This symbol advises the user to read all accompanying literature for safe operation of the unit. Read, retain, and follow all instructions. Heed all warnings. Only connect the power supply cord to an earth grounded AC receptacle in accordance with the voltage and frequency ratings listed under INPUT POWER on the rear panel of this product. WARNING: To prevent damage, fire or shock hazard, do not expose this unit to rain or moisture. Unplug the power supply cord before cleaning the unit exterior (use a damp cloth only). Wait until the unit is completely dry before reconnecting it to power. During operation: Maintain at least 6 inches (15.25 cm) of unobstructed air space around the unit to allow for proper ventilation and cooling of the unit; do not block any vents in the unit chassis. Also, if rack mounted, remove the rack enclosure front and rear covers and leave empty one full rack space above the unit. This product should be located away from heat sources such as radiators, heat registers, or other products that produce heat. This product may be equipped with a polarized plug (one blade wider than the other). This is a safety feature. If you are unable to insert the plug into the outlet, contact an electrician to replace your obsolete outlet. Do not defeat the safety purpose of this plug. Protect the power supply cord from being pinched or abraded. This product should only be used with a cart or stand that is recommended by the manufacturer. The power supply cord of this product should be unplugged from the outlet when left unused for a long period of time, or during electrical storms. This product should be serviced by qualified service personnel when: the power supply cord or the plug has been damaged; or objects have fallen, or liquid has been spilled onto the product; or the product has been exposed to rain; or the product does not appear to operate normally or exhibits a marked change in performance; or the product has been dropped, or the enclosure damaged. Do not drip nor splash liquids, nor place liquid filled containers on the unit. CAUTION: No user serviceable parts inside, refer servicing to qualified personnel only. Fender amplifiers and loudspeaker systems are capable of producing very high sound pressure levels which may cause temporary or permanent hearing damage. Use care when setting and adjusting volume levels during use.

4 Consignes

Scurit

Importantes

Istruzioni

Importanti

Sicurezza
Ce symbole prvient lutilisateur de tensions lectriques dangereuses prsentes dans lappareil.
Questo simbolo indica la presenza di tensione pericolosa all'interno della cassa.

The Fender Bassman amplifier of the 1950s evolved as the pro bassists answer to amplifying a great new invention: the Fender Precision Bass. This powerful rig could easily compete along with the common instrumentation of the day: horns, piano, four-piece drum kit with calfskin heads, and maybe even an electric guitarist with a nice, new 10-Watt amp! By the 60s and 70s, as performance venues grew larger, bassists demanded more and more power and efficiency. Fender and many other manufacturers began to offer amps with bigger speaker cabinets and more output. The original 50-Watt, 4X10, all-tube Tweed Bassman gained popularity with Rock, Country, and Blues guitarists as a great-sounding, reliable, easy to use guitar amp. Even some emerging British amp designers of the day copied its circuitry for use in their most celebrated designs. In the 1990s, Fender released an affordable reissue of the now-collectible 59 Bassman amplifier to massive acclaim. Many players tired of experimenting with preamps, equalizers, and effects racks rediscovered the

To p P a n e l

A. FUSE - Protects the amplifier from electrical faults. Replace a failed fuse ONLY with the type and rating as labeled on the fuse holder. B. POWER - Switches the amplifier on and off. C. STANDBY - Use STANDBY during short breaks instead of switching the amplifier off. By doing this you avoid the warm-up delay upon returning to play and you extend the life span of your tubes. Using STANDBY for 30 seconds when first switching the amplifier on will also extend tube life span. D. POWER INDICATOR - Illuminates when the amplifier is on and receiving power. Unscrew the jeweled cover to access the bulb (type T47) for replacement. E. PRESENCE - Adjusts the ultra-high frequency tone at a point in the circuitry after all the other tone controls to add a special high-end sparkle to the final output sound.
F. MIDDLE - Adjusts the mid-frequency tone. G. BASS - Adjusts the low-frequency tone. Tip: At higher VOLUME settings, lower the BASS setting to reduce speaker flap. H. TREBLE - Adjusts the high-frequency tone. I. VOLUME BRIGHT - Adjusts the loudness of the Bright channel. J. VOLUME NORMAL - Adjusts the loudness of the Normal channel. K. BRIGHT INPUTS - Plug in connection for your guitar. Input 2 provides 6dB less gain than Input 1 for highoutput or pre-amplified instruments. The bright channel boosts treble tones. L. NORMAL INPUTS - Plug in connection for your guitar. Input 2 provides 6dB less gain than Input 1 for high-output or pre-amplified instruments.

Proc. of the 9th Int. Conference on Digital Audio Effects (DAFx-06), Montreal, Canada, September 18-20, 2006
DISCRETIZATION OF THE 59 FENDER BASSMAN TONE STACK David T. Yeh, Julius O. Smith Center for Computer Research in Music and Acoustics (CCRMA) Stanford University, Stanford, CA {dtyeh|jos}@ccrma.stanford.edu
ABSTRACT The market for digital modeling guitar ampliers requires that the digital models behave like the physical prototypes. A component of the iconic Fender Bassman guitar amplier, the tone stack circuit, lters the sound of the electric guitar in a unique and complex way. The controls are not orthogonal, resulting in complicated lter coefcient trajectories as the controls are varied. Because of its electrical simplicity, the tone stack is analyzed symbolically in this work, and digital lter coefcients are derived in closed form. Adhering to the technique of virtual analog, this procedure results in a lter that responds to user controls in exactly the same way as the analog prototype. The general expressions for the continuous-time and discrete-time lter coefcients are given, and the frequency responses are compared for the component values of the Fender 59 Bassman. These expressions are useful implementation and verication of implementations such as the wave digital lter. 1. INTRODUCTION 1.1. Motivation The guitar amplier is an essential component of the electric guitar sound, and often musicians collect several ampliers for their tonal qualities despite the space they occupy. As digital signal processors (DSP) continue to improve in performance, there is great interest in replacing expensive and bulky vacuum tube guitar ampliers with more exible and portable digital models. A digital model of a guitar amplier allows a variety of sounds associated with different ampliers to be selected from a single amplier unit. One company, Line 6 bases its main product line upon this concept, and other companies such as Roland (Boss), Korg (Vox), Harman International (Digitech) have competing products. Most commercially viable digital guitar processing products use simplied models of the distortion and lters to reduce DSP power consumption and reduce manufacturing costs. The distortion is typically a nonlinear transfer curve, accompanied by digital ltering that is manually tuned to match the sound of a famous guitar amplier. With no pressure to produce a commercially successful product, this research takes a different approach. The goal of this research is to see how accurate a sound can be achieved through careful physical modeling of the vacuum tube amplier and to provide a physical basis for the digital model and parameters. Because the tone stack is a passive, linear component, it is a straightforward starting point. 1.2. Properties of the tone stack Commonly found in many guitar ampliers, especially those that derive from the Fender design, the tone stack lters the signal of

Vi C1 R4

(1-t)R1 Vo t R1
Components for 59 Bassman
C1 = 0.25 nF C2 = C3 = 20 nF R1 = 250k R2 = 1M R3 = 25k R4 = 56k

(1-m)R3 C3

Figure 1: Tone stack circuit with component values.
the guitar in a unique and non-ideal way. The user can adjust Treble, Middle, and Bass controls to modify the gain of the respective frequency bands. However, these controls are not orthogonal, and changing some controls affects the other bands in a complex way. The full Bassman schematic can be easily found online [1] and in guitar amplier books. While other guitar ampliers may vary slightly, in the Bassman type designs, the tone stack is found after the preamplier stages and before the phase splitter. In good designs, the tone stack is preceded by a cathode follower to buffer the input and reduce variations in frequency response due to loading. Typically this presents a 1k load to the input and the phase splitter stage presents a 1M load to the output. The Fender 59 Bassman tone stack circuit is shown in Fig. 1. The Treble, Middle, and Bass knobs are potentiometers, which have been modeled here as parameterized resistors. The Treble and Middle controls use linear potentiometers, while the Bass control uses a logarithmic taper potentiometer. In this paper, t and m correspond to the Treble and Middle controls and range in value from [0, 1]. The Bass control, l, also ranges from [0, 1], but is swept logarithmically.

DAFX-1

1.3. Related work Fender Musical Instruments has a patent to simulate various tone stacks using an active analog lter and an interpolation scheme to extract the lter coefcients [2]. Line 6 also models the behavior of the Bassman tone stack as indicated in the BassPODxt manual. However, their implementation is proprietary knowledge. An open source guitar effects plug-in suite for Linux, CAPS [3], uses shelving lters instead of the tone stack. Previous works have analyzed the tone stack using numerical circuit analysis techniques. This involves setting up the nodal equations as a matrix and inverting it or performing Gaussian elimination to nd the solution. For example, the Tone Stack Calculator from Duncan Amps will plot the frequency response of various tone stacks given the control settings [4]. Kuehnel in his book analyzed the mesh equations of the tone stack, using low frequency and high frequency circuit approximations [5]. He also compares these simplied equations to the numerical solutions solved by inverting the matrix of the mesh equations. While the approximations make the circuit analysis more tractable, they do not reduce the order of the equations and do not make the discretization of the lter any easier. Because the tone stack is a third-order passive network of resistors and capacitors (RC), its lter coefcients can be derived and modeled exactly in the digital domain as shown later. The approach taken here is to nd the continuous time transfer function of the circuit analytically and to discretize this by the bilinear transformation. This provides a means of updating the digital lter coefcients based upon changes to the tone controls. The passive lter circuit also is suited to implementation as a wave digital lter (WDF)[6]. This approach can easily model standard components such as inductors, capacitors, and resistors. The analytical form derived here can be used for comparison with and verication of the WDF implementation. 2. DISCRETIZATION PROCEDURE 2.1. Symbolic Circuit Analysis Because this is a relatively simple circuit, it is amenable to exact symbolic analysis by mathematical Computer Aided Design (CAD) software such as Mathematica (Wolfram Research, Inc., Champaign, IL). The lter coefcients can thus be found without any approximations. Performing symbolic nodal analysis on this circuit yields the following input/output transfer function H(s) = Vo (s)/Vi (s), where Vo is the output and Vi is the input as in Fig. 1. H(s) = where b1 = tC1 R1 + mC3 R3 + l(C1 R2 + C2 R2 ) + (C1 R3 + C2 R3 ), b1 s + b2 s2 + b3 s3 , a0 + a1 s + a2 s2 + a3 s3 (1)

b3 = lm(C1 C2 C3 R1 R2 R3 + C1 C2 C3 R2 R3 R4 )
m2 (C1 C2 C3 R1 R3 + C1 C2 C3 R3 R4 ) + m(C1 C2 C3 R1 R3 + C1 C2 C3 R3 R4 ) + tC1 C2 C3 R1 R3 R4 tmC1 C2 C3 R1 R3 R4 + tlC1 C2 C3 R1 R2 R4 ,
a0 = 1, a1 = (C1 R1 + C1 R3 + C2 R3 + C2 R4 + C3 R4 ) + mC3 R3 + l(C1 R2 + C2 R2 ),
2 a2 = m(C1 C3 R1 R3 C2 C3 R3 R4 + C1 C3 R+ C2 C3 R3 ) + lm(C1 C3 R2 R3 + C2 C3 R2 R3 ) m2 (C1 C3 R3 + C2 C3 R3 ) + l(C1 C2 R2 R4 + C1 C2 R1 R2 + C1 C3 R2 R4 + C2 C3 R2 R4 ) + (C1 C2 R1 R4 + C1 C3 R1 R4 + C1 C2 R3 R4 + C1 C2 R1 R3 + C1 C3 R3 R4 + C2 C3 R3 R4 ), a3 = lm(C1 C2 C3 R1 R2 R3 + C1 C2 C3 R2 R3 R4 ) m2 (C1 C2 C3 R1 R3 + C1 C2 C3 R3 R4 ) + m(C1 C2 C3 R3 R4 + C1 C2 C3 R1 R3 C1 C2 C3 R1 R3 R4 ) + lC1 C2 C3 R1 R2 R4 + C1 C2 C3 R1 R3 R4 ,
where t is the Treble (or top) control, l is the Bass (or low) control, and m is the middle control. 2.2. Verication with SPICE circuit simulation To verify the correctness of this expression, Figs. 2 and 3 compare the frequency response with the result from the AC analysis of SPICE1 simulation at the settings t = m = l = 0.5. The plots show an exact match, verifying that Eqn. 1 is a complete and exact expression for the transfer function of the tone stack. SPICE simulation also determined that the frequency response was unaffected by the typical loading of 1k at the input and 1M at the output. 2.3. Discretization by Bilinear Transform The continuous time transfer function was discretized by the bilin1 ear transformation. Substituting s = c 1z1 in (1) using Mathe1+z matica yields H(z) = where B0 = b1 c b2 c2 b3 c3 , B1 = b1 c + b2 c2 + 3b3 c3 , B2 = b1 c + b2 c2 3b3 c3 , B3 = b1 c b2 c2 + b3 c3 , A0 = a0 a1 c a2 c2 a3 c3 , A1 = 3a0 a1 c + a2 c2 + 3a3 c3 , A2 = 3a0 + a1 c + a2 c2 3a3 c3 , A3 = a0 + a1 c a2 c2 + a3 c3.
1 http://bwrc.eecs.berkeley.edu/Classes/IcBook/SPICE/
B0 + B1 z 1 + B2 z 2 + B3 z 3 A0 + A1 z 1 + A2 z 2 + A3 z 3
b2 = t(C1 C2 R1 R4 + C1 C3 R1 R4 ) m2 (C1 C3 R3 + C2 C3 R3 ) + m(C1 C3 R1 R3 + C1 C3 R3 + C2 C3 R3 ) + l(C1 C2 R1 R2 + C1 C2 R2 R4 + C1 C3 R2 R4 ) + lm(C1 C3 R2 R3 + C2 C3 R2 R3 ) + (C1 C2 R1 R3 + C1 C2 R3 R4 + C1 C3 R3 R4 ),

DAFX-2

Magnitude (dB)

0 Magnitude (dB)

Symbolic SPICE
0 -10 -20 -30 --10 --10 -10 10

l=0, m=0

-1000 Frequency (Hz) 10000
100 l=0, m=0.5 t=0 bt t=0
1000 Frequency (Hz) t=0.5 bt t=0.5

10000 t=1 bt t=1

Figure 2: Comparison of magnitude response between analytical expression and SPICE for t = l = m = 0.5.

90 Phase (degrees) -45 Symbolic SPICE

100 l=0, m=1 t=0 bt t=0

100 t=0 bt t=0 t=0.5 bt t=0.5

1000 Frequency (Hz)

100 Frequency (Hz)
Figure 3: Comparison of phase response between analytical expression and SPICE for t = l = m = 0.5.
Figure 4: Comparison of lter magnitude response between original and discretized (fs = 44.1 kHz) lters, l = 0.
0 -10 -20 -30 --10 --10 -t=0 bt t=t=0 bt t=0 l=0.1, m=1000 Frequency (Hz) t=0.5 bt t=0.t=1 bt t=t=0 bt t=0 l=0.1, m=0.1000 Frequency (Hz) t=0.5 bt t=0.t=1 bt t=1 l=0.1, m=t=0.5 bt t=0.Frequency (Hz) t=1 bt t=1 10000
3. ANALYSIS OF RESULTS 3.1. Comparison of continuous- and discrete- time responses Figs. 46 show the discrete- and continuous-time transfer functions compared for various settings of t, m, and l. Each gure shows a different setting of l, and each sub-gure shows a different setting of m. In each plot, the treble control, t, was swept from 0.0001 to 0.5 to 0.9999 and can be distinguished by the corresponding increase in high frequency response. The discretized lter used a sampling frequency of 44.1 kHz as typical for audio systems. The plots for fs = 44.1 kHz show an excellent match through 10 kHz. The discrete and continuous plots are practically indistinguishable, with some deviations at the higher frequencies, as expected with the bilinear transform. Because commercial guitar processing units use a lower sampling rate for cost savings, Figs. 79 show the same plots as above with fs reduced to 20 kHz. These curves deviate slightly more from H(s) at high frequencies, but exhibit the same trends as before. The errors, dened as the difference between the dB values of H(s) and H(z) at each frequency, are plotted in Fig. 10 for fs = 20 kHz and fs = 44.1 kHz (abbreviated as 44k) for the settings of t, m, and l that give the worst case results. The error is only meaningful for frequencies up through fs /2. The curves for t = 0.5, m = 0, b = 1 are characteristic of tone settings that give a high pass response and have error within 0.5 dB for both cases of fs.
Figure 5: Comparison of lter magnitude response between original and discretized (fs = 44.1 kHz) lters, l = 0.1.

DAFX-3

We used c = 2/T , which is ideal for frequencies close to DC.
-10 -20 -30 --10 --10 -t=0 bt t=t=0 bt t=0 l=1, m=t=0.5 bt t=0.Frequency (Hz) t=1 bt t=10 t=0 bt t=0 l=1, m=0.1000 Frequency (Hz) t=0.5 bt t=0.l=1, m=1000 Frequency (Hz) t=0.5 bt t=0.5 10000
0 -10 -20 -30 --10 --10 -t=0 bt t=t=0 bt t=0 l=0.1, m=1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=10 t=0 bt t=0 l=0.1, m=0.1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=l=0.1, m=1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=1 10000

t=1 bt t=1

Figure 6: Comparison of lter magnitude response between original and discretized (fs = 44.1 kHz) lters, l = 1.
Figure 8: Comparison of lter magnitude response between original and discretized (fs = 20 kHz) lters, l = 0.1.
Magnitude (dB) Magnitude (dB)
0 -10 -20 -30 --10 --10 -t=0 bt t=t=0 bt t=0 l=1, m=1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=10 t=0 bt t=0 l=1, m=0.1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=l=1, m=1000 Frequency (Hz) t=0.5 bt t=0.5 t=1 bt t=1 10000

l=0, m=0.5 t=0 bt t=0

t=0.5 bt t=0.5

l=0, m=1 t=0 bt t=0

Frequency (Hz) t=0 bt t=0 t=0.5 bt t=0.5 t=1 bt t=1
Figure 7: Comparison of lter magnitude response between original and discretized (fs = 20 kHz) lters, l = 0.
Figure 9: Comparison of lter magnitude response between original and discretized (fs = 20 kHz) lters, l = 1.

DAFX-4

Frequency (Hz)
t=0 m=0 b=1 fs=44k t=0.5 m=0 b=1 fs=44k t=0 m=0 b=1 fs=20k t=0.5 m=0 b=1 fs=20k
lower order lters. Understanding the poles and zeros of the system, one could make simplifying assumptions, ignoring terms that have little impact on the locations of the poles and zeros. One implementation would be to nd the partial fraction expansion of the transfer function using the expression given and precompute the poles, residues, and direct terms based upon the three-dimensional input space of the tone controls. These terms can be interpolated in the input space and used in the parallel lter structure that arises from the partial fraction expansion. The existence of an analytical expression for the poles and zeros also informs the choice of c in the bilinear transform. The analytical expression allows the computation of frequency domain features such as local maxima or anti-resonance notches to be matched in the discrete-time domain. 4. CONCLUSIONS This work shows that the Fender tone stack can be parameterized exactly in the discrete-time domain and that the bilinear transform provides an outstanding frequency mapping for reasonable sampling rates. The transfer function for the physical tone stack was found as a function of its control parameters and component values using symbolic math software. This analysis provides a formula for updating the digital tone stack coefcients in a way that exactly emulates the physical circuit. The symbolic form of the transfer function also allows easy determination of the poles and zeros of the system and guides the design of a lter with simplied coefcients. Further work remains to factor the expression for the tone stack frequency response and nd a structure with simpler expressions for updating the lter. One possible implementation is the wave digital lter. A real-time implementation of the tone stack is also in progress. 5. ACKNOWLEDGEMENTS David Yeh is supported by the NDSEG fellowship. Thanks to Tim Stilson for help with root loci. 6. REFERENCES [1] Ampwares, 5F6-A schematic, Retrieved June 29th, 2006, [Online] http://www.ampwares.com/ffg/bassman_narrow. html. [2] D. V. Curtis, K. L. Chapman, C. C. Adams, and Fender Musical Instruments, Simulated tone stack for electric guitar, United States Patent 6222110, 2001. [3] T. Goetze, caps, the C Audio Plugin Suite, Retrieved June 29th, 2006, [Online] http://quitte.de/dsp/caps.html. [4] Duncan Amps, Tone stack calculator, Retrieved June 29th, 2006, [Online] http://www.duncanamps.com/tsc/. [5] R. Kuehnel, Circuit Analysis of a Legendary Tube Amplier: The Fender Bassman 5F6-A, 2nd ed. Seattle: Pentode Press, 2005. [Online]. Available: http://www.pentodepress. com/contents.html [6] A. Fettweis, Wave digital lters: Theory and practice, Proc. IEEE, vol. 74, pp. 270327, Feb. 1986.

Figure 10: Error as difference between dB values of H(s) and H(z), for fs = 20 and 44.1 kHz, and the noted tone settings.
The curves for t = 0, m = 0, b = 1 are characteristic of settings that give a low pass response and exhibit a rapidly increasing error as frequency increases because the bilinear transform maps the null at innite frequency to fs /2. The error rises to 3 dB at roughly 6 kHz for fs = 20 kHz, and at 13 kHz for fs = 44.1 kHz. Because of the low pass nature of these responses, the errors occur at frequencies where the magnitude is at least 10-20 dB lower than its peak value, making them perceptually less salient. Also, given that the frequency response of a typical guitar speaker is from 100 Hz to 6000 Hz, the deviations at higher frequencies would be inconsequential. 3.2. Implications of system poles and zeros for lter implementation The plots exhibit the complex dependence of the frequency response upon the tone controls. The most obvious effect is that changes in the Middle control also affect the treble response. The analytical form of the transfer function provides a way to nd the poles and zeros of the system as the settings are varied and gives insight into how the lter could be simplied to facilitate the implementation while maintaining accuracy. Note that the tone stack is an entirely passive circuit composed of resistors and capacitors. This implies that the three poles of this system are all real. There is a zero at DC, leaving a pair of zeros that may be complex depending on the control settings. This also implies that the tone stack cannot be a resonant circuit although the pair of imaginary zeros can set up an anti-resonance as evident in the notch seen in the frequency response plots. Also note from Eqn. (1) that none of the coefcients of the denominator depends on the treble control, t. The treble control therefore does not control the modes of the circuit but only adjusts the position of the zeros. This circuit can be decomposed into a weighted sum of terms that correspond to each mode by the partial fraction expansion. From this perspective, the treble control only affects the weighting of the different modes, but not the pole location of each mode. The poles are controlled exclusively by the bass and middle knobs. This insight suggests possible alternate lter topologies. Instead of implementing the lter directly as a single third-order lter, one could equivalently use series and parallel combinations of

Error (dB)

DAFX-5