Maybe the best part of the day is I get
to share my my hobby I guess my passion

with of a fairly full room full of others
who I hope have at least share part of

my passion for music music instruments and
also especially the piano so

what I want to do today is are sort of
two things first of all I want to talk to

you about some of the some of the history
of the piano some of the ins and

outs of how pianos function
how we listen to them and so

on I also want to to criticize physicists
which I don't and most never do but but

I was as you'll see if I go through this
talk some of our some of the paradigms we

use in physics about simplifying things
and treating things you know there's this

famous quote of Einstein make every
problem as simple as possible but

no simpler Well sometimes physicists
go a little overboard in that

especially with regards as we'll see to
a musical instrument like the piano so

that would be kind of an underlying
theme which I think which you'll see

come back several times through
this talk so what I'm going to

do is begin by giving you a very
general interruption to the piano.

It's overall structure all remind you
of how the strings in the soundboard in

the hammers and so on are all connected to
make this this instrument the piano is in

some ways unique in that we know it's
exactly when it was invented and

we know why it was invented unlike
instruments like well the organ or the or

the violin where we don't quite know who
started it first and and some of it for

the piano we know exactly who start
who who made the first pianos and

we know why you made them and that's an
important part of the story on talk about

then I'll move on and talk about some
interesting aspects of the of piano

acoustics and here's where I'll show you
it's not just the simple physics that we

learn in physics one simple physics and
get us a long way but it can't get us all

the way to a real musical instrument the
one that we're used to listening to that

a piano that was only based on the simple
physics wouldn't be a satisfying piano but

we'll see how to make
a satisfying piano as we go so

some of those aspects include in harmony.

This is a piano with tones they are quite
harmonic as we're used to learning in

physics one will talk about how
piano tone changes with loudness and

why it's extremely important for the
instrument also talk about the variations

of how piano tones vary with time how that
how their time decay how it behaves and

why it's interesting and
a little bit about bass notes and

a psychoacoustic issue known as the
missing fundamental something it was first

noticed by a physicist called home health
so you may have heard of and also a talk

in general about the importance of
psychoacoustics the way we listen to music

in general affects the way the piano was
structured and is is put together and

they're not I've run out of time I want
to talk too much about my own work in

modeling of various parts of the piano and
then I'll wrap up by reminding you

the main message is what we really
learned from this kind of work.

So this is a picture of a typical grand
piano this one sits in my living room it's

this it's a it's a modern well modern It's
about one hundred year old Steinway but

it's and for the purposes of this talk
I'll show lots of photos of this of this

instrument these photos are in my my book
I think this one was lifted to the for

the poster and I should give my daughter
credit she's a photographer and

she took took all the most all the photos
here she did a wonderful job so

there's a typical modern grand piano.

You know the player sits
stands here at the front and

puts on the key person on the keys
the strings run from front.

Front to back and we'll talk more
about that later and the hammers

if you look on inside the hammers you
push on these key levers in the front and

the hammers come up at the strings
from below and then fall back so

the strings get hit from below and they
vibrate in a horizontal plane that runs

away from the player and that's the
general actually probably all knew that

some three of all taken apart your
piano at one time or another.

Now that's a modern grand piano it
has about eighty eight keys it notes

this is the is the oldest extent piano.

Made around seven hundred twenty it was
made in Florence Italy it's interesting

that this that crystal this is Bartolo
male Christopher he's the inventor of this

instrument of this piano of this
instrument it's interesting that is his

he lives he overlapped with a violin
maker named Stradivari who you may have

heard of in Florence isn't too far
from Stradivarius hometown and

also in northern Italy and so this was
a must have been a really fascinating and

very very exciting place still to live
at that time especially culturally

Now this is the only known image of
Christa for in fact this image now

this this painting no longer exists
it was destroyed in World War two and

bombing of of Germany but the image
persists and the reason I showed it here

is that Kristof Henri is holding in his
hands if you blow this up you can see this

piece of paper here he's holding a this
is a diagram of a key part of the piano

now this is his first piano I mentioned
the piano has eighty eight keys

this one has forty nine OK on
almost half the number and

it's really has a lot in common with
the harpsichord But but mainly it but

it's a it's a very different road
different from in the today but

the main the key part of
this instrument is is a.

Mechanical piece that links the hammer
the key levers to the hammers as well see

an important part of this is that when you
push on the piano key you start the hammer

in motion and just before the hammer hits
the strings the key lever escape for

the hammer escape from the key lever is
called the skate Mint So the hammer is

moving freely as his the string is able
to bounce back freely that's a kick and

absolutely crucial part of the piano
This is the diagram of the mechanical

lever system that Christopher invented the
fact that it's in this portrait shows that

he knew exactly how important that was for
the instrument.

And I'll come back to
that several times so

let's talk let's talk about
why why did Krista for you and

vent this piano Why do you get all
the goal is a bigger speak Italian.

OK So
you know what piano means piano piano and

in Italian mean soft and in fact
the early pianos were actually call

piano forte That was one of the terms
Krista for use for his instrument so

piano means soft and Forte means loud
you know from your musical training so

it's it's a soft power but it's an
instrument that can play loud and soft and

now as you may not know that at the time
of the invention of the piano the main

a keyboard instruments or
the organ pipe organ existed and

the and the harpsichord and both of
those instruments have a property and

that if you press the key a little bit or
a lot you get the same amount of volume

the tone is the same whether you press it
hard or soft you you you can't have you

can't get expressive possibilities of
playing loud and saw from note to note and

that's what people want to that's what
Kristof or you want that's what people

like later a few years later Mozart wanted
they wanted they want the expressive

capabilities that come from being able to
change your volume from literally from

Note the note I mean with organs you play
a little bit and you can reach in and

you can pull lever maybe on some
organs and you can then get louder or

change and it's change the sound of things
but you can't change it from note to note

with the piano you can because of
this escapement mechanism that

Christopher invented for me he added the
idea of a hammer to strike the strings and

that came about I think at the time
from people playing with dulcimer so

people knew about hammers hitting
strings from dulcimers but

they didn't know about the idea
of using the key as a lever

through some kind of link is called
the action which I didn't really draw here

that to get the hammer into motion but
then the key falls

away from the hammer the action falls with
a hammer just before it's a string so

the hammer is moving freely if you push
the key lever hard it's going fast because

a little bit is going soft that's that's
the that's the crucial part of the of

Christopher or his invention and
as I said he knew how important this was.

So the rest of this drawing shows you.

Very simplified physics
a physicists conception of a piano

is only a single string of course there
are many more than one in the piano but

the idea is that you have a string
stretched between two and

two held rigidly at two ends you hit it
with a hammer excite waves on the string

we know about standing waves from our
physics introductory physics courses and

those standing waves cause a force on
this bridge which is has to a sound board

that sets the sound board into motion
that acts like a large speaker and

that produces a sound that we hear
a vibrating string by itself produces very

little sound because it moves very little
air the sound board is much larger it's

it's it's a metre or two or three and
across that moves a lot of air and

you know big speakers
can make a lot of sound.

OK so this seems to involve
only very simple physics but

let's let's go on and
see what see what's more here.

OK So let's compare Christopher piano
to the modern piano Steinway but

any modern piano would would do.

There's a difference in technology
of the strings we now have steel

steel strings where they weren't invented
until about eight hundred fifty so

they weren't available to Christa for in
seven hundred twenty he used mainly brass

strings some pianos a time use iron but
they were the they're much different in

composition and much different in strings
so the pianos in his day were much lower

tensions almost an order of magnitude
lower tension than the modern piano.

Or.
The modern hammers are covered with felt I

thought more about that later his were
different focus on one thing here as

I were mentioned he is painted forty nine
notes and modern ones have eighty eight so

the lowest note nowadays on a piano goes
down to it's called it's a zero it's

the lowest key on the piano the far
left and that's got a low frequency of

the fundamental of twenty seven point five
hertz the highest note the far right is

a C. and that has a frequency of a little
over four killers now want to quit so

obviously the piano involved
from four octaves to almost.

Eight Why did it happen well it
happened because people like

Mozart whose pianos had five octaves and
then people like Beethoven who had six and

then Brahms who had seven and
they wanted more and

more know if they wanted
to be able to do more so

the question you might ask is Why do we
stop that's at seven in a few octave bit

octaves when I have more we can put I
mean I can reach farther why not well

the answer in this case has to do with
the way we listen to musical tones it

turns out that if I were to I could easily
make a tone with a much lower frequency

than this I could make a tone that's ten
hearse but when you listen to that ten

hertz tone you would not hear a musical
tone but you hear a series of clicks So

basically it's somewhere around twenty
five heard sort of so is the lower limit

of what we hear is a tone as opposed to
a click right if you if with a ten hertz

click it would be you know clicking
on off it to attend to ten or

ten clicks per second it would sound like
a musical tone so that wouldn't be very

useful to Mozart or Brahms or
anyone or even John Cage probably OK.

So that explains why the piano range
stopped expense extend extending to below

this now somebody in the audience will
be a smart alec and say well what about

a bosun door for if they have extra keys
at the lower end well those keys really

aren't used very much and I'll come back
to what why they why they might why they

I think they are they're a good thing a
little later so whole lot of that question

I'll get that I'll get back to that now
why about the highest No Why not go to

a higher frequency this as you may know
a young person like most of the people in

the audience can hear about twenty kill up
about twenty killer it's OK I probably can

I don't know what my range is now is I'm
sure it's much less than that if you get

older you know your ears can't get your
range up or a part of range goes away or

goes down well the reason for that is even
for a young person who can hear twenty

killer it's the upper limit of what
tones you can hear as a musical tone

ends around five kilohertz or so
that is if you probably know or

if we're talking more.

One of the nice things about musical
tones if you put two together to play

an interval or maybe you play a chord and

they sound good together
because of the way chords and

intervals are are constructed
frequency wise relative to each other.

But if I if I were to so
if I were to play say middle season

is two hundred sixty two Hertz and
I were to play say a G. above that or

see above that I put them together it
sounds good but if I were to play this C.

and then play the next G. above that
you wouldn't be able to hear them

in a musical or you would be able
to hear their musical relationship

that is you could tell that one note
is higher than the other it's wise but

you wouldn't be able to tell if they were
nice interval or not a nice interval your

ability to perceive hitch in a musical
way ends a little above this frequency so

again notes above this I could easily
make I could put them on a piano but

they wouldn't be musically useful so
the way we listen to sound already

has influenced how a piano has been
constructed in design so this and

this has a we'll see another example
later of how the human to the way

humans listen to sounds affects the way
we perceive the sounds from a piano and

other instruments.

OK So
what about let's get back to the physics.

That this picture is a.

Figure but

the the strings on the far right the high
notes the strings here are very short as

you probably know the low end strings
the bass things are much longer and now.

If I suppose I want to take take this I
have a string and I want to then make it

design another string to produce
a note an octave an octave lower so

I'm going to go from say the C.
right right here to a C.

an octave lower Well one way to do that
would be to keep the string composition

the same keep the tension the same and
simply make the string twice as long

as you may know again from your physics
that if I double the length of the string

I double the wave will I
double the wavelength.

Which cuts the frequency in half so
I made an octave after two difference in

the pitch so if I if I keep doing
that I think I can go from one C.

to octave lower to an octave lower octal I
keep reducing the frequency by a factor of

two and this curve is just a just
a one over frequency curve and

that's called with Egger Rian scaling
you may or may not know at the Thag or

is credited with having designed musical
scales long ago as a sympathetic

who had in his spare time talked
about right triangles but

he also as is sort of referred to as
the father of musical scales too.

But if I follow that kind of scaling for
a piano but

I get down to the bottom end I would
be something like six meters long

that would be an awfully long piano it
wouldn't it wouldn't I wouldn't fit in my

living room at least not
with my wife's consent.

So what do we do well it this this dash
curve shows the actual string the links

for this particular piano so
we follow this this but

faggotry a curve pretty well just
a little bit below middle C. S. C.

four and then it be sort of saturate So
what's what's going on here why what how

power you able to saturate How can these
string which isn't which is far shorter

than you might have expected
produce such a low a low note.

And the answer is that another way to make
a string vibrate at a lower frequency

is to make it heavier to increase
its mass per unit length and

the way and so one way to do that would be
just make it thicker I could take my piano

string around middle C. the piano strings
are about a millimeter in diameter so

there are some with thin piece of steel
but if I make it thicker I can make

it vibrated the lower frequency which you
think would be a way to make a low note

the trouble is if I get down here
I have to make the steel string so

thick that it wouldn't be a string
anymore it be like a vibrator bar and

that would not make a good piano so
and in physics terms or

in mechanical engineering terms
there be too much stiffness.

In that string that bar so what piano
makers do in fact this was invented before

pianos were invented for other instruments
is they take a steel core of a string and

they wind copper around it so
thick imagine a steel course of a say

a millimeter to a diameter steel
string with a slinky around around it.

So that Slinky gives it extra mass so

it vibrates slower by Brits
at a lower frequency but

as you know from sleep these are very
flexible so you're at you've added mass

without increasing the stiffness very
much These are called wound strings and

this is the way all all based rings and
all pianos are made and two with and

some of them actually have either two or
even three windings around them to give

you extra mass without without giving
you too much extra stiffness and

that's how you that's how we
violate the thuggery and scaling.

On a piano OK.

Now.

I mention So we need to bond strings
to increase the mass per unit links but

we want to do that in a way that minimizes
the string stiffness does the physics talk

so I get to show some equations here and
what I'll show you is that.

It for a perfectly flexible string
this physicist ideal the fundamental

frequency is some what is some value set
by the tension other things in slings but

it will also vibrate at other higher
frequencies called harmonics and

they they the frequency is a harmonics for

follow a harmonic sequence where just
the frequency of the ends harmonic is N.

times the frequency of the fundamental
OK So that's that's so that's what perfect

strings would do now if I have a stiff
string you are you could not well

physicist can look there's a new term that
comes on here that's a fourth derivative

if you're not a physicist worry about
it the idea is this if I have for a for

a very very flexible string the only thing
that when I say I pluck it or I get it

into motion the thing that brings it back
is the tension in the ring the tension.

If you the restoring force that causes
the wave motion now if I have any real

string has a little bit of stiffness right
if I bend a piece of steel it springs back

by itself so that extra springing
back forces a stiffness force and

that will act to increase the frequency
of vibration now it turns out that

it will increase the frequency of my brace
in more for the higher harmonics and for

the lower harmonics basically because the
higher harmonics have a shorter wavelength

so they're bent more OK So
what happens is the uke

instead of getting this pure harmonic
relationship you get a little D.V.

a deviation from it that depends
upon the harmonic number so

the second harmonic is two times
F one plus a little bit and

the third harmonic is three F one
plus a little bit more and so on.

And these are and so because they are
harmonic anymore they're actually you're

actually called partials so if you have
a conversation with your piano technician

she will refer to them as as
partials if you say can my course.

So this is a this is a measurement of
the IF of how big that effect is for

my piano my my my grand piano this is for

the note A three is the A below
a four forty and and remember for

a perfectly harmonic set of frequencies
it would be F two F three A four if so

what I've done is I've taken the measured
frequencies and divide them by N. so

if it was perfectly harmonic they
would all fall into twenty so

the first harmonic is to twenty
divided by one the second harmonic is

is about four forty divided by
two brings me back and so on so

if this was a harmonics a truly harmonic
sequence every every dot here would fall

on the dotted line a dash line but you see
they don't they go up higher up more and

more and more that's the effect
of string stiffness.

OK And so
this big this is typically a few hertz for

the the fifth or the sixth or
whatever partial of this of this note

now another way to measure that or
display it is.

In terms of a unit called called since and
this is not S. E N S E

as in your physics course but C. and
TS It's a logarithmic unit the important

part about it is that one hundred cents
is the distance from one note to the next

note above it Lemon are no matter where
you are in the musical scale so so just so

this what this says is that that say the
six harmonic is about ten cents too high

or about one tenth of the way to the next
note it gives you a sort of a scale so

it's not huge but it's but
it's quite noticeable in your perception

if you can you can construct either
pianos or you can start artificial

tones that have no stiffness and
compare them to one with this and

you can tell the difference and judging
from the results of listening tests

humans prefer a little bit of stiffness it
gives it makes of the note A little more

interesting in the in the way
that a piano tone is different

in I think merges in say the tone from
the pipe organ a little different tone

call up in Tampa temper.

Now this also besides affecting the
Tambour of a single note this also affects

the way pianos are tuned this is
an effect called octave stretching and

here's what happens.

When when your piano tuner gets to
your piano she tunes to say a for

a say a two to a to B. a certain frequency
and then she wants the next to the note

the next octave up the sound good when
those two notes are played together so

what she will do mine was a she is a she.

Is she will to adjust the note
the a an octave higher so

that its fundamental coincides with
the second harmonic of the lower frequency

If so I have a it say to twenty it has
a second harmonic it's a little above for

forty so she will take a four forty and
make it a little bit higher than for

forty otherwise it wouldn't
sound good together so

what she's actually done is
made an octave which should be.

Factor of two in Picher frequency
a little more than a factor of two.

This is a octave stretching and every
piano is tuned in this way whether you

knew it or
not OK an octave is not an octave OK.

And this is a octave stretching it was
first discussed a lot by a guy named rails

back seventy eighty years ago OK and
what he did is he

measured the tuning of lots of pianos and
then he comes and he found they follow

us on this on the smooth curve so all
I've done is draw in the smooth curve so

this shows you that then what I've
done here is I plotted the tuning of

a typical piano across the keyboard range
from the low end up to the high end and

so over here the scale is in cents so this
means that the lowest note on the piano

is about forty cents lower
than it would have been for

a perfectly harmonically tuned piano and
the high end is about twenty or

twenty five cents higher than it would
have been OK so in the middle things

are pretty good not too much stretching
but there's a lot of stretching noticeable

at the high end a lot at the low
end this is now there's no.

I don't know of anybody who
actually calculated this curve.

But what this curve is is for
typical piano following this tuning curve

makes the best sound best is
a subjective feeling OK but

if you so but this is the way your
piano at home is probably tune and

in fact if you if you've ever
seen your piano tuner or

piano she comes with a little
tuning box very very smart now and

she'll punch in or you have a Steinway M.
She punches that in and that

tells you what curve to follow for the
Steinway M. If you have a bosun door for

She'll pick to put the Bosenova button on
your pretty smart these engineers right.

OK So this is this is a this
is an effect off of stretching

it means that the octaves really
are octaves and this tells you.

That this is a this if there's
one more thing I should mention.

Just because your piano tuner makes.

The fundamental of the upper note mass
the second harmonic of the lower note

there's no way she can make
the higher harmonics all match OK So

she's she's what she's doing
is a grand compromise.

That's why I said that's why I use the
word best the way I meant to use it less

second ago there's no ideal what there's
no way to make all the octo the harmonics

of one note coincide with all
the harmonics or the partials

of the note below it nevermind making
chords and everything else OK so

it's a great again a grand compromise and
this means that no chords will be perfect

no no of course no intervals are perfect
No already no course will be perfect but

this this curve is I could say the grand
compromise of how you tune a piano

now somebody might ask me wait
a second I say what about violins or

play if I'm playing my job all along
with the what the piano how come

my oboe can be perfectly harmonic and
how can a play with something that's so

terrible as a piano and
I don't really know the answer to that.

But I think I know I
think the answer is this

most musical instruments like your oboe or
your singer or your clarinet or

your trumpet whatever they only cover
about two or two and a half octaves and

once you get way into the bass over
any two octave range things aren't so

different than being flat and I So I think
this is how you get away with it not being

offended by the note of when an oboe
plays with the with the piano but

I don't know for sure I never I've
never done the experiments myself.

OK Let's talk a little bit about let's
move away from strings and start talking

about other things like the like the
hammer mechanism I've already mentioned

this is my again my physicist conception
of the escapement I just wrote a little

arrow here I don't want to the whole thing
is too complicated picture in a second but

the idea is that we push
the key lever that pushes

up on the hammer the hammer is moving
up to the string hits this it comes.

It arthritis skate from the action
that's why there's a gap here and

it hits the string it forms a little
wavelet or a little poles and

that Paul's the hammer falls away and

the pulses actually two pulses
travel in opposite directions OK and

then they go back and forth along the
string OK Now what's interesting is that.

Well this picture actually is not quite
correct because this Paul's here this was

the hammer hits the string quite
pretty close to one end and

so the this pulse the one that goes to the
the play at the close of the player will

actually return first and it will return
before the hammer is actually falling away

so it's not just a really quick pulse but
it's a longer post appeared that time for

pulses to return Here's a picture
of the action this is not a very

this is hard to see from this photograph
all I want to impress upon you is

that it's very complicated it's the most
complicated lever I think any physicist

has ever seen OK or every And
if it was ever wants to see OK but

it and it's I don't know how to describe
it other than if I show you a video but

I don't have it here but
I would mention that the piano action was

that we have today was designed to
one hundred fifty years ago or so and

I think there are more patents for how
to make a lever like this than any other

patent with the piano OK or
just there are just hundreds of pads like

everything they can make a better lever.

Now piano hammers.

This is the picture of two hammers this
one's attached to the to the hammer shank

this one's at the she has removed these
are there's a scale it's about two inches

there and this these are basically a.

Wooden.

Felt cover wouldn't mallets OK The core
is made of I think it's this looks like

and usually the shank is made of maple and
this is felt as it's wrapped very tightly

around here you see the this is the
trouble hammered smaller and weighs a lot.

A few grams and this is a bass string
hammer weighs about ten grams it is much

bigger and what the way these are made
is you you get a piece of wood made

in the shape of the hammer head it's
like it's like a long piece and

you wrap the felt around it and
you could like a jelly roll OK so

you get one it's sort of
a match set now Christof or

it didn't have the technology to make
felt this way we felt was knowing that

you could make it in a reproducible way
his hammers are made very differently soon

after Krista fori people started making
hammers with with leather coverings and so

early ones were made with leather until
the felt technology caught up and

I have a lot of pianos with leather
hammers and it's what's remarkable is

a leather hammer sound awful lot like felt
hammers OK and I could talk more about

that I can talk about all these things for
much longer if you want to hear it and

I can talk more about that later
if you want to when I know.

So that's what I am you
know hammer looks like and

you've probably seen that if
you look inside your piano.

Now let's talk about a little bit about
piano hammers and piano strings and this

this title should really be how should
a piano hammer be modeled not string.

Now it's physics we know about well
the piano the piano hammers this is a bury

bad sketch of a piano hammer the idea is
this is the hammer this is the head of

the hammer hitting a string and the hammer
hammer it comes out hits the string and

the hammer the fell compresses a little
bit hits the string then bounces back now

as physicists were all taught it's
ingrained in our in our psyche that

everything that compresses is a is a is
a spring is a linear Spring I mean that's

what I mean by every course we
take it's linear Springs OK So

our first reaction is to model this or

mathematically as a linear spring means
that the force from the hammer is proport

directly proportional to the compression
Z. That's what I mean by linear spring

in those you take a physics course as you
probably have nightmares about this to.

Now if I were a PA If I were
to try to model a real piano.

As a linear spring it would fail
miserably it doesn't it's not linear

at all in fact if it was a linear spring
it would be a disaster for the piano it be

absolute disaster so let me first tell you
what they are they behave and I'll explain

why it would be a disaster here is a plot
isn't something we measured in my lab

measure the hammer force as a function of
the compression and a typical For typical

hammer blow the compression is maybe half
a millimeter maybe a little bit more and

what you see is not linear at all
it's got a lot of curvature and

that curvature is described
approximately by about a cubic law

in some hammers that is into the two point
five summers into the force is no there's

no fundamental number there but it's a lot
bigger than one and what that means is

that at the higher compressions
the hammer is effectively stiffer than

at the low compression towards effectively
soft OK so if I play a note with a of

a very gentle key press and only press the
key to get the hammer going to little bit

I'll only use the soft part of the hammer
characteristic If I press hard

play a loud note I'll get the loud the
the harder part of the characteristic and

if you think about it that's exactly
what we want for musical instrument.

I said before that this is
the pianoforte it means soft loud

I really meant more than that.

When I play a loud note I want
the tone color to change size and

not just the volume I mean think about
it you could you could be playing

music of music in your on your in your car
off your M P three player whatever and

you can turn you can built into piano
music and you can turn the volume way down

you can still tell when the piano is being
played loudly as opposed to being played

soft if you hear the change in tone
color or the way I think about it

if if I'm playing the piano in one room
my wife was saying in the basement

she can still tell when I'm playing
loud or soft not from the volume but

from the tone color same thing is true for
other instruments think about SAIC.

Clarinet or saxophone or if I play a
saxophone loudly the tone color changes or

trumpet right all those things the to me
the key part about the expressiveness

of an instrument is that the tone color
changes when you change volume it's not

just a change in vine that makes it
expressive it's a change in tone color.

And that's what exactly what
these non-linear hammers do for

the piano effect people with the Kristof
Henri's ambers they are non-linear with

just about the same exponent as modern am
and this is why a leather hammer sounds

us about the same as a felt hammer they
have about the same exponent there and

this non-linear area central so
that here's my here's my knock I have how

I forgot the precise business of going on
this is my knock on physicists we are so

obsessed with linear things that here the
non-linearity is an absolutely essential

part of the instrument it wouldn't be
the same without the non-linearity OK so

since Apple where making the physics
too simple and assuming it's linear for

simplicity we've missed
everything important support.

OK So again that's the note to the message
there now let's keep going with this OK

this is the use of measurement again on
the arse my Steinway piano of a change in

the tone color with volume I promised
you this I'm going to show you know

the real measurements so what I did
hear was I played the same note A for

forty three different volume levels
very loud sort of medium and

very soft and then I did an analysis
spectral analysis of the of the signal and

I and I plotted here that the strength
of the different components so

this is the partials of
the fundamental partial number one

second partial third partial Now again
this is not going to again be to criticize

physicists when we draw our our spectra
in our physics classes we think

we always think we tend to think that
the fundamentals are strongest and then

they get they get weaker from their right
it's within that right Paul right and

so here the soft no is indeed
like that the the the.

The fundamental the first partial is
the strongest and then it gets weaker

by about a factor of ten in power and
another factor maybe one hundred power so

it does get weaker as you go up and
up in frequency but

if you look at the loud note the second
partial is actually the strongest one.

OK now it's kind of amazing because we
still hear this as the pitch of the for

of the sound.

But again this this wouldn't happen
this change in the curve wouldn't happen

if we didn't have this nonlinearity
change in tone color so

that the change in tone color is not
a small effect it's a large effect OK And

this is not this is not just special for
this particular note I'll show you some

more results in a second this is a is
a streaming important effect for

many many no misses is a four four is in
the middle of the range and we're going to

see it gets worse while it gets better
as you go down to that in the base.

OK So let's talk more about about this.

Went on to a fun time so let me just come
back to make a couple of quick comments

about something I mentioned quickly but
I pass on over to perhaps too quickly

too fast this is a picture again of that
of that thing of the little wave was going

away from the hammer I mention the hammer
says the string it generates weight pulses

that go opposite directions and the first
wave pulse the one hits is the close and

comes back well as you know physics it
hits this and it gets inverted and even if

you have a physics course you know when
you play when you play jump rope on you

don't do that anymore but when you have
a rope vibrating you know if a wave hits

the end of a string it comes back inverted
so this pulse comes back upside down and

hits the hammer and when it does that the
hammer still in contact with the string

and so it causes a large Rican pression of
the head of the hammer back and forth and

back or so this this is
a picture of the hammer force on

the string as a function of time so
when I play the note

This is when the hammer hits the string
and then you get all these vibe.

Up and down up and
down as these Wait polls go back and

forth back and forth back and
forth between the short end and

back to the hammer so in terms of who
hears it doubly anybody doubly OK So

you know that this is basically
the execution function of our system and

it's going to be this frequency content
which gives us the tone color later OK

it's complicated right and
you see it simple so it's business I would

tend to draw sort of a simple smooth
curve but again I would be wrong and so

again I'm I'm I'm criticizing my own
discipline that's OK You think so

but you want to model a piano sound
you've got to take this into account and

we've got to do that but
I want to time and talk about it today.

OK Let's talk about another
interesting problem.

Where should the hammer strike the string.

Sound simple you see you see here.

That I've got this hammer is pretty
close to one end of the string and

if you go to if you look at my Steinway or

look at most modern pianos the hammer
strikes the string about one seventh or

one eighth of the way along from one end.

Now why is that important.

It's important because of this.

Those partials are different

what are called vibrational mode
vibrational patterns of the string and

the lowest partial is is sketched here
the string just goes up and down together

it's a standing wave and probably all seen
that in many you have the second partial

looks like this part of the string go up
and down and up and down in the center has

what's called a note in the center
there's no motion here at the middle and

the third how much it looks like this
you get like one of ours but it's up and

down in different ways and
you get to knows now the point is this.

If I suppose I hit the string
with a hammer right there

in the middle there's no way I can
excite this kind of this mode and

putting it in the place where doesn't
have any X. and the amplitude.

So if I hit that if they have piano hammer
were to hit in the center it could not

excite the second harmonic second partial
in fact who here is a guitar player.

Have you ever tried to find the center of
the guitar string with your eyes closed.

OK you have OK walk out tell us how
to do it if you close your eyes and

you pluck the string in different places
when you pluck it at exactly the center

all these even harmonics go away and the
tone color changes in a way you can get to

know the answer you know OK OK it's
an honest man appear in front so but

the point is by listening to the tone
color you can tell when the the second and

forth and so on our minds are present or

not present and likewise if I pluck
the string here one third of the way or

two thirds I will not generate the third
in the six and so on so this is

where you hit the string determines again
part of the picture the tone color so for

the piano if you strike the string at the
one one seventh of the way out you won't

generate the seventh in the fourteenth and
so on harmonics so.

And this is this plus going to show
that let me say before I go any farther

I have no idea why eliminating the seventh
harmonic makes a better piano tone

the somebodies decided that it's a good
thing you don't want the seventh of

the harmonics that's
a matter of perception and

how we think about the word best again but
that's that's the way

pianos are made nowadays OK Krista
forced you know actually different but

the modern pianos are all pretty uniform
in that it's about the one seven point for

all the strings if you look at your piano
you'll see that the strike point is

designed very carefully and
here's a plot to show some of that.

To see this carefully I what I
did I actually took the spectrum

of the lowest note on the piano
a zero low very lowest note and

this is the actual the power at different
frequencies as a function of frequency so

every peak here is one of the partials and
you can see that there's a it goes up and

there's a big dip here that's
about the eighth part.

There is no Dipper on the sixteenth
partials is not perfectly clean

we're engines are engineers here not
business but it you can see a definite

pattern that it goes down to around eight
around sixteen around twenty four and

so on so this this is again showing
you this effect of the strike point.

So again where the hammer hits a string
has a big effect on the tone color

OK there's actually no one another
effect I want to show from this plot

if you look down here you see that the
peaks down here get pretty weak down going

down toward the fundamental down
the lowest partial So let's blow it up and

look at that a little bit more.

OK it's the same as
the same plot on the left.

That show just now and I've blown
up this lower part to right here so

this is the this is the spectrum
at the low frequency end

of the lowest note on the piano.

Now I want to I will tell you
that's where the fundamentalists.

I don't see it either OK.

There's a second partially a little bottle
guy there then third fourth fifth and

sixth and you can see that some of the
highest biggest part of the other six and

so parts of these are the hot these
are the stronger partials and

the fundamental is basically missing.

Gone and now this is an effect
that was actually understood and

recognized by Helmholtz a very
famous physicist also very famous

in terms of his physiology I mean he
invented the instrument that that eye

doctors use to look into your eyes is
still used today a very very clever guy or

yes very clever guy.

And he understood this is missing and
then he asked OK Why is it missing or

here's the real mystery
the fundamentals missing OK so

what I don't care but when you
let's talk but the real question is

this when I listen to this note I hear the
pitches being this pitch this twenty seven

Hertz I don't hear it being fifty five or
this or this or that you.

Might have thought that the IF that
the apparent pitch would be the pitch of

the the strongest harmonic but it's not
it's the pitch of this guy who's missing.

So somehow we hear the pitch that isn't
even there that's the mystery and

it's called and it's called the missing
fundamental what it's called.

And the the idea that helmet
was put forward which is

basically still believed today a with
with some important modifications is this

when you here's a here's a very Here's our
back to the fishes a schematic of a part

of a spectrum you can make a spectrum like
this where this is the power as a function

of frequency so I can no fundamental power
that I get power to have three or four or

five so what Hell most pointed
out was he said look these

these partials are separated by F.
one the the value of the fundamental So

maybe what your ear is doing is some kind
of sophisticated spectral analysis which

actually and assigns the pitch to be the
difference between these frequencies and

not the fundamental and
the lowest value and for

low notes that seems to be a pretty
good explanation of most experiments

are there some people are still doing
experiments today with people but

how we perceive tones but that seems
to be explain what's happening and

this explains the the mystery of
the missing fundamental is actually

the mystery to mysteries the first
mystery is why we perceive this to be

the pitch and I kind of explained how
well how most is kind of explained it but

the other mystery is why is it missing in
the first place why do I get why is there

no power in that at that frequency why
why did my piano my good piano I paid so

much money for why did you give
me anything at this at this

you know if I could short change
with a low notes what happens so

that's the second mistake
I want to talk about.

I will get to it before I get
where I can explain that I

thought a little about sound board member
the sound board the far end of the string

attached to the soundboard when the string
vibrates the sound board is pushed up and

down it's like a.

Big speaker so talk about this that let's
talk about the big speaker the soundboard

is a piece of spruce I can't really see
it here it's hidden under the strings.

But it's the same kind of wood that
makes the top plate of a guitar or

the top plate of a violin it's a little
thicker for a piano but it's basically

the same wood this spruce this well is
is the is the would have choice for

this because it's got a very high
elastic constant called the modulus and

in fact the ratio of the Young's modulus
to the density is a figure of merit for

these kinds of vibrating woods and
that and they the figure of merit for

spruce is just as good or
even a little better than steel.

So it's excellent for this purpose.

And this is the soundboard made that is
put the others way it's a piece of spruce

this is the top looking at the top
with all the strings ever taken out of

the piano if you look carefully
you'll see grain running this way and

what happens is there are pieces of spruce
that are that are perhaps maybe six inches

wide depends on the quality the piano
that are glued together edge to edge and

they run from lower right to
upper left that on top of that is

glued to the bridge of the pants
with the strings are attached member

this is the front of the piano where the
keys would be and the strings attached or

go over the bridge like they do like for
a guitar say.

This is the bottom side the spruce is
typically half an inch of an inch thick so

it's pretty thin and so to give it
extra strength in this across the grain

direction there are ribs on the bottom and
this with the ribs look like from below so

this is this is a complicated
enough of thing

that it's not to say it's not
trivial to model in fact.

I want I want to Paris Paul or
anybody here by asking this question but

I'll just pose it how many how many
independent Alas this concepts

are there to describe to the vibrations
of a single piece of wood.

The answer is what he said
to your off by an order.

Magnitude Paul you're a physicist OK.

What has twenty seven different
elastic constants Young's modulus

ratios all these things OK So wood is used
you think physics is complicate you think

electrons are complicated
Wrightwood is complicated.

So so it's but but it can be managed and
I'm sorry for the polar time.

So let's talk about about.

Some were again going to get this.

But OK so about how about of
the House on board vibrates.

This is my cartoon of a soundboard.

The lowest frequency but
you can ask like the string have

had a fundamental frequency that was
lowest frequency it was it was of

give you the pitch of the no there was
a fundamental frequency the soundboards or

things like plates also have a have
different vibrational modes in the lowest

vibrational modes called a breathing
mode just like the way when you breathe

your chest goes out and in and out and
in and that's what I'm trying to show here

this sound board just goes out and
in that's the lowest.

Vibration vibrational of the sound board
the frequency that mode depends upon

the size of the soundboard for a large
grand piano it might be say seventy five

herds so it's toward the low end of
the piano but not but not all the way.

Then the second harmonic are second
actually we're not really harmonic

the second mode is a different mode
where like if I can't do this but

my chest goes out in and out out of phase
like that that's the second mode I try to

draw here and then you get other modes
that are more common to the third mode and

so on and these modes there
are precise modal shapes depend upon

where the ribs are in the bridge and it
gets pretty complicated pretty quickly but

the most important thing to realize is are
two things first of all the sound board

the lowest frequency is about seventy
five hertz and that's important because.

If I try to make something vibrate at
a frequency below its lowest vibrational

frequency it won't move very much and
so you.

You just can't it's very hard
to make the sound board vibrate

at frequencies well below seventy
five Herb's So that's one reason why

the fundamentals missing a twenty seven
Hertz that's way below this one seventy

five it's a couple boxes below it just
doesn't want to vibrate so even though I

have a lot of force in the string
the soundboard won't move very much.

So that's one reason why the fundamental
is very weak or even missing another

reason is that the sound board is
too small to create a frequency

of a tone with a frequency that low the
again these Raise your hand OK Who's it if

you want about antennas No you guys don't
want to buy in tennis anymore do you and.

Well if you learned about antennas you
would know that in general a good intent

of an antenna structure that's good for
certain frequency has to have a size

that's comparable to maybe half
the wavelength of that radiation

who's who's a got it or I will look
I keep looking I can't find people so

the the the wavelength for the frequency

is twenty seven Hertz is a bit more
than ten meters maybe fifteen meters so

it's much much bigger than the sound so
sound was a tiny little thing compared to

the frequency is trying to read compared
to the wavelength is trying to radiate and

that's in the other reason why the the
fundamental is so weak or even missing.

So so I've explained to you why
the fundamentals missing and

that's good OK now.

To get ready for the next next part.

The sound or vibration again it's
a sort of the breathing mode and so

if I look at the breathing mode the sound
board going out and in and out and

in but if I look at a particular point on
the bridge I don't draw a bridge here but

you can imagine that a particular point
that's off center a little bit won't be

measuring moving just out and in it also
be rocking a little bit as a sound right

we were there as my chest out a place off
center will rock a little bit to the side.

And that will be an.

Shortened because when the string
is attached to that point

the end of the string won't just move out
and in it will also rock a little bit and

in physicist language what that will do.

Is let me skip this.

Skipping I'm going to say
they skipped a lot of K..

So one thing.

This is again the equations I
need to show you questions for

the mathematical physicists here.

In physics we're used to dealing with
second order partial differential

equations OK I don't know if I
ever had ever seen a fourth or

differential equation in physics
till I started looking at pianos and

we just don't see them but who are the
civil engineers here raise your hand for

civil engineer come on Raise your hand so

engineers see all is this is thin
plate theory is called spin around for

hundreds of years OK So engineers do have
something office once in a while but

if you want to if you want to model the
vibrations of sound boards you're going to

have a very harmonic kind of structure and
the different the elastic conscience

come in a complicated ways I can talk
about later if you feel like OK so

let's keep going OK let's talk about.

Piano tones and how they how they decay
with time so these four plots show

four different notes let's see this.

Upper left is the lowest C. on the piano
and then we go up a couple of octave C.

three over here there's a C.
six and this is the highest C.

on the piano now what I've done is the
time scale is the same for all these plots

and so it's clear that the low notes last
a lot longer than the high notes and

this one only ended I put I stop to
the no I put the damper down and so

you can see right away at the high notes
decay quickly the low notes to us for

a long time which you already
knew when you play your piano.

Now the reason for that technically it has
to do with the effective impedance and

talk about in impedances for
the string and the soundboard and how they

are the same or different OK So that's it
that's a detail I don't want to go into.

Instead I want to look at
the details of this decay so

what I'm going to do next is plot this
decay but in a different way I'm going

to actually look at the instantaneous
amplitude as a function of time so

I go to this thing to up and
now I crazy and to take that away and

look at the amplitude vs time and it
looks a lot simpler it looks like this so

it's the same curve but
now I'm spotting amplitude vs time and

you can see that it decays this is
a semi log Plaza logarithm over.

Log scale on the left and
linear scale for the horizontal and

it decays roughly like one curve here and

then a different straight line here and
this is this is called a double D.

K. very profound name and what it but
it definitely means that the two different

things are happening on something the DK
is is is has one characteristic or

one character early in the different one
later on and what's happening is this.

Well sketch here here's again
the hammer the string the string

runs from you the player to the back of
the piano the hammer hits the string and

the strings vibrating in this
plane OK now that that direction

of string vibration gets the sound
board moving a lot so you get a lot of

radio a lot of sound generated and the
string dies away role of the quickly but

as the as a soundboard is vibrating
it has this rocking motion so

the end of the string is moving up and
down but also rocking to the side a little

bit so that means you start the string
vibrating in the horizontal plane So

what started out as this kind of vibration
and meaning being a horizontal vibration

now the whores Olla vibration doesn't
move the sound board very much but

somewhere in one and so
it decays much more slowly so

what you've done is converted in technical
language physics language you converted

from one polarization the vertical
one to the horizontal one.

And so and now.

What that does is it means
when you put your piano tone

it dies away quickly at first but
then it will linger for a long.

TIME So if you're playing
an awesome Chopin or whatever and

you it's long passages you have that
a bit of ale ability there which

you wouldn't have if the thing just
decayed away all the way quickly.

OK.

So this is this is so the piano the decay
of NO is a little more complicated you

might have guessed as your
physicist OK as if this is

now it's this this is that the decay for
that lowest note on the piano that has

I didn't mention it it's only
a single string creating the note

as you go I forgot to mention this
earlier I should have I got to excited

as you go up in the can a keyboard some
notes have two strings as you may know and

some even have three so
it's still the same kind of behavior but

now we're going to look at history in
a note this created by two strings.

And what you see is this you get
you get a fantastic eight first and

then you get a slow decay but
you kind of get this oscillation and

if is this would recognize it as
well as a sort of beating effect so

what had what's happening here is when
you first at the string we heard when

the hammer hit first it's two strings they
both go up and down together and he K.

a lot and then they eventually make
this the K. but they start feeding S.

or they get out of phase because they
don't have quite the same frequency and so

it's for a while they're in phase and
then they're out of phase and

then they're in phase so this is this
is an effect called beats so and

you can imagine if I have three
strings it gets more complicated for

a physicist this is very similar to having
two modes that are coupled with some so

we perturbation it's very similar
physics you can write it down.

But this again now this why is this
important if you listen to a piano tone

you can really hear this because sometimes
you hear you play a note and it lingers on

and you hear kind of a little bit of our
bra go OK that's where it comes from.

And that mic and as you probably know
if you listen to your violin friends

play violin they're always ready for

bottle a little bit of Roboto is a good
thing just ask your violinist friends and

that's what that's where you get it
here just from a single note OK.

So the.

I wouldn't I'm Paul.

I Some done I thought I forgot where
I was in my talking I'm sorry so

let's let me just summarize and
sort of recap some of this.

On the face of it a piano
is a very simple thing

right what could be simpler
than a vibrating string.

But there's a lot of physics that involved
here string stiffness is very important

it effects tone color inharmonicity
effects the way a piano is to number

the octave stretching effect
are crucial how piano sounds.

The hammer hammer non-linearity is perhaps
the most important part about the piano

it's it's what makes it expressive this
is the part that Christopher he really

understood even three
hundred some years ago.

The interaction between strings leaves
this very interesting time decay which

isn't just a curiosity but it's really
important for a broader effect which has

make it really gives the tone an extra
richness which which we all like to hear

the bass notes you know or our puzzle
are those are surprise no fundamental but

we still hear the same pitch so again we
see how the importance of us the listener

in the whole in the whole business.

And it's what I mean by I say
psychoacoustics plays a major role in

how we perceive tones and how the piano
is made so to me the bottom line is this

there's a lot of interesting physics
in your piano OK thank you very much.

Of.

A of.

Thank you very much.

Or is.

All.

Right thank you should also.

Be.

It's very well if you will just say.

Well if people who know me know that
I love to criticize engineers and

chemists but I think sometimes
physicists deserve it to you like

I think we have five.

The question was Why don't we make a piano
just out of our computers rather than

out of anything else right.

OK.

I think it's certainly possible a company
like Yamaha are doing things in that

direction not not quite as much
as you think you have the.

No I wouldn't buy I would never
dial violate a physical law

I would never think of doing that.

Well I'll just say one interesting thing
I mean I thought you were Ed you raise

a very interesting point brought to
say one thing one comment about that.

Yamaha makes some very interesting.

Pianos that aren't acoustic piano but
they're they're basically make

sounds into in you know based on equal
on sentence or other things and what it

was interesting to me is the latest ones
they work very hard to give the to give

the Keil every give the action the same
feel as a real piano even though it is not

needed to do that but they also do is they
have vibrators built into the piano so

the bite so the case vibrates the way our
acoustic piano would even has nothing to

do with the way they create sound so
that all they're doing is making it

IS TRYING TO SCHOOL you into thinking it's
a real piano so they decided that it's

important to fool the player to
make him or her comfortable OK so.

It's like we have our ideal
piano in our heads and

we're never going to let it go even though
we don't need these other things OK so

I don't so
maybe the piano is locked into place and

it will never touch some of the parts
of it when I read it I mean.

Yeah I don't.

Know I didn't I didn't say it was
an answer your question I was just making

some comments as my prerogative.

Why they go from whether he covered
hammer covers to tell him recovering so.

I could I could you know
I collect pianos and

I have a lot of pianos with
leather covered hammers.

And it's amazing that the leather
made two hundred years ago

is still very good I think that I think
the chemical way it was made was different

two years ago that is now and
it lasted longer I think the reason is

that the felt was ultimately more durable
when you have higher tensions and

bigger and bigger strings I
think the leather those wear out

although I guess I have you know
they have nice leather coverings and

work well I think it's just a question
of Dura building wants to string tension

got up and it was hard hit that they
were faster you know question over here.

OK.

Yeah I do I well I didn't write that the
player but there on my website you know we

did that I mean we made a whole cup I'm
a whole piano based on you know pushing

the hickey lever and so the calculation
involves this non-linear hammer hitting

the string and then take into account
a complicated way form I showed you and

then at the strings vibrating and gets
the sound board moving and we calculate

the vibrations a soundboard we did that
with all the elastic conscience and

things and then coupling a sound or
motion to the air in creating the you know

the acoustic pressure you hear so
go to my website you'll hear some sounds

in.

That.

Book in the form or do well it's funny my
piano teacher used to always talk to me

about how I push the key
down after after him and

I don't believe that I mean I just
don't believe but but certainly.

Expressiveness I mean a lot of it
is it just is just certainly how

hard you hit push the keys and things
there's have been some really interesting.

Studies of real per piano performance and
a lot of it isn't timing and

having that having the the note come
a little earlier a little late in those

you've probably seen those I mean
not just Chopin type rubato but

either things too and so
that's a big part of it in fact.

The way you make a harpsichord.

Sound more interesting is by even though
the notes are all the same volume is by

zero by the timing so there's a lot a lot
of that I'm not I'm not a great player so

you shouldn't ask me only question and

yes.

It's basically the same physics
question was about about square

square pianos in the green and
then uprights.

Square piano well the earliest pianos were
made in the shape of harpsichords which

had this wing shape we've already seen but
there was also an instrument at the time

I was referred to as you already know this
at that time there was instrument called

the clavichord which was not square but
rectangular and the the little

ones were sort of about this big on
the table maybe this big around and

and they had the strings running sideways
as opposed to front the back and they were

basically the household instrument so
when Mozart would go to give piano lessons

to somebody they would probably play
a square piano in the house and they were

the ones in Mozart's time where some I
could pick it up it's very light and

everything else I could really carry but
I could use that small.

Things are basically the same for that.

The bigger change was when they want
to up right now the square pianos.

They had the same thing happen to them
happen to grand piano if you want to add

more and more nos so the earliest
squares were like four four octaves and

then they started adding bigger and bigger
I have a I have a Steinway square the bass

from eight hundred fifty eight
that has seven octaves and

it weighs I think it weighs a ton I
mean it's the it was it's huge it's got

it's it's very massive and
it wasn't a very good household instrument

anymore at the same time in the early
eighty's hundreds the technology for

uprights was invented and what was really
the key to technology was the action

right the action I didn't talk much
more about it but the action for

a grand piano and for
a square the Hammers come up from below.

Hit the strings and fall back so
you have gravity working for you for

the upright piano the Hammers
hit the string this way and

it had to bounce back she got levers and
keys and all kinds of things and it took

a while for people to invent a good action
and is about eight hundred twenty or so

when the the actually the base of
the uprights today was invented and

after that the square started to go
away so but the physics is so the main

physics differences in the action as a
post anything else back in the back there

I've never looked for that so
the thinking in musical acoustics is

that the phases don't really make much
difference the phases of War of one

partial to the next so
we usually ignore that.

I never study that but I guess that's
what people that's what people think.

Or questions yes.

You want to talk about that.

But the question was this made me
put the question Could people hear

I mention this business
about octave stretching and

I talked about octaves
being a factor too but

there are a lot more ratios that go into
musical scale than just a factor of two.

Now.

Do I start G.'s.

The way we tune pianos today isn't it
is that it is a method called equal

temperament which means there
are basically twelve notes that span

the octave and so we go one twelfth of
the way so that across the octave for

each node but one twelfth of the way and
in a log makes sense OK so

the ratio of C. sharp to C.
is two to the one twelfth and

indeed a C. Sharp another two to the one
twelfth and twelve to the one twelve's or

whatever it is make two right that's what
it has that's equal temperament Now some

people believe that equal temperament
was the worst thing ever invented.

The for the following reason if I play.

The notes C. and I play the note G.

which is supposed to be a perfect fifth or
point I suppose to be a fifth.

If it's a perfect fifth then the ratio
will be precisely three to two so

are three to other OK and that will be
a very nice pleasing ratio in my ears and

I want to hear much beating other than
the higher harmonics higher partials but

the fundamentals won't beat but
equal temperament to the five twelve

divided by two is not three to two OK so.

That that interval seed that she will
beat when you listen to your piano and

some people most of us don't
like that very much and so

OK Mike I can make sure I get
back to your question OK.

So by tuning in equal temperament we give
up the possibility of perfect intervals

and making really beautiful intervals
beautiful walk beautiful chords.

But what we gain is we gain the ability
to play equally bad in any key

OK which some people believe
is a good thing OK And

you can sort of sense how I
feel about it but that's OK OK.

But Equal Temperament equal temperament
was what was used up till about

it depends what you believe at least
Mozart's time I started to change and

then maybe a little even
a little later than that.

Now it's sort of the question
of equal temperament versus

versus other temperaments gets
is sort of a moot question for

the piano because I've already showed you
that even a single note is a harmonic and

the deviations of a single note from
being harmonic are about the same size as

the deviations from temperaments
between equal temperament othersome So

I think tuning a least twenty a modern
equal temperament to an equal temperament

wouldn't really make much difference is
the end OK so I'm not sure about but but

the real reason is you can marshal keys
and you can play you can play in F.R.

minor as well as you play in C.
Major those kinds of it and

that makes it easy for playing other
instruments and other things like that but

but I mean there's been many books written
about it you probably know this by

temperament right there
have been there been many.

Yes.

Why is it that OK well I was
speaking figuratively OK.

I guess I'm not an expert on this there
are people there's lots of these things

I I think I've read about people
doing experiments where they play

one note on one side and
one on the other they and some of the.

Harmonic Motion get put together
not in your single air and

I think also people don't experiment
is not a non-linear effect I mean

it's not just non-linear mixing
I mean that as a double E.

You might think this is non-linear mixing
two fundamental partials mixed to give us

one is not that simple I think it is done
at the higher order of parts I think but

I I think you can look it up
I swear I seem to remember.

Yes.

I forgot to mention that OK the question
was What about the Loki's on the bows and

or for why are they important OK I got
to show up I have to look at my pictures

OK I don't actually own a boat and
or for I can afford it but.

Why do we do that.

OK let's let's do it here.

The lowest note is one
of these pay strings

it's way along the edge of
the soundboard OK now and

it runs over the bridge close to
the edge of the soundboard Now

remember they're breathing mode thing if
I'm trying to make the soundboard move and

I'm pushing on at the edge it's not very
good and so when I'm with the bows and or

fried it throws you haven't seen a post
over some of them have an extra one

of three or four notes on the on the low
end how many is it I don't know if they

have about four the but for no it's leavin
lower than the one on the Stein ways and

every other piano and they actually
give them a different color so

you don't get missed disoriented when
you're playing the piano and but

what they do never mind if you use them or
not what they do is they make a means

a sound one has to be even a little wider
than this one which means this lowest note

is actually closer to a farther from
the edge which is a good thing for it so

that's why so I think adding those
extra four or five most of those never

makes other notes better they serve as
they sort of sacrifice themselves for

the good of the piano and so I think
to me it's that's one of the Course

important things they do so.

This.

OK The question was Has anybody ever
tried to design a key where you could

push not just down but in are now it.

I don't know about that I mean there
is a pedal on the on many pianos

it's called the.

What they want a quarter pedal or if you
see that you know what that OK Out to

explain so some pianists like I like most
grand pianos there's a there's a lever

there's a well there's one there's the one
pedal you all know lift the damper So

the thing vibrates more in
sound sort of more resonant but

there's another pedal which shifts
the entire keyboard over a little bit so

instead of having a hamburger three
strings or two strings it's one own accord

and that's where it comes from right and
the idea there is the you get all you get

first you'll have one string so it changes
the tone color a little bit you also

are playing with a different part of the
hammer which is probably a little softer

than the part that's been you know been
been rough and up by always playing so

it will change so that's a little
bit of what you're saying but

that's that's the only think
it goes in that direction like

that.

A lot of.