There are 3 types of triple tonguing.
The TTK in the Arban book, TKT,KTK (alternating ) in the James Burke "New Directions in Tonguing " book and the rolling Ta-Da-Ga taught in Jacoby's book "Jake's Method".
TTK - TTK, TKT-KTK, & Ta Da Ga are different concepts of tongue useage. There are 3 prevalent triple tonguing concepts. Whereas changing Tu-Tu-Ku into Ta-Ta-Ka , or Da-Da-Ga, or Di-Di-Gi is merely using different syllables. These 4 examples all fall under the Arban TTK . For the Burke double-triple they would be (Tu-Ku-Tu---Ku-Tu-Ku), (Ta-Ka-Ta---Ka-Ta-Ka), (Da-Ga-Da---Ga-Da-Ga) and (Di-Gi-Di---Gi-Di-Gi).
These represent lighter syllables and are quicker to use. The shorter the tongue has to move the faster the tonguing can be. I wrote them in order from heavest - slowest to lightest - quickest. The stroke involving the forward part of the tongue is Tu, Ta, Ti, Da and Di.
The stroke that involves the middle of the tongue is Ku, Ka, Ki, Ga and Gi.
The rolling tongue needs to be Ta Da Ga - Ta Da Ga or Ti Di Gi. So while there are 3 concepts there are alternate syllables to adjust for style of sound or speed.
The alternate syllables work on double tonguing as well Tu-Ku , Ta-Ka, Da-Ga or Di-Gi.
The same set of syllables is useful in single tonguing also. Tu, Ta, Ti, Da and Di. The Ta is the most useful.
To work on speeding up tonguing skills use softer syllables like ta da or di. Also work with a metronome. Start at 1/16 notes @ 1/4 == 60 beats per minute. Every 15 - 20 seconds speed up 10 beats per minute. Make a note of the problem speeds and work on these. Do this with single, double and triple tonguing .