[sdiy] saw vs ramp, audible?

[Mattias Rickardsson]

Donald Tillman <don at till.com> skrev:

> Mattias Rickardsson <mr at analogue.org> wrote:
>
> Donald Tillman <don at till.com> skrev:
>
>> Note that a sawtooth ramping down has all the harmonics in phase with the
>> fundamental.
>>
>
> Yes.
>
> And a sawtooth ramping up has its harmonics alternating in phase (+1,
>> -1/2, +1/3, -1/4,...) from the fundamental.
>>
>
> But are you counting from the midpoint now? Surely they must still be in
> phase if you just flip the sawtooth backwards around its discontinuity?
>
>
> (I had to think about this...)
>
> Good point!  When you play it backwards the even harmonics reverse
> polarity but the odd harmonics don't.  So yeah, it's consistent that way.
>

Not sure how you mean now, but... :-)

Suppose we start a downward-going ramp waveform at its upward-going
discontinuity. This positive jump is caused by all the harmonics having
positive slope. At this point they are all sinewaves going up through a
zero crossing. The sum of those slopes (all of them have derivative 1!) is
a discontinuity.

Halfway through the waveform we come to the point where the ramp slope is
going down through zero. At this point, all the even partials have spent a
whole number of periods, and have another positive zero crossing there. All
the odd partials are halfway through a period, and so they have a negative
zero crossing there.
In summary, partials 1, 3, 5, ... have negative zero crossings and partials
2, 4, 6, ... have positive zero crossings here. (That's why the partials
have alternate signs if this point is taken as the starting point.) The sum
of them is a downwards hill, not a discontinuity.

Trivial perhaps, but anyway. These things are fun to play with using some
math graph tool. :-)
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