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The additional diode does a few things. It gives a little more headroom for dynamics (assuming the gain is not full tilt), and by doing so provides a hotter potential output level. That means one can turn the gain down, the volume up, and provide some reasonable push to an amp or subsequent pedal, whose overdrive you like. That's a little harder to do with a TS-9/808.

Sd-1s are plentiful and cheap. If a person wants to go whole hog, get an SD-1 and turn it into a Timmy...or close enough.
 

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I've heard they we pretty similar a long time ago, never liked how the sd1 reduces the lows kind of like the ds1. I traded mine for an od3 which sounds better than both to my ear, I used to use it after the ds1, that way I could have hi gain with the character of the od3.
 

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The classic, and much-criticized, mid-hump that the TS-9 and SD-1 both have, results from the trimming of bass at the clipping section. The generally-assumed reason for them doing this is because, as I like to say, "a lot of the signal lives in the basement". Because the low end has much greater amplitude, it would get clipped harder than the mids and treble. So, in order to get a consistent degree of clipping across the entire fingerboard, both Boss and Ibanez trimmed back on the bass (below 720hz), so that the low E and A strings would not be much greater amplitude than D and above. Many of the "boutique" pedals derived from the TS-9 change the value of the capacitor that limits the bass content to let more bass through. The tone becomes fuller, but it also does exactly what is expected: the signal gets clipped harder.

Part of the characteristic sound of the SD-1 and TS-9 also comes from where the diodes are located. If they go to ground from the output of the chip, like in the case of the MXR Distortion+ and DOD250, then they clip and certain limit the level, but have no effect on the gain.

Diodes have what is called a forward voltage. When the signal attempting to pass through them is below that voltage, the diodes do not conduct. When the signal reaches that voltage, the diodes let it pass. The guitar signal is not constant, so sometimesthe diodes conduct and sometimes they don't. The "gain" of the chip is dictated by how much of the output goes back to the input. The analogy I like to use is that op-amps are designed to want to go full speed. We set the gain of the chip by controlling how much negative feedback is provided from output back to input. Let a lot of feedback through, and gain is reduced. Provide NO feedback (what chip datasheets often call "open loop gain") and you get the maximum gain the chip is capable of.

So, whenever the signal reaches the forward voltage of the diodes, the diodes allow more negative feedback from output back to input, and gain becomes reduced for as long as the signal is at or above that voltage, which could be mere 10,000ths of a second, if that. But hey, that sounds familiar. In fact, it sounds just like....a compressor. Correct! That's why diodes in the feedback loop, like an SD-1 or TS-9, provide a more "compressed" sound, while diodes going to ground provide a more "ragged" sound, since the gain of the chip is whatever you set it to, and diodes to ground simply lop off the top of the wave.

The Boss OD-1, that came before the SD-1 (and possibly before the TS-9) did not have a tone control. Instead, it used a fixed treble-cut filter, which, to my ears, achieved a more pleasing happy medium than trying to set the SD-1/TS-9 tone control to "the middle". I made myself one, and quite like it. Needs a little more bass, though, so I took care of that.

It's actually not that hard to "improve" an SD-1 or TS-9, tonally. Interested parties hungry for ideas can drop me a PM.
 

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Thanks for the explanation, I didn't know about the similarities of distortions and compressors from a technical point of view but I did know they are basically the same, I recently made a video about how to use a compressor as a distortion:
 
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