What is the waveform with even harmonics called

[Jeremy Shaw-3]

Hello,

What is the name of the waveform which has even harmonics which fall
off at 6dB/octave?

By comparison:

 sawtooth wave = even + odd harmonics falling off at 6dB/octave
 square        = odd harmonics falling off at 6dB/octave
 triangle wave = odd harmonics falling off at 12dB/octave

You can see what it looks like using this matlab/octave code:

# even wave
N = 2000;
t = ([0:1000] / 1000) * 4 * pi;
f = zeros (1,1001);
for k = [1,(1:2:N)+1]
  f = f + sin (k * t) / k;
end
f = f * 0.88678;
plot(t,f)
set(gca,'XTick',0:pi/2:4*pi)
set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'})
grid on
axis normal

Are there any nifty ways to generate this waveform efficiently?

thanks!
- jeremy
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[Charles Henry]

On Mon, Jun 15, 2009 at 3:03 PM, Jeremy Shaw<jeremy@...> wrote:
> Hello,
>
> What is the name of the waveform which has even harmonics which fall
> off at 6dB/octave?

Take the even numbers and divide them by two.  Then you have a full
harmonic series.  So, the even harmonics are just a harmonic series
that is one octave higher.  It's a sawtooth wave plus a sine one
octave down.

There's any number of different functions that will be orthogonal to
the odd-numbered harmonics, but not as likely that it will be one of
these "easy" functions like square, sawtooth, or triangle.

> By comparison:
>
>  sawtooth wave = even + odd harmonics falling off at 6dB/octave
>  square        = odd harmonics falling off at 6dB/octave
>  triangle wave = odd harmonics falling off at 12dB/octave
>
> You can see what it looks like using this matlab/octave code:
>
> # even wave
> N = 2000;
> t = ([0:1000] / 1000) * 4 * pi;
> f = zeros (1,1001);
> for k = [1,(1:2:N)+1]
>  f = f + sin (k * t) / k;
> end
> f = f * 0.88678;
> plot(t,f)
> set(gca,'XTick',0:pi/2:4*pi)
> set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'})
> grid on
> axis normal
>
> Are there any nifty ways to generate this waveform efficiently?
>
> thanks!
> - jeremy
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews, dsp links
> http://music.columbia.edu/cmc/music-dsp
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>

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[Didier Dambrin]

I don't see why waveforms would necessarily need a name.

IMHO to generate any bandlimited (short) waveform efficiently, an inverse
FFT is really the best choice. These days it's very little CPU.

Then once you have direct access to harmonics & phases, you forget
traditional (named) waveforms and you play with sound in a better way.

> Hello,
>
> What is the name of the waveform which has even harmonics which fall
> off at 6dB/octave?
>
> By comparison:
>
> sawtooth wave = even + odd harmonics falling off at 6dB/octave
> square        = odd harmonics falling off at 6dB/octave
> triangle wave = odd harmonics falling off at 12dB/octave
>
> You can see what it looks like using this matlab/octave code:
>
> # even wave
> N = 2000;
> t = ([0:1000] / 1000) * 4 * pi;
> f = zeros (1,1001);
> for k = [1,(1:2:N)+1]
>  f = f + sin (k * t) / k;
> end
> f = f * 0.88678;
> plot(t,f)
> set(gca,'XTick',0:pi/2:4*pi)
> set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'})
> grid on
> axis normal
>
> Are there any nifty ways to generate this waveform efficiently?
>
> thanks!
> - jeremy
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews,
> dsp links
> http://music.columbia.edu/cmc/music-dsp
> http://music.columbia.edu/mailman/listinfo/music-dsp

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[M-.-n]

Sorry to sound like such a noob, but how do you relate the factors of a FFT
that is bound to the sampling rate to one that is related to the harmonic
serie of an oscillator running at a non related frequency ? Do you operate
at the oscillator frequency and then proceed to resampling in the time
domain or is it possible to combine everything in one single operation ?

Thanks
Marc.

-----Original Message-----
From: music-dsp-bounces@...
[mailto:music-dsp-bounces@...] On Behalf Of Didier Dambrin
Sent: mardi 16 juin 2009 0:04
To: A discussion list for music-related DSP
Subject: Re: [music-dsp] What is the waveform with even harmonics called

I don't see why waveforms would necessarily need a name.

IMHO to generate any bandlimited (short) waveform efficiently, an inverse
FFT is really the best choice. These days it's very little CPU.

Then once you have direct access to harmonics & phases, you forget
traditional (named) waveforms and you play with sound in a better way.


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[Andy Farnell]

It's worth remembering that sometimes in sound synthesis
(and other engineering applications) it's the time domain
properties of a wave that we are really interested in.

For example, as a control waveform (say the sawtooth/phasor
used as a display scanline) or if a sound synthesis design
despends upon subsequent non-linear stages (waveshaping).
In these cases we are interested in the shape, so classical
waveform names are really about naming the shape.

a.


I don't see why waveforms would necessarily need a name.

IMHO to generate any bandlimited (short) waveform efficiently, an inverse
FFT is really the best choice. These days it's very little CPU.

Then once you have direct access to harmonics & phases, you forget
traditional (named) waveforms and you play with sound in a better way.

> Hello,
>
> What is the name of the waveform which has even harmonics which fall
> off at 6dB/octave?
>
> By comparison:
>
> sawtooth wave = even + odd harmonics falling off at 6dB/octave
> square        = odd harmonics falling off at 6dB/octave
> triangle wave = odd harmonics falling off at 12dB/octave
>
> You can see what it looks like using this matlab/octave code:
>
> # even wave
> N = 2000;
> t = ([0:1000] / 1000) * 4 * pi;
> f = zeros (1,1001);
> for k = [1,(1:2:N)+1]
>  f = f + sin (k * t) / k;
> end
> f = f * 0.88678;
> plot(t,f)
> set(gca,'XTick',0:pi/2:4*pi)
>

set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'}
)

> grid on
> axis normal
>
> Are there any nifty ways to generate this waveform efficiently?
>
> thanks!
> - jeremy
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews,
> dsp links
> http://music.columbia.edu/cmc/music-dsp
> http://music.columbia.edu/mailman/listinfo/music-dsp

--------------------------------------------------------------------------------




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05:54:00

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[Didier Dambrin]

You would indeed compute a shape somewhat larger (depending on your
interpolation method) than a phase at the samplerate, & resample.

The only flaw with this design is if you want to pitchbend, you may have to
recompute bandlimited shape. But instead, I'd suggest processing at twice
the samplerate, bandlimiting at 1.5 nyquist, leaving you a +-6 semitones
pitch bending.

> Sorry to sound like such a noob, but how do you relate the factors of a
> FFT
> that is bound to the sampling rate to one that is related to the harmonic
> serie of an oscillator running at a non related frequency ? Do you operate
> at the oscillator frequency and then proceed to resampling in the time
> domain or is it possible to combine everything in one single operation ?
>
> Thanks
> Marc.
>
> -----Original Message-----
> From: music-dsp-bounces@...
> [mailto:music-dsp-bounces@...] On Behalf Of Didier Dambrin
> Sent: mardi 16 juin 2009 0:04
> To: A discussion list for music-related DSP
> Subject: Re: [music-dsp] What is the waveform with even harmonics called
>
> I don't see why waveforms would necessarily need a name.
>
> IMHO to generate any bandlimited (short) waveform efficiently, an inverse
> FFT is really the best choice. These days it's very little CPU.
>
> Then once you have direct access to harmonics & phases, you forget
> traditional (named) waveforms and you play with sound in a better way.
>
>
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews,
> dsp links
> http://music.columbia.edu/cmc/music-dsp
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[James Chandler Jr]

On Jun 17, 2009, at 5:50 AM, padawan12@... wrote:

> It's worth remembering that sometimes in sound synthesis
> (and other engineering applications) it's the time domain
> properties of a wave that we are really interested in.
>
> For example, as a control waveform (say the sawtooth/phasor
> used as a display scanline) or if a sound synthesis design
> despends upon subsequent non-linear stages (waveshaping).
> In these cases we are interested in the shape, so classical
> waveform names are really about naming the shape.

Thanks. Made me think about 'turning the idea inside out'-- I wonder  
if anyone has explored the mangling of LFO control waveforms via phase  
shifters? Seems an obvious idea but I don't recall trying it or  
reading about it.

Run the stereotypical triangle or sawtooth LFO thru phase shifting, to  
scatter the harmonics. The altered modulation time patterns might be  
interesting. One could even feed the LFO waveform thru an auto-
sweeping phase shifter so that the LFO wave shape would continually  
'evolve'.

James
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[Didier Dambrin]

Even there you certainly can't rely on names because what's offered as a
sawtooth in synths is quite often not a pure sawtooth, just something that
sounds alike.

> It's worth remembering that sometimes in sound synthesis
> (and other engineering applications) it's the time domain
> properties of a wave that we are really interested in.
>
> For example, as a control waveform (say the sawtooth/phasor
> used as a display scanline) or if a sound synthesis design
> despends upon subsequent non-linear stages (waveshaping).
> In these cases we are interested in the shape, so classical
> waveform names are really about naming the shape.
>
> a.
>
>
> I don't see why waveforms would necessarily need a name.
>
> IMHO to generate any bandlimited (short) waveform efficiently, an inverse
> FFT is really the best choice. These days it's very little CPU.
>
> Then once you have direct access to harmonics & phases, you forget
> traditional (named) waveforms and you play with sound in a better way.
>
>
>
>
>
>> Hello,
>>
>> What is the name of the waveform which has even harmonics which fall
>> off at 6dB/octave?
>>
>> By comparison:
>>
>> sawtooth wave = even + odd harmonics falling off at 6dB/octave
>> square        = odd harmonics falling off at 6dB/octave
>> triangle wave = odd harmonics falling off at 12dB/octave
>>
>> You can see what it looks like using this matlab/octave code:
>>
>> # even wave
>> N = 2000;
>> t = ([0:1000] / 1000) * 4 * pi;
>> f = zeros (1,1001);
>> for k = [1,(1:2:N)+1]
>>  f = f + sin (k * t) / k;
>> end
>> f = f * 0.88678;
>> plot(t,f)
>> set(gca,'XTick',0:pi/2:4*pi)
>>
> set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'}
> )
>> grid on
>> axis normal
>>
>> Are there any nifty ways to generate this waveform efficiently?
>>
>> thanks!
>> - jeremy
>> --
>> dupswapdrop -- the music-dsp mailing list and website:
>> subscription info, FAQ, source code archive, list archive, book reviews,
>> dsp links
>> http://music.columbia.edu/cmc/music-dsp
>> http://music.columbia.edu/mailman/listinfo/music-dsp
>
>
> --------------------------------------------------------------------------------
>
>
>
>
> No virus found in this incoming message.
> Checked by AVG - www.avg.com
> Version: 8.5.364 / Virus Database: 270.12.70/2177 - Release Date: 06/15/09
> 05:54:00
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> dupswapdrop -- the music-dsp mailing list and website:
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> links
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[Andy Farnell]

Even there you certainly can't rely on names because what's offered as a
sawtooth in synths is quite often not a pure sawtooth, just something that
sounds alike.


Absolutely, I seem to remember an 'expansion board' or some kind of ROM
I once had for my Roland 1080, nothing but variations on sawtooths
from different analogue synthesisers!

My distinction is between the sound, and the use, if you like. For
instance, when I specify a phasor, there is no subjectivity
in that name, it's like invoking a Platonic shape - I expect
a perfectly linear unipolar function. The 'ideal' triangle wave
is also just that... ideal, and so forth.

a.

> It's worth remembering that sometimes in sound synthesis
> (and other engineering applications) it's the time domain
> properties of a wave that we are really interested in.
>
> For example, as a control waveform (say the sawtooth/phasor
> used as a display scanline) or if a sound synthesis design
> despends upon subsequent non-linear stages (waveshaping).
> In these cases we are interested in the shape, so classical
> waveform names are really about naming the shape.
>
> a.
>
>
> I don't see why waveforms would necessarily need a name.
>
> IMHO to generate any bandlimited (short) waveform efficiently, an inverse
> FFT is really the best choice. These days it's very little CPU.
>
> Then once you have direct access to harmonics & phases, you forget
> traditional (named) waveforms and you play with sound in a better way.
>
>
>
>
>
>> Hello,
>>
>> What is the name of the waveform which has even harmonics which fall
>> off at 6dB/octave?
>>
>> By comparison:
>>
>> sawtooth wave = even + odd harmonics falling off at 6dB/octave
>> square        = odd harmonics falling off at 6dB/octave
>> triangle wave = odd harmonics falling off at 12dB/octave
>>
>> You can see what it looks like using this matlab/octave code:
>>
>> # even wave
>> N = 2000;
>> t = ([0:1000] / 1000) * 4 * pi;
>> f = zeros (1,1001);
>> for k = [1,(1:2:N)+1]
>>  f = f + sin (k * t) / k;
>> end
>> f = f * 0.88678;
>> plot(t,f)
>> set(gca,'XTick',0:pi/2:4*pi)
>>
>

set(gca,'XTickLabel',{'0','pi/2','pi','3pi/2','2pi','5pi/2','3pi','7pi/2','4pi'}

> )
>> grid on
>> axis normal
>>
>> Are there any nifty ways to generate this waveform efficiently?
>>
>> thanks!
>> - jeremy
>> --
>> dupswapdrop -- the music-dsp mailing list and website:
>> subscription info, FAQ, source code archive, list archive, book reviews,
>> dsp links
>> http://music.columbia.edu/cmc/music-dsp
>> http://music.columbia.edu/mailman/listinfo/music-dsp
>
>
>

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[Jeremy Shaw-3]

So, in summary,

At first it seems unusual that we have names for simply defined wave forms such as:

 sawtooth wave = even + odd harmonics falling off at 6dB/octave
 square        = odd harmonics falling off at 6dB/octave
 triangle wave = odd harmonics falling off at 12dB/octave

But that we don't have a counterpart to square and triangle for even
harmonics. But, upon further examination it makes more sense. The
above waveforms (as well as sine)

 1. sound fairly distinct
 2. are close to other fairly primative shapes (squares, triangles, circles)
 2. are not trivially generated by simple additions of two existing waveforms (ie, not easy to create on a two oscillator synth)

The waveform with even harmonics falling off at 6dB/octave has none of
these properties:

 1. it sounds like a sawtooth with a sine-wave sub-osc an octave down
 2. it looks really funny
 3. it is trivial to recreate using a normal two oscilator synth

And, in fact, when creating the matlab code, my code initially had a
bug where it did not include the 'fundamental' frequency, and so I
just got out a sawtooth wave. :p

As for wanting a name -- it makes it easier to search on google for a
waveform that has a name versus one that does not. But, in this case,
it seems like a very uninteresting waveform anyway.

I was curious about it, because I have seen many people claim that
people prefer distortion that provides even order harmonics instead of
odd order harmonics. So, it seemed amiss that there was not a common
waveform with only even order harmonics...

(Actually, I did create a different wave with even order harmonics --
namely the fully-rectified sine wave).

Thanks!
- jeremy



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[Didier Dambrin]

> So, in summary,
>
> At first it seems unusual that we have names for simply defined wave forms
> such as:
>
> sawtooth wave = even + odd harmonics falling off at 6dB/octave
> square        = odd harmonics falling off at 6dB/octave
> triangle wave = odd harmonics falling off at 12dB/octave
>
> But that we don't have a counterpart to square and triangle for even
> harmonics. But, upon further examination it makes more sense. The
> above waveforms (as well as sine)
>
> 1. sound fairly distinct

triangle isn't much interesting, sounds like a slighty saturated sine

> 2. are close to other fairly primative shapes (squares, triangles,
> circles)

that's pretty much the only reason IMHO

> 2. are not trivially generated by simple additions of two existing
> waveforms (ie, not easy to create on a two oscillator synth)

square can be computed out of saws

> The waveform with even harmonics falling off at 6dB/octave has none of
> these properties:
>
> 1. it sounds like a sawtooth with a sine-wave sub-osc an octave down
> 2. it looks really funny
> 3. it is trivial to recreate using a normal two oscilator synth
>
> And, in fact, when creating the matlab code, my code initially had a
> bug where it did not include the 'fundamental' frequency, and so I
> just got out a sawtooth wave. :p
>
> As for wanting a name -- it makes it easier to search on google for a
> waveform that has a name versus one that does not. But, in this case,
> it seems like a very uninteresting waveform anyway.
>
> I was curious about it, because I have seen many people claim that
> people prefer distortion that provides even order harmonics instead of
> odd order harmonics. So, it seemed amiss that there was not a common
> waveform with only even order harmonics...
>
> (Actually, I did create a different wave with even order harmonics --
> namely the fully-rectified sine wave).
>
> Thanks!
> - jeremy
>
>
>
> --
> dupswapdrop -- the music-dsp mailing list and website:
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[richarddobson]

Didier Dambrin wrote:
..
>
> triangle isn't much interesting, sounds like a slighty saturated sine
>


But useful: ~very~ important as a modulation source (LFO), and where (as
with the other waveforms) the absence of wiggles may often be valuable
musically. Waveforms are not just to be listened to directly. As control
signals the geometric waveforms have a key role, while all manner of
extended exotica (smoothly-varying random waveforms, self-modulated
waveforms etc) also matter. Sometimes, classic geometric "perfection" is
required; at other times the absence of predictability (or simply a more
subtle organic quality) is sought.

Richard Dobson




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[Didier Dambrin]

Yes but as LFO or for other purpose we're talking about something else, you
wouldn't want to generate them bandlimited in those cases. Same for FM
modulators.

>> triangle isn't much interesting, sounds like a slighty saturated sine
>>
>
>
> But useful: ~very~ important as a modulation source (LFO), and where (as
> with the other waveforms) the absence of wiggles may often be valuable
> musically. Waveforms are not just to be listened to directly. As control
> signals the geometric waveforms have a key role, while all manner of
> extended exotica (smoothly-varying random waveforms, self-modulated
> waveforms etc) also matter. Sometimes, classic geometric "perfection" is
> required; at other times the absence of predictability (or simply a more
> subtle organic quality) is sought.
>
> Richard Dobson
>
>
>
>
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[Andy Farnell]

> Yes but as LFO or for other purpose we're talking about something else

Yes, exactly, we are talking about something else: Those cases where time
domain features are more significant to an operation than spectra.

> you  wouldn't want to generate them bandlimited in those cases. Same for FM
modulators.

Why not? There are cases where a specific shape is required AND a bandlimited
spectrum is desirable.

Perhaps it's helpful to distinguish 'bandlimited' from 'Fourier (additvely)
constructed. All additively constructed signals are bandlimited, but not
all bandlimited signals are constructed additively. There are other ways,
such as piecewise polynomial construction, that can be shown to give predictable
shapes and bandwidths. When used as a modulator, for example as AM in a fast autopanner, the absence of sharp corners avoids clicks (non-bandlimited modulation
artifacts), and the shape (say triangle) gives the desired behaviour.

As Perl programmers like to say, there's more than one way to do it. :)

a.

>> triangle isn't much interesting, sounds like a slighty saturated sine
>>
>
>
> But useful: ~very~ important as a modulation source (LFO), and where (as
> with the other waveforms) the absence of wiggles may often be valuable
> musically. Waveforms are not just to be listened to directly. As control
> signals the geometric waveforms have a key role, while all manner of
> extended exotica (smoothly-varying random waveforms, self-modulated
> waveforms etc) also matter. Sometimes, classic geometric "perfection" is
> required; at other times the absence of predictability (or simply a more
> subtle organic quality) is sought.
>
> Richard Dobson
>
>
>
>
> --
> dupswapdrop -- the music-dsp mailing list and website:
> subscription info, FAQ, source code archive, list archive, book reviews,
> dsp links
> http://music.columbia.edu/cmc/music-dsp
> http://music.columbia.edu/mailman/listinfo/music-dsp

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[spambait1000006]

This thread got me playing with waveforms and I got to looking at a
square wave
sum( sin(2pi n x)/n , n=1,3,5... )
but with the sine->cosine
sum( cos(2pi n x)/n , n=1,3,5... )
effectively, a 90 degree or pi/2 radian phase shift of each term. This
should be a "sound alike" waveform for a square wave. You get a waveform
that diverges that looks a little like this:

.  |           |
. | |         | |
./   \       /   \
.------------------------
.      \   /       \   /
.       | |         | |
.        |           |

and managed to calculate a closed form for it:
ln((1+cos(2 pi x))/(1-cos(2 pi x))
(Does this waveform have a name? If not I suggest L7 :-)

Clearly, going from a bounded signal to an unbounded signal isn't
very useful, but it does suggest one question, is it possible to go
the other way? Are there bounded sound-alikes for impulse or other
unbounded waveforms? It seems /possible/ that if you re-arrange the
phases of the sine waves that make up any signal of finite power that
you should be able to create a bounded signal.

This could be useful since for a naive digital aproximate impulse (one
or the max value for the 1st sample of a period, zero for all others)
there isn't much power in the wavefrom since it is zero most of the time,
while no other waveform of the same peak value can have more power
than a square wave, so you might be able to pack more power in a
"sound alike" waveform.
For example, for a band limited triangle like
sum( cos(2pi n x)/n^2 , n=1,2,3 )
you can reduce the peak value a bit by turning the last term over:
sum( cos(2pi n x)/n^2 , n=1,2 )-cos(2pi 3 x)/9
without changing the sound (if there is no wave shaping down the line!)
so you could multiply it by G>1 and keep the same max amplitude for a
little more power.

CR
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[spambait1000006]

Just thought of an amusing class of "sound alikes" A square wave of
frequncy omega and a sawtooth of freq 2 omega, (both the same amplitude,
one synced so one makes a -1 to 1 transition while the other goes 1 to -1)
add to a sawtooth of freqency of omega, but, since they have no partials
in common, we can add them in any (fixed) phase relation and have a
waveform that can look quite a bit different yet sound the same.

   CR

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[Martin Eisenberg]

spambait1000006@... wrote:

> ln((1+cos(2 pi x))/(1-cos(2 pi x))
> (Does this waveform have a name? If not I suggest L7 :-)

Depending on the richness of your taxonomic system,
"arctanh-waveshaped cosine" may count as a name ;)


Martin

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[Scott Nordlund]

> Clearly, going from a bounded signal to an unbounded signal isn't
> very useful, but it does suggest one question, is it possible to go
> the other way? Are there bounded sound-alikes for impulse or other
> unbounded waveforms? It seems /possible/ that if you re-arrange the
> phases of the sine waves that make up any signal of finite power that
> you should be able to create a bounded signal.

I think for an impulse you'd want a chirp signal, though this might sound a bit odd at low frequencies.

A similar "odd impulse" waveform would also be useful.

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[spambait1000006]

On Thu, 9 Jul 2009, Scott Nordlund wrote:
> A similar "odd impulse" waveform would also be useful.

odd/even odd or odd-ball odd?

anyway, a band limited impulse wave

sum( cos(2*n*pi*x), n=1..m) = -1/2 + sin((2*n+1)*x)/sin(pi*x)

and similarly a band limited "gnomeulse" wave

sum( sin(2*n*pi*x), n=1..m) = [ cos(pi*x) - cos(pi*(2*n+1)*x)]/sin(pi*x)

which for m=4 is about 20% shorter but would require one more lookup or
calculatin of a cosine and should integrate up to the L7 wave (Martin:
mostly the name was to have something to call it in my notes and as a
joke, but I think L7 would look better on a button than arctanh-waveshaped
cosine anyway :-)

   CR
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[Andy Farnell]

Parabolic waves are rather useful in natural sound synthesis
and make appearances in many situations, perhaps for geometrical
reasons and as good approximations of other behaviour. Examples
include the ideal pressure signature of a raindrop, the edge
effect of a rotating fan blade, piston pulses in an engine
model, and the sound of distant thunder (where atmospheric
effects seem to amount to integration of the N-wave signature).
(which can be considered a single cycle of a periodic bipolar
sawtooth)

andy

> Charlie Morrow wrote :
> The > sawtooth has all harmonics present and a 1/x roll-off (-6dB
> / oct) to its frequency spectrum.  The parabola has a 1/(x^2)
> roll-off (-12dB / oct) and also has all harmonics present.
>
> I have never seen this waveform on the menu of any analogue
> or digital synthesiser.  

I included the even harmonics 1/(N*N) in the Seer Systems Reality software
synthesizer in 1997 and nicknamed it the "Missing Link".  1/(N*N) waveforms are
more useful in modulation synthesis. The lack of demand for this waveform is
the similar to absence of celebrity for the triangle:  both are rather
uninteresting on their own and are more dynamically simulated with a low pass
filter on either the sawtooth or square wave.



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