The epoch of TAI is 1961-01-01T20:00:00 UT2(OP)

Because of actions at the Bureau International de l'Heure

As recorded in issues of Bulletin Horaire

Other web pages about Bulletin Horaire

After several individual laboratories with cesium atomic chronometers started constructing atomic time scales the CCDS meeting in 1961 suggested that BIH should construct an international version. BIH complied and produced tables of Temps Atomique Integre based on all of the available atomic chronometers.

The USNO atomic time scale was called A.1 and its value had been set equal to UT2(Washington) as of 1958-01-01. The initial BIH atomic time scale was based on as many atomic chronometers as were available starting in 1955, and by 1958 there were 5, so BIH named their time scale A5. BIH followed the lead of USNO and set their atomic time scale A5 equal to UT2(BIH) as of 1958-01-01T20:00:00.

The second CCDS meeting happened 1961-04-11/12. Recommendation 2 of that meeting authorized BIH to publish their atomic time scales.

By 1961 there were 9 cesium atomic chronometers available for comparison. BIH created another time scale A9 based on all of those. While creating A9 BIH reset the values of their time scales for all 9 chronometers.
For 1961-01-01T20:00:00 UT2(OP) we see (UT2 - Heure demi-definitive_Paris(AllBIHatomics)) = (60 - 58.5601) = -1.4399 s
As reference with the tables below also note that
For 1961-01-05T20:00:00 UT2(OP) we see (UT2 - Heure demi-definitive_Paris(A9)) = (60 - 58.5554) = -1.4446
For 1961-01-10T20:00:00 UT2(OP) we see (UT2 - Heure demi-definitive_Paris(A9)) = (60 - 58.5495) = -1.4505

In mid 1962 BIH had received and computed the data from all the observatories in 1961, and they published their values of P_i. This allowed BIH to convert the epoch of all the atomic time scales from Heure demi-definitive_Paris to Heure definitive.
At 1961-01-01T20:00:00 UT2(BIH) we see P_Paris = +0.0276 s
Heure definitive = Heure demi-definitive_Paris + P_Paris
For 1961-01-01T20:00:00 UT2(BIH) we have Heure definitive(AllBIHatomics) = -1.4399 + 0.0276 = -1.4123 s
As reference with the table below also note that
At 1961-01-05T20:00:00 UT2(BIH) we see P_Paris = +0.0280 s
At 1961-01-10T20:00:00 UT2(BIH) we see P_Paris = +0.0281 s

By 1964 BIH realized that some atomic chronometers were much better than others, and they constructed A3 based on the best 3. BIH set A3 equal to A5 and A9 at the epoch when they were synchronized.
This put (UT2 - Heure definitive(AllBIHatomics)) = -1.4123 s at the epoch 1961-01-01T20:00:00 UT2(BIH).
In the process of tabulating A3 they re-interpolated all the data for 5 day intervals at UT2 = 20:00:00
Heure definitive(A3) = Heure demi-definitive_Paris(A9) - 𝝙9 + P_Paris
For 1961-01-05T20:00:00 UT2 we see (60 - 58.5834) = -1.4166 = -1.4446 - 0.0000 + 0.0280
For 1961-01-10T20:00:00 UT2 we see (60 - 58.5775) = -1.4225 = -1.4505 - 0.0001 + 0.0281
So although 1961-01-01 is missing in the following table we can see that A3 matches A9 in the days immediately following.

After Anna and Nicholas Stoyko retired from BIH Bernard Guinot reinterpolated A3 from 20h UT2 to 0h UT2.
Subsequently BIH changed A3 to become a weighted average of the best available cesium atomic chronometers.
In 1971 when the CGPM approved the creation of TAI it was based on A3.

At the epoch 1961-01-01T20:00:00 UT2(BIH) the Heure definitive offset currently recognized as UT2(BIH) - A3 = -1.4123 s
So the offset between Universal Time and the TAI that we now have was not set at 1958-01-01 because all BIH atomic time scales were reset to this same fixed offset at 1961-01-01T20:00:00 UT2(OP).

More details coming soon: In the above table there is a jump of 1.6 ms on 1962-01-01 because that is the date of the switch from the FK3 catalog to the FK4 catalog. So the value of A3 at epoch 1961-01-01T20:00:00 UT2(OP) does not correspond to values of UT2 that would have been quoted a year or more later.
Unanswered question: When Guinot re-interpolated A3 from 20h UT2 to 0h UT2 did he compensate for the 1.6 ms discontinuity at 1962-01-02?

Also, in the glimpse of the table above note that 𝝙9 increases notably. This is saying that the rates of A9 and A3 differed notably. That rate difference was eventually recognized as a problem and it is why the rate of TAI was changed by 1e-12 on 1977-01-01.