this video is sponsored by CuriosityStream!
hey, welcome to 12tone! music has lots of
different scales but, as any music theorist
could tell you, there's no best scale, no
perfect scale for every situation.
the right scale to use will always depend
on what you're trying to do.
but what if we look the other direction?
sure, there's no truly perfect scale, but
is there a least perfect one?
well… yeah.
kinda.
it depends.
(tick, tick, tick, tick, tock)
so I suppose I should define my terms here,
because I'm actually using "perfect" in a
pretty specific, technical way: I'm talking
about the presence of perfect intervals.
these are intervals defined by really simple
frequency ratios, and they form the basis
of the Western idea of consonance.
they come in three main flavors: the perfect
octave, the perfect fifth, and the perfect
fourth, so when I ask for the least perfect
scale, what I'm really looking for is the
scale that includes as few of these as possible.
or, actually, let's make things simpler: we're
not gonna worry about octaves.
most Western scales are based on what's called
octave equivalency anyway, which means that
notes an octave apart are considered effectively
the same note, so this and this are both G#.
we could try to build a scale without octave
equivalency, but then we have to get into
tuning theory, and this was supposed to be
a scale theory video, so we're gonna say octaves
are fine.
we're just concerned with perfect 5ths and
4ths.
and we don't even really have to worry about
4ths. like, let's take our G# from earlier.
if we add the note a perfect 5th above that
we get D#, but thanks to octave equivalency,
adding this G# gives us this one for free,
which is a perfect 4th above the D#. this
means that for every perfect 5th in your scale,
you also have a 4th, and vice versa, so we
only have to count one, and since 5ths are
a little more consonant, I'm just gonna look
at those.
so, ok, rules set.
now which scale has the fewest perfect 5ths?
well, depends how you define scales, but if
you use my preferred definition, which is
just any collection of notes with a root,
then the answer is… this.
(bang) so that's boring.
we can try refining our scale definition,
say, by using William Zeitler's version which
says a scale can't have gaps larger than a
major 3rd, but that just leaves us with this:
(bang) which isn't much better.
the problem is that the question encourages
us to use as few notes as possible, since
we start with zero perfect 5ths, and then
every note we add runs the risk of creating
one and ruining everything.
but using so few notes leaves us with something
that doesn't feel very scale-like in practice,
even if it technically fits our definition.
so is there a way around that?
well, I'm glad you asked, because yes, of
course there is: imperfections.
this term has been used to mean a lot of different
things in music, but the version I'm talking
about comes, again, from William Zeitler:
basically, it's a way of measuring how many
perfect 5ths are missing from a scale.
in Zeitler's system, a note in a scale is
an imperfection if the scale doesn't also
contain the note a perfect 5th above it, so
if we looked at, say, the major scale: (bang)
we'd see that, of the 7 unique notes, 6 of
them can be paired with their perfect 5th
in the scale: (bang) leaving just 1 imperfection,
here.
on the other hand, something like melodic
minor (bang) has four perfect 5ths (bang)
leaving us with 3 imperfections, making it
a less perfect scale.
there's only one scale with no imperfections,
the chromatic scale (bang) which is just the
name we give to the collection with all the
notes.
every other scale has at least 1, but I'm
pretty sure we can do better than that.
looking at it in terms of imperfections means
we no longer have an incentive to use fewer
notes, because adding them increases our score,
as long as those notes don't make perfect
5ths. so with that in mind, let's reframe
our initial question: instead of asking which
scale has the fewest perfect 5ths, let's ask
which one has the most imperfections.
now, I could just run some numbers and tell
you the answer, but it's more fun if we try
to build it ourselves.
first things first, we're gonna need a root.
we've been using G#, so let's stick with that.
now, adding this note immediately rules out
a couple others.
like, we know our scale can't have an D# now,
because then our G# has a perfect 5th, which
means it's wasted.
but if we can't have D#, then suddenly A#
becomes really tempting: after all, it's a
perfect 5th above a note that we already know
we're leaving out, so it doesn't have anything
to spoil.
however it can still be spoiled, by adding
a F, so now we have to leave that out too,
which means C becomes cheaper to add, which
then rules out G, and if we keep going like
this we wind up with… (bang) *sigh* god
damn it.
so what we have here is a very well-known
scale in music theory circles called the whole-tone
scale, because all the notes are a whole step
apart.
now, don't get me wrong: the whole-tone scale
is a fascinating structure, and it's been
used to great effect by all sorts of artists
from classical composers to jazz soloists,
and sure, it has 6 imperfections, which is
more than any other scale in standard tuning,
but I wanted to find something new.
the whole-tone scale already gets enough attention,
it doesn't need this title too, so I'm gonna
do what every good theorist does when presented
with a boring answer: I'm gonna change the
question.
the whole-tone scale has 6 notes, but many
of the most popular western scales have 7,
so what if we make that a requirement?
what's the least perfect scale with 7 notes?
well, that's a harder question to answer.
or I guess it's easy: there is none.
first place is a 21-way tie.
turns out, throwing away the correct answer
makes finding the best wrong answer tricky,
but that's ok, we can sort through this.
like I said, there's 21 different 7-note scales
that each contain 5 imperfections, but fortunately
for us, they all fall into one of three groups.
the first group works like this: we take our
whole-tone scale from before (bang) and just
add a note into one of these gaps. let's say…
here.
(bang) now, because we'd carefully balanced
it such that every missing note was a perfect
5th above and below a note in the scale, this
winds up spoiling one of our imperfections
and also not being one itself, leaving us
with 5 total, but hey, 5 still isn't a bad
score.
the major scale would kill to have 5 imperfections.
or, I mean, it probably wouldn't, but locrian
might.
anyway, the whole-tone scale has 6 different
gaps, each of which we can put a note in,
and then there's also the option of making
the added note the root (bang) which gives
us 7 scales.
our next group is even simpler.
here, check it out: (bang) turns out, if we
cram 7 notes as close together as we possibly
can, only the top two wind up perfect.
we can rotate this around, shifting some notes
from the bottom of the octave to the top (bang)
but in all these cases, what we have is a
tritone where one side is entirely filled
in and the other is completely empty.
we can position that tritone in 7 different
places in the octave, which gives us 7 more
scales for a total of 14.
still, we haven't really found anything interesting,
have we? these have all felt pretty obvious.
but we're not done yet.
we've got one group left, and to find this
one, we have to start with that last group.
let's take our string of half-steps again,
but this time we'll make it so the root is
in the center of the bunch: (bang) now, where
are our perfect 5ths?
well, the top of this bottom part is the root
of one of them, and the bottom of this top
part is the 5th of the other.
but the rest of the scale is just a mess of
half-steps, so we can take these tips and
slide them toward each other (bang) and they
just shift which note they're paired with.
we've broken two perfect 5ths but created
two more, so we're left with something that
still has 7 notes, still has 5 imperfections,
and actually looks like a real scale.
it even meets Zeitler's definition.
and, just like before, any of these notes
could be the root, which makes another family
of 7 different scales, for our full 21.
I should note, for the record, that this isn't
the first time I've seen this scale: it was
actually sent to me on twitter a while back
by a composer named Petra, who'd been playing
around in a similar area and decided to call
it the crater scale in honor of this big hole
in the middle.
given that, as far as I know, it doesn't have
any other name, and given that "crater scale"
sounds awesome, I think I'm gonna call it
that too.
now, technically speaking, any of these scales
are equally good answers: I mean, they all
have 5 imperfections.
but I think the point of thought experiments
like this is to find something exciting, and
by far the most exciting result for me is
this last one.
the whole-tone scale has been thoroughly explored,
so adding a note in the middle of it still
puts us in familiar territory, and while I'd
probably argue that a string of half-steps
spanning a tritone does technically count
as a scale, it's certainly not much of one,
but the crater scale? that feels like a real
thing.
it has a structure to it, it's not immediately
obvious, and most of all, it just sounds cool.
it feels like something you could actually
write music with, even though it's incredibly
imperfect.
speaking of perfection, I made a video about
what I think is the perfect TV show intro
over on Nebula.
boom, nailed that transition.
anyway, if you've ever wondered why I use
a penny farthing bicycle to represent the
number 6, or if you already know, then you
might want to check out my video on the british
cult classic The Prisoner.
it's one of my favorite shows, and when Nebula
started a series about the best TV intro sequences,
I knew which one I needed to talk about.
wait, I should probably explain: Nebula is
a streaming service launched by some of the
YouTube's top educational creators, including
Braincraft, Polyphonic, and Hbomberguy, and
for some reason I'm involved too?
I still don't know why they let me tag along.
but anyway, we built it as a place to experiment
outside of YouTube's restrictive, algorithm-driven
ecosystem, and one of the coolest experiments
so far is a series called Working Titles,
where each episode a different creator breaks
down their favorite TV intro sequence.
it's fascinating to see how everyone approaches
the problem: Patrick Willems did a shot-by-shot
breakdown of the 90s X-men cartoon, Polyphonic
talked about the themes of the Game Of Thrones
sequence, and Mia Mulder looked at how Battlestar
Galactica's opening set it apart from other
sci fi shows.
and me?
well, I talked about dichotomies.
a lot.
it's kinda my thing.
if you want to check out my Prisoner video,
or the rest of Working Titles, or the rest
of Nebula for that matter, we've partnered
with CuriosityStream to work out an amazing
deal: if you sign up for CuriosityStream with
the link in the description, not only do you
get a free month of premium access to their
whole library of amazing documentaries, but
as long as you remain a member there, you
also get a Nebula account completely free.
no additional charge, ever, and since CuriosityStream
has annual plans starting under 20 buck a
year, that's a lot of great educational content
for not a lot of money.
and hey, thanks for watching, thanks to our
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