I am not a music qualified; therefore some of the observations I may make about 'sangati' may not be very correct. Pardon for such errors.
Sri tyAgarAja uses 'sangati' with reference to music in the following two kRtis - nAda sudhA rasam - Arabhi
svaramulArunnokaTi ghaNTalu vara rAgamu kOdaNDamu
dura naya dESyamu tri guNamu nirata gati SaramurA
sandarbhamu gala giramulurA
dhara bhajana bhAgyamurA tyAgarAju sEvincu (nAda)
is a technial term and I have provided the following quotes in my translation -
sangati - Tyagaraja's outstanding contribution to the advancement of the kriti format was the introduction of the sangati (lit. coming together) - a set of variations on a theme, gradually unfolding the melodic potential of the musical phrase. Largely set in the madhyamakala (middle tempo), Tyagaraja's kritis were more appropriate for the modern concert paddhati. As vidwan G. N. Balasubramaniam observed during his presidential address at the 32nd conference of the Music Academy in 1958, nearly 60% of Tyagaraja's kritis are composed in this kala and this speed "seems to be the best for both the lay and the learned listeners." â€“ Source http://www.sawf.org/newedit/edit05132002/musicarts1.asp
In the kRti hari dAsulu - yamuna kalyANi
the following words occur - sangatigAnu
pongucu vIdhulakEgucu merayucu (hari)
Here, IMHO, sangati is not a technical term.
In regard to the kRti 'dArini telusukoNTi' - the following is pallavi
dArini telusukoNti tripura
sundari ninnE SaraNaNTi
However, reputed musicians have made a mess of this pallavi by converting 'dArini' into 'darini' in order just to create sangati. They sing 'darini telusukoNTi tripura sundarini
telusukoNTi' wherein 'darini' has no meaning. Therefore, there is no sangati here and musicians for their convenience made such sangatis which militate against the kRti.
There are many such examples. The kRti 'rAma nannu brOvA rAvEmakO' and 'graha balamEmi' wherein the sangatis created by musicians have totally distorted the meanings. I have discussed such issues in the thread 'Rendering of kRtis'.
But who will bell the cat? And the show goes on.
As violinlover has commented it is very doubtful whether Sri tyAgarAja had deliberately introduced sangatis in his kRtis. In any way we the mortals were not his audience - it is Lord Sri rAma - We should ask Lord rAma whether He is fond of sangatis or bhakti.