And integrating gives parabolic wave

(2*x-1)^2/2-ln(cosh(a*(2*x-1)))/(a*tanh(a))

which can be used to get variable duty cycle triangle waves.
When the formula for sawtooth is defined so that x goes from -1 to 1  
it is rather simple

x-tanh(a*x)/tanh(a)

and parabolic wave is then

1/2*x^2-ln(cosh(a*x))/(a*tanh(a))

Other sigmoids could be used in the place of tanh I guess.

Kalle


Quoting Victor.Lazzarini@nuim.ie:

> I don't think so. Interesting.
>
> Victor
>
> ----- Original Message -----
> From: kalleaho@mappi.helsinki.fi
> Date: Monday, March 28, 2011 11:54 pm
> Subject: [Csnd] A remarkable formula
> To: csound@lists.bath.ac.uk
>
>> Hi,
>>
>> I got inspired by Victor's tanh() ideas and found this
>> remarkable formula
>>
>> 2*x-1-tanh(a*(2*x-1))/tanh(a)
>>
>> If x is a phasor going from 0 to 1, this gives an approximate
>> sawtooth
>> with roughly 2*a harmonics, when a is about 3.5 or greater. Try it!
>>
>> Is this known?
>>
>> Kalle
>>
>>
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>             https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body
>> "unsubscribe csound"
>>
>
> Dr Victor Lazzarini, Senior Lecturer, Dept. of Music,
> National University of Ireland, Maynooth
>
>
>
> Send bugs reports to the Sourceforge bug tracker
>             https://sourceforge.net/tracker/?group_id=81968&atid=564599
> Discussions of bugs and features can be posted here
> To unsubscribe, send email sympa@lists.bath.ac.uk with body  
> "unsubscribe csound"
>
>




Send bugs reports to the Sourceforge bug tracker
            https://sourceforge.net/tracker/?group_id=81968&atid=564599
Discussions of bugs and features can be posted here
To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe csound"