question about periodicity

I know I asked this question before, but it would be good to
review. Basically, I've been thinking about periodicity and the
harmonic series on a vibrating string, since we've been discussing
the *inharmonic* piano strings.

Basically, the question is why it is that in a *harmonically
vibrating* string, the accompaning vibrations are in simple integer
ratios.

If I recall, Paul Erlich mentioned it had something to do, I think,
with the fact the the periodic vibrations "divide down" to other
integer proportions, or something like that, but I really could use
a little review on this.

Thanks!

Joseph P.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> Basically, the question is why it is that in a *harmonically
> vibrating* string, the accompaning vibrations are in simple integer
> ratios.

if the string repeats its motion exactly 440 times per second, then
all the frequency components within the string's vibration must
repeat themselves exactly 440 times per second. so the only allowed
frequency components are those that cycle an integer number of times
within 1/440th of a second, namely 440 Hz, 880 Hz, 1760 Hz,
etc. . . . any frequency components in-between would look different
at a point in time t and at a point in time t+1/440, thus spoiling
the supposed periodicity.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

/tuning/topicId_44196.html#44211

<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > Basically, the question is why it is that in a *harmonically
> > vibrating* string, the accompaning vibrations are in simple
integer
> > ratios.
>
> if the string repeats its motion exactly 440 times per second, then
> all the frequency components within the string's vibration must
> repeat themselves exactly 440 times per second. so the only allowed
> frequency components are those that cycle an integer number of
times
> within 1/440th of a second, namely 440 Hz, 880 Hz, 1760 Hz,
> etc. . . . any frequency components in-between would look different
> at a point in time t and at a point in time t+1/440, thus spoiling
> the supposed periodicity.

***Thanks, Paul. But what happens to the 3:1 or 1320 Hz? Wouldn't
that also be a part of it?

And, now I beginning to wonder what an *integer* is anyway... Maybe
that's a "Tuning Math" entry... (I'm assuming it's a "real" rather
than an "unreal...! number :)

Joseph P.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> And, now I beginning to wonder what an *integer* is anyway... Maybe
> that's a "Tuning Math" entry... (I'm assuming it's a "real" rather
> than an "unreal...! number :)

A number like -2, 0, 7, 53 etc. Number theorists sometimes call these
"rational integers" because they hold to strange beliefs such as that
integers can be irrational numbers, and that the square root of -1
would be an example of one. Normal people believe in neither idea.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

/tuning/topicId_44196.html#44226

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > And, now I beginning to wonder what an *integer* is anyway...
Maybe
> > that's a "Tuning Math" entry... (I'm assuming it's a "real"
rather
> > than an "unreal...! number :)
>
> A number like -2, 0, 7, 53 etc. Number theorists sometimes call
these
> "rational integers" because they hold to strange beliefs such as
that
> integers can be irrational numbers, and that the square root of -1
> would be an example of one. Normal people believe in neither idea.

***Thanks, Gene. I guess I was asking more what makes an integer
seem so *finite?* Is that a "real" number. Didn't somebody say that
an integer like 3 is really 3.00000000...??

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
>
> /tuning/topicId_44196.html#44211
>
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
> >
> > > Basically, the question is why it is that in a *harmonically
> > > vibrating* string, the accompaning vibrations are in simple
> integer
> > > ratios.
> >
> > if the string repeats its motion exactly 440 times per second,
then
> > all the frequency components within the string's vibration must
> > repeat themselves exactly 440 times per second. so the only
allowed
> > frequency components are those that cycle an integer number of
> times
> > within 1/440th of a second, namely 440 Hz, 880 Hz, 1760 Hz,
> > etc. . . . any frequency components in-between would look
different
> > at a point in time t and at a point in time t+1/440, thus
spoiling
> > the supposed periodicity.
>
> ***Thanks, Paul. But what happens to the 3:1 or 1320 Hz? Wouldn't
> that also be a part of it?

yes, sorry i skipped that one.

> And, now I beginning to wonder what an *integer* is anyway...
Maybe
> that's a "Tuning Math" entry... (I'm assuming it's a "real" rather
> than an "unreal...! number :)

i should have said "whole number" instead of integer. the whole
numbers are 1, 2, 3, 4, 5, . . . etc. "integers" also includes 0, -
1, -2, -3, -4, . . . as well as the whole numbers.