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hey, welcome to 12tone! let me ask you a question: do these two notes sound good together?
how about these two? would you describe one pair as more pleasant than the other? if you're
like most people in Western society, you probably picked the second one as sounding better,
but why?
what is it about those notes that makes them feel so much nicer than the first ones?
I mean, you could just say it's because you like them better, but I'm a music theorist
so that's nowhere near enough unnecessary detail for me.
fortunately, I'm not the only one who feels that way, and this question even made its
way to one of the most famous mathematicians of all time, Leonhard Euler.
back in the Baroque period, our model was fairly straightforward: you had consonances,
which sounded good together, like the octave or the perfect 5th, and you had dissonances,
which sounded bad together, like the tritone or the minor 2nd. every interval was one of
these two things, and which group it fell into told you how you were allowed to use
it.
this is great if you like composing with a million rules, but in reality there's a lot
more than two levels of dissonance, and in his book Tentamen Novae Theoriae Musicae,
or An Attempt at a New Theory of Music, Euler wanted to find a better system.
first, though, a little background.
when you hear a note, what you're really hearing is a sound wave with a specific frequency,
and when you hear two notes together, your ear effectively calculates the ratio between
the two in order to determine the interval. as a general rule, the simpler that ratio
is, the more consonant the interval will sound.
but that's not really an answer, 'cause it's not clear what it means to be simple.
I mean, which is more consonant, 5:6 or 4:7? it's hard to say, which means we're gonna
need a better definition.
when you're trying to solve a math problem, which is how Euler was approaching this, it's
often best to start by making observations.
once you know a few things you can start to recognize patterns, which help you figure
out more things and hopefully point you towards your solution.
in Euler's case, his observations started like this: the most consonant interval between
two notes is the 1:1 ratio, or unison, where they're both playing the exact same thing.
Euler assigns this ratio the title Primus Suavitatus Gradum, or first order of softness.
the next most consonant interval is the 1:2 ratio, or octave, to which Euler assigns the
second order of softness.
side note: orders of softness are like golf.
lower is better.
anyway, after that come the 1:3 ratio, which is an octave plus a perfect fifth, and the
1:4 ratio, which is two octaves.
Euler decided that these were equally consonant, and assigned them both to the third order
of softness.
now we have just enough data points to pick out a pattern: if unison is the first order,
one octave is the second, and two octaves is the third, then we can reasonably assume
that three octaves will be the fourth order, four octaves the fifth, and so on.
every time we double the frequency the order goes up by 1, and this makes a fair amount
of sense.
of course octaves will always sound pretty good together, but as they get further apart,
they start to lose that sort of octave-y cohesion that makes them so special.
this leads us to our first rule: if the ratio is a power of 2, then the order is that ratio's
exponent, or how many 2s it has, plus 1, so if we took say, 1:64, we could conclude that
since 64 is 2^6, the interval would be of order 6+1, or 7. we don't have to play it,
or compare it to other sounds: we know just by looking at the numbers.
that tells us how consonant octaves are, but we already knew that.
what about other intervals?
well, let's keep making observations.
the next ratio we haven't looked at is 1:5, which is two octaves plus a major 3rd, and
Euler feels like this is actually less consonant than the 1:8 ratio even though it's smaller.
since we just saw that 1:8 is of order 4, that means 1:5 must be order 5, and now we're
starting to see a second pattern form.
when we put in 1:2, we got order 2, 1:3 gave us order 3, and 1:5 gave us order 5. this
is important because these numbers are all prime, which means they're only divisible
by themselves and 1. for our purposes, though, it also means that we can't build them out
of smaller ratios.
like, take 1:4. we can make that by stacking a 1:2 ratio on top of another 1:2 ratio, but
with something like 1:5 we can't do that, which gives us our second rule: if the ratio
is prime, the order is just that same prime number again, so 1:7 is order 7, 1:11 is order
11, and so on.
but so far, all we've done is identify corner cases.
what about all the ratios that aren't primes or powers of 2? can we generalize these ideas
to figure out how good any two notes will sound together?
well, we've got a couple rules now, so the next thing we can do is try to find a formula
that fits both.
that's exactly what Euler did, and he came up with… this, which we now call the Euler
Gradus Function.
it looks a little scary, but the concept is actually pretty straightforward.
what we're doing is splitting the ratio up into its prime factors, or the smaller ratios
that we built our big one out of, then for each of those, we're subtracting 1 and multiplying
it by its exponent.
then we add those all together, add an extra 1 on top of it for good luck, and boom: we've
got an order of softness.
still confused?
ok, let's try an example.
what's the order of softness for the 1:20 ratio?
20 is equal to 2 times 2 times 5, or 2^2 times 5^1. we start by taking the 2, subtracting
1 to get 1, then multiplying it by its exponent, 2, which gives us a total of 2. then we take
the 5, subtract 1 off that to get 4, multiply that by its exponent, 1, and get 4. we add
2 plus 4 to get 6, then add this extra 1 from up front and find that the 1:20 ratio has
the 7th order of softness, just like the 1:64 ratio from earlier.
a quick review of this formula shows that it does in fact meet both of our previous
rules: if we put in a power of 2, we wind up multiplying its exponent by 1 and then
adding another 1, just like we wanted, and if we put in a prime, we subtract 1, multiply
by 1, then add 1, which will give us the same number we started with, so this seems like
a pretty good answer, but there's one thing missing: what about complex ratios? so far
we've been pretending the first number is always 1, but those ratios spiral out of control
really fast.
in practice, a lot of our intervals are within an octave of each other, like 2:3 or 5:6,
so our theory is kinda useless if if can't handle those too.
fortunately, we've actually already learned how to approach them: you just build them
out of other ratios.
like, to build a 1:20 ratio we went up by 1:5, then up by 1:4.
but if we instead go up by 1:5, then down by 1:4, we get the 4:5 ratio, or the major
3rd, instead.
Euler decided that direction doesn't matter, though: they're built from the same ratios,
so they have the same order, so he made a simple rule: once you've simplified your ratio
as much as possible, you just multiply the two numbers together and plug that in, so
for 2:3 we just use 6, 3:5 is 15, and so on.
we can even do this with more than two notes at once, like with the 4:5:6 ratio, or major
triad, although it gets a bit more complicated.
basically, instead of multiplying all the numbers together, we find what's called the
least common multiple, which is just the smallest multiple that they all share.
in the case of the 4:5:6 ratio, that's 60, and if we run that through the formula we
see that the major triad has a softness of 9.
interestingly, the minor triad's 10:12:15 ratio has the same least common multiple,
and thus the same softness.
score one for the harmonic dualists, I guess.
so how does this all add up? which intervals are actually the most consonant?
well, I put all the common ones through the formula, and generated a list that I don't
feel like writing out, but I'll put it on the side here.
at a glance, the list looks pretty good to me: it puts things like the octave and perfect
5th on top and the tritone and minor 2nd on the bottom, and it shows a lot more nuance
in the middle than the old binary consonance-dissonance system did.
if, instead of messing around with all this math stuff, you'd just asked me to make my
own list, I probably would've come up with something pretty similar to this.
that's a good sign: it means the formula does a good job reflecting reality, or at least
reflecting my experience, so we can use it to make predictions about things we don't
already know.
for instance, I rarely encounter the 4:7 ratio, but if we run it through the formula we see
it's order 9, which means we'd expect it to be about as consonant as a minor 7th.
but there's still some problems with the system, especially as we get to bigger numbers.
for instance, the 200:299 ratio is order 45 so it should sound really, really dissonant,
and if our ears were precise enough maybe it would, but in practice it just sounds like
a slightly flat perfect 5th. this also highlights the problem of using this system for modern
music: we don't actually tune to these ratios anymore.
modern systems are all slightly off, but our ears can make the adjustment so that even
though the modern major 3rd is actually closer to a 50:63 ratio, we can still hear and appreciate
it like it's 4:5.
so is Euler's system perfect?
no.
nothing ever is, but it's a pretty good guess, and if you're looking to get a better sense
of how two notes will sound together, you could certainly do a lot worse.
this video was a little mathier than most, so if you were having trouble keeping up you
might want to check out this video's sponsor, Brilliant! specifically, you should check
out their course on number theory, which goes much deeper than I did into things like prime
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don't get me wrong, I love explaining things, but if you're trying to really wrap your head
around a new concept, sometimes it helps to try it yourself.
plus it's a lot of fun.
so if you want to develop your skills or just learn more about Brilliant, you can go to
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