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    Building, troubleshooting and testing of these amplifiers should only be
    performed by someone who is thoroughly familiar with
    the safety precautions around high voltages.

6DJ8 vs 6N1P (6H1N)

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Using an LM317 as a current source has a couple of caveats.
I use the LM317 in that way, but have to watch how I use it.

The minimum recommended current of an LM317 is 10 mA (it requires current to run the internal circuitry). So for instance, you could bias up a pair of cathode coupled triodes in a phase splitter at 5 mA each (10 mA total). Any less than that, the current source may not be high impedance (no longer a current source).

With a negative voltage return to the current source, when you turn off the B+, and when the cathodes are still warm, the current source will still pull cathode current. The current will no longer be from the plates, it will be from the grids. You do not want 5 or 10 mA of grid current.
So, I use a larger grid stopper resistor, say 3.9 k Ohms. The resistance value is a compromise of bandwidth, and maximum grid current at power down sequence.
Consider the Cgp x (1 + u) and Cg input in parallel, versus Rgrid stopper for the bandwidth you will get.
 
How a tube sounds has a lot to do with the circuitry around the tube.

Each tube type has one (or more,depending on the application) optimal operating conditions, that may not be optimal for another tube type.

What impedance drives the tube, what is the bias voltage and plate voltage, what is the plate load impedance (i.e. plate load resistor in parallel with the grid return resistor of the following stage), what are the tube element capacitances, etc.?

A 6N1P has these approximate ratings:
u = 35, rp = 7,700 Ohms, Transconductance 4,500 micro mhos
A typical application might have a bias of 2V.

A 6SN7 has these approximate ratings
u = 20, rp = 7,700 Ohms, Transconductance = 2,600 micro mhos
A typical application might have a bias of 8V.

A 6DJ8 has these approximate ratings
u = 33, rp = 2640 Ohms, Transconductance = 12,500 micro mhos
A typical application might have a bias of 1.3V.

How they sound will also depend on how much voltage drives the grid, how much voltage is required to drive the next stage's grid, etc.

I do not see the 6SN7 and 6N1P as being very equivalent.
And the 6DJ8 / 6922 are not very equivalent to the 6N1P.
 
Take a look at this, black is Voskhod 6n1p-ev red is Telefunken ecc88/6dj8. It tells the whole story. This is real measurements from my utracer.
 

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...The minimum recommended current of an LM317 is 10 mA (it requires current to run the internal circuitry). So for instance, you could bias up a pair of cathode coupled triodes in a phase splitter at 5 mA each (10 mA total). Any less than that, the current source may not be high impedance (no longer a current source).

In practise the situation with LM317 as a CCS is not that bad.
According to the manufacturer the min. load current is 3.5 mA (typically).
In addition in all the application data the sense resistor R1 = 240 ohms
which gives 5 mA load current.

I have used LM317 as a CCS with 6N1P-EV LTP at currents less than 5 mA with perfect operation.
 
LM317 Use it any way you want, but "your mileage may vary"

Link .PDF data sheet Texas Instruments LM317 Revised September 2016:

http://www.ti.com/lit/ds/symlink/lm317.pdf

6.3 Recommended Operating Conditions:
Minimum output current 0.01 A (10 mA)
Vi - Vo input to output differential Voltage MIN 3V

6.5 Electrical Characteristics:
Minimum load current (in order) to maintain regulation TYP 3.5 mA, MAX 10 mA

I have personally been tripped up by not paying attention to these specs in my circuits.
Now I know what I have to pay attention to.
 
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