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oscillators.lib
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//############################## oscillators.lib ######################################
// This library contains a collection of sound generators. Its official prefix is `os`.
//
// The oscillators library is organized into 9 sections:
//
// * [Wave-Table-Based Oscillators](#wave-table-based-oscillators)
// * [Low Frequency Oscillators](#low-frequency-oscillators)
// * [Low Frequency Sawtooths](#low-frequency-sawtooths)
// * [Alias-Suppressed Sawtooth](#alias-suppressed-sawtooth)
// * [Alias-Suppressed Pulse, Square, and Impulse Trains](#alias-suppressed-pulse-square-and-impulse-trains)
// * [Filter-Based Oscillators](#filter-based-oscillators)
// * [Waveguide-Resonator-Based Oscillators](#waveguide-resonator-based-oscillators)
// * [Casio CZ Oscillators](#casio-cz-oscillators)
// * [PolyBLEP-Based Oscillators](#polyblep-based-oscillators)
//
// #### References
// * <https://github.com/grame-cncm/faustlibraries/blob/master/oscillators.lib>
//########################################################################################
/************************************************************************
************************************************************************
FAUST library file, GRAME section
Except where noted otherwise, Copyright (C) 2003-2017 by GRAME,
Centre National de Creation Musicale.
----------------------------------------------------------------------
GRAME LICENSE
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation; either version 2.1 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, write to the Free
Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
02111-1307 USA.
EXCEPTION TO THE LGPL LICENSE : As a special exception, you may create a
larger FAUST program which directly or indirectly imports this library
file and still distribute the compiled code generated by the FAUST
compiler, or a modified version of this compiled code, under your own
copyright and license. This EXCEPTION TO THE LGPL LICENSE explicitly
grants you the right to freely choose the license for the resulting
compiled code. In particular the resulting compiled code has no obligation
to be LGPL or GPL. For example you are free to choose a commercial or
closed source license or any other license if you decide so.
************************************************************************
************************************************************************/
ma = library("maths.lib");
ba = library("basics.lib");
fi = library("filters.lib");
si = library("signals.lib");
declare name "Faust Oscillator Library";
declare version "1.6.0";
// This library contains platform specific constants
pl = library("platform.lib");
//======================Oscillators based on mathematical functions===============
//
// Note that there is a numerical problem with several phasor functions built using the internal
// `phasor_imp`. The reason is that the incremental step is smaller than `ma.EPSILON`, which happens with very small frequencies,
// so it will have no effect when summed to 1, but it will be enough to make the fractional function wrap
// around when summed to 0. An example of this problem can be observed when running the following code:
//
// `process = os.phasor(1.0, -.001);`
//
// The output of this program is the sequence 1, 0, 1, 0, 1... This happens because the negative incremental
// step is greater than `-ma.EPSILON`, which will have no effect when summed to 1, but it will be significant
// enough to make the fractional function wrap around when summed to 0.
//
// The incremental step can be clipped to guarantee that the phasor will
// always run correctly for its full cycle, otherwise, for increments smaller than `ma.EPSILON`,
// phasor would initially run but it'd eventually get stuck once the output gets big enough.
//
// All functions using `phasor_imp` are affected by this problem, but a safer
// version is implemented, and can be used alternatively by setting `SAFE=1` in the environment using
// [explicit sustitution](https://faustdoc.grame.fr/manual/syntax/#explicit-substitution) syntax.
//
// For example: `process = os[SAFE=1;].phasor(1.0, -.001);` will use the safer implementation of `phasor_imp`.
//=================================================================================
//=========================Wave-Table-Based Oscillators===================================
// Oscillators using tables. The table size is set by the
// [pl.tablesize](https://github.com/grame-cncm/faustlibraries/blob/master/platform.lib) constant.
//========================================================================================
// Global parameter to use the safer version of `phasor_imp`, but which
// could be used in other functions as well.
SAFE = 0; // 0: use the faster version, 1: use the safer version
//-----------------------`(os.)sinwaveform`------------------------
// Sine waveform ready to use with a `rdtable`.
//
// #### Usage
//
// ```
// sinwaveform(tablesize) : _
// ```
//
// Where:
//
// * `tablesize`: the table size
//------------------------------------------------------------
sinwaveform(tablesize) =
sin(float(ba.period(tablesize)) * (2.0 * ma.PI) / float(tablesize));
//-----------------------`(os.)coswaveform`------------------------
// Cosine waveform ready to use with a `rdtable`.
//
// #### Usage
//
// ```
// coswaveform(tablesize) : _
// ```
//
// Where:
//
// * `tablesize`: the table size
//------------------------------------------------------------
coswaveform(tablesize) =
cos(float(ba.period(tablesize)) * (2.0 * ma.PI) / float(tablesize));
// Possibly faster version using integer arithmetic
phasor_env(freq, N) = environment {
//------- GLOBAL PARAMS
nbits = 31;
tablesize = 1<<N;
accuracy = int(nbits - N);
mask = (1<<nbits)-1;
inc(step) = int(tablesize * step * (1<<accuracy));
//------- LAMBDA DSP CASE
lambda(inc_op) = (inc_op : &(mask)) ~ _ : >>(accuracy) : /(tablesize);
//------- MINIMAL CASE
hsp(0,0) = lambda(+(inc(freq/ma.SR)'));
//------- GENERAL CASE
hsp(reset,phase) = lambda(select2(hard_reset,+(inc(freq/ma.SR)),inc(phase)))
with {
hard_reset = (1-1')|reset;
};
};
declare phasor_env author "Pierre Mascarade Relano, Maxime Sirbu, Stéphane Letz";
// Generic phasor with `reset` and `phase` parameters to be specialised in concrete use-cases.
phasor_imp(freq, reset, phase) = (select2(hard_reset, +(incr(SAFE)), phase) : ma.decimal) ~ _
with {
incr_aux = freq/ma.SR;
// Faster but less accurate version
incr(0) = incr_aux;
// To make sure that the incremental step is greater or equal to EPSILON or
// less than or equal to -EPSILON to avoid numerical problems.
// A frequency of 0Hz can still be used to freeze the phasor.
incr(1)= (freq != 0) * ba.if(freq < 0, min(-1.0 * ma.EPSILON, incr_aux), max(ma.EPSILON, incr_aux));
// To correctly start at `phase` at the first sample
hard_reset = (1-1')|reset;
};
// Possibly faster version using integer arithmetic
// phasor_imp(freq, reset, phase) = phasor_env(freq, 16).hsp(reset, phase);
// Version to be used with tables
phasor_table(tablesize, freq, reset, phase) = phasor_imp(freq, reset, phase) : *(float(tablesize));
//-----------------------`(os.)phasor`------------------------
// A simple phasor to be used with a `rdtable`.
// `phasor` is a standard Faust function.
//
// #### Usage
//
// ```
// phasor(tablesize,freq) : _
// ```
//
// Where:
//
// * `tablesize`: the table size
// * `freq`: the frequency in Hz
//
// Note that `tablesize` is just a multiplier for the output of a unit-amp phasor
// so `phasor(1.0, freq)` can be used to generate a phasor output in the range [0, 1[.
//------------------------------------------------------------
phasor(tablesize, freq) = phasor_table(tablesize, freq, 0, 0);
//-----------------------`(os.)hs_phasor`------------------------
// Hardsyncing phasor to be used with a `rdtable`.
//
// #### Usage
//
// ```
// hs_phasor(tablesize,freq,reset) : _
// ```
//
// Where:
//
// * `tablesize`: the table size
// * `freq`: the frequency in Hz
// * `reset`: a reset signal, reset phase to 0 when equal to 1
//---------------------------------------------------------
declare hs_phasor author "Mike Olsen, revised by Stéphane Letz";
hs_phasor(tablesize, freq, reset) = phasor_table(tablesize, freq, reset, 0);
//-----------------------`(os.)hsp_phasor`------------------------
// Hardsyncing phasor with selectable phase to be used with a `rdtable`.
//
// #### Usage
//
// ```
// hsp_phasor(tablesize,freq,reset,phase)
// ```
//
// Where:
//
// * `tablesize`: the table size
// * `freq`: the frequency in Hz
// * `reset`: reset the oscillator to phase when equal to 1
// * `phase`: phase between 0 and 1
//---------------------------------------------------------
declare hsp_phasor author "Christophe Lebreton, revised by Stéphane Letz";
hsp_phasor(tablesize, freq, reset, phase) = phasor_table(tablesize, freq, reset, phase);
//-----------------------`(os.)oscsin`------------------------
// Sine wave oscillator.
// `oscsin` is a standard Faust function.
//
// #### Usage
//
// ```
// oscsin(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
oscsin(freq) = rdtable(tablesize, sinwaveform(tablesize), int(phasor(tablesize,freq)))
with {
tablesize = pl.tablesize;
};
//-----------------------`(os.)hs_oscsin`------------------------
// Sin lookup table with hardsyncing phase.
//
// #### Usage
//
// ```
// hs_oscsin(freq,reset) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
// * `reset`: reset the oscillator to 0 when equal to 1
//---------------------------------------------------------
declare hs_oscsin author "Mike Olsen";
hs_oscsin(freq,reset) = rdtable(tablesize, sinwaveform(tablesize), int(hs_phasor(tablesize,freq,reset)))
with {
tablesize = pl.tablesize;
};
//-----------------------`(os.)osccos`------------------------
// Cosine wave oscillator.
//
// #### Usage
//
// ```
// osccos(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
osccos(freq) = rdtable(tablesize, coswaveform(tablesize), int(phasor(tablesize,freq)))
with {
tablesize = pl.tablesize;
};
//-----------------------`(os.)hs_osccos`------------------------
// Cos lookup table with hardsyncing phase.
//
// #### Usage
//
// ```
// hs_osccos(freq,reset) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
// * `reset`: reset the oscillator to 0 when equal to 1
//---------------------------------------------------------
declare hs_osccos author "Stéphane Letz";
hs_osccos(freq,reset) = rdtable(tablesize, coswaveform(tablesize), int(hs_phasor(tablesize,freq,reset)))
with {
tablesize = pl.tablesize;
};
//-----------------------`(os.)oscp`------------------------
// A sine wave generator with controllable phase.
//
// #### Usage
//
// ```
// oscp(freq,phase) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
// * `phase`: the phase in radian
//------------------------------------------------------------
oscp(freq,phase) = oscsin(freq) * cos(phase) + osccos(freq) * sin(phase);
//-----------------------`(os.)osci`------------------------
// Interpolated phase sine wave oscillator.
//
// #### Usage
//
// ```
// osci(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
osci(freq) = s1 + d * (s2 - s1)
with {
tablesize = pl.tablesize;
i = int(phasor(tablesize,freq));
d = ma.decimal(phasor(tablesize,freq));
s1 = rdtable(tablesize+1,sinwaveform(tablesize),i);
s2 = rdtable(tablesize+1,sinwaveform(tablesize),i+1);
};
//-----------------------`(os.)osc`------------------------
// Default sine wave oscillator (same as [oscsin](#oscsin)).
// `osc` is a standard Faust function.
//
// #### Usage
//
// ```
// osc(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
osc = oscsin;
//-----------------------`(os.)m_oscsin`------------------------
// Sine wave oscillator based on the `sin` mathematical function.
//
// #### Usage
//
// ```
// m_oscsin(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
m_oscsin(freq) = lf_sawpos(freq) : *(2*ma.PI) : sin;
//-----------------------`(os.)m_osccos`------------------------
// Sine wave oscillator based on the `cos` mathematical function.
//
// #### Usage
//
// ```
// m_osccos(freq) : _
// ```
//
// Where:
//
// * `freq`: the frequency in Hz
//------------------------------------------------------------
m_osccos(freq) = lf_sawpos(freq) : *(2*ma.PI) : cos;
// end GRAME section
//########################################################################################
/************************************************************************
FAUST library file, jos section
Except where noted otherwise, The Faust functions below in this
section are Copyright (C) 2003-2022 by Julius O. Smith III <[email protected]>
([jos](http://ccrma.stanford.edu/~jos/)), and released under the
(MIT-style) [STK-4.3](#stk-4.3-license) license.
The MarkDown comments in this section are Copyright 2016-2017 by Romain
Michon and Julius O. Smith III, and are released under the
[CCA4I](https://creativecommons.org/licenses/by/4.0/) license (TODO: if/when Romain agrees)
************************************************************************/
//===============================Low Frequency Oscillators===============================
// Low Frequency Oscillators (LFOs) have prefix `lf_`
// (no aliasing suppression, since it is inaudible at LF).
// Use `sawN` and its derivatives for audio oscillators with suppressed aliasing.
//==================================================================
//--------`(os.)lf_imptrain`----------
// Unit-amplitude low-frequency impulse train.
// `lf_imptrain` is a standard Faust function.
// #### Usage
//
// ```
// lf_imptrain(freq) : _
// ```
// Where:
//
// * `freq`: frequency in Hz
//------------------------------------------------------------
lf_imptrain(freq) = lf_sawpos(freq)<:-(mem)<0; // definition below
//--------`(os.)lf_pulsetrainpos`----------
// Unit-amplitude nonnegative LF pulse train, duty cycle between 0 and 1.
//
//
// #### Usage
//
// ```
// lf_pulsetrainpos(freq, duty) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
// * `duty`: duty cycle between 0 and 1
//------------------------------------------------------------
lf_pulsetrainpos(freq,duty) = float(lf_sawpos(freq) <= duty);
//pulsetrainpos = lf_pulsetrainpos; // for backward compatibility
//--------`(os.)lf_pulsetrain`----------
// Unit-amplitude zero-mean LF pulse train, duty cycle between 0 and 1.
//
// #### Usage
//
// ```
// lf_pulsetrain(freq,duty) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
// * `duty`: duty cycle between 0 and 1
//------------------------------------------------------------
lf_pulsetrain(freq,duty) = 2.0*lf_pulsetrainpos(freq,duty) - 1.0;
//--------`(os.)lf_squarewavepos`----------
// Positive LF square wave in [0,1]
//
// #### Usage
//
// ```
// lf_squarewavepos(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//------------------------------------------------------------
lf_squarewavepos(freq) = lf_pulsetrainpos(freq,0.5);
// squarewavepos = lf_squarewavepos; // for backward compatibility
//--------`(os.)lf_squarewave`----------
// Zero-mean unit-amplitude LF square wave.
// `lf_squarewave` is a standard Faust function.
//
// #### Usage
//
// ```
// lf_squarewave(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//------------------------------------------------------------
lf_squarewave(freq) = 2.0*lf_squarewavepos(freq) - 1.0;
// squarewave = lf_squarewave; // for backward compatibility
//--------`(os.)lf_trianglepos`----------
// Positive unit-amplitude LF positive triangle wave.
//
// #### Usage
//
// ```
// lf_trianglepos(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//------------------------------------------------------------
lf_trianglepos(freq) = 1.0-abs(saw1(freq)); // saw1 defined below
//----------`(os.)lf_triangle`----------
// Zero-mean unit-amplitude LF triangle wave.
// `lf_triangle` is a standard Faust function.
//
// #### Usage
//
// ```
// lf_triangle(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//------------------------------------------------------------
declare lf_triangle author "Bart Brouns";
declare lf_triangle licence "STK-4.3";
lf_triangle(freq) = 2.0*lf_trianglepos(freq) - 1.0;
//================== Low Frequency Sawtooths ====================
// Sawtooth waveform oscillators for virtual analog synthesis et al.
// The 'simple' versions (`lf_rawsaw`, `lf_sawpos` and `saw1`), are mere samplings of
// the ideal continuous-time ("analog") waveforms. While simple, the
// aliasing due to sampling is quite audible. The differentiated
// polynomial waveform family (`saw2`, `sawN`, and derived functions)
// do some extra processing to suppress aliasing (not audible for
// very low fundamental frequencies). According to Lehtonen et al.
// (JASA 2012), the aliasing of `saw2` should be inaudible at fundamental
// frequencies below 2 kHz or so, for a 44.1 kHz sampling rate and 60 dB SPL
// presentation level; fundamentals 415 and below required no aliasing
// suppression (i.e., `saw1` is ok).
//=====================================================================
//-----------------`(os.)lf_rawsaw`--------------------
// Simple sawtooth waveform oscillator between 0 and period in samples.
//
// #### Usage
//
// ```
// lf_rawsaw(periodsamps) : _
// ```
//
// Where:
//
// * `periodsamps`: number of periods per samples
//---------------------------------------------------------
lf_rawsaw(periodsamps) = (_,periodsamps : fmod) ~ +(1.0);
//-----------------`(os.)lf_sawpos`--------------------
// Simple sawtooth waveform oscillator between 0 and 1.
//
// #### Usage
//
// ```
// lf_sawpos(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//
//---------------------------------------------------------
declare lf_sawpos author "Bart Brouns, revised by Stéphane Letz";
declare lf_sawpos licence "STK-4.3";
lf_sawpos(freq) = phasor_imp(freq, 0, 0);
//-----------------`(os.)lf_sawpos_phase`--------------------
// Simple sawtooth waveform oscillator between 0 and 1
// with phase control.
//
// #### Usage
//
// ```
// lf_sawpos_phase(freq, phase) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
// * `phase`: phase between 0 and 1
//---------------------------------------------------------
declare lf_sawpos_phase author "Bart Brouns, revised by Stéphane Letz";
declare lf_sawpos_phase licence "STK-4.3";
lf_sawpos_phase(freq,phase) = phasor_imp(freq, 0, phase);
//-----------------`(os.)lf_sawpos_reset`--------------------
// Simple sawtooth waveform oscillator between 0 and 1
// with reset.
//
// #### Usage
//
// ```
// lf_sawpos_reset(freq,reset) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
// * `reset`: reset the oscillator to 0 when equal to 1
//
//---------------------------------------------------------
declare lf_sawpos_reset author "Bart Brouns, revised by Stéphane Letz";
declare lf_sawpos_reset licence "STK-4.3";
lf_sawpos_reset(freq,reset) = phasor_imp(freq, reset, 0);
//-----------------`(os.)lf_sawpos_phase_reset`--------------------
// Simple sawtooth waveform oscillator between 0 and 1
// with phase control and reset.
//
// #### Usage
//
// ```
// lf_sawpos_phase_reset(freq,phase,reset) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
// * `phase`: phase between 0 and 1
// * `reset`: reset the oscillator to phase when equal to 1
//
//---------------------------------------------------------
declare lf_sawpos_phase_reset author "Bart Brouns, revised by Stéphane Letz";
declare lf_sawpos_phase_reset licence "STK-4.3";
lf_sawpos_phase_reset(freq,phase,reset) = phasor_imp(freq, reset, phase);
//-----------------`(os.)lf_saw`--------------------
// Simple sawtooth waveform oscillator between -1 and 1.
// `lf_saw` is a standard Faust function.
//
// #### Usage
//
// ```
// lf_saw(freq) : _
// ```
//
// Where:
//
// * `freq`: frequency in Hz
//---------------------------------------------------------
declare saw1 author "Bart Brouns";
declare saw1 licence "STK-4.3";
saw1(freq) = 2.0 * lf_sawpos(freq) - 1.0;
lf_saw(freq) = saw1(freq);
//================== Alias-Suppressed Sawtooth ====================
//-----------------`(os.)sawN`--------------------
// Alias-Suppressed Sawtooth Audio-Frequency Oscillator using Nth-order polynomial transitions
// to reduce aliasing.
//
// `sawN(N,freq)`, `sawNp(N,freq,phase)`, `saw2dpw(freq)`, `saw2(freq)`, `saw3(freq)`,
// `saw4(freq)`, `sawtooth(freq)`, `saw2f2(freq)`, `saw2f4(freq)`
//
// #### Usage
//
// ```
// sawN(N,freq) : _ // Nth-order aliasing-suppressed sawtooth using DPW method (see below)
// sawNp(N,freq,phase) : _ // sawN with phase offset feature
// saw2dpw(freq) : _ // saw2 using DPW
// saw2ptr(freq) : _ // saw2 using the faster, stateless PTR method
// saw2(freq) : _ // DPW method, but subject to change if a better method emerges
// saw3(freq) : _ // sawN(3)
// saw4(freq) : _ // sawN(4)
// sawtooth(freq) : _ // saw2
// saw2f2(freq) : _ // saw2dpw with 2nd-order droop-correction filtering
// saw2f4(freq) : _ // saw2dpw with 4th-order droop-correction filtering
// ```
//
// Where:
//
// * `N`: polynomial order, a constant numerical expression between 1 and 4
// * `freq`: frequency in Hz
// * `phase`: phase between 0 and 1
//
// #### Method
// Differentiated Polynomial Wave (DPW).
//
// ##### Reference
// "Alias-Suppressed Oscillators based on Differentiated Polynomial Waveforms",
// Vesa Valimaki, Juhan Nam, Julius Smith, and Jonathan Abel,
// IEEE Tr. Audio, Speech, and Language Processing (IEEE-ASLP),
// Vol. 18, no. 5, pp 786-798, May 2010.
// 10.1109/TASL.2009.2026507.
//
// #### Notes
// The polynomial order `N` is limited to 4 because noise has been
// observed at very low `freq` values. (LFO sawtooths should of course
// be generated using `lf_sawpos` instead.)
//-----------------------------------------------------------------
declare sawN author "Julius O. Smith III";
declare sawN license "STK-4.3";
// --- sawN for N = 1 to 4 ---
// Orders 5 and 6 have noise at low fundamentals: MAX_SAW_ORDER = 6; MAX_SAW_ORDER_NEXTPOW2 = 8;
MAX_SAW_ORDER = 4;
MAX_SAW_ORDER_NEXTPOW2 = 8; // par cannot handle the case of 0 elements
sawN(N,freq) = saw1l : poly(Nc) : D(Nc-1) : gate(Nc-1)
with {
Nc = max(1,min(N,MAX_SAW_ORDER));
clippedFreq = max(20.0,abs(freq)); // use lf_sawpos(freq) for LFOs (freq < 20 Hz)
saw1l = 2*lf_sawpos(clippedFreq) - 1; // zero-mean, amplitude +/- 1
poly(1,x) = x;
poly(2,x) = x*x;
poly(3,x) = x*x*x - x;
poly(4,x) = x*x*(x*x - 2.0);
poly(5,x) = x*(7.0/3 + x*x*(-10.0/3.0 + x*x));
poly(6,x) = x*x*(7.0 + x*x*(-5.0 + x*x));
p0n = float(ma.SR)/clippedFreq; // period in samples
diff1(x) = (x - x')/(2.0/p0n);
diff(N) = seq(n,N,diff1); // N diff1s in series
factorial(0) = 1;
factorial(i) = i * factorial(i-1);
D(0) = _;
D(i) = diff(i)/factorial(i+1);
gate(N) = *(1@(N)); // delayed step for blanking startup glitch
};
//------------------`(os.)sawNp`--------------------------------
// Same as `(os.)sawN` but with a controllable waveform phase.
//
// #### Usage
//
// ```
// sawNp(N,freq,phase) : _
// ```
//
// where
//
// * `N`: waveform interpolation polynomial order 1 to 4 (constant integer expression)
// * `freq`: frequency in Hz
// * `phase`: waveform phase as a fraction of one period (rounded to nearest sample)
//
// #### Implementation Notes
//
// The phase offset is implemented by delaying `sawN(N,freq)` by
// `round(phase*ma.SR/freq)` samples, for up to 8191 samples.
// The minimum sawtooth frequency that can be delayed a whole period
// is therefore `ma.SR/8191`, which is well below audibility for normal
// audio sampling rates.
//
//-----------------------------------------------------------------
declare sawNp author "Julius O. Smith III";
declare sawNp license "STK-4.3";
// --- sawNp for N = 1 to 4 ---
// Phase offset = delay (max 8191 samples is more than one period of audio):
sawNp(N,freq,phase) = sawN(N,freq) : @(max(0,min(8191,int(0.5+phase*ma.SR/freq))));
//------------------`(os.)saw2, (os.)saw3, (os.)saw4`--------------
// Alias-Suppressed Sawtooth Audio-Frequency Oscillators of order 2, 3, 4.
//
// #### Usage
//
// ```
// saw2(freq) : _
// saw3(freq) : _
// saw4(freq) : _
// ```
//
// where
//
// * `freq`: frequency in Hz
//
// ##### References
// See `sawN` above.
//
// #### Implementation Notes
//
// Presently, only `saw2` uses the PTR method, while `saw3` and `saw4` use DPW.
// This is because PTR has been implemented and tested for the 2nd-order case only.
//
//------------------------------------------------------------------
saw2 = saw2ptr; // "faustlibraries choice"
saw3 = sawN(3); // only choice available right now
saw4 = sawN(4); // only choice available right now
//---------------------------`(os.)saw2ptr`---------------------------
// Alias-Suppressed Sawtooth Audio-Frequency Oscillator
// using Polynomial Transition Regions (PTR) for order 2.
//
// #### Usage
//
// ```
// saw2ptr(freq) : _
// ```
//
// where
//
// * `freq`: frequency in Hz
//
// ##### Implementation
//
// Polynomial Transition Regions (PTR) method for aliasing suppression.
//
// ##### References
//
// * Kleimola, J.; Valimaki, V., "Reducing Aliasing from Synthetic Audio
// Signals Using Polynomial Transition Regions," in Signal Processing
// Letters, IEEE , vol.19, no.2, pp.67-70, Feb. 2012
// * <https://aaltodoc.aalto.fi/bitstream/handle/123456789/7747/publication6.pdf?sequence=9>
// * <http://research.spa.aalto.fi/publications/papers/spl-ptr/>
//
// ##### Notes
//
// Method PTR may be preferred because it requires less
// computation and is stateless which means that the frequency `freq`
// can be modulated arbitrarily fast over time without filtering
// artifacts. For this reason, `saw2` is presently defined as `saw2ptr`.
//
//--------------------------------------------------------
declare saw2ptr author "Julius O. Smith III";
declare saw2ptr license "STK-4.3";
// specialized reimplementation:
saw2ptr(freq) = y with { // newer PTR version (stateless - freq can vary at any speed)
p0 = float(ma.SR)/float(max(ma.EPSILON,abs(freq))); // period in samples
t0 = 1.0/p0; // phase increment
p = ((_<:(-(1)<:_,_),_) <: selector1,selector2) ~(+(t0)):!,_;
selector1 = select2(<(0)); // for feedback
selector2 = select2(<(0), (_<:_,(*(1-p0):+(1)):+), _); // for output
y = 2*p-1;
};
//----------------------`(os.)saw2dpw`---------------------
// Alias-Suppressed Sawtooth Audio-Frequency Oscillator
// using the Differentiated Polynomial Waveform (DWP) method.
//
// #### Usage
//
// ```
// saw2dpw(freq) : _
// ```
//
// where
//
// * `freq`: frequency in Hz
//
// This is the original Faust `saw2` function using the DPW method.
// Since `saw2` is now defined as `saw2ptr`, the DPW version
// is now available as `saw2dwp`.
//--------------------------------------------------------
declare saw2dpw author "Julius O. Smith III";
declare saw2dpw license "STK-4.3";
saw2dpw(freq) = saw1(freq) <: * <: -(mem) : *(0.25'*ma.SR/freq);
//------------------`(os.)sawtooth`--------------------------------
// Alias-suppressed aliasing-suppressed sawtooth oscillator, presently defined as `saw2`.
// `sawtooth` is a standard Faust function.
//
// #### Usage
//
// ```
// sawtooth(freq) : _
// ```
//
// with
//
// * `freq`: frequency in Hz
//--------------------------------------------------------
sawtooth = saw2; // default choice for sawtooth signal - see also sawN
//------------------`(os.)saw2f2, (os.)saw2f4`--------------------------------
// Alias-Suppressed Sawtooth Audio-Frequency Oscillator with Order 2 or 4 Droop Correction Filtering.
//
// #### Usage
//
// ```
// saw2f2(freq) : _
// saw2f4(freq) : _
// ```
//
// with
//
// * `freq`: frequency in Hz
//
// In return for aliasing suppression, there is some attenuation near half the sampling rate.
// This can be considered as beneficial, or it can be compensated with a high-frequency boost.
// The boost filter is second-order for `saw2f2` and fourth-order for `saw2f4`, and both are designed
// for the DWP case and therefore use `saw2dpw`.
// See Figure 4(b) in the DPW reference for a plot of the slight droop in the DPW case.
//--------------------------------------------------------
declare saw2f2 author "Julius O. Smith III";
declare saw2f2 license "STK-4.3";
// --- Correction-filtered versions of saw2: saw2f2, saw2f4 -----
saw2f2 = saw2dpw : cf2 with {
cf2 = fi.tf2(1.155704605878911, 0.745184288225518,0.040305967265900,
0.823765146386639, 0.117420665547108);
};
declare saw2f4 author "Julius O. Smith III";
declare saw2f4 license "STK-4.3";
saw2f4 = saw2dpw : cf4 with {
cf4 = fi.iir((1.155727435125014, 2.285861038554662,
1.430915027294021, 0.290713280893317, 0.008306401748854),
(2.156834679164532, 1.559532244409321, 0.423036498118354,
0.032080681130972));
};
//=========Alias-Suppressed Pulse, Square, and Impulse Trains============
// Alias-Suppressed Pulse, Square and Impulse Trains.
//
// `pulsetrainN`, `pulsetrain`, `squareN`, `square`, `imptrainN`, `imptrain`,
// `triangleN`, `triangle`
//
// All are zero-mean and meant to oscillate in the audio frequency range.
// Use simpler sample-rounded `lf_*` versions above for LFOs.
//
// #### Usage
//
// ```
// pulsetrainN(N,freq,duty) : _
// pulsetrain(freq, duty) : _ // = pulsetrainN(2)
//
// squareN(N,freq) : _
// square : _ // = squareN(2)
//
// imptrainN(N,freq) : _
// imptrain : _ // = imptrainN(2)
//
// triangleN(N,freq) : _
// triangle : _ // = triangleN(2)
// ```
//
// Where:
//
// * `N`: polynomial order, a constant numerical expression
// * `freq`: frequency in Hz
//====================================================================
//------------------`(os.)impulse`--------------------------------
// One-time impulse generated when the Faust process is started.
// `impulse` is a standard Faust function.
//
// #### Usage
//
// ```
// impulse : _
// ```
//--------------------------------------------------------
impulse = 1-1';
//------------------`(os.)pulsetrainN`--------------------------------
// Alias-suppressed pulse train oscillator.
//
// #### Usage
//
// ```
// pulsetrainN(N,freq,duty) : _
// ```