hey welcome to twelve-tone let me ask
you something how many different notes
are there wait that sounds familiar I
think I already asked that in our video
on quarter tones but quarter tones are
just the beginning there's so many more
possible answers out there it would be
borderline irresponsible of me to stop
after just one video so let's generalize
systems like quarter tones or the normal
twelve note one work by taking the
octave then breaking it up into a bunch
of evenly spaced chunks this is called
equal temperament and it's the reason
modern music can be as harmonically
complex as it is the system we use is
called twelve-tone equal temperament or
twelve tet for short and other systems
like quarter tones that use the same
approach are generally called Tet
systems or ito systems for equally
divided octaves but I like Ted better
but you can't just pick any number you
want and build a system out of it
well okay you definitely can but you
probably shouldn't twelve notes isn't
just an arbitrary choice we use it
because it sounds good and not every TED
system well so how do we figure out
which ones will work well here we turn
to something called the harmonic series
these are the frequency ratios we used
to define what intervals are supposed to
sound like for instance where you hear
two notes an octave apart the higher one
is hitting your ear exactly twice as
often as the lower one whereas in a
perfectly tuned perfect fifth you'd
expect a ratio of three to two but in
practice it's not not quite anyway in
twelve tenth we don't have an interval
that perfectly matches that ratio but we
do have this which is a stunningly good
approximation so good in fact that it
sounds exactly the same even to
well-trained ears the numbers aren't
quite right but real people just can't
tell other intervals though aren't quite
so lucky the minor seventh should be a
nine to five ratio but the best we can
put together is this but how do we even
know that it's off we're dealing with
really tiny distances here smaller than
anything we normally interact with our
smallest standard interval is a
half-step but for this we're going to
have to break out a much more precise
unit descent this is pretty simple $0.01
equals exactly 1/100 of a half step and
it's such a small distance that it's
almost impossible to hear the smallest
difference humans can generally notice
is about five to ten cents
although even that can be really hard so
if they sounded the same don't worry
about it
anywhere we can use
to describe how closely we're able to
match our ideal intervals our perfect
fit is only about two cents off from
where we want it where's the minor
seventh it's about 18 cents away but of
course not all intervals are created
equal either the perfect fit is a much
bigger deal than the minor seventh so if
we have to choose one we'd prefer an
accurate fit in general a good system
should prioritize matching the intervals
that matter most musicians tend to have
a reasonable intuition for what those
might be the octave course is paramount
and things like the perfect fifth
perfect fourth and major third are also
pretty obviously big deals but since
we're getting mathematical anyway why
not try to be a bit more rigorous a good
place to start would be the major scale
wait sorry that's wrong that's actually
not the major scale it's just the
closest we can get with the twelve tip
system here's the real one this scale is
one of the most important devices in all
of modern music theory so we can
reasonably assume that the most
important intervals are going to show up
inside it from there we just need to
figure out which one's of those are the
biggest deals there's a couple
approaches to this with names like
Euler's greatest function and Helmholtz
as roughness function but suffice it to
say we can do some math to them and
produce a list for most least important
or more precisely consonant depending on
which formula we use we get slightly
different answers but they both agree on
the top five and since they also agree
with what most musicians would guess if
you ask them it seems like a pretty good
place to start so now that we know which
intervals we care most about we need to
find the Tet system that gets us the
closest but there's one more concern
left sighs if we have say a thousand
notes per octave we're going to have
really good approximations of pretty
much any ratio you can dream up but can
you imagine playing a piano with a
thousand keys it's completely
impractical and besides with only about
1.2 cents per step you wouldn't even be
able to hear most of the differences so
in addition to being accurate we want
our system to use as few notes as
possible there's really no sense using
13 notes when 12 works better anyway if
we put all those concerns together a
couple standouts emerge for instance 7
does a pretty good job considering how
few notes it has and of course 12
is excellent but if we look at slightly
larger ones we find even better options
nineteen Tet gives us a major sixth
that's less than a single cent away from
where it should be along with a slightly
better major third as well scaling up
higher systems like 41 Ted and 53 Ted
offer some really precise tuning options
although they're starting to get a bit
unwieldy and if we throw out the major
scale completely and just try to
directly approximate the harmonic series
31 Tet is absolutely incredible
of course these results will vary
slightly based on your choice of
methodology so if you want to read more
on the specific approach I'm basing this
off of there's a link in the description
to the paper I'm working from there's
also one last interesting revelation
there we talked about working from the
major scale but if we instead base our
calculations off the minor scale it
turns out 12 notes isn't even all that
good 9/10
is both smaller and more accurate so why
do we use 12 notes if there's so many
other options I think fundamentally 12
is just good enough it approximates most
of the things that matter pretty well
without taking up too much space of
course there's also historical reasons
we were using 12 notes long before we
were using equal-tempered systems some
of the old methods we use they're
actually pretty incredible too and I'm
sure we'll be back to talk more about
this soon until then thanks for watching
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