A Computational Model of the Hammond Organ Vibrato/Chorus Using Wave Digital Filters
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A Computational model of the Hammond Organ Vibrato/Chorus Using Wave Digital Filters Werner, K. J., Dunkel, W. R., & Germain, F. G. (2016). A Computational model of the Hammond Organ Vibrato/Chorus Using Wave Digital Filters. In P. Rajmic, F. Rund, & J. Schimmel (Eds.), Proceedings of the 19th International Conference on Digital Audio Effects (pp. 271–278). [54] DAFx. http://dafx16.vutbr.cz/proceedings.html Published in: Proceedings of the 19th International Conference on Digital Audio Effects Document Version: Publisher's PDF, also known as Version of record Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal Publisher rights Copyright 2016 The Authors Published in the Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16) General rights Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact [email protected]. Download date:28. Sep. 2021 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 2 [1 ··· 6] y 2 [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 2 [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 [2 ··· 6] are connected.