I think that in this (and most other) contexts the broad definition of the tritone makes sense, which says that a tritone is an interval spanned by six semi-tones.
So with this definition both augmented fourth and diminished fifth qualify as a tritone.
Whether that note in the blues scale is written as a #4 or
a b5 usually depends on the direction of the melody: #4 ascends, b5 descends.
I believe that the reason why in the blues scale this note is more frequently called b5 is that the #4 is a very common note in many genres, and in many contexts it doesn't sound bluesy at all, e.g. if used melodically over a V/V chord (i.e. in C major over a D7 resolving to G7).
Also take the lydian scale: it has a #4 but there's no blues character whatsoever.
Unlike the #4, the b5 (descending) does have that bluesy character.
So the blues character of that note is mainly perceived if it coexists with the unaltered 4, and then it would often descend to the 4 as a b5.
Coexistence with the 5 (without an unaltered 4) does usually not generate any bluesy character (e.g., lydian).
music.stackexchange.com/users/9724/Matt L.
To be 100% exact to the name 'tritone', it would be #4.
Because, if we ascend 3 tones from the root, we have #4, and not b5.
The three tones would be C-D, D-E,
E-F#.
Although, from what I have understood, it depends on the progression.
Usually, the #4/b5 tritone would descend a semitone.
For instance, a common progression would be Gb7-F (maj7 or 7).
Since we are in F major, a cadence like F#7-Fmaj7 is really rare (not that it doesn't exist).
The most common use of the tritone is on the tritone substitution of the V7 chord, with the bII7 chord.
It wouldn't be nice to see #I 7.
But to sum up, I don't think any of the two names is wrong.
You can use whichever suits you better.
music.stackexchange.com/users/7356/Shevliaskovic
but why is b5 favoured?
I don't know if there can be a definitive answer, but I would say that probably it's because b5 is commonly used in chord names, while #4 is not.
Being used to always call that interval b5 in the context of chords, one will naturally tend to use the same name when talking about the note itself as well.
music.stackexchange.com/users/19915/MarcoB
Matt L.'s answer is an excellent one.
If you are asking "what should I call it, and when should I call it that?
" here's my experience.
In heptatonic (7 note scales) each number gets used once, you always call it a #4 if its the fourth note in the scale, e.g. in Lydian, or for a minor example, in the 4th mode of harmonic minor (essentially Dorian with a #4).
If its the fifth note of a heptatonic scale you must call it a b5.
The Half Diminished scale (aka Locrian mode) is a good example.
As to what to call it in the Blues scale, I see it coming down to genre more than anything else.
In Rock, Blues, and Metal its always the b5.
I see Jazz players, websites, and theory books calling it the sharp four more and more.
If you are in doubt and at a jam session with Jazz players , just call the note the sharp 4 , with Rock or Blues guys/gals call it the flat five.
With Metalheads I might call it the devil's tone, or diabolus in musica, just because its more fun to.
Jazz is predominantly Major; the #4 (with no 4) gets used quite a bit because it is not dissonant against the 3 in a major chord (but the natural 4th is).
There are virtually no scales in rock where you have to call it a sharp four, the only exception that comes to mind is the Double Harmonic scale (typically used for Dick Dale inspired Surf rock).
music.stackexchange.com/users/20000/Jay Skyler
The whole POINT of a tritone is its ambiguity!
But, in a system of harmonic analysis built on the "pile of thirds" model, I can see why a modified 5th is preferred to a modified 4th.
I find insistence that the indisputably aurally flattened b10 must be labelled #9 rather harder to take!
music.stackexchange.com/users/17003/Laurence Payne
Generally, in Blues and jazz, the oft-used tritone (C to F#/Gb) is referred to as a b5. Why is this a better term than #4? In the Blues scale, there will always be two notes of the same name, but why is b5 favoured? Possibly to match the b3 and b7?
The Tritone - is it a #4 or a b5?
music.stackexchange.com
<theory><notation>
music.stackexchange.com/questions/31952
music.stackexchange.com/users/2808/Tim
the reason why in the blues scale this note is more frequently called b5 may be that the #4 is a very common note in many genres, and in many contexts it doesn't sound bluesy at all, e.g. if used melodically over a V/V chord (i.e. in C major over a D7 resolving to G7). It may be because b5 is commonly used in chord names, while #4 is not. In Rock, Blues, and Metal its always the b5.  The reason this is referred to as a b5 is probably more related to genre and convention than any particular rule.
The reason this is referred to as a b5 is probably more related to genre and convention than any particular rule.