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<span style="font-size: 14pt;">Thanks Tom and Andy,</span>
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<div><span style="font-size: 18.6667px;">The subtle point here is perhaps that we were discussing a crossfade. This is necessarily time-varying to begin with, so LTI is out. Moreover, in this context, linearity might refer to the crossing over of amplitude
determining straight lines (traditional crossfading). Or, since superposition is necessarily involved, non-linearity in the sense of the generation of new frequency components, beyond any modulation concerns, might be the </span><span style="font-size: 18.6667px;">issue. </span></div>
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<div><span style="font-size: 18.6667px;">Overall, the pioneer and reigning expert on waveshaping issues is Ian. Recently, I reviewed the status of this somewhat neglected study. </span></div>
<div><span style="font-size: 18.6667px;"> </span></div>
<div><span style="font-size: 18.6667px;"> </span><a href="http://electronotes.netfirms.com/EN228.pdf" class="OWAAutoLink" id="LPlnk697247" previewremoved="true" style="font-size: 18.6667px;">http://electronotes.netfirms.com/EN228.pdf</a></div>
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<div><span style="font-size: 18.6667px;">- Bernie</span></div>
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<div><span style="font-size: 18.6667px;"> </span><span style="font-size: 18.6667px;"> </span></div>
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<div id="x_divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" color="#000000" style="font-size:11pt"><b>From:</b> Tom Wiltshire <tom@electricdruid.net><br>
<b>Sent:</b> Friday, September 15, 2017 7:01 AM<br>
<b>To:</b> Bernard Arthur Hutchins Jr<br>
<b>Cc:</b> Ian Fritz; synth-diy@synth-diy.org<br>
<b>Subject:</b> Re: [sdiy] Walsh Functions/EN S-008</font>
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<div class="PlainText">Hi Bernie,<br>
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What I meant was that if the phase of the harmonics is the same in both the start and destination waveforms (As an example, take a ramp and a square) then the amount of each harmonic during the crossfade is linear. So in our example, if you look at the amount
of the 2nd harmonic, it decreases linearly as we crossfade to a square.<br>
<br>
This is *not* true if you choose waveforms that have differing phases for the harmonics.<br>
<br>
Sorry, this stuff is easier to explain with pictures!<br>
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Regards,<br>
Tom<br>
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<br>
==================<br>
Electric Druid<br>
Synth & Stompbox DIY<br>
==================<br>
<br>
> On 15 Sep 2017, at 03:24, Bernard Arthur Hutchins Jr <bah13@cornell.edu> wrote:<br>
> <br>
> Tom - you said:<br>
> <br>
> " For this reason I’ve been looking at the way shifting the phase of the harmonics in a waveform affects the crossfade between one waveform and another - suddenly something that was previously a simple linear crossfade becomes non-linear and more interesting,.
. . . "<br>
> <br>
> Why are you saying "non-linear". It would break a LTI requirement (but only slightly if done relatively slowly, as most crossfades are). It would be the time varying aspect that is violated, not linearity. And this is already broken in any crossfade (a
differing mix at time progresses). What am I missing?<br>
> <br>
> - Bernie<br>
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