- Thanks Allan! But... I still don't understand how to use this

information. I'm using Microsoft Excel to compute the values.

'exp' means 'e' raised to the powers 0.085 to -0.062, right?

Can you please explain how the numbers '-48' to '48' fit into

the formula, and how I use the range from '0.085' to '-0.062'?

Does that decrement in some way?

When I use the Excel formula EXP(x), where 'x' is the decremented

range from 0.085 to -0.062, I get ratios with cents-values that

are quite a bit larger than those on the graph, and the the

graph of these values is linear, which is not the case with the

values given in the Rhodes manual. And, the 'horizontal axis'

numbers 48 to -48 are not part of the calculation.

???

Thanks.

-monz

--- In tuning@egroups.com, Allan Myhara <amyhara@m...> wrote:

> http://www.egroups.com/message/tuning/13863

>

> Hello!

> A rare opportunity for me to contribute here! If you

> re-label the horizontal axis as -48, -36, -24, -12, 0, 12, 24,

> 36, 48 then you can use exponential functions and an accurate

> formulation is exp^0.085 - exp^-0.062. I hope this is useful

> to you!

>

> " Monz" wrote:

> > > The following values are taken from the stretch-tuning chart

> > > in the _Rhodes Keyboard Instruments USA Service Manual_,

> > > (c) 1979, which came with my Rhodes electric piano.

> >

> >

> > Figuring that someone familiar with calculus might recognize

> > the key to the formula quickly by seeing a graph of the values,

> > I made one and uploaded it to:

> >

> > http://www.egroups.com/files/tuning/monz/rhodes.jpg

-monz

http://www.ixpres.com/interval/monzo/homepage.html - From: " Monz" <MONZ@...>

> Thanks Allan! But... I still don't understand how to use this

I just looked at what I sent. Sorry, the formula should have been

> information. I'm using Microsoft Excel to compute the values.

>

> 'exp' means 'e' raised to the powers 0.085 to -0.062, right?

e^0.085x - e^-0.062x. I neglected to type the variable, x. Yes, e raised

to those powers, the products of the constants and the variable. I

didn't mean to imply a range of values. There are two exponentials in

the function: e^0.085x minus e^-0.062x. Simply feed it the numbers -48

to 48.

> Can you please explain how the numbers '-48' to '48' fit into

--

> the formula, and how I use the range from '0.085' to '-0.062'?

> Does that decrement in some way?

>

> When I use the Excel formula EXP(x), where 'x' is the decremented

> range from 0.085 to -0.062, I get ratios with cents-values that

> are quite a bit larger than those on the graph, and the the

> graph of these values is linear, which is not the case with the

> values given in the Rhodes manual. And, the 'horizontal axis'

> numbers 48 to -48 are not part of the calculation.

>

> ???

Bye for now

Allan Myhara

Winnipeg, Manitoba, Canada - --- In tuning@egroups.com, Allan Myhara <amyhara@m...> wrote:
> http://www.egroups.com/message/tuning/13879

Thanks, Allan! It fits the Rhodes numbers like a glove!!

>

> I just looked at what I sent. Sorry, the formula should have

> been e^0.085x - e^-0.062x. I neglected to type the variable, x.

> Yes, e raised to those powers, the products of the constants

> and the variable. I didn't mean to imply a range of values.

> There are two exponentials in the function: e^0.085x minus

> e^-0.062x. Simply feed it the numbers -48 to 48.

If one numbers the variable _x_ from 1 to 88, as piano keys,

then the precise formula in Excel format is:

=EXP(0.085*(x-48))-EXP(-0.062*(x-48))

In Excel, 'EXP()' means 'e^'.

Would you mind explaining to us how you derived it, step by

step? I need it in 'remedial calculus 001'.

In case anyone's wondering why I wanted this: be on the lookout

for a new retuning of an old piece I did originally in 12-tET.

It features a piano timbre, so it's going to be retuned in a

stretch pseudo-JI.

I need the formula so that after I calculate the amount of

MIDI pitch-bend to add or subtract from the 12-tET MIDI-note

number, I can simply plug these in as the variables in Allan's

formula to get the amount of additional pitch-bend needed for

the proper amount of stretch.

-monz

http://www.ixpres.com/interval/monzo/homepage.html - Monz wrote:

> > I just looked at what I sent. Sorry, the formula should have

I thought it should do quite well.

> > been e^0.085x - e^-0.062x. I neglected to type the variable, x.

> > Yes, e raised to those powers, the products of the constants

> > and the variable. I didn't mean to imply a range of values.

> > There are two exponentials in the function: e^0.085x minus

> > e^-0.062x. Simply feed it the numbers -48 to 48.

> Thanks, Allan! It fits the Rhodes numbers like a glove!!

> If one numbers the variable _x_ from 1 to 88, as piano keys,

I really didn't have to apply any calculus (although exponentials are

> then the precise formula in Excel format is:

>

> =EXP(0.085*(x-48))-EXP(-0.062*(x-48))

>

> In Excel, 'EXP()' means 'e^'.

>

> Would you mind explaining to us how you derived it, step by

> step? I need it in 'remedial calculus 001'.

quite common in calculus). At first I thought of approximating it with

polynomials, but I soon realized that the horizontal axis was labelled

arbitrarily and that the shape of the graph was an exponential function

on both sides of the graph from the zero crossing point from which the

graph trends negative to the left and positive to the right. An

exponential function, on its own, never decreases to zero nor is ever

negative, it can only get really small as its exponent gets negatively

big. (But it can be forced negative by multiplying it by -1, in other

words by putting a minus sign in front of it.) So, setting the variable

to -48, the positive term, with its negative exponent, could be ignored

as I adjusted the constant in the exponent of the negative term, with

its positive exponent, to make the negative term agree with the left end

of the graph. Then, setting the variable to +40, I followed the same

procedure with the positive term, ignoring the contribution of the

negative term. After finding suitable constants, I plugged in the

different variable values of -36 through +36 into the two term function

to see if it matched the numbers indicated by the graph, and lo and

behold, it looked nice. Of course this is a just-so-story, things like

this are not reasoned out quite so smoothly!

> In case anyone's wondering why I wanted this: be on the lookout

--

> for a new retuning of an old piece I did originally in 12-tET.

> It features a piano timbre, so it's going to be retuned in a

> stretch pseudo-JI.

>

> I need the formula so that after I calculate the amount of

> MIDI pitch-bend to add or subtract from the 12-tET MIDI-note

> number, I can simply plug these in as the variables in Allan's

> formula to get the amount of additional pitch-bend needed for

> the proper amount of stretch.

Bye for now

Allan Myhara

Winnipeg, Manitoba, Canada