[music-dsp] family of soft clipping functions ... crossed with ... Modular Synthesis Language Moselle Alpha Release Today.
As an illustration of my newly-released software I dug through the
recent archive for something that would be easy and fun to implement
in Moselle, and came across this post from forum stalwart Robert
Bistow-Johnson.
The following is a Moselle program (or patch) that implements the
first half-dozen functions in Robert's series of soft-clippers. MIDI
general controller 1 selects which clipping function to use, while
general controller 2 selects the waveform to input. (The oscillator
is spec'd as an LFO as the LFO doesn't scale its amplitude with pitch,
which saved me a bit of typing.) Each of the functions is repeated in
the Oscilloscope module so they can all be seen simultaneously, even
though only one is heard at a time.
See a screenshot of the Oscilloscope with waveforms and harmonic
spectrum at:
http://moselle.invisionzone.com/uploads/gallery/album_1/gallery_1_1_12533.gif
The self-contained Moselle IDE is available for no-cost download at:
http://moselle.invisionzone.com/index.php?/files/file/2-moselle-alpha-release
The following is literally the Moselle patch that produced the
screenshot (and woke up my wife while I was jamming with it).
[LFO]
Waveform = Select( General2, Sawtooth, Triangle, Sine)
Frequency = Voice:Frequency
[Voice]
Mono = IF( LFO -1, -1,
LFO 1, 1,
Select( General1,
LFO,
(LFO - 1 * LFO^3/3
)*3/2,
(LFO - 2 * LFO^3/3 + 1 * LFO^5/5
)*15/8,
(LFO - 3 * LFO^3/3 + 3 * LFO^5/5 - 1 * LFO^7/7
)*35/16,
(LFO - 4 * LFO^3/3 + 6 * LFO^5/5 - 4 * LFO^7/7 + 1 * LFO^9/9
)*315/128,
(LFO - 5 * LFO^3/3 + 10 * LFO^5/5 - 10 * LFO^7/7 + 5 * LFO^9/9 -
LFO^11/11 )*3465/1280
)
[Scope]
Probe1 = LFO
Probe2 = (LFO - 1 * LFO^3/3
)*3/2
Probe3 = (LFO - 2 * LFO^3/3 + 1 * LFO^5/5
)*15/8
Probe4 = (LFO - 3 * LFO^3/3 + 3 * LFO^5/5 - 1 * LFO^7/7
)*35/16
Probe5 = (LFO - 4 * LFO^3/3 + 6 * LFO^5/5 - 4 * LFO^7/7 + 1 *
LFO^9/9 )*315/128
Probe6 = (LFO - 5 * LFO^3/3 + 10 * LFO^5/5 - 10 * LFO^7/7 + 5 *
LFO^9/9 - LFO^11/11 )*3465/1280
Probe7 = Voice:Mono
Start = LFO:SyncOut
Stop= LFO:SyncOut
FreqHint= LFO:Frequency
at the last AES in NYC, i was talking with some other folks (that likely
hang out here, too) about this family of soft clipping curves made outa
polynomials (so you have some idea of how high in frequency any
generated images will appear).
these are odd-order, odd-symmetry polynomials that are monotonic from -1
x +1 and have as many continuous derivatives as possible at +/- 1
where these curves might be spliced to constant-valued rails.
the whole idea is to integrate the even polynomial (1 - x^2)^N
x
g(x) = integral{ (1 - v^2)^N dv}
0
you figger this out using binomial expansion and integrating each power
term.
normalize g(x) with whatever g(1) is so that the curve is g(x)/g(1) and
splice that to two constant functions for the rails
{ -1x = -1
{
f(x) = { g(x)/g(1) -1 = x = +1
{
{ +1 +1 = x
you can hard limit (at +/- 1) before passing through this soft clipper
and it still works fine. but it has some gain in the linear region
which is g(0).
if you want to add some even harmonic distortion to this, add a little
bit of
(1 - x^2)^M
to f(x) for |x| 1 and it's still smooth everywhere, but there is a
little DC added (which has to be the case for even-symmetry distortion).
M does not have to be the same as N and i wouldn't expect it to be.
you can think of f(x) as a smooth approximation the sign or signum
function
sgn(x) = lim f(a*x)
a - +inf
or
sgn(x) = lim f(x)
N - +inf
which motivates using this as a smooth approximation of the sgn(x)
function as an alternative to
(2/pi)*arctan(a*x) or tanh(a*x). from the sgn(x) function, you can
create smooth versions of the unit step function and use that for
splicing. as many derivatives are continuous in the splice as possible.
and it satisfies conditions of symmetry and complementarity that are
useful in our line of work.
u(x) = 1/2 * (1 + sgn(x)) =approx 1/2*(1 + f(x))
you can run a raised cosine (Hann) through this and get a more flattened
Hann. in some old stuff i wrote, i dubbed this window:
w(x) = 1/2 + (9/16)*cos(pi*x) - (1/16)*cos(3*pi*x)
as the Flattened Hann Window but long ago Carla Scaletti called it the
Bristow-Johnson window in some Kyma manual. i don't think it deserves
that label (i've seen that function in some wavelet/filterbank lit since
for half-band filters). you get that window by running a simple Hann
through the biased waveshaper:
1/2 * ( 1 + f(2x-1) )
with N=1. you will get an even more pronounced effect (of smoothly
flattening the Hann) with higher N.
below
Re: [music-dsp] family of soft clipping functions.
On 03/11/2013, robert bristow-johnson r...@audioimagination.com wrote: the point is that if you upsample, then soft-clip, then LPF, and finally downsample back to the original sample rate, you need only prevent the aliases from getting back into your *original* baseband. it doesn't matter that *some* of the images have folded over and become non-harmonic aliases, just as long as they do not survive into the final output. i don't know how better to explain it, without a drawing. Oh, hang on, I think I get it. So an nth-order polynomial expands the spectrum produced by n times. Let's say a 3rd-order poly turns a bandwidth of 22kHz into one of 66kHz. What you're saying is that you only need to upsample a 44.1kHz signal by 2, do the shaping, and run a low pass at the original 22kHz - because whilst everything above 44.1kHz will fold back, it'll only fold back down to frequencies above 22.3kHz and leave your original 0-22kHz bandwidth alone, right? (But, say, a 4th-order poly would require upsampling by more than 2, otherwise it'd produce harmonics up to 88kHz, which would fold back into your desired bandwidth.) Have I understood properly? -- 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
Re: [music-dsp] family of soft clipping functions.
I think I've been caught out on the html email thing as well, I wonder
how many posts have gone completely missing that I've sent? Here is
one I sent 5 days ago, sorry if this is a double up, I checked the
archives but couldn't find anything:
Hi Robert,
Thanks very much for the post! I plotted the shapes and noticed that
as the order increased the maximum output value decreased. I find for
audio use it's nice to have the maximum output remain constant, for
example at +-1 like a tanh. This can easily be added to your equations
by scaling the x^2 term by the appropriate amount, lets call it a,
so the final formula is:
f[x, a, n] = Integrate[(1 - a v^2)^n, {v, 0, x}]
So then the input bounds will change from -1, 1 to some other value
say -x1 to x1, and then there are two variables a and x1 that can
be solved by the following two equations:
f'[x1, a, n] == 0 and f[x1, a, n] == 1
The first few solutions to these equations are:
3rd order with x1 = 3/2, f3(x) = x - (4 x^3)/27
5th order with x1 = 15/8, f5(x) = x - (128 x^3)/675 + (4096 x^5)/253125
7th order with x1 = 35/16, f7(x) = x - (256 x^3)/1225 + (196608
x^5)/7503125 - (16777216 x^7)/12867859375
9th order with x1 = 315/128, f9(x) = x - (65536 x^3)/297675 +
(536870912 x^5)/16409334375 - (17592186044416 x^7)/6838508054109375 +
(72057594037927936 x^9)/872422665003003515625
By comparing the results for lots of values of n I noticed that these
are the same solutions that you get if you have f[x] = x + a(3) x^3 +
... + a(2n+1) x^(2n+1) and solve for the a coeffs and x1 using the
system of equations f[x1] == 1, f'[x1]==0, ..., f'n[x1]==0.
Enjoy!
Andy
--
cytomic - sound music software
--
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subscription info, FAQ, source code archive, list archive, book reviews, dsp
links
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Re: [music-dsp] family of soft clipping functions.
On 11/2/13 6:58 PM, Wen Xue wrote: But, soft-clipping is not going to change periodicity, is it? it should not. that's why we don't want non-harmonic components (aliases) to survive the soft-clipping process. the point is that if you upsample, then soft-clip, then LPF, and finally downsample back to the original sample rate, you need only prevent the aliases from getting back into your *original* baseband. it doesn't matter that *some* of the images have folded over and become non-harmonic aliases, just as long as they do not survive into the final output. i don't know how better to explain it, without a drawing. bestest, r b-j So if you soft-clip a sine wave, be it polynomial or not, the outcome is periodical at the same period, so contains only perfect harmonics. It cannot behave in the folded alias way one usually suspect. On 02/11/2013 06:36, robert bristow-johnson wrote: just to be clear. the general rule is that an Nth-order polynomial can generate images at frequencies up to the Nth multiple of the frequency of the original baseband image. it is sufficient to oversample by a factor of (N+1)/2 to prevent any of these generated images from potentially folding back into the baseband. e.g. 3rd-order softclipping requires upsampling by a factor of 2. another e.g. 7th-order softclipping requires upsampling by a factor of 4 to avoid any folded aliases from contaminating the original baseband. -- 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
Re: [music-dsp] Family of soft clipping functions.
If you are just looking for a minimally noticable softclip, with analog sound, just two three-order distortions in series is optimal. With a crossfade on 1 and 2, and a threshold on knee-depth, it really gives a sound many wants, and is very optimal, in terms of computing power required aswell. This I have already done in my softclip plugin. Peace Be With You. -- Ove Karlsen, www.ovekarlsen.com -- 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
Re: [music-dsp] family of soft clipping functions.
But, soft-clipping is not going to change periodicity, is it? So if you soft-clip a sine wave, be it polynomial or not, the outcome is periodical at the same period, so contains only perfect harmonics. It cannot behave in the folded alias way one usually suspect. Xue On 02/11/2013 06:36, robert bristow-johnson wrote: just to be clear. the general rule is that an Nth-order polynomial can generate images at frequencies up to the Nth multiple of the frequency of the original baseband image. it is sufficient to oversample by a factor of (N+1)/2 to prevent any of these generated images from potentially folding back into the baseband. e.g. 3rd-order softclipping requires upsampling by a factor of 2. another e.g. 7th-order softclipping requires upsampling by a factor of 4 to avoid any folded aliases from contaminating the original baseband. -- 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
Re: [music-dsp] family of soft clipping functions.
On 10/30/13 4:24 PM, Theo Verelst wrote: For whoever follows this, I thought and worked a bit about the clipping idea, and especially I was thinking about the harmonic behavior of the clipping function of N degree, so i tried using Maxima, and found there are distinct harmonics added to a sine wave passing through it. just to be clear. the general rule is that an Nth-order polynomial can generate images at frequencies up to the Nth multiple of the frequency of the original baseband image. it is sufficient to oversample by a factor of (N+1)/2 to prevent any of these generated images from potentially folding back into the baseband. e.g. 3rd-order softclipping requires upsampling by a factor of 2. another e.g. 7th-order softclipping requires upsampling by a factor of 4 to avoid any folded aliases from contaminating the original baseband. -- r b-j r...@audioimagination.com Imagination is more important than knowledge. -- 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
Re: [music-dsp] family of soft clipping functions.
On 2013-11-01, robert bristow-johnson wrote: just to be clear. the general rule is that an Nth-order polynomial can generate images at frequencies up to the Nth multiple of the frequency of the original baseband image. Quite so. So in addition, if you want to really keep it clean of aliasing artifacts, you typically want to oversample your nonlinear system as many times over as is the highest polynomial order you're using. Then there are at at least two troublesome things one the way. First, how overdriven transformers, coils, capacitors and tubes/transistors work. All of them tend to saturate, and when they do, the effect tends to be of ridiculously high order. We're not talking aabout second order, here, but something like 200 before we get to the proper curve even in the steady state, memoryless curve. And secondly, in the right proportion, purposeful aliasing can actually sound pretty good and just right. I used it to good effect once in my youth, at least. :) -- Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 -- 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
Re: [music-dsp] family of soft clipping functions.
Hi For whoever follows this, I thought and worked a bit about the clipping idea, and especially I was thinking about the harmonic behavior of the clipping function of N degree, so i tried using Maxima, and found there are distinct harmonics added to a sine wave passing through it. Maybe this follows from the definition, but anyhow maxima can simplify the sine terms in the clipped signal and make them explicit components, for instance for N=6: http://www.theover.org/Dsp/Softclip/softclip3.png Might be interesting as a frequency limited/controlled envelope as well, such that the combined spectrum of the signal and the envelope shape through the VCA multiplication has a (added) spectrum which remains Nyquist correct. T.V. -- 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
Re: [music-dsp] family of soft clipping functions.
robert bristow-johnson wrote: at the last AES in NYC, i was talking with some other folks (that likely hang out here, too) about this family of soft clipping curves made outa polynomials... Th script you supplied worked fine with the Open Source (and free) Octave as well, and gives this graphical result: http://www.theover.org/Dsp/Softclip/softclip1.png T.V. -- 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
Re: [music-dsp] family of soft clipping functions.
This reminds me of experimenting with polynomials as an amplitude enveloping
function for a soft synthesiser. There was something rather alluring about the
idea of a one-line-of-code amplitude envelope - unfortunately it made creating
the envelopes pretty tiresome, and when I thought about the performance I
realised it was probably slower to do that maths than the few conditionals you
would use for a standard piecewise envelope. So basically a rubbish idea unless
you have some strange need to have your amplitude envelope be a single line of
code :I
-Original Message-
From: music-dsp-boun...@music.columbia.edu
[mailto:music-dsp-boun...@music.columbia.edu] On Behalf Of robert
bristow-johnson
Sent: 29 October 2013 01:56
To: music-dsp@music.columbia.edu
Subject: [music-dsp] family of soft clipping functions.
at the last AES in NYC, i was talking with some other folks (that likely hang
out here, too) about this family of soft clipping curves made outa polynomials
(so you have some idea of how high in frequency any generated images will
appear).
these are odd-order, odd-symmetry polynomials that are monotonic from -1 x
+1 and have as many continuous derivatives as possible at +/- 1 where these
curves might be spliced to constant-valued rails.
the whole idea is to integrate the even polynomial (1 - x^2)^N
x
g(x) = integral{ (1 - v^2)^N dv}
0
you figger this out using binomial expansion and integrating each power term.
normalize g(x) with whatever g(1) is so that the curve is g(x)/g(1) and splice
that to two constant functions for the rails
{ -1x = -1
{
f(x) = { g(x)/g(1) -1 = x = +1
{
{ +1 +1 = x
you can hard limit (at +/- 1) before passing through this soft clipper and it
still works fine. but it has some gain in the linear region which is g(0).
if you want to add some even harmonic distortion to this, add a little bit of
(1 - x^2)^M
to f(x) for |x| 1 and it's still smooth everywhere, but there is a little DC
added (which has to be the case for even-symmetry distortion).
M does not have to be the same as N and i wouldn't expect it to be.
you can think of f(x) as a smooth approximation the sign or signum
function
sgn(x) = lim f(a*x)
a - +inf
or
sgn(x) = lim f(x)
N - +inf
which motivates using this as a smooth approximation of the sgn(x)
function as an alternative to
(2/pi)*arctan(a*x) or tanh(a*x). from the sgn(x) function, you can
create smooth versions of the unit step function and use that for
splicing. as many derivatives are continuous in the splice as possible.
and it satisfies conditions of symmetry and complementarity that are
useful in our line of work.
u(x) = 1/2 * (1 + sgn(x)) =approx 1/2*(1 + f(x))
you can run a raised cosine (Hann) through this and get a more flattened
Hann. in some old stuff i wrote, i dubbed this window:
w(x) = 1/2 + (9/16)*cos(pi*x) - (1/16)*cos(3*pi*x)
as the Flattened Hann Window but long ago Carla Scaletti called it the
Bristow-Johnson window in some Kyma manual. i don't think it deserves
that label (i've seen that function in some wavelet/filterbank lit since
for half-band filters). you get that window by running a simple Hann
through the biased waveshaper:
1/2 * ( 1 + f(2x-1) )
with N=1. you will get an even more pronounced effect (of smoothly
flattening the Hann) with higher N.
below is a matlab file that demonstrates this as a soft clipping function.
BTW, Olli N, this can be integrated with that splicing theory thing we
were talking about approximately a year ago. it would define the
odd-symmetry component. we should do an AES paper about this. i
think now there is nearly enough meat to make a decent paper. before
i didn't think so.
FILE: softclip.m
line_color = ['g' 'c' 'b' 'm' 'r'];
figure;
hold on;
x = linspace(-2, 2, 2^18 + 1);
x = max(x, -1); % hard clip for |x| 1
x = min(x, +1);
for N = 0:10
n = linspace(0, N, N+1);
a = (-1).^n ./ (factorial(N-n) .* factorial(n) .* (2*n + 1));
a = a/sum(a);
y = x .* polyval(fliplr(a), x.^2); % stupid MATLAB puts the coefs
in the wrong order
plot(x, y, line_color(mod(N,length(line_color))+1) );
end
hold off;
have fun with it, if you're so inclined.
L8r,
--
r b-j r...@audioimagination.com
Imagination is more important than knowledge.
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links
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http
Re: [music-dsp] family of soft clipping functions.
robert bristow-johnson wrote:
x
g(x) = integral{ (1 - v^2)^N dv}
0
you figger this out using binomial expansion and integrating each power
term.
Maybe a nice worksheet of (wx)Maxima, the FOS algebraic manipulation
programs:
http://www.theover.org/Dsp/Softclip/softclip2.png
I thought G1() would be close to the second order Taylor approximation
of the cos(x), but that isn't accurate...
Regards,
Theo V.
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links
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