<p>Exactly 2/7 of time is &quot;weekend&quot;,  There is no exact alignment of the lunar orbit to the length of a week, so a full moon is equally &quot;likely&quot; to be at the weekend</p>
<p>So 2/7 of full moons occur at the weekend. As the synodic month is about 29.5 days there are about 1245 New Moons per century, so you'd get about 356 on the weekend.  This full moon (10th Sept) is on Saturday, and next full moon, 9th October 2022 is on Sunday (at my location)</p>
<p>However the moon appears full for a couple of days on either side of the moment at which its longitude is 180° relative to the Sun. This being the case, the moon will appear full at the weekend if it is exactly full on any day between Thursday and Tuesday - which means that most months will have a moon that looks full at the weekend.</p>
<hr />
<p>Now to think about this in a different way, in a century there are 5218 weekends (±1) and 356 are &quot;full moon weekends&quot;,  so about 6.8% of weekends are full moon weekends, or to put it another way 1 weekend in 14.66 is a full moon weekend.</p>
<p>On average, you need to wait for 14.66 weeks before you get another full moon weekend.  <strong>But</strong> this is an average: the pattern of full moon weekends is not at all random.  In fact what tends to happen is that there is one full moon weekend, followed four weeks later by second one.  Then a gap of more than 14 weeks, before the pattern almost repeats.</p>
<p>So (in my location) there is a full moon on <em>Saturday</em> 10th Sept and <em>Sunday</em> 9th Oct.  Then it proceeds: Tuesday in Nov, Thursday in Dec, Friday in Jan, then <em>Sunday</em> 5th February.  That is 17 weeks after the October full moon weekend.</p>
<p>Thus, final answers:  2/7 of full moons are at the weekend. 2/29.530588 = 6.8% of weekends have a full moon.</p>

CC BY-SA 4.0
2022-09-10T19:25:01.953Z
"2/7 of full moons occur at the weekend". That is non-intuitive for me. Are you saying the length of the synodic month does not matter here? If the moon phases would require 25 days, 31 or whatever, it would be the same then?
Mark
CC BY-SA 4.0
2022-09-10T19:32:23.063Z
Yes.  Imagine you have a bag with 200 red counters and 500 blue ones.  You spread them out in a line and take every 25th counter.  You'd get 2/7 red and 5/7 blue.  What if you took every 31st counter?  You'd get 2/7 red and 5/7 blue.  It is the same with moon phases.
James K
CC BY-SA 4.0
2022-09-10T19:49:37.067Z
@Mark That number from Wikipedia is the *mean* synodic month length. The [Moon's motion is rather complex](https://en.wikipedia.org/wiki/Lunar_theory), so there's quite a bit of variation in the actual month length. Fred Espenak says [here](http://astropixels.com/ephemeris/phasescat/phasescat.html) that "the length of any one synodic month can vary from 29.26 to 29.80 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit".
PM 2Ring
CC BY-SA 4.0
2022-09-10T19:57:37.390Z
Though if the synodic month was 31 days, there would be fewer full moons in a century, and so fewer full moons on the weekend.
James K
CC BY-SA 4.0
2022-09-10T23:40:30.720Z
Nice, though unsourced answer. The equal probability assumption is intuitive and we all assume it's true, but I'd love to see a supporting source for it. I haven't a clue to where to look for one though, so I've just asked min Math SE [histogramming phases between a periodic function and another periodic, quasiperiodic or almost-periodic function with irrational period relationship.](https://math.stackexchange.com/q/4528938/284619)
uhoh
CC BY-SA 4.0
2022-09-11T15:47:28.463Z
This answer cuts the Gordian knot of the problem. Well done.
Wastrel
CC BY-SA 4.0
2022-09-11T15:48:33.580Z
The length of the synodic month does not matter to the questions of whether a randomly-chosen full moon occurs on a weekend or whether a randomly chosen weekend will have a full moon, but it *does* matter to questions such as the OP expressed in comments on the Q (since this answer was posted): how many weeks must one wait between weekends with full moons?
John Bollinger
CC BY-SA 4.0
2022-09-11T16:15:44.870Z
@Mark yes, the length of the lunar month doesn't matter over a long period of time. The concept you want is [Ergodicity](https://en.wikipedia.org/wiki/Ergodicity). Over a short period of time it's false (e.g. the answer to "if it is a full moon this weekend, what is the probability that it will be a full moon next weekend?" is zero), but averaged over a longer period of time, every moon phase occurs on every day of the week with equal probability.
hobbs
CC BY-SA 4.0
2022-09-11T22:32:12.463Z
@hobbs A comment under my new Math SE question linked above points to https://en.wikipedia.org/wiki/Equidistribution_theorem Now I have to figure out (or better yet, ask about) how the two concepts overlap. Are all Ergodic systems subject to the equidistribution system? Are all system to which the equidistribution theorm applies necessarily ergotic? I learned a new word today, yay! (ergotic)
uhoh
CC BY-SA 4.0
2022-09-11T22:46:01.567Z
@uhoh Are you actually asking for a source that 2/7 of all days are weekends?  Or a source that says that the lunar cycle is not synchronised to calendar weeks? (For example, if a the lunar cycle were EXACTLY 4 calendar weeks, then either ALL full moons would be at the weekend, or NONE of them would).
Dawood ibn Kareem
CC BY-SA 4.0
2022-09-11T22:49:00.310Z
@DawoodibnKareem no, that's a mischaracterization via oversimplification
uhoh
CC BY-SA 4.0
2022-09-11T23:06:55.153Z
So which part of the answer do you think needs additional sourcing?
Dawood ibn Kareem
CC BY-SA 4.0
2022-09-12T00:50:16.313Z
BTW, that Fred Espenak link I posted earlier is from his six millennium catalog of Moon phases, if someone wants to write a scraper and do some stats...
PM 2Ring
CC BY-SA 4.0
2022-09-12T21:09:55.347Z
@DawoodibnKareem you didn't `@` me but I think your comment is directed at me. Motion of the Sun-Earth-Moon system is not a perfect random number generator. It's a complex system so those of us who care rather than wave our hands give it some thought. See the links in my previous comments; there's a nice body of mathematical work on this topic. JamesK frequently writes completely unsourced answers which is antithetical to "good Stack Exchange answer" writing. At least a reference to ergoticity and/or a link to the equidistribution theorem in Wikipedia would be fine.
uhoh
CC BY-SA 4.0
2022-09-12T21:12:03.813Z
@DawoodibnKareem  I understand that most folks are fine with "It's just common sense" but we shouldn't just target the lowest common denominator when writing Stack Exchange answers.
uhoh
CC BY-SA 4.0
2022-09-10T19:16:22.320Z
2023-01-26T07:23:40.090Z
2023-01-26T07:23:40.090Z
James K
<p>Assuming the average properties of the moon's orbit apply out to infinity, we could easily assume the answer is 2/7.  But, over any finite period of time, a perfectly even distribution would be highly unlikely.</p>
<p>The below Python code uses <a href="https://rhodesmill.org/pyephem/" rel="nofollow noreferrer">PyEphem</a> to find each full moon.  I choose to start from 1600 to avoid any confusion over the change to the Gregorian calendar, and then go about just as far forward as I did backwards to 2800.</p>
<p>Results:
Total Full Moons: 14841</p>
<pre><code>DOW       #    %
Monday    2116 14.25779934
Tuesday   2123 14.30496597
Wednesday 2122 14.29822788
Thursday  2117 14.26453743
Friday    2125 14.31844215
Saturday  2115 14.25106125
Sunday    2123 14.30496597

Weekend   4238 28.55602722
</code></pre>
<p>A perfectly even distribution would have resulted in 28.5714% weekend full moons.</p>
<p>The code (on most platforms &quot;pip install pyephem&quot; will install the ephem dependency).</p>
<pre><code>#!/usr/bin/python
import ephem
from datetime import datetime

days=[0,0,0,0,0,0,0]
count=0
fm = ephem.next_full_moon('1600/1/1')
d=datetime.strptime(str(fm),&quot;%Y/%m/%d %H:%M:%S&quot;)

while d.year&lt;2800:
    days[d.weekday()]+=1
    
    fm = ephem.next_full_moon(fm)
    d=datetime.strptime(str(fm),&quot;%Y/%m/%d %H:%M:%S&quot;)
    count+=1

print(count)
print(days)
</code></pre>

CC BY-SA 4.0
2022-09-12T21:06:45.557Z
Well, certainly 4,238 of anything can't be distributed evenly into 7 buckets :)
hobbs
CC BY-SA 4.0
2022-09-13T10:45:02.313Z
@hobbs The total number of samples is 14841, not 4238; although your point still stands, since that is also not divisible by 7. However, it's still clear that the distribution isn't quite even; the "perfect" distribution would be six days with 2120 full moons, and one with 2121 (2120 * 7 + 1 = 14841). Using percentages just makes things more confusing, since 100 isn't divisible by 7.
IMSoP
CC BY-SA 4.0
2022-09-13T13:39:29.297Z
@hobbs, not sure what you're getting at.  The fact that an arbitrary finite interval has about a 1/7 chance of being impossible to distribute evenly just re-emphasizes my point.
Greg Miller
CC BY-SA 4.0
2023-01-26T07:22:55.883Z
You probably need to tell us which time zone you used for your calculations, as that certainly affects the distribution (because the time zone determines when midnight occurs).
Toby Speight
CC BY-SA 4.0
2022-09-12T17:38:24.547Z
2022-09-12T17:38:24.547Z
Greg Miller
<p>Some further info on the actual distribution pattern.</p>
<p>The synodic month and the week length is descended by 1.53059 days, so one month later the full moon comes 1.53059 days later in the week.</p>
<p>That means here are two types of full moon weekends. The first type is where the full moon hits the weekend, but one month later, 1.53059 days is enough to bring it past midnight Sunday. The <strong>isolated full moon weekend</strong>.</p>
<p>In the other type, the full moon falls early enough in the weekend that 1.53059 days is still Sunday, meaning there will be a <strong>full moon weekend pair</strong> 4 weeks apart.</p>
<p>The same applies to the wait for the 1.53059 days to work itself through the 5 weekdays. This is has to be exactly 12 or 16 weekends in a row without a full moon.</p>
<p>Full moon weekend pairs have some constraints though. The earliest in the weekend the second full moon of the pair can fall is Sunday afternoon, so the next full Moon can be on a Tuesday on the earliest.<br />
<em>full moon weekend pairs are followed and preceded by the short waiting period of 12 weekends without a full moon</em>.</p>
<p>Pattern:</p>
<ul>
<li>12 weekends without a full moon</li>
<li>Full moon weekend!</li>
<li>3 weekends without a full moon</li>
<li>Full moon weekend!</li>
<li>12 weekends without a full moon</li>
<li>1 to 4 repetitions of an isolated full moon weekend, interspaced by either the 12 weekend or 16 weekend waiting period</li>
<li>goto 1</li>
</ul>
<p><a href="https://i.sstatic.net/fmK3Y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fmK3Y.png" alt="full moon weekend" /></a></p>


CC BY-SA 4.0
2022-09-12T20:59:56.190Z
+1 nicely presented.
JohnHoltz
CC BY-SA 4.0
2022-09-13T05:16:09.550Z
All the answers are very valuable and interesting, but this one it the most close to my actual question. Sorry if it wasn't so clear.
Mark
CC BY-SA 4.0
2022-09-12T17:56:18.280Z
2023-01-26T08:31:45.377Z
2023-01-26T08:31:45.377Z
SE - stop firing the good guys
<p>Wikipedia says that a synodic month has a length 29 d 12 h 44 min and 2.9 s.
I'm interested to find out how often a full moon (but actually, it's the same for any phase) happens during a weekend.</p>
<p>I thought:</p>
<ul>
<li>a weekend lasts 48 h</li>
<li>a week has 168 h</li>
<li>a synodic month has 708.75 h</li>
</ul>
<p>But I'm not able to calculate when the full moon happens inside the window of the 48 h we call &quot;weekend&quot;.</p>

CC BY-SA 4.0
2022-09-10T21:50:23.073Z
Are you asking "What fraction of full  moons occur on the weekend", or "What fraction of weekends have a full moon".  The answer will be different.  Or are you asking something else?
James K
CC BY-SA 4.0
2022-09-11T09:53:54.603Z
@JamesK I can rephrase my question in this way: how much weeks (in average) should I wait to have another full moon on the weekend?
Mark
CC BY-SA 4.0
2023-01-26T07:13:57.077Z
Is Friday evening not part of the weekend for you?  That's unfortunate.
Toby Speight
<moon-phases>
How often does a full moon happen on the weekend?
Mark
# How often does a full moon happen on the weekend?

(Asked by: username_0 on 2022-09-10)

Wikipedia says that a synodic month has a length 29 d 12 h 44 min and 2.9 s.
I'm interested to find out how often a full moon (but actually, it's the same for any phase) happens during a weekend.

I thought:

* a weekend lasts 48 h
* a week has 168 h
* a synodic month has 708.75 h

But I'm not able to calculate when the full moon happens inside the window of the 48 h we call "weekend".

**Tags:** moon-phases

> **Comments:**
> **username_1**: Are you asking "What fraction of full  moons occur on the weekend", or "What fraction of weekends have a full moon".  The answer will be different.  Or are you asking something else?  
> **username_0**: @username_1 I can rephrase my question in this way: how much weeks (in average) should I wait to have another full moon on the weekend?  
> **username_2**: Is Friday evening not part of the weekend for you?  That's unfortunate.  

---

## 3 answers

### Answer by username_11 on 2022-09-12. Score: 7 | Accepted answer

Some further info on the actual distribution pattern.

The synodic month and the week length is descended by 1.53059 days, so one month later the full moon comes 1.53059 days later in the week.

That means here are two types of full moon weekends. The first type is where the full moon hits the weekend, but one month later, 1.53059 days is enough to bring it past midnight Sunday. The **isolated full moon weekend**.

In the other type, the full moon falls early enough in the weekend that 1.53059 days is still Sunday, meaning there will be a **full moon weekend pair** 4 weeks apart.

The same applies to the wait for the 1.53059 days to work itself through the 5 weekdays. This is has to be exactly 12 or 16 weekends in a row without a full moon.

Full moon weekend pairs have some constraints though. The earliest in the weekend the second full moon of the pair can fall is Sunday afternoon, so the next full Moon can be on a Tuesday on the earliest.  
*full moon weekend pairs are followed and preceded by the short waiting period of 12 weekends without a full moon*.

Pattern:

* 12 weekends without a full moon
* Full moon weekend!
* 3 weekends without a full moon
* Full moon weekend!
* 12 weekends without a full moon
* 1 to 4 repetitions of an isolated full moon weekend, interspaced by either the 12 weekend or 16 weekend waiting period
* goto 1

[![full moon weekend](https://i.sstatic.net/fmK3Y.png)](https://i.sstatic.net/fmK3Y.png)

> **Comments:**
> **username_12**: +1 nicely presented.  
> **username_0**: All the answers are very valuable and interesting, but this one it the most close to my actual question. Sorry if it wasn't so clear.  

---

### Answer by username_1 on 2022-09-10. Score: 22

Exactly 2/7 of time is "weekend", There is no exact alignment of the lunar orbit to the length of a week, so a full moon is equally "likely" to be at the weekend

So 2/7 of full moons occur at the weekend. As the synodic month is about 29.5 days there are about 1245 New Moons per century, so you'd get about 356 on the weekend. This full moon (10th Sept) is on Saturday, and next full moon, 9th October 2022 is on Sunday (at my location)

However the moon appears full for a couple of days on either side of the moment at which its longitude is 180° relative to the Sun. This being the case, the moon will appear full at the weekend if it is exactly full on any day between Thursday and Tuesday - which means that most months will have a moon that looks full at the weekend.

---

Now to think about this in a different way, in a century there are 5218 weekends (±1) and 356 are "full moon weekends", so about 6.8% of weekends are full moon weekends, or to put it another way 1 weekend in 14.66 is a full moon weekend.

On average, you need to wait for 14.66 weeks before you get another full moon weekend. **But** this is an average: the pattern of full moon weekends is not at all random. In fact what tends to happen is that there is one full moon weekend, followed four weeks later by second one. Then a gap of more than 14 weeks, before the pattern almost repeats.

So (in my location) there is a full moon on *Saturday* 10th Sept and *Sunday* 9th Oct. Then it proceeds: Tuesday in Nov, Thursday in Dec, Friday in Jan, then *Sunday* 5th February. That is 17 weeks after the October full moon weekend.

Thus, final answers: 2/7 of full moons are at the weekend. 2/29.530588 = 6.8% of weekends have a full moon.

> **Comments:**
> **username_0**: "2/7 of full moons occur at the weekend". That is non-intuitive for me. Are you saying the length of the synodic month does not matter here? If the moon phases would require 25 days, 31 or whatever, it would be the same then?  
> **username_1**: Yes.  Imagine you have a bag with 200 red counters and 500 blue ones.  You spread them out in a line and take every 25th counter.  You'd get 2/7 red and 5/7 blue.  What if you took every 31st counter?  You'd get 2/7 red and 5/7 blue.  It is the same with moon phases.  
> **username_3**: @username_0 That number from Wikipedia is the *mean* synodic month length. The [Moon's motion is rather complex](https://en.wikipedia.org/wiki/Lunar_theory), so there's quite a bit of variation in the actual month length. Fred Espenak says [here](http://astropixels.com/ephemeris/phasescat/phasescat.html) that "the length of any one synodic month can vary from 29.26 to 29.80 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit".  
> **username_1**: Though if the synodic month was 31 days, there would be fewer full moons in a century, and so fewer full moons on the weekend.  
> **username_4**: Nice, though unsourced answer. The equal probability assumption is intuitive and we all assume it's true, but I'd love to see a supporting source for it. I haven't a clue to where to look for one though, so I've just asked min Math SE [histogramming phases between a periodic function and another periodic, quasiperiodic or almost-periodic function with irrational period relationship.](https://math.stackexchange.com/q/4528938/284619)  
> **username_5**: This answer cuts the Gordian knot of the problem. Well done.  
> **username_6**: The length of the synodic month does not matter to the questions of whether a randomly-chosen full moon occurs on a weekend or whether a randomly chosen weekend will have a full moon, but it *does* matter to questions such as the OP expressed in comments on the Q (since this answer was posted): how many weeks must one wait between weekends with full moons?  
> **username_7**: @username_0 yes, the length of the lunar month doesn't matter over a long period of time. The concept you want is [Ergodicity](https://en.wikipedia.org/wiki/Ergodicity). Over a short period of time it's false (e.g. the answer to "if it is a full moon this weekend, what is the probability that it will be a full moon next weekend?" is zero), but averaged over a longer period of time, every moon phase occurs on every day of the week with equal probability.  
> **username_4**: @username_7 A comment under my new Math SE question linked above points to https://en.wikipedia.org/wiki/Equidistribution_theorem Now I have to figure out (or better yet, ask about) how the two concepts overlap. Are all Ergodic systems subject to the equidistribution system? Are all system to which the equidistribution theorm applies necessarily ergotic? I learned a new word today, yay! (ergotic)  
> **username_8**: @username_4 Are you actually asking for a source that 2/7 of all days are weekends?  Or a source that says that the lunar cycle is not synchronised to calendar weeks? (For example, if a the lunar cycle were EXACTLY 4 calendar weeks, then either ALL full moons would be at the weekend, or NONE of them would).  
> **username_4**: @username_8 no, that's a mischaracterization via oversimplification  
> **username_8**: So which part of the answer do you think needs additional sourcing?  
> **username_3**: BTW, that Fred Espenak link I posted earlier is from his six millennium catalog of Moon phases, if someone wants to write a scraper and do some stats...  
> **username_4**: @username_8 you didn't `@` me but I think your comment is directed at me. Motion of the Sun-Earth-Moon system is not a perfect random number generator. It's a complex system so those of us who care rather than wave our hands give it some thought. See the links in my previous comments; there's a nice body of mathematical work on this topic. username_1 frequently writes completely unsourced answers which is antithetical to "good Stack Exchange answer" writing. At least a reference to ergoticity and/or a link to the equidistribution theorem in Wikipedia would be fine.  
> **username_4**: @username_8  I understand that most folks are fine with "It's just common sense" but we shouldn't just target the lowest common denominator when writing Stack Exchange answers.  

---

### Answer by username_9 on 2022-09-12. Score: 4

Assuming the average properties of the moon's orbit apply out to infinity, we could easily assume the answer is 2/7. But, over any finite period of time, a perfectly even distribution would be highly unlikely.

The below Python code uses [PyEphem](https://rhodesmill.org/pyephem/) to find each full moon. I choose to start from 1600 to avoid any confusion over the change to the Gregorian calendar, and then go about just as far forward as I did backwards to 2800.

Results:
Total Full Moons: 14841

```
DOW       #    %
Monday    2116 14.25779934
Tuesday   2123 14.30496597
Wednesday 2122 14.29822788
Thursday  2117 14.26453743
Friday    2125 14.31844215
Saturday  2115 14.25106125
Sunday    2123 14.30496597

Weekend   4238 28.55602722

```

A perfectly even distribution would have resulted in 28.5714% weekend full moons.

The code (on most platforms "pip install pyephem" will install the ephem dependency).

```
#!/usr/bin/python
import ephem
from datetime import datetime

days=[0,0,0,0,0,0,0]
count=0
fm = ephem.next_full_moon('1600/1/1')
d=datetime.strptime(str(fm),"%Y/%m/%d %H:%M:%S")

while d.year<2800:
    days[d.weekday()]+=1
    
    fm = ephem.next_full_moon(fm)
    d=datetime.strptime(str(fm),"%Y/%m/%d %H:%M:%S")
    count+=1

print(count)
print(days)

```

> **Comments:**
> **username_7**: Well, certainly 4,238 of anything can't be distributed evenly into 7 buckets :)  
> **username_10**: @username_7 The total number of samples is 14841, not 4238; although your point still stands, since that is also not divisible by 7. However, it's still clear that the distribution isn't quite even; the "perfect" distribution would be six days with 2120 full moons, and one with 2121 (2120 * 7 + 1 = 14841). Using percentages just makes things more confusing, since 100 isn't divisible by 7.  
> **username_9**: @username_7, not sure what you're getting at.  The fact that an arbitrary finite interval has about a 1/7 chance of being impossible to distribute evenly just re-emphasizes my point.  
> **username_2**: You probably need to tell us which time zone you used for your calculations, as that certainly affects the distribution (because the time zone determines when midnight occurs).  

---

