The beat: a derived unit of time for the COMMON day

1 Introduction

The COMMON calendar divides the day into 100 centidies and 1,000 millidies. A third subdivision has been used in the clock's precision mode but has lacked a name: the hundredth of a millidie, one hundred-thousandth of a day. This note names it the beat and gives it a definition that can be stated, checked, and reproduced.

The second is the SI base unit of time, defined by the caesium-133 hyperfine transition. The beat is not a second and is not an SI unit. It is a derived unit of the civil day, in the same family as the day itself. The distinction matters: a definition that starts from the day gives an exact decimal fraction of a day; a definition that starts from the atom gives an integer count of atomic periods. The two cannot both be exact at once, because one hundred-thousandth of a day is not an integer multiple of the caesium period. We choose the day, and we state the consequence.

The note is organized as follows. Section 2 fixes the day. Section 3 defines the beat. Section 4 derives its caesium count. Section 5 explains why that count is fractional. Section 6 describes the dial realization. Section 7 discusses the design trade-off, and Section 8 concludes.

2 The COMMON day

The COMMON day is the nominal civil day of 24 hours:

The clock is computed from wall-clock fields, not from elapsed milliseconds since midnight. On a daylight-saving transition the displayed day fraction jumps with the local clock; it is not stretched or compressed to fit a 23- or 25-hour civil day. The beat is therefore a clock unit, not a monotonic elapsed-time unit. This is the same convention used by the centidies and millidies readouts.

The day is a non-SI unit accepted for use with the SI, with the symbol d and the exact value 86,400 s [1, Table 8]. We use that value throughout. ISO 80000-3 gives the names and symbols for time quantities and units [2].

3 Definition of the beat

3.1 Name and symbol

The unit is the beat, plural beats, symbol bt. The symbol is written in lower case, without a period, and is not pluralized: 1 bt, 100 bt. The name is a mnemonic rather than a musical quantity; one beat is a fixed duration, not a tempo.

3.2 Exact value

One beat is one hundredth of a millidie (symbol md), or one hundred-thousandth of a day:

The fraction 108/125 is exact. The decimal 0.864 is exact in decimal notation but not in binary floating point; implementations should use the rational form or integer millisecond arithmetic when a boundary matters.

Proposition 1. One beat is exactly 108/125 s. Proof. By (1) and the definition, 1 bt = 86,400/100,000 s = 108/125 s. □

3.3 Frequency

The corresponding frequency is

3.4 Definition in the form of the SI

The definition can be stated in the form used by the SI for the second:

The beat, symbol bt, is the COMMON unit of time. It is defined by taking the fixed numerical value of the caesium frequency Δν_Cs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9,192,631,770 when expressed in the unit Hz, and by fixing the length of the day at 86,400 s. One beat is exactly 1/100,000 of a day.

The first clause is the SI definition of the second [1, §2.3.1]. The second clause fixes the day. The beat is the quotient. This is the sense in which the beat is “defined like the second”: it uses the same fixed constant, but it adds the day as a second defining condition.

4 Relation to the caesium standard

The SI second is defined by the fixed numerical value of the caesium frequency:

Equivalently,

Substituting (5) into (2) gives the beat in terms of the caesium frequency:

The number of caesium periods in one beat is therefore

For a whole day,

Proposition 2. One beat is exactly 198,560,846,232/25 caesium periods. Proof. By (4) and Proposition 1, N_Cs = (108/125) × 9,192,631,770 = 992,804,231,160/125 = 198,560,846,232/25. □

The symbols cd and md are coined for this note; they are not SI symbols.

5 Why the caesium count is fractional

The denominator 125 in (2) is 5³. The caesium frequency is 9,192,631,770 Hz; it is divisible by 5 but not by 25. After simplification the count is 198,560,846,232/25, whose denominator is 25. No integer count of caesium periods can equal one beat exactly.

This is not a defect. It is the expected consequence of defining a unit by the day rather than by the atom. The day itself is exactly 86,400 s, an integer number of seconds and therefore an integer number of caesium periods; the IAU system likewise uses a day of 86,400 s [3]. The beat is a rational fraction of a second whose denominator does not divide the caesium frequency, so its count is rational but not integral.

The nearest integer counts are 7,942,433,849 and 7,942,433,850 periods. The first is short by 0.28 period, or about 30.5 ps per beat; the second is long by 0.72 period, or about 78.3 ps per beat. Over a day of 100,000 beats the difference would be about 3.05 µs or 7.83 µs. Either choice would make the beat an integer number of atomic periods, but the day would no longer be exactly 86,400 s. For a civil clock tied to the calendar, the day is the quantity that must stay fixed. We therefore keep the fractional count.

6 Realization on the dial

The COMMON dial's fine hand makes one turn per 100 beats, or 86.4 s. Each of the 100 fine ticks is one beat. The COMMON implementation also reports decimal seconds, defined here as 1/1,000,000 of a day (86.4 ms), so one beat is ten decimal seconds:

The implementation derives the day fraction from the wall-clock fields and uses integer arithmetic at the boundaries, so the displayed value truncates at the selected scale rather than rounding ahead: whole millidies by default, whole beats in extra-precision mode. The same definition is used by the live readout, the extra-precision mode, and the converter. The dial realization is described at new-time.org/explore#unit.

7 Discussion

The beat is deliberately a derived unit. It does not replace the second, and it does not compete with the SI base unit. It subdivides the COMMON day in the same way that the day itself subdivides the year. The second remains the base unit; the day remains the bridge between the atomic standard and the civil calendar.

The beat also fits the uniform structure of the International Fixed Calendar. A 28-day month is exactly 2,800,000 beats, a 7-day week is exactly 700,000 beats, and the thirteen months contain 36,400,000 beats. Year Day adds exactly 100,000 beats, so a common year is 36,500,000 beats and a leap year 36,600,000. Because every month has the same length, the beat count of a month does not vary, unlike the 28–31-day months of the Gregorian calendar [4]. The calendar fixes the day count; the beat fixes the day's subdivision. The two are independent, but their regularity is complementary.

The name is a mnemonic. A day of 100,000 beats is 69.4 beats per minute, close to a resting human pulse. The coincidence is useful for intuition but is not part of the definition. The definition is the fraction 1/100,000 of 86,400 s.

The alternative — defining the beat by an integer number of caesium periods — would make the beat a base-like atomic unit, but it would break the exact relation to the day. We reject that alternative because the COMMON clock is a civil clock: its first obligation is to the day, not to the atom. The fractional caesium count is the honest record of that choice.

8 Conclusion

The beat is the COMMON unit of time: exactly 108/125 s = 0.864 s, or 1/100,000 of a day. One hundred beats make a millidie. Its caesium count is 7,942,433,849.28 periods, exact but fractional. The definition uses the SI second and the fixed day, and it leaves the second where it belongs: as the base unit of time.