Abstract
Almost every wristwatch contains a crystal vibrating at $32768$ Hz. This paper asks what fixed that number——the answer is not the physics of quartz but divisibility in binary. No new mathematical theorem and no new law is claimed.
First, $32768=2^{15}$, and fifteen stages of division give exactly $1$ Hz (Section 2).
Second, this is the core of the paper. The neighbouring $32000$ Hz divides by two only eight times and leaves $125$ (Section 2).
Third, three demands pull in different directions. Low frequency for power, high frequency for resolution, and a power of two for the divider (Section 3).
Fourth, only the last of the three picks one point out of the band (Section 3).
Fifth, keeping to one second a day demands $\mathbf{11.5741}$ ppm. A $20$ ppm crystal drifts $1.7280$ seconds a day and $631.1520$ a year (Section 4).
Sixth, the separator is whether the number was fixed by nature or by the implementation (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——tuning-fork quartz resonators, binary division, and the ppm notation for frequency tolerance are all standard. We do not build the physics of quartz——piezoelectricity, tuning-fork vibration modes, and the parabolic temperature characteristic are not entered. They are mentioned and no more. We design no circuit——oscillator and divider implementation and load-capacitance trimming are not entered. We do not treat time standards——atomic clocks, Coordinated Universal Time and leap seconds are not entered. They belong to Paper 319. We do not adjudicate history——who adopted this frequency and when is not entered. We do not treat temperature compensation——ovens and compensated implementations are not the subject. Relation to earlier papers: Paper 306 showed how many “base” units there are to be a promise——$32768$ belongs to the same category. But the promise in 306 was “into how many”, while the promise here is “at which value”. Paper 319 showed a calendar to be three approximations stacked——there human numbers were fitted to a natural period; here a natural vibration is fitted to a human number. The direction is reversed. Paper 95 separated convention from fact——$32768$ stands on the side of convention, while a crystal vibrating at $32768$ Hz stands on the side of fact. One number stands on both sides. Paper 316 showed the same integer returning from three different counts——here one integer sits at the crossing of three different demands. What is added is actually dividing the neighbouring candidates to show they do not come out, separating which of the three demands fixes the value to a point, matching ppm to drift per day and per year, and putting the separator on nature against implementation.
1. Notation and the Question
Setting: A watch must manufacture one second. A crystal is made to vibrate, its vibrations are counted, and $1$ Hz comes out. The counting is binary division——each flip-flop stage halves the frequency.
Observation 1.1 Division can only make halves. Making $1/3$ or $1/5$ takes extra circuitry——the cheapest thing is a number that halves down to exactly $1$.
Observation 1.2 The frequency actually used is $32768$ Hz. Nearly every wristwatch and many microcontroller sub-clocks carry this value——why not $32000$, or $33000$?
Observation 1.3 The way of measuring is fixed first. Frequency tolerance in ppm, a day of $86400$ seconds, a year of $365.25$ days——drift is proportional to tolerance, so multiplication suffices.
Question: Was this number fixed by the physics of quartz, or by the convenience of a circuit?
2. Halve Repeatedly and Only $32768$ Lands on $1$
Calculation 2.1: Halve $32768$ repeatedly (computed in this paper).
| Stage | Frequency [Hz] | Stage | Frequency [Hz] |
|---|---|---|---|
| $0$ | $32768$ | $8$ | $128$ |
| $1$ | $16384$ | $9$ | $64$ |
| $2$ | $8192$ | $10$ | $32$ |
| $3$ | $4096$ | $11$ | $16$ |
| $4$ | $2048$ | $12$ | $8$ |
| $5$ | $1024$ | $13$ | $4$ |
| $6$ | $512$ | $14$ | $2$ |
| $7$ | $256$ | $\mathbf{15}$ | $\mathbf{1}$ |
Observation 2.2 Exactly fifteen stages give $1$ Hz. Because $32768=2^{15}$——fifteen flip-flops in a row, with no division and no remainder.
Calculation 2.3: The same for the neighbouring candidates (computed in this paper).
| Frequency [Hz] | Times divisible by $2$ | Remainder | Reaches $1$ Hz |
|---|---|---|---|
| $32000$ | $8$ | $\mathbf{125}$ | no |
| $32767$ | $0$ | $32767$ | no |
| $\mathbf{32768}$ | $\mathbf{15}$ | $\mathbf{1}$ | yes |
| $33000$ | $3$ | $4125$ | no |
| $16384$ | $14$ | $1$ | yes |
| $65536$ | $16$ | $1$ | yes |
Observation 2.4 $32000$ Hz looks round and does not divide. $32000=2^8\times125$, and $125$ remains——being round in decimal and dividing in binary are entirely separate things.
Observation 2.5 $32767$ is one less than $32768$ and does not divide even once. It is odd——a neighbour one apart sits at the opposite extreme in binary.
Observation 2.6 But $16384$ and $65536$ divide as well. These are $2^{14}$ and $2^{16}$——so “divides in binary” alone does not settle on $32768$. Three candidates remain.
3. The Core: Three Demands Pull, and Only the Last Fixes a Point
Setting: Choosing a frequency, there are three demands.
| Demand | Direction | Reason |
|---|---|---|
| Power consumption | lower is better | oscillation and division cost roughly go as frequency |
| Resolution and stability | higher is better | finer ticks, and a smaller, steadier crystal |
| Divides in binary | be a power of two | the divider needs no extra circuitry |
Observation 3.1 This is the core of the paper. The first two settle only a band——somewhere in the tens of kilohertz, and no more. What fixes a point is the third alone.
Calculation 3.2: Compare the three remaining candidates (computed in this paper).
| Frequency | Divider stages | One tick [ms] | Standing |
|---|---|---|---|
| $2^{14}=16384$ | $14$ | $0.061035$ | better on power, half the resolution |
| $\mathbf{2^{15}=32768}$ | $\mathbf{15}$ | $\mathbf{0.030518}$ | the value adopted |
| $2^{16}=65536$ | $16$ | $0.015259$ | better on resolution, twice the power |
Observation 3.3 $32768$ stands in the middle of the binary staircase. A step down eases the power, a step up raises the resolution——less the answer to an optimisation than the middle of a staircase with only three steps on it.
Observation 3.4 And $32768$ Hz sits above the upper edge of hearing. A factor of $1.6384$ above the usual $20$ kHz——had $2^{14}=16384$ Hz been chosen, it would have fallen inside the audible range.
Observation 3.5 To be stated honestly——this paper does not claim $32768$ is optimal. The weights on the three demands depend on the battery and on the application——what is claimed is only that what narrowed the value to a point was the binary demand. Why $2^{15}$ rather than $2^{14}$ does not follow from these numbers.
Observation 3.6 Apply the division of Paper 95. “A crystal vibrates at $32768$ Hz” is a fact; “choose $32768$ Hz” is a convention——one number stands on both sides. And the convention was fixed first, with the fact (the crystal’s dimensions) made to follow.
4. How Far It Drifts
Calculation 4.1: Frequency tolerance against clock drift (computed in this paper).
| Tolerance [ppm] | Drift per day [s] | Drift per year [s] |
|---|---|---|
| $20$ | $\mathbf{1.7280}$ | $\mathbf{631.1520}$ |
| $5$ | $0.4320$ | $157.7880$ |
| $0.5$ | $0.0432$ | $15.7788$ |
Observation 4.2 A common $20$ ppm crystal drifts $631.1520$ seconds a year. About $10.5$ minutes——however exactly the divider is $2^{15}$, the crystal’s own tolerance survives untouched.
Calculation 4.3: Read the other way, what tolerance keeps a day to one second (computed in this paper)?
$$\frac{1\ \text{second}}{86400\ \text{seconds}}\times10^6=\mathbf{11.5741}\ \text{ppm}$$Observation 4.4 $11.5741$ ppm. A common $20$ ppm part does not reach it——the claim “within a second a day” demands a selected $5$ ppm part or temperature compensation.
Observation 4.5 Binary plays no part here at all. That the divider comes out exactly at $2^{15}$ and how many ppm the crystal drifts are wholly separate matters——a property of integers gives no accuracy whatever.
Observation 4.6 Here is the parting from Paper 319. Paper 319 wrote that a calendar is three approximations stacked——there the natural period (a year) was not a divisible number, so the human side stacked approximations. Here it runs the other way: the human side fixed the clean number $2^{15}$ first and made nature (the crystal) fit it.
5. The Separator: Was the Number Fixed by Nature or by the Implementation?
Observation 5.1 The separator——whether the number is one that turned out that way on measurement, or one placed there because it was convenient.
| Example | Fixed by | Can it be changed | |
|---|---|---|---|
| Fixed by nature | a year of $365.2422$ days | nature | no |
| Fixed by implementation | $32768$ Hz | circuit convenience | yes |
Observation 5.2 The clue is how clean the number is. $365.2422$ is all remainder, and $32768$ is exactly $2^{15}$——nature almost never hands over an exact power of two.
Observation 5.3 The separator can be moved. Allow extra circuitry in the divider and any frequency can produce $1$ Hz——$32768$ is needed because one wants to build cheaply, not as a matter of principle. Where power is plentiful, other frequencies are in fact used.
Observation 5.4 To be stated honestly——the crystal side has constraints too. A tuning-fork crystal’s frequency is set by its dimensions, so the band of tens of kilohertz is fixed by physics——what is claimed is that binary picked one point inside the band, not that the band itself is arbitrary.
Observation 5.5 Where this stands relative to Paper 306. Paper 306 wrote that how many “base” units there are is a promise——that promise was “into how many”, while this one is “at which value”. Under one word “promise”, fixing a count and fixing a value are different acts.
Observation 5.6 One last thing——this is one of the most widely carried powers of two in ordinary life. Most people wear it on a wrist and nobody thinks of it as $2^{15}$——and once seen, the most exactly determined thing inside the watch is this integer, and not the frequency of the crystal.
6. The Census and the Close
- $32768=2^{15}$ (Calculation 2.1): fifteen stages give exactly $1$ Hz.
- $32000$ divides eight times and leaves $125$ (Calculation 2.3): decimal roundness is other.
- $32767$ does not divide once (Observation 2.5): opposite extreme, one apart.
- $16384$ and $65536$ divide too (Observation 2.6): binary alone does not settle it.
- Two of the three demands settle only a band (Observation 3.1): binary fixes the point.
- One tick is $0.030518$ ms (Calculation 3.2): twice that at $2^{14}$, half at $2^{16}$.
- $1.6384$ above the edge of hearing (Observation 3.4): $2^{14}$ would have been audible.
- $631.1520$ seconds a year at $20$ ppm (Calculation 4.1): about $10.5$ minutes.
- $11.5741$ ppm for one second a day (Calculation 4.3): $20$ ppm does not reach.
- Binary gives no accuracy at all (Observation 4.5): integers and tolerance are separate.
Closing: Nearly every wristwatch holds a crystal at $32768$ Hz, and that number is $2^{15}$——fifteen flip-flop stages give exactly $1$ Hz, with no division and no remainder. The neighbouring $32000$ Hz looks round in decimal but is $2^8\times125$ and leaves $\mathbf{125}$. One less, $32767$ is odd and does not divide even once. Yet dividing in binary does not by itself settle on this value——$2^{14}=16384$ and $2^{16}=65536$ divide as well. Three demands act on the choice: power wants it low, resolution and stability want it high, and binary wants a power of two. The first two settle only a band, somewhere in the tens of kilohertz; what narrows the value to a point is the third alone. $32768$ stands in the middle of that binary staircase——a step down eases power, a step up raises resolution. One tick is $0.030518$ milliseconds, and $32768$ Hz sits $1.6384$ above the upper edge of hearing——had $2^{14}$ been chosen it would have fallen inside the audible range. This paper does not claim $32768$ is optimal; it claims that what fixed the value to a point was the binary demand. And binary gives no accuracy whatever——however exactly the divider is $2^{15}$, the crystal’s own tolerance survives. A $20$ ppm crystal drifts $\mathbf{1.7280}$ seconds a day and $\mathbf{631.1520}$ a year, about $10.5$ minutes. Keeping to one second a day demands $\mathbf{11.5741}$ ppm, which a $20$ ppm part does not reach. A property of integers and a magnitude of drift are wholly separate matters. One thing separates them——whether the number turned out that way on measurement, or was placed there for convenience. The clue is how clean it is——a year of $365.2422$ days is all remainder, and $32768$ is exactly $2^{15}$; nature almost never hands over an exact power of two. Paper 319 wrote that a calendar is three approximations stacked——there the natural period was not divisible so the human side stacked approximations. Here it runs the other way: the human side fixed the clean number $2^{15}$ first and made the crystal’s dimensions follow. To be said honestly——a tuning-fork crystal’s frequency is set by its dimensions, so the band of tens of kilohertz is fixed by physics. What is claimed is that binary picked one point inside the band, not that the band itself is arbitrary. One last thing——this is one of the most widely carried powers of two in ordinary life. Most people wear it on a wrist and nobody thinks of it as $2^{15}$. And once seen, the most exactly determined thing inside the watch is this integer, and not the frequency of the crystal.
References
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- International Electrotechnical Commission (2008). IEC 60122: Quartz Crystal Units of Assessed Quality. Geneva: IEC.
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). How Many “Base” Units There Are Is a Promise [Paper 306]. Zenodo. 10.5281/zenodo.22172149
- Yamagishi, Y. (2026). The Same Integer Comes Back from Three Different Counts [Paper 316]. Zenodo. 10.5281/zenodo.22208229
- Yamagishi, Y. (2026). A Calendar Is Three Approximations Stacked [Paper 319]. Zenodo. 10.5281/zenodo.22208933
Reference
Yamagishi, Y. (2026). The Number $32768$ Inside a Wristwatch Was Not Chosen by Physics [Paper 336]. YOI: https://www.yamagishiyuuki.com/yoi/336
DOI: 10.5281/zenodo.22681913
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 336. This paper is a preprint and has not been peer reviewed.