# producing odd&even harmonics

Started by November 2, 2005
```
Hi,

This is a little bit a repost but i have some new information now.

I'd like to produce harmonics in a signal.  Using the chebychev
polynomials, you can produce individual overtones (1,2,3th, 4th, ...
harmonic ) where
T1(x) = 1; T2(x) = x
T_n(x) = 2xT_{n-1} - T_{n-2}.

But how to  produce the undertones?  Can i base myself on aliasing for
this effect? how pls?  Help appreciated!

thnx&bye,
on4cko

```
```hadji elea wrote:
>
>
> Hi,
>
> This is a little bit a repost but i have some new information now.
>
> I'd like to produce harmonics in a signal.  Using the chebychev
> polynomials, you can produce individual overtones (1,2,3th, 4th, ...
> harmonic ) where
>   T1(x) = 1; T2(x) = x
>   T_n(x) = 2xT_{n-1} - T_{n-2}.
>
> But how to  produce the undertones?  Can i base myself on aliasing for
> this effect? how pls?  Help appreciated!

I've never actually heard of using Chebychev polynomials for creating
harmonics, but then I've never really wanted to created harmonics before...

As for undertones, I don't believe there will be any simple way to
generate then other than perhaps looking into a software PLL
implementation.  However if you have many input frequencies and you want
to create the undertones of each of these then I'd think it'd be very
much trickier.  Harmonics shouldn't be too hard, just squaring the
signal and then filtering out the harmonics that you desire.

If squaring isn't sufficient, then use some exponential function to get
the best harmonic response desired, and filter out those not wanted.
Remember that an exponential can be expressed as a power series which
would give all harmonics desired if appropriately configured.
```
```in article Ff2af.329\$xD6.16264@news.xtra.co.nz, Bevan Weiss at
kaizen__@NOSPAMhotmail.com wrote on 11/02/2005 07:27:

>>
>> I'd like to produce harmonics in a signal.  Using the chebychev
>> polynomials, you can produce individual overtones (1,2,3th, 4th, ...
>> harmonic ) where
>> T1(x) = 1; T2(x) = x
>> T_n(x) = 2xT_{n-1} - T_{n-2}.
>>
>
> I've never actually heard of using Chebychev polynomials for creating
> harmonics,

oh, they're built for that!

.--------------------------.
|        N                 |
x(t) = cos(w*t) ----->|   y = SUM a_n * T_n(x)   |----->  y(t)
|       n=0                |
'--------------------------'

N
y(t) = SUM a_n * cos(n*w*t)
n=0

ain't that cool?

now, once you reduce the amplitude of the input sinusoid, then other, less
predictable things happen.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```"Bevan Weiss" <kaizen__@NOSPAMhotmail.com> wrote in message
news:Ff2af.329\$xD6.16264@news.xtra.co.nz...
>>
>>
>> Hi,
>>
>> This is a little bit a repost but i have some new information now.
>>
>> I'd like to produce harmonics in a signal.  Using the chebychev polynomials,
>> you can produce individual overtones (1,2,3th, 4th, ... harmonic ) where
>>   T1(x) = 1; T2(x) = x
>>   T_n(x) = 2xT_{n-1} - T_{n-2}.
>>
>> But how to  produce the undertones?  Can i base myself on aliasing for this
>> effect? how pls?  Help appreciated!
>
> As for undertones, I don't believe there will be any simple way to generate
> then other than perhaps looking into a software PLL implementation.  However
> if you have many input frequencies and you want to create the undertones of
> each of these then I'd think it'd be very much trickier.  Harmonics shouldn't
> be too hard, just squaring the signal and then filtering out the harmonics
> that you desire.

Maybe some type of pitch shifter followed by a low-pass filter would work for
"undertones" (aka sub-harmonics)?  I never tried it myself, just thinking out
loud.  I know there are lots of commercial products that do this, usually
marketed toward bass players.  For example:
http://filters.muziq.be/model/ebs/octabass.  The description seems to imply a
dividing circuit and claims to deal with 2-3 note chords.

```
```Any value in pitch shifting techniques to create a subharmonic?  Just
downshift 1 octave maybe.  And upshift to create normal harmonics.
Remix the created components in the desired ratio.

Such a system would be inherently polyphonic and give output amplitudes
which linearly track the input component amplitudes, desirable for audio
work.  It would avoid the difficulties & delays of pitch recognition.

Caveat: I have not done it myself.
Jim A.

```