A Computational Model of the Hammond Organ Vibrato/Chorus Using Wave Digital Filters

Home , Hammond organ

Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

A COMPUTATIONAL MODEL OF THE HAMMOND ORGAN VIBRATO/CHORUS USING WAVE DIGITAL FILTERS

Kurt James Werner, W. Ross Dunkel, and François G. Germain Center for Computer Research in Music and Acoustics (CCRMA), Stanford University 660 Lomita Drive, Stanford, CA 94305, USA [kwerner, chigi22, francois]@ccrma.stanford.edu

ABSTRACT to get each port’s polarity correct and to simplify the calculation of node voltages, we review the derivation of wave-digital polarity We present a computational model of the Hammond tonewheel inverters and illustrate their systematic use. organ vibrato/chorus, a musical audio effect comprising an LC Although the vibrato/chorus has not been studied in the vir- ladder circuit and an electromechanical scanner. We model the tual analog context, there exists extensive related work on mod- LC ladder using the Wave Digital Filter (WDF) formalism, and eling other aspects of the complex and pleasingly idiosyncratic introduce a new approach to resolving multiple nonadaptable lin- sound of the Hammond organ. For the practicing musician, a se- ear elements at the root of a WDF tree. Additionally we formal- ries of five Sound on Sound articles (beginning with [9]) details ize how to apply the well-known warped Bilinear Transform to how to mimic each sub-system of the Hammond from tonewheel WDF discretization of capacitors and inductors and review WDF to Leslie speaker using standard synthesis tools. [10] points out the polarity inverters. To model the scanner we propose a simpli- difficulty of emulating the vibrato/chorus using a standard digital fied and physically-informed approach. We discuss the time- and chorus. Numerous commercial emulations known as “clonewheel frequency-domain behavior of the model, emphasizing the spectral organs” have been released over the years. Academic papers have properties of interpolation between the taps of the LC ladder. covered various aspects of the Hammond sound. Pekonen et al. [11] propose efficient models of the organ’s basic apparatus including 1. INTRODUCTION tonewheels draw-bars. More abstractly, a novel “Hammondizer” effect by Werner and Abel [12] imprints the sonic characteristics 1 The Hammond tonewheel organ’s vibrato/chorus (Fig. 1, Table 1) of the organ onto any input audio, extending effect processing [13] is a crucial ingredient of its unique sound. Its sonic character is within a modal reverberator framework [14]. An important part highly valued by musicians, having even been made into a gui- of the organ’s sound, the Leslie rotating speaker [15] has been tar effect [2]. The vibrato/chorus consists of an LC ladder circuit the subject of the majority of Hammond-related academic work. (Fig. 1) and an electromechanical “scanner” [3], with three user- Its simulation has been tackled using a perceptual approach [16], selectable “vibrato” (V1, V2, V3) and “chorus” (C1, C2, C3) set- modulated and interpolated delay lines [17,18]2, Doppler shift and tings. In this paper, we introduce a model of the Hammond organ amplitude modulation [19, 20], a measurement-based black box vibrato/chorus comprising a Wave Digital Filter (WDF) [4] model approach [21], and spectral delay filters [11]. of the LC ladder circuit and a simplified model of the scanner. The paper is structured as follows. Section 2 details the Ham- WDF theory was originally developed to facilitate the design mond vibrato/chorus. Section 3 presents a simplified model of the of digital filters based on analog ladder prototypes [5]. In that scanner. Section 4 presents a WDF model of the LC ladder circuit. context, the low coefficient sensitivity of these prototypes leads Section 5 characterizes these models. to attractive numerical properties in the WDF. Recent years have seen an expansion of the use of WDFs into new fields including virtual analog circuit modeling [6]. Interestingly, ladder topologies 2. REFERENCE SYSTEM DESCRIPTION also show up in electro-mechanical equivalent circuit models of the torsional modes of spring vibration relevant to spring reverb This section details the Hammond Organ vibrato/chorus, which in- units [7], another effect common in Hammond organs. cludes a LC ladder circuit (Fig. 1, bottom, Section 2.1) and an elec- Modeling the Hammond organ LC ladder as a WDF presents tromechanical “scanner” apparatus (Fig. 1, top, Section 2.2). The an issue that suggests an extension to WDF theory, and an oppor- gray box on Fig. 1 represents a bank of switches that connect the tunity to discuss finer points of polarity handling and reactance tap node voltages v1 ··· v19 on the ladder to the terminals t1 ··· t9 discretization. First, the ladder circuit has two non-adaptable lin- on the scanner. The setting (V1/V2/V3/C1/C2/C3) controls these ear elements (a voltage source and a switch), one more than clas- switches according to Table 2. sical WDF methods can handle. To address this, we extend the In principle, the LC ladder serves the same purpose as the de- method of [8] to the case of multiple linear nonadaptable elements lay line in a standard digital chorus effect [22]. The LC ladder at the root of a WDF tree. Second, the circuit’s 36 reactances cre- differs from a delay line in that the LC ladder is not strictly uni- ate magnitude responses with numerous salient features. We apply directional and that it filters as it delays a signal. This filtering the well-known frequency-warped bilinear transform to the wave- features pronounced non-uniform passband ripples and a lowpass digital capacitor and inductor to help control magnitude response cutoff that depends on the inductor and capacitor values. matching. Finally, polarity bookkeeping of port connections and On the other hand, the scanner serves the same purpose as in- the 19 outputs of the LC ladder is non-trivial. Since it is essential terpolation in a standard digital modulated-delay effect [22]. Stan-

1We study the version used in late-model Hammond B-3s [1] 2https://ccrma.stanford.edu/ jos/pasp/Leslie.html

DAFX-271 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

vout

t1 t2 t3 t4 t5 t6 t7 t8 t9 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18

+ R1+ R2+ R3+ R4+ R5+ R6+ − Rt C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 vin R1− R2− R3− R4− R5− R6−

Rc vibrato (closed) chorus (open)

Figure 1: Vibrato/Chorus Schematic.

vx vy v19 Table 1: Component values.

Lx Ly Name value units ··· ··· ··· Rc 22 kΩ

+ + + + + + R 27 kΩ + Rx+ 1+ − R1− 68 kΩ vin vD vx,l vx,r vy,l vy,r vt Rt Cx Cy R2+ 56 kΩ Rx− − − − − − − R3+ 39 kΩ ··· ··· ··· R2−, R3− 0.15 MΩ Rc R4+ 33 kΩ R5+ 18 kΩ R6+ 12 kΩ ×6 ×12 R4− ··· R6− 0.18 MΩ x ∈ [1 ··· 6] y ∈ [7 ··· 18] L1 ··· L18 500 mH C1 ··· C17 0.004 µF C18 0.001 µF Figure 2: Vibrato/Chorus Schematic Partitioned. Rt 15 kΩ dard digital linear interpolation has a well-known lowpass charac- This highly structured circuit is partitioned into four subcir- 3 teristic [18] that digital audio effect designers often try to avoid cuits as shown in Fig. 2. The ﬁrst subcircuit includes vin, Rc, and by using, e.g., allpass interpolation [18]4. Ironically, the scanner of the switch and presents a port “D” to the rest of the circuit. the Hammond Organ vibrato/chorus essentially implements linear The second subcircuit has 6 stages indexed by x ∈ [1 ··· 6]: interpolation—meaning it does not have an allpass characteristic. inductor Lx, capacitor Cx, and voltage divider pair Rx+ and Rx−. The tap node voltage vx is the output of each stage. Each stage Table 2: Taps for different depth settings. presents a left-facing (“x, l”) and right-facing (“x, r”) port to the rest of the circuit. Ports “D” and “1, l” are connected and the 5 depth t1 t2 t3 t4 t5 t6 t7 t8 t9 port pairs “(k + 1), l” and “k, r”, k ∈ [2 ··· 6] are connected. V1/C1 v1 v2 v3 v4 v5 v6 v7 v8 v9 The third subcircuit has 12 stages indexed by y ∈ [7 ··· 18]: V2/C2 v1 v2 v3 v5 v7 v9 v11 v12 v13 inductor Ly and capacitor Cy. The tap node voltage vy is the V3/C3 v v v v v v v v v 1 2 4 7 10 13 16 18 19 output of each stage. Each stage presents a left-facing (“y, l”) and right-facing (“y, r”) port to the rest of the circuit. Ports “6, r” and “7, l” are connected and the 11 port pairs “(k + 1), l” and “k, r”, 2.1. LC Ladder Circuit k ∈ [7 ··· 17] are connected. The fourth subcircuit is simply the termination resistance Rt The input signal is represented as an ideal voltage source vin. 19 that presents port t to the rest of the circuit and has the tap node LC ladder stages are composed of inductors L1 ··· L19, capac- voltage v19 as an output. Ports “18, r” and “t” are connected. itors C1 ··· C19, and voltage divider pairs Rk+ and Rk−, k ∈ [1 ··· 6]. A termination resistor Rt ends the ladder. A switch controls whether Rc is shorted or not. Electrical component values for 2.2. Scanner Device the circuit are given in Table 1 [1]. The vibrato scanner consists of a moving rotor with node volt- 3https://ccrma.stanford.edu/~jos/pasp/ age vout that cyclically scans a stack of keystone-shaped output Fractional_Delay_Filtering_Linear.html plates across 16 ﬁxed stacks of identical plates arranged in a cir- 4https://ccrma.stanford.edu/~jos/pasp/First_ cle. At any given time, 2 of these 16 stacks partially overlap the Order_Allpass_Interpolation.html rotor stack, forming two capacitors whose capacitances are pro-

DAFX-272 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

9 4.1. WDF Tree (Subcircuits 2–4) 8 7 To model the LC ladder circuit, we derive the WDF structure of its 6 circuit. Fig. 4 shows the partitioned schematic rearranged to high- 5 light the underlying topology (with polarities labeled) and Fig. 5 gain 4 shows the resulting WDF structure. 3 The 6 stages in the second subcircuit can be decomposed into 2 standard WDF one-ports (R , R , L , and C ) and adaptors 1 x− x+ x x 0 1 (Sx, Sx0, Px, and Px0). The presence of the two inverters Ix and 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π Ix0 warrants explanation. We have already assigned polarities to the ports that connect stages, and series and parallel adaptors have angle (Radians) inherent port polarities. Inverters Ix reconcile the discrepancy between the polarity of the right-facing port of each parallel adaptor Figure 3: Gain applied to each scanner terminal t1 ··· t9. Px and the left-facing port of P(x+1)0 or S7. Inverters Ix0 simplify the extraction of node voltages vx, which are calculated by combining port voltages across resistors Rc and Rx− as portional to the overlapping area between each fixed stack and the rotor stack. Conceptually, these two capacitances form a time- 1 v = v + v = (a + b + a + b ) , (1) varying capacitive voltage divider, which crossfades between the x c x− 2 c c x− x− node voltages of these 2 stacks. The 16 fixed plate stacks are con- where v , a , and b are the port voltage across, incident wave, and nected to the 9 terminals t1 ··· t9 which sets their respective node c c c voltages to the node voltages of the corresponding terminals. As reflected wave at resistor Rc in the first subcircuit. the rotor undergoes a complete rotation, it “scans” from t1 to t9 The 12 stages in the third subcircuit can be decomposed into and back. The “there-and-back-again” form of the tap gains pro- standard WDF one-ports (Ly and Cy) and adaptors (Sy and Py). duces various cyclic vibrato effects. Again inverters Iy reconcile the discrepancy between the polarity of the right-facing port of each Py and the left-facing port of Sy+1 or Rt. Node voltages vy in this subcircuit are extracted by 3. SCANNER MODEL combining the port voltages of resistor Rc and the left-facing port voltage of each stage vy,l as In this section, we propose a simplified model of the scanner. Since 1 the scanner capacitances are small compared to the ladder capac- vy = vc + vy,l = (ac + bc + ay,l + by,l) . (2) itors, it is reasonable to assume that they don’t load the ladder 2 circuit. Therefore, we are justified in modeling the LC ladder The fourth subcircuit is decomposed simply into a WDF resis- and scanner separately. The scanner takes the 19 node voltages tor Rt. Node voltage v19 is extracted by v1 ··· v19 as input and produces a single output voltage vout. At any given time, two plates overlap the rotor plate, creating 1 v = v + v = (a + b + a + b ) . (3) two distinct capacitors. We assume that only these two capacitors 19 c t 2 c c t t have a non-negligible contribution to the output voltage. Under that assumption, those two form a capacitive voltage divider. To re- 4.2. Root with Multiple Elements flect this, we model the scanner output as a crossfade between two node voltages vα and vβ according to vout = η vα +(1−η)vβ . The Reference circuits such as the Hammond organ vibrato/chorus cir- mixing coefficient η ∈ [0.0, 1.0] varies with the two capacitances cuit commonly include multiple nonadaptable elements (linear and Cα and Cβ as η = Cα/(Cα + Cβ ). Assuming circular symmetry nonlinear). Trying to accomodate multiple nonadaptable elements of each plate and ignoring the small gap between stacks, Cα + Cβ in a standard WDF connection tree causes unavoidable delay-free is a constant. Since each capacitance is theoretically proportional loops which leads to computability problems. to the overlap area, the gain is modeled as a simple function of Historically, algorithm designers commonly use one of two the rotor angle, according to Fig. 3. The variation of rotor angle tactics to ameliorate these issues. One tactic is to alter the ref- over time follows the one in the physical organ, where the rotor is erence circuit to make the structure computable. It is common to driven at a constant rate of ≈ 6 Hz by a synchronous motor [3]. approximate ideal voltage sources as resistive voltage sources with small series resistances and to approximate ideal current sources as resistive current sources with large parallel resistances. The same 4. WDF MODEL OF LC LADDER principle can be used to approximate short circuits or the closed state of switches as small resistances and to approximate open cir- In this section, we detail WDF simulation of the LC ladder circuit. cuits or the open state of switches as large resistances. Further- The structure for subcircuits two, three, and four is outlined in Sec- more, certain nonlinear elements can be reasonably approximated tion 4.1. The first subcircuit contains multiple nonadaptable linear by linearizing them with controlled sources and immitances [6,23]. elements and cannot be handled with standard WDF techniques. A second tactic is to alter the WDF by introducing fictitious unit We introduce a new approach to resolving this issue in Section 4.2 delays to resolve delay-free loops. Fettweis used this approach [5] and apply it to the LC ladder in Section 4.3. In Section 4.4 we before developing reflection-free ports [24], and it is still common use the frequency-warped bilinear transform for WDF discretiza- in virtual analog [25, 26]. Of course, altering the reference circuit tion of capacitors and inductors; in Section 4.5 we review WDF through these tactics introduces error (e.g. dissipation, dispersion) polarity inverters which are essential for proper bookkeeping. and can have adverse effects on stability.

DAFX-273 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

Lx Cx Ly Cy

3 RA vA vD RD 6

vin vD vx,l −+ −+ vx,r vy,l −+ −+ vy,r vt Rt

··· ··· ··· + + iA iD +

+ 1 −

− − − − e − e − + + + + + + + + A D − − − − + + − − + + − − − − ··· ··· ··· −+ 4 RB vB 0

+ − + R +

− + − x+ + iB eB − +− −+ − 2

Rc Rx− iC − + vC

RC + ×6 ×12 − 5 x ∈ [1 ··· 6] y ∈ [7 ··· 18] eC (a) Highlight WDF structure (b) Thévenin equivalents attached.

Figure 4: LC ladder Schematic, rearranged towards WDF.

Lx Cx Ly Cy

× vin Px0 Rt vin bD ax,l bx,r ay,l by,r at

··· ··· ··· + b b

+ Sx A0 D

− − + + Px − − ··· ··· ··· aA0 bA aD aD bx,l ax,r by,l ay,r bt Ix Sy Py Iy R R bB0 aB + Sx0 − Rx+ aB0 bB Ix0 aC bC Rc Rx− Rc

eC ×6 ×12 x ∈ [1 ··· 6] y ∈ [7 ··· 18] × (a) WDF structure (b) Root Waves.

Figure 5: WDF structure of LC ladder.

In [8], Werner et al. propose a method for handling multiple waves aroot and inputs xroot to reflected waves broot by nonlinearities that does not resort to these tactics. All of the nonlinearities are grouped as sub-elements of a WDF structure at the broot = Φaroot + Ψxroot , (5) root of the WDF tree. Inside that structure, and after proper modifi- where Φ and Ψ embody the wave-domain behavior of the linear cation of the circuit graph, those elements end up being connected elements. a and b are related to the a and b by to each other through a complex R-type adaptor that also inter- root root i i faces those elements to the rest of the circuit. The method of [27] aroot = bi and ai = broot . (6) is used to solve for the scattering behavior of this R-type adaptor. Because of the non-adaptable nature of the root elements, the Combining (4), (5), and (6) and solving for be yields response of the root adaptor structure from the perspective of the b = Γa + Θx with rest of the tree forms an implicit loop that we can resolve using ei- e e root −1 (7) ther a tabulated solution [8] or an iterative solution [28,29]. These Γ = ΘS12 + S22 , Θ = S21 (I − ΦS11) Ψ,. approaches extends readily to nonadaptable linear elements, but is unnecessarily complex. Here we propose a novel more efficient 4.3. WDF Root (Subcircuit 1) approach for the case of multiple nonadaptable linear elements. Consider a complex root topology with “external” incident Here, we apply the theory developed in Section 4.2 to the first sub- waves ae and reflected waves be facing the rest of the circuit and circuit of the Hammond vibrato/chorus. The first subcircuit con- “internal” incident waves ai and reflected waves bi facing the non- tains two non-adaptable elements, a voltage source and a switch. adaptable linear elements, related by the scattering relationship As a result, those two elements need to be grouped at the root of the WDF structure following the method outlined in Section 4.2, bi S11 S12 ai connecting them through an R-type adaptor [27] with incident and = (4) be S21 S22 ae reflected waves

> >> > >> The vector of nonadaptable linear elements relates the incident a = ai ae and b = bi be , (8)

DAFX-274 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

0 1 2 3 4 5 6 A B C D      0 GB 0 0 0 −GB 0 0 −1 0 −1 0 v 0 0 1  0 GA + GD 0 −GA 0 0 −GD 0 0 0 −1  v 1   0       2  0 0 GC 0 0 −GC 0 0 −1 0 0  v 2   0      0  3  0 −GA 0 GA 0 0 0 1 0 0 0  v 3    4  −GB 0 0 0 GB 0 0 0 1 0 0  v 4   0      0  5  0 0 −G 0 0 G 0 0 0 1 0  v 5 =   C C    0  6  0 −GD 0 0 0 0 GD 0 0 0 1  v 6        eA  A  −1 0 0 1 0 0 0 0 0 0 0  jA        eB  B  0 0 −1 0 1 0 0 0 0 0 0  jB    C  −1 0 0 0 0 1 0 0 0 0 0  jC   eC  D 0 −1 0 0 0 0 1 0 0 0 0 jD eD

Figure 6: MNA matrix. Red and blue cells respectively show examples of resistor and voltage source “stamps” [27].

“external” wave vectors control the magnitude response’s match to the reference domain, we apply the well-known frequency-warped bilinear transform to > > ae = aC aD and be = bC bD , (9) the wave-digital capacitor and inductor. WDFs involve one-port ideal linear reactances: the capacitor and “internal” wave vectors (of capacitance C) and inductor (of inductance L). Their current– > > voltage relationships are: ai = aA aB and bi = bA bB . (10) Cv˙(t) = i(t) and v(t) = Li˙(t) (16) Using the method of [27], we can solve for the scattering matrix S that relates the incident waves a and reflected waves b as b = Sa. where v is the port voltage, and i is the port current. Their corre- To do so, we attach instantaneous Thévenin port equivalents to sponding Laplace transforms are: each of the ports A ··· D (Fig. 4b) and confront Modified Nodal Analysis (the MNA system for Fig. 4b is shown in Fig. 6) with the CsV(s) = I(s) and V(s) = LsI(s) . (17) standard voltage wave definition, yielding Plugging in the standard WDF voltage-wave definitions

S = I + 2 0 R X−1 0 I> (11) a = v + Ri and b = v − Ri (18) R where R = diag ([R ··· R ]) is the diagonal matrix of port re- parameterized by arbitrary port resistance yields continuous- A D H(s) = B(s)/A(s) sistances and X is the MNA system matrix. time transfer functions : The vector of nonadaptable linear elements includes the volt- 1 − RCs R − Ls HC (s) = and HL(s) = (19) age source vin and the switch, which relate “root” wave vectors 1 + RCs R + Ls > > To simulate the system, we discretize reactive elements to ob- a = a 0 a 0 and b = b 0 b 0 . (12) root A B root A B tain H(z−1) = B(z−1)/A(z−1) for each. WDFs commonly 2 1−z−1 As before, port connections enforce aroot = bi and ai = broot. use the bilinear transform (BLT) [4], which substitutes T 1+z−1 The ideal voltage source vin has the wave-domain relationship for s in H(s) to form H(z−1) (T is the sampling period). The BLT’s desirable numerical properties include transfer function or- b 0 [n] = a 0 [n] + 2 v [n]. (13) A A in der preservation, unconditional stability, and passivity in the WDF An ideal switch has the wave-domain relationship domain, but it suffers from a well-known frequency distortion [18]. A common extension to the BLT is the warped (or generalized) ( BLT which is identical except T is replaced by T 0 [30] so as to −1 open switch −1 bB0 [n] = w aB0 [n] , w = . (14) 2 1−z s +1 closed switch substitute T 0 1+z−1 for . This degree of freedom is used to alter the BLT’s frequency distortion and ensure that, by selecting T 0 In the context of (5), (13)–(14) define properly, one continuous-time frequency Ω0 is mapped correctly, −jΩ0T 0 i.e., H(jΩ0) = H(e ). The coefficient T that achieves the −1 0 2 0 v Φ = , Ψ = , x = in . (15) correct mapping is given by: 0 w 0 0 root 0 0 T = 2 tan (Ω0T/2) /Ω0 . (20) Plugging (15) and (11) into (7) yields b , solving the root topology e The warped BLT has the same desirable numerical properties as with multiple nonadaptable linear elements. the BLT. Since it is not common in the WDF context, we briefly develop warped BLT discretization of WDF one-port reactances. 4.4. Frequency-Warped One-Port Linear Reactances One-port linear reactances have a first-order continuous-time Having solved the issue of realizing the WDF, we now turn our transfer function, so the warped BLT yields a first-order transfer attention to discretization schemes for its reactances. The LC lad- function in discrete time with z-transform −1 −1 der’s 36 reactances combine to create magnitude responses with −1 B(z ) β0 + β1z H(z ) = −1 = −1 . (21) numerous salient features, especially a sharp lowpass cutoff. To A(z ) α0 + α1z

DAFX-275 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

For a capacitor C and inductor L, these coefﬁcients are: 5. RESULTS

0 0 T T Here we discuss some results that characterize our model of the C: β0 = α1 = − R , β1 = α0 = + R (22) 2C 2C Hammond vibrato/chorus, including the impulse and magnitude 2L 2L L: β = −α = R + , β = −α = R − . (23) responses of each tap in the LC ladder (Section 5.1), a study on 0 1 T 0 1 0 T 0 the spectral aspects of scanner interpolation (Section 5.2), and the To eliminate delay-free loops, all one-port leaf elements of a response to a single sinusoid (Section 5.3). These results reveal a variety of effects, including delay-length modulation, phaser-like WDF require adaptation: picking a value of R that satisﬁes β0 = 0. The port impedances that adapt a capacitor and inductor are effects, amplitude modulation, and modulated comb ﬁlter effects.

0 0 RC = T /(2C) and RL = 2L/T (24) 5.1. Impulse and Magnitude Responses of LC Ladder which yield discretized transfer functions Figs. 8 and 9 show the impulse and magnitude responses at each tap v ··· v under two different WDF discretizations compared H (z−1) = z−1 H (z−1) = −z−1 . 1 19 C and L (25) to a reference “ground truth” SPICE simulation. Interestingly, the discretized transfer functions of the capacitor In Fig. 8, we use a sampling rate of fs = 44100 Hz, with the and inductor do not depend on C, L, or T 0. However, all of these capacitors and inductors discretized using the standard BLT with 0 −5 do affect their adapted port resistance. no frequency warping, i.e., T = T = 1/fs ≈ 2.2676 × 10 . In Fig. 9, use use instead a warped BLT with T 0 chosen to match the frequency Ω0 = 7075 Hz, approximately the passband edge of the ladder, yielding T 0 ≈ 2.2724 × 10−5 (20). a1 b2 − i2 + In the time domain plots, it can be seen that the LC ladder ap- v1 v2 + + i1 − − proximates a delay line. In theory, LC ladders have an idealized b a2 1 total delay time of pP L × P C [32], meaning ≈ 0.85 ms for (a) Kirchhoff domain. (b) Wave domain. the Hammond vibrato/chorus. It can be seen in the SPICE simulations that the impulse is delayed and “smeared” progressively as it Figure 7: WDF 2-port series adaptor / inverter. travels down the line, and indeed experiences ≈ 0.85 ms of delay by tap 19. To understand the complex nature of this smearing, we turn to the magnitude response. 4.5. Wave-Digital Inverter In the magnitude response, the lowpass characteristic of the LC ladder is apparent. In the SPICE simulations, the passband We saw above that wave-digital polarity inverters must necessarily edge frequency is ≈ 7075 Hz. The amount of attenuation in the be employed for proper bookkeeping of port connection polarity stopband depends on tap index: v1 has no attenuation, and the and to simplify the calculation of node voltages. Here, we review slope increases as tap index increases. Notice that in the simula- the derivation of those inverters. tion using the unwarped BLT, dc is matched perfectly, while fre- Consider two connected ports 1 and 2 with port voltages v1 quency distortion builds up as frequency increases. Specifically, and v2 and port currents i1 and i2; these ports can be connected in the passband edge is depressed by almost 500 Hz compared to the two ways. In the Kirchhoff domain, a two-port parallel connection SPICE simulation. Using the warped BLT, 7075 Hz is matched is characterized by v1 = v2 and i1 = −i2 and a two-port series perfectly. While matching the passband edge may be preferable connection by i1 = i2 and v1 = −v2. Plugging in the standard due to its perceptual salience, a mismatch remains for the rest of WDF voltage wave definition (18) yields a scattering relationship the magnitude response, most noticeably between dc and the passband edge. While the passband has dozens of features, the warped b s s a 1 = 11 12 1 , (26) BLT can only match one. Notice that, back in the time domain, the b s s a 2 21 22 2 frequency warpings of different discretizations manifests as dif- | {z } S ferent smearings. Alternatively, applying 4× oversampling is an effective though expensive way to achieve good agreement from parameterized by the two port resistances R and R . These two- 1 2 dc to the passband edge. port adaptors scatter according to " # ( − R1−R2 2λR1 −1 series 5.2. Magnitude Response of Scanner Model Interpolation S = R1+R2 R1+R2 , λ = (27) 2λR2 R1−R2 +1 parallel R1+R2 R1+R2 Fig. 10 shows the magnitude response of scanner model interpolation between terminals for the V1 (Fig. 10a), V2 (Fig. 10b), and and are both rendered reflection-free by setting R1 = R2: V3 (Fig. 10c) settings (using the unwarped BLT). dB markings are shown on the color axis. The horizontal axis represents the scanner s11 s12 0 λ = . (28) angle θ. At the vertical markings with tap indices labeled under- s21 s22 λ 0 neath, the scanner is exactly on one of the terminals. Between tap Notice that the reflection-free two-port parallel connection is sim- indices are interpolations between them. ply a normal WDF port connection [4] with each incident wave In addition to providing a time-varying delay, the ladder circuit equal to the opposite reflected wave. The two-port series connec- and scanner impart complex spectral coloration. First, the sharp tion inverts the reflected wave from each port to form the inci- passband edge is modulated slightly over the course of each vi- dent wave at the other port; it is in fact the wave-digital inverter brato cycle. The passband ripples also follow complex trajecto- (Fig. 7) [27, 31]. ries during each cycle. Since the ripples are relatively deep (many

DAFX-276 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

0.5 1 0.5 1 2 2 1 −12 1 −12 0 3 0 3 2 2 −0.5 −36 4 −0.5 −36 4 3 3 −1 5 −1 5 4 4 −60 6 −60 6 −1.5 5 7 −1.5 5 7 −2 6 −84 8 −2 6 −84 8 7 7 −2.5 9 −2.5 9 8 −108 10 8 −108 10 −3 9 11 −3 9 11 −3.5 10 −132 12 −3.5 10 −132 12 −4 11 13 −4 11 13 tap voltage (V) tap voltage (V) magnitude (dB) magnitude (dB) 12 −156 14 12 −156 14 −4.5 −4.5 13 15 13 15 −5 −5 14 −180 16 14 −180 16 17 17 −5.5 15 −5.5 15 18 18 16 −204 16 −204 −6 19 −6 19 17 17 −6.5 −228 −6.5 −228 18 SPICE 18 SPICE −7 −7 19 WDF 19 Warped WDF −252 −252 −7.5 −7.5 0 0.5 1 1.5 2 2.5 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 1000 2000 3000 4000 5000 6000 7000 8000 time (ms) frequency (Hz) time (ms) frequency (Hz) (a) Impulse response. (b) Magnitude response. (a) Impulse response. (b) Magnitude response.

Figure 8: Responses of the LC ladder, using unwarped BLT. Figure 9: Responses of the LC ladder, using warped BLT.

6. CONCLUSION around 6 dB and some larger), they create an audible phaser-like effect on broadband input signals. The voltage dividers R and k− In this study on modeling the Hammond organ vibrato/chorus, we R , k ∈ 1 ··· 6, produce amplitude modulation during each cy- k+ introduced new theoretical tools enabling the inclusion of multi- cle of θ [10]. Table 3 shows the gain of each stage’s divider. ple linear nonadaptable elements at the root of a WDF tree, applied the well-known frequency-warped bilinear transform to the Table 3: Voltage divider gains (in dB) at each tap. derivation of wave-digital capacitors and inductors, and illustrated the systematic use of wave-digital polarity inverters. Although be- tap 1 2 3 4 5 6 7 ··· 19 yond the scope of this paper, the complex spectral properties and gain −2.9 −2.8 −2.0 −1.5 −0.83 −0.56 0 frequency-dependent vibrato of the Hammond organ vibrato/chorus deserve further study (cf. the complexities of vocal vibrato, including “spectral modulation” [33]). Since the Hammond vibrato/chorus approximates a delay line it is not surprising that the magnitude response of the scanner interpolation exhibits comb-filter-like features. Assuming the idealized 7. ACKNOWLEDGMENTS delay time discussed earlier, linear interpolation would produce notches halfway through each crossfade at frequencies dictated by Thanks to Jonathan Abel for helpful discussions on ladder circuits. the separation between taps, at the locations indicated by × sym- bols. The actual minima in the Hammond response are very close 8. REFERENCES to these notches, as predicted. [1] “Service manual—models: A, A-100, AB, BA, BC, BCV, BV, B2, B3, C, CV, C2, C-2G, C3, D, DV, D-100, E, G, GV, RT, RT-2, RT- 3,” Tech. Rep. HOOO-OOO495, Nov. 1987. 5.3. Sinusoid Study [2] S. Hammond, “Analog Outfitters scanner review,” Premier Guitar, Dec. 18 2015. [3] J. M. Hanert, “Electrical musical apparatus,” Aug. 14 1945, U.S. We study a single 1760 Hz (A6 in scientific pitch notation) sinu- Patent No. 2,382,413. soidal input for the three vibrato depth settings (Fig. 11). Notice [4] A. Fettweis, “Wave digital filters: Theory and practice,” Proc. IEEE, that the V1 setting produces the narrowest vibrato, the V2 setting vol. 74, no. 2, pp. 270–327, 1986. produces a medium vibrato, and the V3 setting produces the widest [5] A. Fettweis, “Digital filters structures related to classical filter net- vibrato. Notice also that each setting produces a differently shaped works,” Archiv Elektronik Übertragungstechnik (AEÜ), vol. 25, pp. vibrato. The vibrato width and shape are a consequence of the dif- 79–89, 1971. ferent tap spacings of each setting; the time-varying phase shift [6] G. De Sanctis and A. Sarti, “Virtual analog modeling in the wave- digital domain,” IEEE Trans. Audio, Speech, Language Process. for a given frequency, which manifests as frequency modulation, (TASLP), vol. 18, no. 4, pp. 715–727, 2010. is proportional to the time derivative of its group delay [19]. The [7] H. E. Meinema and W. C. Johnson, H. A. .and Laube Jr., “A new combination of amplitude modulation and frequency modulation reverberation device for high fidelity systems,” J. Audio Eng. Soc. is visible as 6 × 16 Hz spaced sidebands around the main signal. (JAES), vol. 9, no. 4, pp. 284–326, 1961.

DAFX-277 Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5–9, 2016

(a) V1 Setting. (b) V2 Setting. (c) V3 Setting.

Figure 10: Magnitude response at various interpolations. Recall Fig. 3 and Table 2.

[8] K. J. Werner, V. Nangia, J. O. Smith III, and J. S. Abel, “Resolving wave digital filters with multiple/multiport nonlinearities,” in Proc. 18th Int. Conf. on Digital Audio Effects (DAFx), Trondheim, Norway, Nov. 30 – Dec. 3 2015. [9] G. Reid, “Synthesizing tonewheel organs,” Sound on Sound (SOS), Nov. 2003. [10] G. Reid, “Synthesizing Hammond organ effects: Part 1,” SOS, Jan. 2004. [11] J. Pekonen, T. Pihlajamäki, and V. Välimäki, “Computationally efficient Hammond organ synthesis,” in Proc. 14th DAFx, Paris, France, Sept. 19–23 2011. [12] K. J. Werner and J. S. Abel, “Modal processor effects inspired by Hammond tonewheel organs,” Appl. Sci., vol. 6, no. 7 (Special Issue “Audio Signal Processing”), 2016. [13] J. S. Abel and K. J. Werner, “Distortion and pitch processing using a (a) V1 setting. (b) V2 setting. (c) V3 setting. modal reverb architecture,” in Proc. 18th DAFx, Trondheim, Norway, Nov. 30 – Dec. 3 2015. [14] J. S. Abel, S. Coffin, and K. S. Spratt, “A modal architecture for Figure 11: Studying a single 1760 Hz sinusoid. artificial reverberation with application to room acoustics modeling,” in Proc. 137th AES, Los Angeles, CA, Oct. 9–12 2014. [15] D. J. Leslie, “Rotatable tremulant sound producer,” Nov. 29 1949, [26] J. Pakarinen and M. Karjalainen, “Enhanced wave digital triode U.S. Patent No. 2,489,653. model for real-time tube amplifier emulation,” IEEE TASLP, vol. [16] R. Kronland-Martinet and T. Voinier, “Real-time perceptual sim- 18, no. 4, pp. 738–746, 2010. ulation of moving sources: Application to the Leslie cabinet and [27] K. J. Werner, J. O. Smith III, and J. S. Abel, “Wave digital filter adap- 3D sound immersion,” EURASIP J. Audio, Speech, Music Process., tors for arbitrary topologies and multiport linear elements,” in Proc. 2008, Article ID 849696. 18th DAFx [17] J. Smith, S. Serafin, J. Abel, and D. Berners, “Doppler simulation , Trondheim, Norway, Nov. 30 – Dec. 3 2015. and the Leslie,” in Proc. 5th DAFx, Hamburg, Germany, Sept. 26–28 [28] M. J. Olsen, K. J. Werner, and J. O. Smith III, “An iterative approach 2002. to resolving wave digital filters with grouped nonlinearities,” in Proc. [18] J. O. Smith III, Physical Audio Signal Processing for Virtual Musical 19th DAFx, Brno, Czech Republic, Sept. 5–9 2016. Instruments and Audio Effects, Online book, 2010 edition. [29] W. R. Dunkel, M. Rest, K. J. Werner, M. J. Olsen, and J. O. Smith III, [19] S. Disch and U. Zölzer, “Modulation and delay line based digital “The Fender Bassman 5F6-A family of preamplifier circuits—a wave audio effects,” in Proc. 2nd DAFx, Trondheim, Norway, Dec. 9–11 digital filter case study,” in Proc. 19th DAFx, Brno, Czech Republic, 1999. Sept. 5–9. [20] P. Dutilleux, M. Holters, S. Disch, and U. Zölzer, Modulators and [30] F. G. Germain and K. J. Werner, “Design principles for lumped model demodulators, chapter 3, pp. 83–99, 2011, appears in [34]. discretization using Möbius transforms,” in Proc. 18th DAFx, Trond- [21] J. Herrera, C. Hanson, and S. Abel, J. “Discrete time emulation of heim, Norway, Nov. 30 – Dec. 3 2015. the Leslie speaker,” in Proc. 127th Conv. Audio Eng. Soc. (AES), [31] S. D’Angelo and V. Välimäki, “Wave-digital polarity and current New York, NY, USA, Oct. 9–12 2009, Conv. Paper 7925. inverters and their application to virtual analog audio processing,” [22] J. Dattorro, “Effect design part 2: Delay-line modulation and cho- in IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP), Kyoto, rus,” JAES, vol. 45, no. 10, pp. 764–768, Oct. 1997. Japan, Mar. 2012, pp. 469–472. [23] K. J. Werner, W. R. Dunkel, M. Rest, M. J. Olsen, and J. O. Smith [32] Rhombus Industries Inc., “Passive delay line design considerations,” III, “Wave digital filter modeling of circuits with operational am- Tech. Rep. APP1_PAS, Huntington Beach, CA, Jan. 1998. plifiers,” in Proc. European Signal Process. Conf. (EUSIPCO), Bu- [33] R. Maher and J. Beauchamp, “An investigation of vocal vibrato for dapest, Hungary, Aug. 29 – Sept. 2 2016. synthesis,” Appl. Acoust., vol. 30, pp. 219–245, 1990. [24] A. Fettweis and K. Meerkötter, “On adaptors for wave digital filters,” [34] U. Zölzer, Ed., DAFX: Digital Audio Effects, John Wiley & Sons, IEEE Trans. Acoust., Speech, Signal Process., vol. 23, no. 6, 1975. second edition, 2011. [25] R. C. D. de Paiva, J. Pakarinen, V. Välimäki, and M. Tikander, “Real-time audio transformer emulation for virtual tube amplifiers,” EURASIP J. Advances Signal Process., 2011.

DAFX-278