[sdiy] Generating acyclic waveforms?

[cheater cheater]

BTW,
here are some interesting reads on wikipedia that are to the topic if
anyone's interested:

http://en.wikipedia.org/wiki/Harmonic_series_(music)
http://en.wikipedia.org/wiki/Inharmonicity
http://en.wikipedia.org/wiki/Otonality_and_Utonality
http://en.wikipedia.org/wiki/Anharmonicity
http://en.wikipedia.org/wiki/Pseudo-octave
http://en.wikipedia.org/wiki/String_resonance_(music)
http://en.wikipedia.org/wiki/Nonlinear_resonance
http://en.wikipedia.org/wiki/Duffing_equation
http://en.wikipedia.org/wiki/Anharmonicity#Examples_in_Physics

The last link might have a clue or two:

> In fact, virtually all oscillators become anharmonic when
> their pump amplitude increases beyond some threshold,
> and as a result it is necessary to use nonlinear
> equations of motion to describe their behavior.

How does this apply to the typical synth VCO?

Also the well known fact is mentioned that a pendulum, for big
displacements, starts becoming an anharmonic oscillator. I assume this
bread-and-butter experiment must have been simulated in analog
electronics during the analog computing era. Has anyone come across a
simulation like this?

This could be of use:
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0T-4H3SBWH-5&_user=10&_coverDate=07%2F31%2F1972&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1260157201&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=23a5de9589a6c05ff330c181061795fc

(Has anyone got access to this whitepaper?)

Cheers
D.

On Sun, Mar 21, 2010 at 17:58, cheater cheater <cheater00 at gmail.com> wrote:
> Hi guys,
> I was lately considering acyclic waveforms for use in music synthesis,
> and I'm wondering about techniques of realizing them in synthesizers.
>
> Acyclic waveforms are interesting for the purpose of making the sound
> less static. An acyclic waveform will interact with nonlinearities in
> such a manner that the timbre will change ever so slightly over time.
> It is also good for making certain harmonic tricks easier or possible
> at all: an acyclic waveform hides problems in pitch articulation; on
> the other hand, something like a stretched-harmonic waveform could be
> used to play music compatible with the piano.
>
> Most people here know that in piano timbre the partials are sine waves
> that are not spaced like in synth waveforms. That is, in the
> synthesizer, if the note's fundamental frequency is F, then harmonics
> will be at 1F, 2F, 3F, 4F etc. In a stretched-harmonic waveform, the
> frequencies will for example be of the form 1sF, 2sF, 3sF, 4sF, and so
> on, where s is the stretching factor. For s=1 we have the usual
> harmonic series. For s which is not an integer and not a specially
> chosen rational number, we have a probability of 100% of generating an
> acyclic waveform. Of course, the spacing of harmonics in the piano is
> more complex than that, but this is the first approximation.
>
> Here's what researchers had to say about acyclic waveforms:
>
>> Inharmonicity is not necessarily unpleasant. In 1962, research
>> by Harvey Fletcher and his collaborators indicated that the spectral
>> inharmonicity is important for tones to sound piano-like. They
>> proposed that inharmonicity is responsible for the "warmth" property
>> common to real piano tones.[2] According to their research
>> synthesized piano tones sounded more natural when some
>> inharmonicity was introduced.[3] In general, electronic instruments
>> that duplicate acoustic instruments must duplicate both the
>> inharmonicity and the resulting stretched tuning of the original
>> instruments.
>>
>> 2. Acoustical Society of America - Large grand and small upright
>> pianos by Alexander Galembo and Lola L. Cuddy]
>> http://www.acoustics.org/press/134th/galembo.htm
>> 3. Matti Karjalainen (1999). "Audibility of Inharmonicity in String
>> Instrument Sounds, and Implications to Digital Sound Systems"
>> http://www.acoustics.hut.fi/~hjarvela/publications/icmc99.pdf
>
> (taken from http://en.wikipedia.org/wiki/Inharmonicity )
>
> Of course we've since learnt better than to try and immitate the piano
> timbre with synths. It's a stupid want and will never happen; if you
> want a piano, buy a piano. But I think it is a very worthwhile
> consideration which can be used to greatly enhance the timbre of
> synthesizers, and can also make synthesizers musically compatible with
> accoustic instruments which, in my opinion, differ most from what we
> can timbrally do with synthesizers because of their stretched-harmonic
> nature.
>
> What are some ways of generating acyclic waveforms - either in
> stretched-harmonic spectra, or just generally acyclic?
>
> Of course, trivially enough, noise is an acyclic waveform. (Unless
> you're using a certain 80s synth that I forgot the name of...)
>
> Additive approaches are one way of doing this. Pros are that
> absolutely everything can be controlled about the waveform. Cons are
> that these approaches are computationally intensive, the functionality
> difficult to expose to the user easily, and generally this is a
> digital technique, which is not always desirable. Of course, doing an
> additive engine that only transforms between waveforms (by using the
> harmonic levels from cyclic oscillators, i.e.
>
> One way of generating an acyclic waveform is - I think - taking a
> usual oscillator core and adding a small amount of noise to the reset
> comparator's reference voltage. This is only a theory, I haven't been
> able to try it. What should happen is that the waveform will still
> look like the normal waveform (for triangle and ramp), and the
> segments will still be linear, but the peaks will happen either sooner
> or later than they should. Think of it as something similar to hard
> sync, except because of the balanced nature of noise statistically the
> pitch doesn't change: this means that segments will be shorter or
> longer (effectively being higher or lower in pitch), but as we measure
> more and more cycles the oscillator is more and more in-tune. The pros
> of this approach are that it could be easy to implement in most
> existing designs; that it is well behaved enough to let people
> understand it; that it animates the waveform while not really altering
> it that much. The cons are that it does not go very far from where we
> started out, leaving a lot of space for exploration via other
> techniques. Of course, it does not generate stretched tuning.
>
> One way to generate stretched tuning in the analogue domain is to use
> a frequency shifter on a normal oscillator. I anticipate this is going
> to be expensive to do and failure prone: the process might not be that
> well behaved. However, I'm not sure what's involved in this. Are there
> good precision frequency shifters that you guys could recommend? The
> frequency would only need to be shifted by a few hz, much less than
> one semitone. I will leave out the pros and cons for now because I
> simply don't know much about frequency shifters, so I'll leave this up
> to people more knowledgable like me.
>
> Chorus and (large, as in more than 5) oscillator stacks are some more
> or less valid way of making acyclic waveforms. In this approach - as I
> understand it - the power of the harmonics of the oscillator(s) is
> centered around the frequencies F, 2F, 3F, ... and the further we move
> away from the exact frequency nF the less power there is in the
> spectrum of the resultant waveform.
>
> But the lingering question is: is there a way to generate an acyclic
> waveform algorithmically in the digital domain without using additive
> approaches, and more importantly, is it possible to create an
> inherently acyclic oscillator in the analogue domain, without
> resorting to secondary processing such as frequency shifting,
> chorusing, or stacking?
>
> And as a follow up question: is it possible to create an inherently
> stretched-harmonic oscillator in analog? Is there a specialized
> algorithm in digital that can do this? We are either talking about
> something that generates waveforms with partials on frequencies n*a*F,
> n*a^b*F, or some other thing similar to that.
>
> Thanks
> D.
>