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Today, I will talk about tunings and temperaments. In particular, I’d like to expose why different tuning systems were used, and how they were constructed. Next time, we will merge everything we will have seen today along with the consonance calculations we saw in a previous post.

But first of all, we need a few definitions…

Units for intervals

Everyone who is familiar with acoustics, even remotely, know that the octave is the interval between a pitch of a given frequency $f_0$, and another pitch at the double frequency $2f_0$. For example, given an A4 on the piano, at 440Hz, we will find (in first approximation) the A5 at 880 Hz, the A6 at 1760 Hz, and so on. Human perception of pitch is thus roughly logarithmic, and the interval size between two pitches of different frequencies $f_1$ and $f_2$ can be expressed by the ratio $f_2/f_1$.

It may be more convenient however to express the different pitches in a linear scale, so that the interval size could be calculated by a simple difference of values, as we are accustomed to with, say, temperatures. This is the case of the widely known MIDI standard, which associates to a given frequency $f$ a number $p$ defined as

$p = 69+12\text{log}_2(f/440),$

so that $p=60$ for the C4 pitch of the piano.

More generally, given any logarithmic (base 2) transformation of type $p=a+b\text{log}_2(f)$ the interval size between two frequencies $f_1$ and $f_2$ expressed as their values $p_1$ and $p_2$ will be

$d=b\text{log}_2(f_2/f_1) = b\text{log}_2(f_2)-b\text{log}_2(f_1) = p_2-p_1.$

Just as we wanted, a simple difference.

The choice of the value $b$ is arbitrary. A common choice is $b=1200$, the resulting unit of interval size being called the cent. Thus, an octave contains 1200 cents. This is the unit we are going to use for the rest of this post. You may have heard of the savart as another unit, which corresponds to $b=1000$.

As a direct application of what we have just seen, consider the 12-tone equal temperament (also known as 12-TET). In this temperament, the octave is equally divided in 12 semi-tones. Thus, if the octave corresponds to 1200 cents, every semi-tone in 12-TET is equal to 100 cents. Taking C as our reference point, we thus obtain the following table for each pitch class:

 C C# D Eb E F F# G G# A Bb B 0 100 200 300 400 500 600 700 800 900 1000 1100

The threshold of perceptible difference, otherwise known as the just noticeable difference, can be quite different among individuals, notably between trained musicians and others. It is generally admitted that everyone can perceive a 25 cents difference.

Perfect intervals

You may have heard some people complain about the equal temperament, in particular that its thirds are not perfect. Well, the thirds are indeed not perfect, although this does not necessarily mean that we should get rid of 12-TET. But we have to define first what a perfect third is, or more generally, what we mean by a perfect (or pure) interval.

Again, I am sure that you know that a harmonic sound for a given fundamental frequency $f_0$ contains partials in its spectrum at frequencies being multiples of $f_0$. So, the spectrum of that sound can be very schematically represented by the following picture:

Now, if we have a second harmonic sound being produced at twice the fundamental frequency, its partials will match those of the first sound, as shown below:

Remember from our last post that this partial matching is likely to be perceived as consonant. It is indeed no surprise that the octave is generally perceived as a very consonant interval.

Do we have any other cases of such partial matching ? Consider a sound being produced at $(3/2)f_0$:

The ratio 3/2 is that of a perfect fifth. We then get the perfect fourth, for a harmonic sound being produced at $(4/3)f_0$:

And then, the perfect major third, at $(5/4)f_0$:

And finally, we will introduce the perfect minor third, at $(6/5)f_0$:

We will stop here for the purpose of this post, but we could have continued for smaller and smaller ratios. For example, the major tone is defined at the ratio 9/8, while the minor tone is defined at the ratio 10/9, and so on.

Notice that this pure intervals can all be expressed as ratios of small integers, and often of the form $(n+1)/n$. Without any knowledge about the nature of harmonic sounds, spectras, or partials, this phenomenon had nevertheless been remarked in the Antiquity.

Can you now express these intervals in cents ? (take some time to get accustomed to the conversion between ratios and cents).

Let’s see if you found the same values. We get:

• The octave at 1200 cents (of course)
• The perfect fifth at $1200\text{log}_2(3/2)=701,955$ cents. From now on, we will round the values to the nearest integer; for example, our perfect fifth will be 702 cents.
• The perfect fourth at 498 cents
• The perfect major third at 386 cents
• The perfect minor third at 316 cents

Let’s compare these intervals with those of the 12-TET system. The fifth in 12-TET is at 700 cents, so it’s only off by 2 cents, a difference which is hardly, if not at all, perceptible. The fourth is at 500 cents: again, not perceptible. The major third, however, is at 400 cents: this is a 14 cents difference with the perfect major third. At this point, the difference may be noticed, and the same goes for the minor third in 12-TET, off by 16 cents as compared to the perfect minor third. Given that the great majority of Western music is built on triadic harmony, i.e. the systematic use of major/minor triads which contains both major and minor thirds, this explains the complaints against 12-TET.

So, why not using pure intervals to build a scale, so that all triads would sound good ?

The Pythagorean tuning

The pure major fifth seems like a good starting point for building our new scale. So, the process will be as follows. We start on C, and we add a perfect major fifth to it. We get G at 702 cents. We then add a perfect major fifth to this new pitch, to get D at 702+702=1404 cents, which, by octave equivalence, is 204 cents. We repeat this operation until we have determined all 12 notes. In this manner, we will obtain the following table of interval size for each note, relative to C. This is what we call the Pythagorean tuning.

 C C# D Eb E F F# G G# A Bb B 0 114 204 294 408 498 612 702 816 906 996 1110

Note: there may be a difference between the figures in this Table, and what you may find on the Internet or in some books. In most cases, this depends on the first note you use to build your scale.

We can see the difference between equal temperament and the Pythagorean tuning in a more graphical way. This is the twelve notes of equal temperament, equally distributed on the circle of semi-tones.

And this is the Pythagorean tuning, with the new intervals which we have determined.

Let’s see how it sounds: we are going to hear a series of fifths played on each note, starting on A. First, equal temperament:

And next, in Pythagorean tuning. Now, I haven’t quite figured how exactly my piano is tuned, but the tuning it uses definitely starts on another note. Thus there will be some discrepancy between the result you’re about to hear and our following discussion, but this is not that important. I can assure you that you are going to hear very clearly what is important with the Pythagorean tuning (or else, that means you could listen to Xenakis all day).

Wait a minute. What was that third chord we heard ? Wasn’t the Pythagorean tuning supposed to be entirely made of perfect fifths ? How come it is sounding so bad ?

The answer to that second question is: no.

Let’s see what we have. Between C and G, 702 cents:

Let’s try another: between D and A, 702 cents:

So far, so good. We can do that for any note. For example, G#: between G# and Eb, 702 cents:

Hold on. This isn’t right. What we have instead is 678 cents, not 702:

No wonder it’s sounding bad: it’s off by 24 cents from a perfect fifth ! This is what we call a wolf interval. But where did we go wrong ?

It’s quite simple, really. We’ve constructed the Pythagorean tuning by stacking pure fifths, and I said earlier “We repeat this operation until we have determined all 12 notes”. That would probably induce you into thinking that by stacking twelve pure fifths, you’d come back where you started (which is true with equal-tempered fifths). This is not possible, by an easy mathematical argument: $(3/2)^{12} = 129.746$, whereas seven octaves would be equal to 128. The difference is the ratio $(3/2)^{12}/128 = 1.01364$, which is equal to 23.46 cents. With the rounding error, that’s precisely the interval between G# and Eb in our tuning. This difference is called the Pythagorean comma.

In addition, the Pythagorean tuning is particularly bad for thirds. You can notice in the table above that C-E = 408 cents, off by 22 cents (!) from the pure major third, while C-Eb = 294 cents, off by 22 cents too from the pure minor third.This difference of 22 cents (exactly, 21.51 cents) is called a syntonic comma.

So, if you are a professional musician living in the Middle Ages during the era of the Notre-Dame school, you probably doesn’t bother much with this, and Pythagorean tuning is perfect for you. That’s because the treatises of that time are telling you that the only consonant intervals are the octave, the fifths, and the fourths. The music you would sing would consist of polyphonies made with those intervals. You merely have to avoid the wolf interval, otherwise you’re ok.

But if you are living in the Renaissance, new rules for harmony are starting to emerge. The order of the different intervals is being re-examined and triadic harmony (major/minor triads) starts to impose itself in musical practice. Here are our usual twelve major triads in 12-TET:

and here are those triads in Pythagorean tuning:

Definitely not good. So, what do we do ?

The quarter-comma meantone temperament

We want a new system which allows us to play triads with intervals as pure as possible. The major third C-E in Pythagorean tuning is off by 32 cents. Notice that it is obtained by stacking four perfect fifths of 702 cents. The idea here is to temperate those fifths, so that the major third obtained by stacking would be a pure one. In other words, we will diminish slightly the size of the fifths so that we get closer to 386 cents. Since the difference we want to abolish is a syntonic comma (21.51 cents), and that the major third is obtained by stacking four fifths, it suffices to remove one quarter of a syntonic comma (5.37 cents) from the pure fifth, and to stack them to get our new system. We thus get the quarter-comma meantone temperament, whose table of interval size is given below.

 C C# D Eb E F F# G G# A Bb B 0 76 193 310 386 503 579 697 773 890 1007 1083

And in a graphical form:

It really looks promising ! The fifth C-G is 697 cents, so that’s hardly different than a pure fifth. And the major third is now a pure one !

Let’s hear what a series of fifths looks like:

I know what you are going to say: “Come on ! That last fifth was really bad ! What was the point of all this ?”

Well, yes, that fifth was bad. That’s because we are using an interval size for the fifth which still doesn’t close the loop to an octave after stacking it twelve times. So, while most fifths are ok in this temperament, for example the C-G fifth:

we still have one wolf interval, here G#-Eb, at 737 cents (!), off by 35 cents (!!) from the pure fifth:

The good news is that a higher number of triads are really sounding great in this system:

Still, some of them are sounding weird. Remember though that we are now playing music from the Renaissance and Baroque era, so tonality is well-established. That means that, for a given key, we are using only certain degrees, hence certain triads and not all of them. Consequently, certain keys are well adapted to meantone temperament, while others are not. Look at this video to see the difference. The organist starts with quarter-comma meantone temperament (probably using a different starting note than I did), and play in F minor, which is particularly bad in his system. He then switches to equal temperament, in which all keys are equal.

In some treatises of that era, you will find comments about the different emotions (as subjective as that can be) of the different keys, owing to the unequal temperament being used. You also have to remember that tuning an church organ was at that time a complicated operation (and not what we can do now with electronics), which would only be done once or twice a year. Thus, you couldn’t retune everything everytime you wanted to change keys. This also limited the modulations you could write in your music: you couldn’t depart very far from your basic tonality, what with the risk of having some very bad triads.

I love those tuning videos. Here is another one, with various tunings:

Check out this entire recital on a meantone organ, with a taste of various keys (don’t miss the chaconne of Pachelbel in F minor at 37:50)

Can we yet improve on the quarter-comma meantone temperament ?

The Werckmeister III temperament

Andreas Werckmeister was an organist and music theorist of the Baroque era, who proposed many different tuning systems. The idea of the Werckmeister III temperament is to temper only four fifths by a quarter of a syntonic comma, namely C-G, G-D, D-A, and B-F#, and to leave the rest of the fifths as pure ones. We thus get the following table of interval sizes:

 C C# D Eb E F F# G G# A Bb B 0 90 192 294 390 498 588 696 792 888 996 1092

And in a graphical form:

Neither of the thirds are pure, but the major thirds are closer to the pure ones than equal temperament. In addition, as you can hear below, it seems we have access to a larger range of keys, though some chords still sound a little weird.

Here are all the fifths:

And in the following series of videos, you can compare the same piece being played in equal temperament, quarter-comma meantone temperament, and Werckmeister III.

With the advent of Bach, and his Well-Tempered Clavier, the plethora of tuning systems which existed during the Baroque era was progressively replaced with equal-temperament, which allowed for all keys to be used, and distant modulations to be composed*. Late 19-th century music would certainly be different if not for equal temperament. In the 20-th century, many composers revisited different tuning systems. There is no place here to talk in detail about what would probably fill three or four PhD thesis, so let me conclude this post with the Well-Tuned Piano by La Monte Young:

And if you’re feeling curious about his tuning system, check out the paper of Kyle Gann, who cracked the tuning system that La Monte Young wanted to keep secret.

• K. Gann, “La Monte Young’s The Well-Tuned Piano”, Perspectives of New Music, 31 (1), 1993, pp. 134-162