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I dug out these two old things I had lying around, an acoustic simulator and an external clip-on preamp, and tried each one (separately) between the guitar and amp. They made a really big difference in the right direction. (Sorry about the huge pics.)

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The cutoff frequency of a 22n tone cap and a 250k pot is 29Hz and with a 500k pot it is an octave lower so for all practical purposes the fully open tone pot is just a resistor to ground in parallel with the volume pot. The actual value of the capacitor has no discernible effect until the tone is turned well down.
This implies that a pickup with 500k tone and volume is effectively loaded with 250k at full volume. When the volume is full, the pickup is directly connected to the cable capacitance and amp input capacitance and resistance. It is this loading that brings the pickup resonance down from its unloaded value around 7kHz to the region around 2-3 kHz where it contributes to the characteristic pickup tone.
As the volume is turned down three things happen.
1) The pickup gradually loses the extra capacitive loading so that the resonant frequency moves back towards the unloaded value. The 2-3 kHz peak rapidly disappears.
2) The effective output resistance of the guitar rises because the volume pot is no longer just in parallel with the pickup but the top part of it is in series with the output.
3) The higher output resistance forms a low pass filter with the cable and amp capacitance.
These are the reasons why there is a noticeable loss of treble as the volume is turned down a little.
The significance of item 3 is that while a higher value volume pot will give a stronger resonant peak or more treble at full volume, it will cause a more rapid loss of treble as the volume is turned down a little.
Taking away the tone pot would allow a 250k volume pot to be used. This would leave the full volume response as before but would reduce the high frequency loss as the volume is turned down. Adding a 250k No Load tone pot would leave the full tone, full volume response unaltered. The only disadvantage seems to be the loss of treble control in the very top part of the tone setting where the resonant peak is gradually reduced. The tone would step initially from peaked to flat, the rest of the range working as usual. It is important to note that this disadvantage occurs only when the volume is at full.
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For many of us in jazz the loss of treble when the volume is reduced from 10 to about 8 is a feature, not a bug.
Those who don't want that can use an RC treble bleed circuit.
Recommandations for Hollowbodies for $600 and under?
Today, 05:20 AM in Guitar, Amps & Gizmos