question from math moron

Well... I certainly hope this will be a "math intensive" list... and
people won't necessarily be frightened by a little such acumen on
this
list!

However, from the "math moron" of this group, I have to ask a
question... and I know it won't be TOO OT (or, hopefully, "outre")
for this list.

Paul mentioned that the terms "logarithmic" and "exponential" are
OPPOSITES.

I find this intriguing! Could I have a quick "dummy refresher??"
_________ ____ __ __
Joseph Pehrson

Joseph,

exp(log(x)) = x

log(exp(x)) = x

They undo one another.

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:

http://www.egroups.com/message/harmonic_entropy/3

> Joseph,
>
> exp(log(x)) = x
>
> log(exp(x)) = x
>
> They undo one another.

You know... believe it or not, that's coming back a little bit now!
___________ ___ __ _
Joseph Pehrson

[Joseph Pehrson:]
>>Well... I certainly hope this will be a "math intensive" list... and
>>people won't necessarily be frightened by a little such acumen on
>>this
>>list!
>>
>>However, from the "math moron" of this group, I have to ask a
>>question... and I know it won't be TOO OT (or, hopefully, "outre")
>>for this list.
>>
>>Paul mentioned that the terms "logarithmic" and "exponential" are
>>OPPOSITES.
>>
>>I find this intriguing! Could I have a quick "dummy refresher??"

[Paul E:]
>Joseph,
>
>exp(log(x)) = x
>
>log(exp(x)) = x
>
>They undo one another.

Correct, but doesn't really explain the functions.

Joe, I gave a stab at explaining logs and exponents at:

http://www.egroups.com/message/tuning/4284

One quick excerpt:
>
> Logs are a mathematical way of relating doubling to linear motion.
> Exponents (powers) go the other way, relating linear motion to doubling.
>

In using the word "doubling", I am taking slight liberty with the notion
of natural logs, based upon e (about 2.718281828), but conceptually it's
the same game.

In that same post, I try to describe how to convert cents to frequency
ratios and back using a hand calculator.

JdL

--- In harmonic_entropy@egroups.com, "John A. deLaubenfels"
<jdl@a...>
wrote:

http://www.egroups.com/message/harmonic_entropy/5

Thank you, John! Your earlier post is a terrific one and has a
fantastically humorous opener.

I am saving it in my "permanent" collection. It's very helpful!

______________ ___ __ _
Joseph Pehrson

--- In harmonic_entropy@egroups.com, "John A. deLaubenfels"
<jdl@a...>
wrote:

http://www.egroups.com/message/harmonic_entropy/5

John... you'll be happy to learn that I got my calculator to all the
things you mention in your early post (!!)

Actually, that much I had reviewed before.... ratios to cents, etc.
It's in the very back of the Bachus "Acoustics" book I have been
studying again.

Thanks!
___________ ___ __ _ _
Joseph Pehrson

Joe, glad you enjoyed the opening scene of that page! We nerdy guys
may never have had this happen, and may never HAVE it happen, but the
fantasy never dies, eh? Of course, impressing a beautiful woman through
mathematical tours de force does strain even the credulity of a silly
dream, perhaps!

Glad, too, that you're "calculator proficient." Is it clear what's
going on with logs and exponents? You know, the basic relationship,
as exemplified by the piano and its linear vs. exponential nature?
Once that's clear, then the fact that logs and exps are the inverse of
each other becomes obvious. Just going through the transform forward,
backward.

JdL

--- In harmonic_entropy@egroups.com, "John A. deLaubenfels"
<jdl@a...>
wrote:

http://www.egroups.com/message/harmonic_entropy/8

>
> Glad, too, that you're "calculator proficient." Is it clear what's
> going on with logs and exponents? You know, the basic relationship,
> as exemplified by the piano and its linear vs. exponential nature?
> Once that's clear, then the fact that logs and exps are the inverse
of each other becomes obvious. Just going through the transform
forward,backward.
>
> JdL

Ummm. Well, I get the part of "discrete" counting of the keyboard
keys and how the actual pitches are MULTIPLICATION...but how that
SPECIFICALLY applies to logarithms still is rather foggy...

I understand the "exponent" part such as

2^(12/1200)
2^(11/1200)
2^(10/1200)

etc...

Or am I getting that wrong, too??

Gee... this list looks like "review math," not "Harmonic Entropy"
yet...

Well, maybe the pot will boil later...
________ ___ __ _ _
Joseph Pehrson

[I wrote:]
>>Glad, too, that you're "calculator proficient." Is it clear what's
>>going on with logs and exponents? You know, the basic relationship,
>>as exemplified by the piano and its linear vs. exponential nature?
>>Once that's clear, then the fact that logs and exps are the inverse
>>of each other becomes obvious. Just going through the transform
>>forward, backward.

[Joseph Pehrson:]
>Ummm. Well, I get the part of "discrete" counting of the keyboard
>keys and how the actual pitches are MULTIPLICATION...but how that
>SPECIFICALLY applies to logarithms still is rather foggy...

>I understand the "exponent" part such as

>2^(12/1200)
>2^(11/1200)
>2^(10/1200)
>
>etc...
>
>Or am I getting that wrong, too??

Well... if you're walking down the keys on a piano, the ratios would
be represented by:

2^(12/12)
2^(11/12)
2^(10/12)

Where the amount inside the parentheses is "number of octaves"
represented by an interval, and the whole expression calculates the
frequency ratio between the notes. The expressions you've got correctly
calculate frequency ratios for much smaller intervals (12 cents, 11
cents, etc.).

Going the other way, we can ask questions like, "A frequency ratio of
7.5 represents how many octaves?" The answer:

log2(7.5) = log(7.5) / log(2.0) ~= 2.9069 octaves

Check: 2^2.9069 = 7.5

The calculations seem different largely because the calculator has no
"log to base 2" button, but does allow a direct computation of the
inverse, "2 to the power".

>Gee... this list looks like "review math," not "Harmonic Entropy"
>yet...

>Well, maybe the pot will boil later...

I think Paul will forgive us for straying slightly from the stated
purpose of this list, right Paul? And I'm sure Paul is getting set to
"boil" out some H.E. stuff.

JdL

--- In harmonic_entropy@egroups.com, "John A. deLaubenfels"
<jdl@a...>
wrote:

> Well... if you're walking down the keys on a piano, the ratios would
> be represented by:
>
> 2^(12/12)
> 2^(11/12)
> 2^(10/12)
>
> Where the amount inside the parentheses is "number of octaves"
> represented by an interval, and the whole expression calculates the
> frequency ratio between the notes.

Whoops... that's what I was looking for...

> Going the other way, we can ask questions like, "A frequency ratio
of 7.5 represents how many octaves?" The answer:
>
> log2(7.5) = log(7.5) / log(2.0) ~= 2.9069 octaves
>
> Check: 2^2.9069 = 7.5
>
> The calculations seem different largely because the calculator has
no "log to base 2" button, but does allow a direct computation of
the
> inverse, "2 to the power".
>

Oh, I see (I think)... why the logs are used to go the "other way."

Well, a few more examples and some practice and then "yippie do"...
I think I'll be "getting it."

THANKS!
______________ ___ __ __ _ _
Joseph Pehrson

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> Joseph,
>
> exp(log(x)) = x

where log(x) is defined!! log(x) for x real is defined for x > 0. For
x complex, IIRC, it's defined everywhere except on the negative real
axis, and there are principal values to worry about.

>
> log(exp(x)) = x

exp(x) is defined everywhere, so this one works.

BTW, log(x) in the old days referred to the log base 10 of x and ln
(x) to the log base e. The above equations use log(x) to refer to
base e logs.

>
> They undo one another.

--- In harmonic_entropy@egroups.com, "John A. deLaubenfels"
<jdl@a...>
wrote:
> Joe, glad you enjoyed the opening scene of that page! We nerdy guys
> may never have had this happen, and may never HAVE it happen, but
>the
> fantasy never dies, eh? Of course, impressing a beautiful woman
>through
> mathematical tours de force does strain even the credulity of a
>silly dream, perhaps!
> JdL

Believe it or not, I actually did impress a beautiful woman with
tuning math the other day! Of course it is a lot easier in a
mathematics department where people actually appreciate this stuff.
On another note, I attend the Dynamics Days conference every year
that
I can, and it is interesting (but not so surprising) that in the
poster sessions if an exhibit has any musical or sound content, a big
crowd of physicists and mathematicians will gather round. Diana Dabby
is always a big draw with her classical "continuation" algorithms.
She
builds an attractor for a composers style and lets her algorithm
complete a piece from some arbitrary point. It is surprisingly
effective.

(Oops!! Sent this first to the tuning list by mistake! Oh well,
they'll figure it out...)

[John Starrett:]
>Believe it or not, I actually did impress a beautiful woman with
>tuning math the other day!

You rogue! She loved it when you used the words "two to the POWER",
eh?

[John S:]
>On another note, I attend the Dynamics Days conference every year that
>I can, and it is interesting (but not so surprising) that in the
>poster sessions if an exhibit has any musical or sound content, a big
>crowd of physicists and mathematicians will gather round. Diana Dabby
>is always a big draw with her classical "continuation" algorithms.
>She builds an attractor for a composers style and lets her algorithm
>complete a piece from some arbitrary point. It is surprisingly
>effective.

That's very interesting - does she have a web page, do you know?

JdL