I'd like to be able to change the frequency of a phasor in realtime without getting that annoying scratchy sound caused by changes being made mid-phase. I tried using a [samphold] object triggered by the output of the phasor, but it doesn't work, probably because it creates a feedback loop.

Is there any way to only allow the frequency to be changed when the phase wraps around?

I'm not sure that the annoying scratchy sound is caused as you say. More likely it's aliasing that you will get whenever you frequency modulate a phasor. The best way around this is to use a band limited sawtooth instead of a phasor.

Any instantaneous change in frequency will be heard abruptly, so you really want the smoothest change possible throughout the duration of each cycle. You could control the phasor frequency with a smoothed audio rate signal [sig~]-[lop~ 3]-[phasor~].

To answer the last question. Make your own phasor with [vline~] and [metro]. The slope remains constant between bangs.

Thanks Obiwan - I will probably end up using the vline~ and metro method.

However, surely if it was possible to change the frequency of a phasor~ at exactly the beginning of its phase, then the speed should change without any kink in the shape of the wave?

If you have a monophonic synth set up to only re-trigger notes if the current key is released, pressing subsequent keys without releasing will cause abrupt changes in frequency but without annoying artifacts - is it a given that abrupt freq changes will usually cause these effects?

Also, what is the purpose of the lowpass 3 filter in the smoothed audio rate signal? how does it work to smooth the signal?

Yeah, it's possible to do exactly as you require for slow moving signals, by sending a [0( to the right [phasor~] inlet and forcing its phase to 0 on a transition. Problem is that this operation is block-accurate not sample-accurate so for fast signals comparable with the audio block size some dirty clicking will happen. [vline~] on the other hand can construct nice accurate time domain lines that aren't block quantised.

The lowpass used with [sig~] is a little trick to slew the control signal and make it more like an analog synthesiser. You could also use [line~] here with a 5-20ms follow time.

A signal that moves fast contains higher frequencies than a signal that moves slowly.

Let's say we have a signal that moves from 0 to 3 in 4 samples.
A perfectly linear function like a wire outputs the same number that
goes in, so for an input that goes {0, 1, 2, 3 } the output
would also go {0, 1, 2, 3}. Easy.

Now if we replace the wire with a function that makes the output be the average of the current value and the last two values, what happens? A low pass filter works by averaging together previous inputs, so it has to have some kind of memory. If the signal was at 0 to begin with then let's assume previous memory locations will contain 0.

out[0] = (0 + 0 + 0 )/3 = 0 same as before
out[1] = (1 + 0 + 0 )/3 = 0.33 now our output moves more slowly
out[2] = (2 + 1 + 0 )/3 = 1 trying to catch up
out[3] = (3 + 2 + 1 )/3 = 2 the averaging acts like an 'inertia'
out[4] = (3 + 3 + 2 )/3 = 2.66
out[5] = (3 + 3 + 3 )/3 = 3 finally we get there

It took us 6 steps to get to where we would have got in 4 steps.

This way of thinking about filters makes easy sense, but to really to understand you can read about how filters work below, but you need to sit down and scratch your chin over the maths. Millers explanation is in Argand/pole form which I find more difficult, Smiths examples include more block diagrams and C code that are more instructive to programmers imo.

http://crca.ucsd.edu/~msp/techniques/latest/book-html/node140.html
http://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_I.html