of well-temperaments, and arguably the only “unbiased” or “universal” one. It is actually equivalent to tempering all fifths by 1/12th of a Pythagorean comma, and as close as makes no effective difference to tempering all fifths by 1/12 of the (slightly smaller) syntonic comma. So 1/11 comma meantone is functionally equivalent, and 1/12 Pythagorean comma meantone exactly so.
The luthier Vincenzo Galilei (father of Galileo) was perhaps the first to actually apply an approximate ET12 in instrument building, when he calculated fret spacings based on the ratio 18/17=1.058823529…, a fairly decent rational approximation to 2^(1/12)=1.059463094.
Ears used to the clean major thirds of quarter-comma meantone balked at first: also, 12-TET is not so easy to tune correctly with the naked ear. Despite the common misconception that everybody since Bach used equal temperament, other forms of well-temperament did not leave common practice until well into the 19th Century, but eventually 12-TET did become the Western standard for fixed-pitch instruments. Other well-temperaments have seen a modest revival in the HIP (“historically informed performance”) movement, particularly for harpsichord and organ tunings.
People with relative pitch may claim that in 12-TET, keys lose their “character”. To people with absolute pitch, they still have distinct sounds — though I have often asked myself a “cicken or egg” question here. For example, do I think of D minor as a “pensive, cerebral key” because it sounds like that (to someone with absolute pitch), or because I’ll forever associate it with Bach’s Art of Fugue BWV1080?

How ‘Well-Tempered’ was Bach’s Clavier?

Many people mistakenly assume J. S. Bach invented 12-TET. Of course he did not, nor was he even the first to write a composition exploiting it — that would have been Johann Caspar Fischer . Bach was however the first to write a major cyclical work, of transcendent musical value no less, that absolutely requires some form of well-temperament — and in doing so certainly hastened its adoption.
There is a scholarly consensus nowadays that Bach used not 12-TET but one or more well-temperaments, though it is not clear which. Bradley Lehman, in an article in Early Music, claimed that the ornament of the title page of the Well-Tempered Clavier actually encoded Bach’s own favored well-temperament [ and ], while a harpsichordist has recently argued [] that the temperament was in fact just the tempérament ordinaire described in Diderot’s Encyclopédie.

Equal temperament: blessing or curse?

Paraphrasing Winston Churchill about democracy: 12-TET is the worst possible solution for tuning fixed-pitch instruments…. except for all the others that have been tried.
On modern electronic instruments, when performing tonal music that also goes easy on modulation, one could in principle play in bespoke temperaments for each key. However, 12-TET is at this point so ingrained that people with fine musical hearing may actually consider just intonation or a favorably located well-temperament as ‘off’, even though it is objectively more in tune! Yet, unequal temperaments pop up in the strangest places — such as some guitarists slightly tuning down their B string in order to get just-intoned major thirds.
Allow me to end this post with one of my favorite Bach preludes played in two different temperaments on the same piano: the first time in Young temperament, the second time in modern 12-TET. Enjoy!