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hey, welcome to 12tone!
lemme show you a magic trick.
here's a perfectly ordinary major scale (bang) and like any good magician, I'm gonna cut
it in half.
if we just draw a line straight through the middle here, we see that there's four notes
on either side.
now if we look at the structures of those notes, we see the bottom half starts with
a whole step, then another whole step, then a half step, while the top half starts with
a whole step, then another whole step, then a half step.
wait, that sounds familiar.
we've taken a non-symmetrical scale and chopped it up into two identical mini-scales, and
if that's not your idea of a great magic trick then… yeah, ok, fair enough, but still,
it's pretty cool.
so what's going on?
well, these groups of four notes are called tetrachords, which literally means "four notes"
because names are hard.
still, goofy name or not, tetrachords are a really powerful tool for building scales:
you can take any seven-note scale, add in the octave, and then chop it up into a lower
tetrachord and an upper tetrachord.
because the root and the octave are fixed points, we know where the lower tetrachord
starts and where the upper one ends, which means all we need to do is choose which ones
we're gonna use and voila, we've made a whole scale.
for instance, as we saw, using two of these (bang) makes the major scale, so we call it
the major tetrachord.
again, names are hard.
but minor (bang) isn't quite so simple.
if we look at the lower tetrachord, the half-step is in the middle, whereas in the upper one
it's right at the beginning.
this means that minor scales require two different kinds of tetrachords in order to construct.
the lower one is generally called the minor tetrachord, because it's the one that feels
more characteristically minor.
the other one is sometimes called the upper minor tetrachord, but I prefer to just call
it Phrygian.
that way it gets to have a bit more of its own identity.
these three together make up the simplest group of tetrachords, but there's plenty of
scales we can't make with just them.
like, harmonic minor (bang) has this big leap in the middle of its upper tetrachord, which
doesn't exist in anything we've looked at so far.
here, we've got a brand new kind of tetrachord: instead of having two whole steps and a half
step, it's got two half steps and then a minor third.
this version, with the leap in the middle, is often called the harmonic tetrachord, because
it comes from harmonic minor.
in theory, we could also put the leap at the bottom (bang) or the top (bang) but you don't
tend to see that in actual scales, so we don't really have names for them, at least that
I could find.
anyway, now seems like a good time to talk about the Greeks, because they invented this
tetrachord stuff a long time ago.
like, a really long time ago.
they trace their roots back to at least the work of the ancient greek theorist Aristoxenus,
a student of Aristotle.
there's a lot of really fun, really complicated tuning stuff going on with the Greek tetrachords,
but for now I'm just gonna approximate them into modern tuning 'cause what I really want
to focus on is the structures.
if you like tuning math, though, it's well worth looking up the details.
or you could just wait a bit, I'm sure I'm gonna make a whole video about Greek tuning
theory at some point.
back to Aristoxenus, though, he defined three different varieties of tetrachords, which
he called genera, based on the intervals they're made of.
in all three genera, the top and bottom notes are always a perfect fourth apart: the only
things that changed were the two notes in the middle.
so in the diatonic genus, we have this (bang) where the intervals are a half-step and two
whole steps.
this resembles our phrygian tetrachord from earlier, and by rotating the intervals around
we can find the major and minor ones too.
in effect, they're all variations of the same basic pattern.
we can also see that in the harmonic tetrachord, which is related to Aristoxenus's chromatic
genus.
(bang) here, the tetrachord is dominated by a minor third that takes up more than half
the space, and then the rest of it, called the pyknon, is split into two half-steps.
pyknon is greek for "condensed", because the minor third is actually bigger than the entire
rest of the tetrachord put together, so the other two intervals are just sort of squeezed
in there.
and this gets even worse in Aristoxenus's third group, the enharmonic genus (bang) where
the leap has grown into a major third, leaving the pyknon as just a single half-step.
now you might think there's no way to split that in half, but fortunately, we can bust
out a thing called quarter-tones, which we've talked about before.
these are basically just half a half-step, which is unsurprisingly useful for splitting
a half-step in half. ...half.
anyway, we can see the influence of Aristoxenus's tetrachords in the more modern system: the
enharmonic genus is largely gone because we don't use quarter-tones very often these days,
but the diatonic genus became the major, minor, and phrygian tetrachords, and the chromatic
genus became the harmonic one.
but his system assumed that every scale should have notes a perfect fourth above and below
the root, and while most of the popular ones still do, some are a little less cooperative.
for example, Lydian (bang) has an augmented fourth, and Altered (bang) has a diminished
one.
this means we need to name new tetrachords in order to accommodate these scales, and
we might as well name them after the scales they come from, so the Lydian tetrachord (bang)
is all whole steps, and the Altered tetrachord (bang) is like the phrygian one but smaller.
as another brief aside, the Lydian tetrachord is crucial to understanding something I wanted
to talk about in the Jacob Collier video: super-ultra-hyper-mega-meta-Lydian.
this is a scale he's talked about that's basically just a stack of Lydian tetrachords forever.
now, if we do this the way we've been doing, with a fixed octave, we get this (bang) which
is called the whole tone scale 'cause it's all whole steps.
but Jacob doesn't do that.
instead, he fixes the perfect 5th, so that each tetrachord starts a half-step above the
end of the last one, creating this (bang) where we actually overshoot the octave.
Lydian has a famously bright sound, and Jacob argues that by stacking Lydian tetrachords
like this, you can keep that brightness going forever, creating, in effect, the ultimate
major scale.
kinda.
moving back to normal music, you can find all sorts of new scales just by combining
different tetrachords. in this video, we've talked about six: Major, Minor, Phrygian,
Lydian, Harmonic, and Altered.
with just those, you can make 36 different scales, and that number only goes up if you
include things like the chromatic variations or even the enharmonic ones. and other cultures
have developed even more tetrachords, like this Arabic one (bang) that employs an interval
called a neutral second, but as always it can be dangerous to just drop non-Western
ideas into Western models because you lose a lot of the cultural context.
still, though, the real strength of tetrachords is that they give you a simple way to tell
what the scale will sound like, as long as you know the sounds of its component parts.
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bye!
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