Below is my interpretation of the cool partial-shifting method Robert has proposed. (I'm just trying to understand, no claim to originality.)
As far as I understand it, the method obtains, from a harmonic waveform, a family of new waveforms where the partials are selectively pitch-shifted, but remain harmonic. It doesn't smoothly vary the pitch thru inharmonic frequencies. *Basic sonic waveform* to be manipulated: a finite, weighted,* harmonic* sum of k sinusoids, which for simplicity I'll assume all cross 0 at time 0. So, the 1-periodic waveform F(t) = Sum over i = 1, ..., k: a_i * sin( w_i t) where each w_i is a positive integer mult. of 2 pi. (We'll see where this gets used.) *Desired:* a way to selectively shift some of the harmonics of our waveform. Let q_1, ..., q_k be "shift-susceptibility" parameters for the k harmonics; each may be positive, negative, or zero, but they need to be integer multiples of 2 * pi. *2-D Wavetable:* store for "all" pairs (t, y) in the unit square [0, 1]^2, the value G(t, y) = Sum over i = 1, ..., k: a_i * sin( w_i * t + q_i * y ) Ignoring time- and value-quantization issues here, as well as y-quantization. (Observe that for fixed y-value, the waveform over t is just a selectively phase-shifted version of F.) *Parameter to choose:* an *integer* value c giving overall pitch-shift amount desired. (One could have more tuning parameters at the cost of a higher-dimensional table G.) *Synthesis lookup rule: *at time T > 0 (a real value) the digital oscillator outputs the value Val(T) = G(T mod 1, (cT) mod 1) which, we note, equals = Sum over i = 1, ..., k: a_i * sin(w_i * (T mod 1) + q_i * (cT mod 1)) = Sum over i = 1, ..., k: a_i * sin((w_i + c * q_i) * T) ---the last equality using the fact that w_i and c*q_i are both integer mults of 2 *pi. ...so that, under this setting to c, the i'th partial has been frequency-shifted by an amount c*q_i from its original frequency, w_i. On Tue, Mar 13, 2018 at 11:05 AM, Risto Holopainen <ebel...@ristoid.net> wrote: > > > imagine it's two-dimensional vector synthesis like a Prophet VS. one > > dimension is some other timbre parameter with a minimum and a maximum > > (no wrap around). > > > > so, in the other dimension, imagine having say, 6 identical wavetables > > except the 2nd harmonic is offset by 60 degrees in phase between > > adjacent wavetable vector points in that dimension. all other > > harmonics are exactly the same. so as you crossfade from wavetable 0 > > to 1, that 2nd harmonic advances 60 degrees, as you crossfade from > > wavetable 1 to 2, the 2nd harmonic advances another 60 degrees. > > wavetable 6 and wavetable 0 are exactly the same. as you crossfade > > from wavetable 5 to 6 you're advancing the final 60 degrees back to > > the original phase of wavetable 0. > > > That makes sense, I'll have to try it. Six wavetables for the detuned > partial seems like a good number, and I can see that you would not want > too few of them. But what's the reasoning behind how many to use? > > > now, if all of the other harmonics remain the same phase for all 6 > > wavetables, moving around between them does not detune those > > harmonics. but if you go around that circle (in the positive > > direction) one complete loop, the 2nd harmonic made one more cycle > > than it would have otherwise if the vector location was stationary. > > if you whip around that loop 50 times per second, the 2nd harmonic > > will be detuned higher by 50 Hz. if you whip around that loop in the > > opposite direction, you will be detuning that 2nd harmonic lower in > > frequency. > > > > the application where this might be useful might be with piano tones > > or some other natural instrument with sharpened higher harmonics (like > > above the 9th or 12th harmonic). it's a different (and cheaper) way > > of doing it than employing what they call "group additive synthesis" > > where the higher harmonics are put into a different set of wavetables > > and run in a different wavetable oscillator that runs at a slightly > > sharp fundamental. > > > > On the other hand, I find group additive synthesis conceptually simpler > when dealing with inharmonic partials. > > What is the maximum detuning a partial can have with the wavetable > method? Intuitively I would guess it's the same as the fundamental > frequency, so the harmonic k could be tuned down to (k-1) or up to (k+1) > at most, is that right? > > Risto > > > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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