Lisa and I will be leaving on the 9th, but luckily we'll be back in the spring!
As my friends are learning about my current obsession with tuning systems, they're starting to ask interesting questions I don't know the answers to.
For example, Michael Fourman asked me: if harmonies coming from simple fractions are so natural, do any bird or whale songs feature such harmonies?
It turns out an Australian bird called the pied butcherbird has long been a favorite of many composers! Jean-Michel Maujean figured out the frequency ratios that appear in the songs of this bird. He found the 4 most common ratios are close to
He notes that
0.607 is close to going down a major sixth (3/5),
0.745 is close to going down a perfect fourth (3/4),
0.815 is kinda close to going down a major third (4/5),
1.34 is close to going up a perfect fourth (4/3).
His work looks good — but he shouldn't have bothered comparing the ratios to 12-tone or 18-tone equal temperament. Equal temperament is a system developed for keyboard instruments in the late 1700s. It would be amazing if the birds used this!
Maujean also has a nice review of the literature on harmonies in bird songs, so I should dig into it:
But I get the feeling that most birds don't sing with frequency ratios that are simple fractions. What's up with these other birds?
Without any electronic equipment, piano tuners can tell if two strings are vibrating at almost but not quite the same frequency. They do it by listening for 'beats': pulsations in loudness.
How does this work? If you add two sine waves of slightly different frequencies, say \(\sin(\omega t)\) and \(\sin(\omega' t)\), they will add up and be loud for a while, but then drift out of synch and cancel out for a while. Then they'll drift back into synch and get loud again, etc.
There's even a formula for this: $$ \sin(\omega t) + \sin(\omega' t) = 2 \sin\left(\frac{(\omega + \omega')t}{2}\right) \cos\left(\frac{(\omega - \omega')t}{2}\right) $$ We get a sine wave whose frequency is the average \((\omega + \omega')/2\) slowly pulsing because it's multiplied by a cosine wave with the low frequency \((\omega - \omega')/2\).
I got the animated gif from here:
So maybe you didn't really need to learn those identities like $$ \sin(\alpha) + \sin(\beta) = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) $$ when you were a kid in trig class — but they can sure be illuminating at the right moment.
So far I've focused on the quarter-comma meantone system in its mathematically beautiful, symmetrical form above. Today I'll say more about the scale as actually played: how we trim it down from a 13-note scale to a more practical 12-note scale, and what are the intervals between notes in the resulting scale.
The scale above has:
You can see this tiny gap between F♯ and G♭. Both these notes are versions of the 'tritone' in the scale of C major. F♯ is called the 'augmented fourth' and G♭ is called the 'diminished fifth'. Unless we build a keyboard with split keys — as some people actually have, but they never caught on — one of these notes has got to go!
There are two options. The most popular is to remove the diminished fifth:
The other is to remove the augmented fourth:
Either approach gives a big bad 'wolf fifth', which must be avoided. But what does the resulting scale actually look like? What are the intervals between neighboring notes?
To understand this let's go back to the start, our original 'circle of fifths' with 13 notes:
When we rearrange the notes, listing them in order, we get a 'star of fifths':
Now let's add arrows showing the intervals between neighboring notes:
Except for the lesser diesis, the neighboring notes are all separated by two kinds of interval:
We worked out these wacky numbers in my December 31st diary entry. It may seem weird to have two sizes of semitone, but they sound fine. The problem, to repeat myself, is the lesser diesis between the augmented 4th and the diminished 5th. But when we remove either one of these notes, something nice happens!
This is the cool part. On December 31st we saw this relation:
So, when we remove either the augmented 4th or diminished 5th, the lesser diesis combines with one of the chromatic semitones adjacent to it to give an extra diatonic semitone!
If we remove the diminished 5th, we get this scale:
It's not completely symmetrical, but it's still quite pretty — and now the lesser diesis has been banished. Most of the fifths sound good, but there's wolf fifth between the augmented fourth (F♯) and the minor second (C♯), so you should avoid this. The semitones alternate between chromatic and diatonic... except for two diatonic semitones in a row between F♯ and A♭, and between B and C♯.
Best of all, this scale has lots of just major thirds — though one fewer than in the mathematically beautiful 13-note version of the scale. Let's figure out where they are. Last time we noticed this relation:
This implies that
But 5/4 is the frequency ratio of a just major third! So we get a just major third whenever we go up two chromatic semitones and two diatonic semitones. So the just major thirds are the blue arrows here:
It's a bit random-looking, thanks to how we broke the symmetry. If there were a just major third from F♯ to B♭ the pattern of blue arrows would be symmetrical. But there's not: there's only a just major third from G♭ to B♭, and we've eliminated G♭ from this scale.
But still, this scale has lots of just major thirds! And that was the main goal.
I've said a lot about quarter-comma meantone and its great properties. It's almost time to start exploring the vast realm of 'well-tempered' tuning systems that flourished starting around 1690.
But there's one more thing I want to say about quarter-comma meantone. If you use this system, there are some advantages to having your scale start at D rather than C! For example, Wikipedia presents quarter-comma meantone starting from D here:
It's not an arbitrary decision! This confused me at first, but Matt McIrvin straightened me out and I think I get it now. It's all about white keys versus black keys on the piano — or harpsichord, or clavichord, or organ.
The same ideas also apply to Pythagorean tuning or just intonation, but I'll illustrate them with quarter-comma meantone.
If we straighten out the circle of fifths shown above, putting C at the middle, we get this picture:
There are 13 notes here, since we need an odd number of notes if we want one to be in the middle. Thus, I'm writing F♯ and G♭ as two separate notes, even though some tuning systems consider them as the same. Of course in the picture at the top of this blog article they were different.
Notice how asymmetrical this picture is. To emphasize the asymmetry I've marked the flat notes in red and the sharp ones in blue. Though C is in the middle, it's much closer to the flat notes than the sharp ones!
You may complain that any flat note can be rewritten as a sharp one. That's true. So here's a more precise way to make my point. Say a note is an accidental if it has either a flat sign or a sharp sign. Then: though C is in the middle, there are more accidentals to its left than to its right!
But here's the weird part. If we straighten out the circle of fifths putting D at the middle, this asymmetry evaporates:
Puzzle. Why is this true? Why, even though the white notes on a piano form a major scale starting at C, does
equal
only when you start at D?
Of course a 'why' question can have many different answers, including that's just how things are! But there are a few enlightening answers to this question. I'll just mention that I'm not concerned here with our conventions concerning letter names for notes.
Yes, it's odd that we say the white notes on a piano form a major scale starting with C rather than something more logical like A. That convention ultimately goes back to a decision made by Boethius shortly after 500 AD, long before pianos or harpsichords existed:
But if we changed our letter names for notes, my puzzle would persist, with different names for things.
Now to the point. All along in my discussion of quarter-comma meantone I've started my scales at C. This puts all the problems connected to the tritone at the bottom of this circle, between F♯ and G♭:
But there are more accidentals on the left side of this circle than on the right! There are five on the left and just one on the right.
If instead we start our scale at D, this asymmetry disappears:
Now we've got 3 flat notes on the left of the circle and 3 sharps on the right.
This has implications for practical music. Basically, we'd like to hide the lesser diesis 128/125, or the wolf fifth that appears when we eliminate the lesser diesis, as deeply as possible among the black notes. That will make the scales that are mainly white notes sound better.
Even if you haven't followed all the details, I hope you've seen that there's a lot of richness to the tuning systems I've discussed so far. They kept musicians happy until around 1690. But then a large number of 'well-tempered' systems burst onto the scene, which exploited the basic principles we've seen so far in new ways. So that's what I want to talk about next!
The tuning system called 'quarter-comma meantone’ dominated western keyboard music from about 1550 to roughly 1690. The reason: it has very nice thirds and fifths in many different keys!
But as I keep saying, every tuning system has problems: like lumps in the carpet, the best you can do is move the problems around. Quarter-comma meantone achieves its greatness by completely flattening out the carpet except for one big lump: a single highly dissonant 'wolf fifth'. Alas, this utterly spoils keys where that fifth is important — or other chords using the note that creates that wolf fifth, which is F♯ in the chart above.
As Baroque musicians became increasingly interested in switching between keys, there was pressure to find tuning systems where the lumps in the carpet were more evenly spread out. But interestingly, they did not embrace equal temperament, where the lumps are spread out as evenly as possible.
It's not that equal temperament was unknown! Apparently, musicians wanted some keys to have truly beautiful fifths and thirds, and weren't willing to sacrifice that beauty and purity to make all keys sound equally good — or bad. Thus, they invented compromise systems, called well temperaments, in which each key has its own personality, but all sound reasonably good.
You can see these personalities discussed in Christian Schubart's Ideen zu einer Aesthetik der Tonkunst, written in 1806:
C Major. Completely pure. Its character is: innocence, simplicity, naïvety, children's talk.C Minor. Declaration of love and at the same time the lament of unhappy love. All languishing, longing, sighing of the love-sick soul lies in this key.
D♭ Major. A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying. Consequently, only unusual characters and feelings can be brought out in this key.
C# Minor. Penitential lamentation, intimate conversation with God, the friend and help-meet of life; sighs of disappointed friendship and love lie in its radius.
D Major. The key of triumph, of Hallejuahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key.
It's with great effort that I resist listing all 24 keys! You'll just have to visit the link to see which key "tugs at passion as a dog biting a dress", and which has "pious womanliness and tenderness of character".
When equal temperament took over in the early 1800s, all this diversity was flattened, although the reputations of the different keys persisted for quite some time. Some consider this flattening a tragedy; others say it opened the doors to Beethoven and jazz. Perhaps both are true.
But what were these well tempered systems, exactly? What were their distinct advantages? Here things get much more complicated and murky. For example:
The biggest advertisement for well-tempered tuning systems was Bach's The Well-Tempered Clavier. In 1722 and then again in 1742, he wrote a piece in each of the 12 major and 12 minor keys, to illustrate the flexibility of well temperament — and presumably to showcase how different keys had different personalities. But which well tempered system was he actually using?
Nobody knows! We don't have Bach's words on this topic, and despite a vast amount of scholarship nobody has been able to pin it down. Serious musicologists have even spent serious time studying a doodle in Bach's manuscript of The Well-Tempered Clavier, hoping it contains an encoded description of his tuning system! There's no proof that it does.
For a fun but insightful introduction to the controversy, watch this:
For more, try this:
This seems to be the best in-depth survey of the subject:
I haven't read most of it, but it looks to be a treasure chest of information on well temperaments, their history and their mathematics. And frankly, I find that much more interesting than the futile quest to figure out what Bach was thinking.
There are a lot of interesting well tempered systems. People wrote a lot about them when they were invented, and much more since. So we are not reduced to decoding Bach's squiggles to understand well temperaments. The hard part, at least for this mathematician, is figuring out their governing principles.
In my efforts, I've been helped immensely by this website:
He describes about 30 different tuning systems using circular diagrams — a method that I've decided to copy in my blog articles here. He is less interested in the math than I am. But he is more more efficient at explaining tuning systems than other sources, and covers other topics: for example, he describes how to tune a harpsichord in all these systems!
Instead of covering well-tempered systems chronologically, I'll start with the ones I find easiest to explain. I'll try to cover some of the most important ones, but certainly not all. Many are named after people like Werckmeister, Kirnberger and Vallotti, while some have descriptive names like sixth-comma meantone.
The danger is getting lost in the undergrowth of these tuning systems and not seeing the forest for the trees. So before diving in, I'll start by surveying some of the mathematical principles that seem to be at work. You could see these between the lines of what I've written so far, but I want to be a bit more explicit.
For example, what's really going on with these weird numbers:
and this relationship:
Why have these funny things mattered so much in the history of tuning systems?
Last time I ended with a question: why are certain numbers close to 1 so important in tuning systems? It helps to understand a bit about this before we plunge into the study of well temperaments. It turns out that in some sense western harmony evolved one prime at a time, so let's look at the subject that way.
If all the frequency ratios in our tuning system were powers of 2:
life would be very simple. Multiplying a frequency by 2 raises its pitch by an octave, so the only chords we could play are those built out of octaves. Not much music could be made! But there'd be no difficult decisions, either.
In Pythagorean tuning, also called 3-limit tuning, we generate all our frequency ratios by multiplying powers of 2 and powers of 3:
This is more exciting. While the frequency ratio of 2 is an octave, that of 3/2 is called a just perfect fifth. So now we can use octaves and fifths to build other intervals (that is, frequency ratios).
But in fact, any positive real number can be approximated arbitrarily well by numbers of the form 2^{i} · 3^{j}, so we have an embarrassment of riches: more intervals than we really want! To bring the system under control, we take some number of the form 2^{i} · 3^{j} that's really close to 1 and act like it is 1.
I examined the options in an earlier post and got a list of 'winners' according to some precise criterion. A couple of early winners are
and
These would be important in scales with 5 or 7 notes, but western music holds out for a much better one, called the Pythagorean comma:
This is important for a 12-tone scale, because it means that if we go up 12 fifths, multiplying the frequency by 3/2 each time, it's almost the same as going up 7 octaves.
But not quite! There are many ways of dealing with this problem. In Pythagorean tuning we absorb the problem by dividing one of our fifths by the Pythagorean comma, turning it into an unpleasant 'wolf fifth':
For example:
But we can spread the inverse of the Pythagorean comma around the circle of fifths any way we like, and different ways give different tuning systems.
For example, in equal temperament we spread it completely evenly around the circle of fifths, using the equal tempered fifth everywhere:
This is not an example of 3-limit tuning because it uses irrational numbers! But it's an obvious way to solve the problem of the Pythagorean comma which emerges in 3-limit tuning. More interesting solutions tend to involve the next prime number.
In 5-limit tuning we generate all our frequency ratios by multiplying powers of 2, 3 and 5:
Equivalently, we build them using the octave (2), the just perfect fifth (3/2) and the just major third: 5/4.
There are some new simple fractions close to 1 that you can build with 2, 3 and also 5. The most important is the syntonic comma:
This shows up when you try to reconcile the perfect fifth and the major third. If you go up four just perfect fifths, you boost the frequency by a factor of (3/2)^{4} = 81/16 = 5.0625, which is a bit more than a major third and two octaves, namely 5/4 × 2^{2} = 5. The ratio is the syntonic comma.
As we'll see in future episodes, this realization is fundamental to many well tempered tuning systems. We've already seen the grand-daddy of these systems: quarter-comma meantone. It's not well tempered itself, but fixing its main flaw leads to well tempered systems. In quarter-comma meantone, we divide most of our fifths by the fourth root of the syntonic comma, which gives lots of just major thirds, shown in blue below:
So, this scale has many 'quarter-comma fifths' with a frequency ratio of (3/2)σ^{-1/4}. Going around the whole circle and multiplying 12 of these quarter-comma fifths would give 125, which is not quite the 128 we need to go up 7 octaves. So we need to take one of these quarter-comma fifths and multiply it by 128/125. The resulting 'wolf fifth' sounds terrible — and this is what well temperaments seek to cure.
The number 128/125 is an important fraction close to 1 built from just the primes 2 and 5. It's called the lesser diesis:
It's not only a power of 2 divided by a power of 5, but also a power of 2 divided by a power of 10. You've bumped into it if you've ever wondered why people often use 'kilobyte' to mean 1024 bytes, not 1000.
From the Pythagorean comma, syntonic comma and lesser diesis we can generate other fractions close to 1 built from the primes 2, 3 and 5. For example, I've already discussed the product of the syntonic comma and lesser diesis, and also their ratio.
But when we study well temperaments, more important will be the Pythagorean comma divided by the syntonic comma. Called the schisma, this fraction is very close to 1:
I'll talk about it more next time.
It's also important to note that the lesser diesis is not independent from the Pythagorean comma and syntonic comma. We've already seen today that going up a fifth twelve times is the same as going up 7 octaves divided by the Pythagorean comma. Now we're seeing that going up (3/2)σ^{-1/4} twelve times is the same as going up 7 octaves times the lesser diesis. So, we have
or
We've already this in a slightly different way before.
Due to this relation there must be other fractions close to 1, built only from powers of the primes 2, 3, and 5, that are independent from p, σ and δ. In fact, we've already seen four such fractions appearing as the sizes of semitones in just intonation:
The ratios of these semitones include the syntonic comma, the lesser diesis, and also their product the greater diesis and their ratio the diaschisma!
But these semitones are not extremely close to 1. The smallest, the lesser chromatic semitone, is 25/24 ≈ 1.041666. So there must be interesting examples of fractions built from 2, 3 and 5, independent of the syntonic and Pythagorean commas, and much closer to 1. On Mastodon I asked for examples built solely from the primes 3 and 5, and a bunch of people helped me out. Here are some of the first few winners:
The main thing to notice here is that we need fractions with impractically large numerators and denominators to get closer than the large diatonic semitone, 27/25 = 1.08. These fractions won't play a role in well temperaments.
I won't say much about primes after 5 now. But they've been studied in music theory at least since Ptolemy, and the compositions of Ben Johnston really run wild with them. For a tiny bit about the virtues of the prime 7, read my post on the harmonic seventh chord.
The facts I've crudely laid out above must be part of an elegant general theory of approximating the number 1 by fractions built from powers of a specified set of primes, and how to build scales from these fractions. Done systematically, this could be of interest not just to music theorists but even pure mathematicians. But I will not explore this now, since my goal was merely to recall some facts needed to understand the explosion of well temperaments starting around 1690!
Next time I'll digress slightly into Kirnberger's discovery of a tuning system with frequency ratios built only from the primes 2, 3, and 5 that comes extremely close to equal temperament. This is not a practical system, but it relies on an utterly astounding coincidence, and more importantly it highlights the role of the schisma, which will keep showing up in other systems.
It was discovered by Bach's student Johann Kirnberger, and it has a frequency ratio absurdly close to 1:
It arose naturally in Kirnberger's attempt to find a tuning system close to equal temperament with only rational frequency ratios. But it relies on a mathematical miracle: a coincidence so eye-popping that a famous expert in black hole physics wrote a paper trying to explain it!
Throughout my discussion of tuning systems, we've repeatedly encountered two glitches in the fabric of music called 'commas':
The first shows up when you try to build a scale from fifths, while the second shows up when you try to have lots of nice fifths and major thirds.
They are quite close together, so their ratio is even closer to one! It has a cool name: it's called the schisma. I'll abbreviate it with the Greek letter chi:
The schisma is a kind of meta-glitch: a glitch between glitches! It may seem like a mere curiosity, since two pitches whose frequency ratio is a schisma sound the same to everyone. But precisely for this reason, it plays a role in some well tempered tuning systems.
You see, sometimes when you're building a tuning system you need a Pythagorean comma to make your circle of fifths close up nicely, but to get a really nice major third you use the syntonic comma instead. When you do this, you're off by a schisma! And like a lump in the carpet, this schisma has to go somewhere. It's so small that it scarcely matters where you put it. If you're not extremely careful in tuning, your notes are probably off by more than a schisma anyway. But mathematically, it's there.
In future episodes, I'll show you examples of how this happens in some well-known tuning systems. But today I want to show you a mind-bending, completely crazy way that Kirnberger used the schisma.
Let's get started!
As we saw in our study of Pythagorean tuning, going up 12 just perfect fifths takes you up a bit more than 7 octaves. Their ratio is the Pythagorean comma:
As a result, if we divide the just perfect fifth (that is, 3/2) by the 12th root of the Pythagorean comma, we get the equal tempered fifth (that is, 2^{7/12}), whose 12th power is exactly 7 octaves. This correction, the 12th root of the Pythagorean comma,
is called a grad. I'll call it γ for short:
So, what I'm saying is that if we divide 3/2 by the grad we get 2^{7/12}, which is the equal tempered fifth:
Now for the eye-popping coincidence. The grad
is remarkably close to the schisma:
Look at that! For no obvious reason, they match to almost seven decimal places!
But unlike the grad, the schisma is rational. This let Kirnberger create a tuning system very close to equal temperament but with rational frequency ratios. His idea was to use a circle of fifths where instead of using the equal tempered fifth
which is irrational, we use 3/2 divided by a schisma
which is rational. They are remarkably close!
The quantity 3/2χ is called the schismatic fifth:
We can try to build a circle of fifths using the schismatic fifth instead of the equal-tempered fifth. But there's a slight problem. Actually, 'slight' is an overstatement: it's a nearly infinitesimal problem.
If we go up 12 schismatic fifths we don't go up exactly 7 octaves. We go up a microscopic amount more, since
and the number (γ/χ)^{12} is microscopically more than one. Since this number was discussed by Kirnberger, it's called the atom of Kirnberger. I'll call it α for short:
Let's work out what it equals! Remember that the grad is the 12th root of the Pythagorean comma, so
and turning the crank on the old calculator:
Using these ideas, Kirnberger created a tuning system called rational equal temperament. It's very close to equal temperament, but with only rational numbers as frequency ratios. To do this, he used 11 schismatic fifths and one schismatic fifth divided by the atom of Kirnberger. Just for fun, I'll call the last an atomic fifth:
I don't know where Kirnberger put the atomic fifth, but I'll follow the common tradition of putting problems right after the tritone, which is F♯ in the key of C:
Compare this to equal temperament:
Nobody can hear the difference, so Kirnberger's rational equal temperament is not used in music. But it sheds light on the interaction between the Pythagorean comma and syntonic comma, and that is important for the well tempered scales we'll be seeing next.
It also raises a math puzzle: why is the grad so close to the schisma? The physicist Don Page, famous for his work on black hole thermodynamics, has written a paper exploring this:
Since the schisma is already a meta-glitch, the Kirnberger kernel is a meta-meta-glitch! He shows that the closeness of this number to 1 is equivalent to a number of other coincidences, notably
He then wrestles this coincidence down to a fact involving only integers, which he tries to explain using properties of the hyperbolic tangent function! He is much better at these things than me, so if you enjoyed my article you should definitely take a look at his.
Next time I'll get back to business and talk about well tempered tuning systems — starting with three more practical systems developed by Kirnberger.
Even more amazing, China's population decreased by 2 million last year — double the decrease in the year before. This is being painted as a 'crisis', and yes it comes with plenty of problems, but bigger is not always better. Anyway, it's happening.
If our civilization collapses, extraterrestrial archeologists can look at this and be impressed. Three satellites following the Earth in an equilateral triangle, each 2.5 million kilometers from the other two. Each contains two gold cubes in free-fall. The satellites accelerate just enough so they don't get blown off course by the solar wind. The gold cubes inside feel nothing but gravity.
Lasers bounce between each cube and its partner in another satellite, measuring the distance between them to an accuracy of 20 picometers: less than the diameter of a helium atom! This lets the satellites detect gravitational waves — ripples in the curvature of spacetime — with very long wavelengths, and correspondingly low frequencies.
It should see so many binary white dwarfs, neutron stars and black holes in the Milky Way that these will be nothing but foreground noise. More excitingly, it should see mergers of supermassive black holes at the centers of galaxies as far as... the dawn of time, or whenever such black holes were first formed. (The farther you look, the older things you see.)
It may even be able to see the 'gravitational background radiation': the thrumming vibrations in the fabric of spacetime left over from the Big Bang. These gravitational waves were created before the hot gas in the Universe cooled down enough to become transparent to light. So they're older than the microwave background radiation, which is the oldest thing we see now.
It's called LISA — the Laser Interferometric Satellite Antenna. And we're in luck: ESA has just decided to launch it in 2035.