[LEAPSECS] Terminology question

Michael Deckers michael.deckers at yahoo.com
Wed Mar 10 17:01:50 EST 2010

On 2010-03-10 17:54, Steve Allen wrote:

> On Wed 2010-03-10T16:05:28 +0000, Michael Deckers hath writ:

> > On 2010-03-09 17:34, Zefram wrote:

> > > Apparently not. I'm inclined to use the phrase "faux linear time".

> > > These timescales are really encodings of UTC-wise broken-down time, but

> > > they sufficiently resembles a linear timescale that it's commonly mistaken

> > > for one. I think it's worth reminding people clearly of the reality.

> >

> > Perhaps "piecewise linear" may also be appropriate -- it is a

> > common term in math. TAI - UTC (after 1972) may then be called

> > a "step function", which is a "piecewise constant" function.


> Except that to say that any available form of UTC is linear

> is to invalidate its use as a precision time scale.

"piecewise linear" is not "linear" -- it is a mathematical way
to say "faux linear". More precisely, it can express the fact
that between any two successive leap second events (and also
after the last such event), a POSIX time scale, and also UTC
looks like a linear time scale. (Actually, the term would
even apply to UTC before 1972.)


> Read BIPM's Circular T.

> http://www.bipm.org/jsp/en/TimeFtp.jsp?TypePub=scale

> Follow the link to BIPM's TT(BIPM09)


> Plot the numbers. It's not linear.

You are right in that "linear" requires a qualification, as
in "linear function of ..". And so does "piecewise linear":
UTC would be a piecewise linear function of TAI, and of
TT(TAI), but not of TT, nor of TT(BIPM), TCB, TDB, UT1,...

Some nice graphs with d(TAI - TT(BIPM)) are in


> All precision time scales must employ empirical lookup tables.


> Leap seconds are just one form of lookup table, and the total number

> of table entries for UTC is smaller than for the other scales.

Yes, the relationship between UTC and TAI is simple.
I also suggest "piecewise linear" as a description of the
relationship: there are intervals of TAI values on which
UTC is linear in TAI, and arguably such intervals cover
the range of all TAI values (except possibly during positive
leap seconds).


> If the question simply wants "generic" UTC, then from the formal

> definition we can only expect a precision of 1 millisecond. In that

> case I could say "linear within the 1 ms precision of the formal

> specification of UTC as defined by ITU-R TF.460", but according to

> that as soon as a computer system clock deviates by more than 1 ms

> it can no longer claim to be UTC. In that case what is it?

If I understand you correctly: my computer gives me UTC(my_computer)
and I can convert that easily to TAI(my_computer), with the usual
lookup table. The error and the uncertainty with respect to UTC
and TAI are the same. I did not want to suggest that UTC(my_computer)
was a linear function of UTC. It certainly isn't!

I do not understand how the formal definition of UTC limits its
precision to 1 ms. UTC can be determined with the same
uncertainty as TAI.


> This is the vocabulary issue which triggers me to answer "mu", for

> many of the questions seem to want an answer which is far simpler

> than required to describe (or even in denial of) reality.

Hai, you are right, an easy term should not cover up a complex

POSIX systems use different ways to represent UTC values around
a positive leap second in a time_t value, and the details
are not so easy to find out. In the few cases where I (thought I)
understood the scheme, "piecewise linear function" of TAI was a
valid description. Also, the time scale UTS proposed by Markus Kuhn
for use in information and communication systems is a "piecewise
linear" function of TAI in the exact mathematical sense
(see [http://www.cl.cam.ac.uk/~mgk25/uts.txt]).

Thank you for your detailed comments!

Michael Deckers.

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