When we began our blogosphere journey through the holidays of the year, we more or less dove into the holidays without explaining too much about why we chose to celebrate these particular eight, other than to share our commitment to choosing holidays that continued to be meaningful and not merely traditional.  Our family started with the Winter Solstice with its obvious significance amongst our Northern Ancestors on both sides of the Atlantic Ocean.  The lengthening nights was a reason for trepidation and longest night that marked the return of the Sun was a reason for celebration, and even now we can celebrate both the experiences of our Ancestors and the astronomical significance of the days.  The Quarter Holidays, the Winter and Summer Solstices and the Spring and Autumn Equinoxes, are all similarly astronomically and traditionally significant, but what about the Cross-Quarters, the holidays halfway between the Solstices and Equinoxes: Imbolc (Feb 2), Beltane (May 1), Lunasa (Aug  1), and Samhain (Oct 31)?

These four holidays are a familiar part of the Wheel of the Year.  In the deepest winter, we celebrate Imbolc and the hope for new life; at Beltane or May Day, we celebrate fertility and new life; at the height of summer, we celebrate Lunasa, and at Samhain or Halloween, we remember the dead in anticipation of winter.  In some traditions, Beltane and Samhain are the two most significant holidays.  So, these holidays are not trivial.  What is interesting about the Cross-Quarter Sabbats is that they have significance beyond tradition in the natural course of the seasons of our planet.  In fact, in addition to marking the halfway points between the Solstices and Equinoxes, they track the annual cycle of the hottest and coldest times of the year. Let’s see how this works.

First, let’s look at the Quarter Sabbats (Yule, Ostara, Litha, Mabon).  These are the Solstices and Equinoxes. They mark the days with the shortest daylight, with the longest daylight, and with daytime exactly as long as nighttime.  Celebrating makes sense on these special days in the yearly daylight cycle.¹  On the Wheel of the Year, they provide four equally spaced holidays: the four points of the year when divided by light level (see diagram).

Imagine slicing the year into 8 sections, between each of the 8 Sabbats.  In terms of light, the 1^st section (Yule to Imbolc) is just like the 8^th section (Samhain to Yule) since they both are equally dark, bordering on the darkest day of the year (Yule or the Winter Solstice).  So if I listed the 8 sections according to light level, they might go like this (going through the year, starting at the Winter Solstice) — dark, dim, light, bright //Summer Solstice// bright, light, dim, dark //Winter Solstice// Repeat.

In terms of temperature, one might expect the temperature to be also the same for the 1st section and the 8th section, but surprisingly, they are not. Just as the hottest time of the day is several hours after noon (and not the time when the sun is highest), the hottest and coldest times of the year are not the Solstices, but rather a date about a month and a half later: Imbolc and Lunasa. As you may have noticed, it takes a while for our Earth to heat up or cool down in response to the hours of sunlight, so the yearly temperature (thermal) cycle lags the daylight cycle.²

You can see this by noting that the Equinoxes, though they have the same amount of sunlight, are very different in temperature (March is much cooler than Sept).  Depending on local terrain, this lag will usually be between 4 and 7 weeks.  So, while the Quarter Sabbats are the “peaks” and “equal points” of the daylight cycle, the Cross-Quarter Sabbats can be the “peaks” and “equal points” of the thermal cycle – thus Imbolc commemorates the coldest time of the year, the peak of winter, or the Winter Thermstice³, making Beltane the Spring Equitherm, etc.

This is likely what gave us the origin of the Cross-Quarters.  They mark  both the boundaries and midpoints of four seasons.⁴  These dates are seasonally significant in terms of temperature.  The temperature cycles matter greatly for Northern agriculture.  Each year, our crops depend on the cycle of cold snows of winter that melt into nourishing water for the new seedlings in the Spring that grow strong and fruitful in the sweltering heat of Summer.  Finally, this cycle ends with the Harvest during the cooler, soothing temperatures of autumn, leading once again into the bitterness of winter.

So the Cross-Quarter Sabbats are the peaks (-stice) and equipoints (equi-) of the thermal cycle, and the Quarter Sabbats are the peaks (-stice) and equipoints (equi-) of the daylight cycle.  What an deeply interwoven system, with each Sabbat simultaneously representing a cardinal point of a cycle, as well as the boundary to a quarter section (season) of the other cycle⁴!

The result of the interplay of these two cycles is complicated, and cool.  You’ll want to refer to the combined Wheel of the Year to follow along for this next part.

Remember that the temperature cycle lags by about 1/8^th of the year, so a year goes like this for temperature: cold //Imbolc// cold, cool, warm, hot, //Lunasa// hot, warm, cool, repeat.  So putting that together with the daylight cycle, going from the first through the 8^th sections, you get: //Winter Solstice// dark & cold, dim & cold, light & cool, bright & warm, //Summer Solstice// bright & hot, light & hot, dim & warm, dark & cool, repeat.  So, each 1/8^th section of the year is unique – no two are alike!  I love it!

For instance, for family campfires, it helps to have warm temperatures, but not really late sunsets (more time in the dark), so the section from Lunasa to Mabon or Samhain works well.  So where would it be nice to have a meteor shower?  Of course, somewhere near the same time, so that it is warm enough out to stay out late at night, but so that it gets dark before getting too late.  And we are so lucky in the Northern Hemisphere that the Perseid meteor (~August 13) falls right in this season.  Our friends in the Southern Hemisphere aren’t out of luck though, because the Geminid meteor shower is almost as nicely placed, at ~ December 14 (just look to the point opposite the date to see the seasons in the Southern Hemisphere).  Or, say, cross country skiing – though fun at any time in the winter, the long days of late February give both sufficiently cold temperatures, with days as long as those in October.  And so on. This turns each season, each month, even each week into a special gift, not to be repeated until another year has gone by!

As Pagans, many of us (whether Naturalistic Pagans or not) also already have traditional ways to celebrate each of these Sabbats, as well as our own personal traditions we may add to each one.  While the dates of the Quarter Sabbats are uniform regardless of location (reversed in the Southern Hemisphere), local terrain shifts the thermstices and equitherms a little from place to place.  Nonetheless, our family celebrates these on their traditional dates – that way we both celebrate them with other Pagans around the globe, as well as giving us yet another connection to our Ancestors, who lived in a wide range of different places.  The traditional dates also preserve symmetry.⁴  Of course, if your location on our Earth has very different cycles⁵ (such as in the tropics), much of your Wheel of the Year may be different – which could give very different Sabbats/holidays.  After all, for most us Pagans, our Sabbats are one of many threads connecting us to our place on Earth.  For me, the Wheel of the Year, interwoven with meaning, keeps me connected, helping me appreciate the glory and joy of each part of the year.

May you be touched by each of our beautiful seasons —

— Jon (& Heather) Cleland Host

Edit:   Want to learn about how we celebrate any of the 8 holidays?  Here are links to each of them.

Winter Solstice (Yule)

Winter Thermstice (Imbolc)

Spring Equinox (Ostara)

Spring Equitherm (Beltane)

Summer Solstice (Litha)

Summer Thermstice (Lunasa)

Fall Equinox (Mabon)

Fall Equitherm (Samhain)

Scary math/etymology footnotes:

1.  For locations outside the tropics, the cycle of daylight over the year is close to a shifted sine(x) function, so after building the equation, this gives:

where y = minutes of daylight on a given day, t_W = minutes of daylight on the Winter Solstice,  t_R = time sub range (equals the difference in daylight minutes between the Summer and Winter Solstices), and x = the number of days since the Winter Solstice.  Thus, the amplitude of this sine wave daylight cycle varies with latitude, with the maximum change in daylight time (maximum amplitude) at either pole of our Earth.  This close approximation ignores small factors like the ellipticity of our orbit, the ellipsoidal shape of the Earth, refraction, and so on.   The formula above gives the black line in graph below.  The blue line shows the actual minutes of daylight as listed by the US Government (NOAA).  Wow, the pure sine wave is very close to the actual (the blue line is nearly on top of the black line)!  So all those other little factors don’t make much difference.   If you’d like to try your hand at working out the formula, go for it – the answer I worked out for my location here in Michigan is y = 536 + 195.5*Sin(2π*(x)/365.25-π/2).

2.  One would expect the temperature cycle to be less orderly.  Two factors among many include the highly variable local terrain, as well as the highly variable daily weather.  So, imagine my surprise when, upon plotting the temperature cycle, it also gave a nearly perfect sine wave!  In the graph below, the actual recorded temperature averages give the orange line, and the approximation of the thermal cycle using a pure sine wave gives the dotted line.  While not quite as close a match as that for the daylight cycle, you can see that the orange line is nearly on top of the black line.  While at first glance one might think it would be a simple cosine, that doesn’t work at all (try it!).  Building the equation for the thermal cycle gives:

where y_T = average temperature on a given day, T_W = average temperature on the Winter Thermstice,  T_R = temperature sub range (equals the difference in average temperature between the Summer and Winter Thermstices), F = a constant shift factor to give the lag for the thermal cycle, and x = the number of days since the Winter Solstice (because if we counted from the Thermstice, then all four lines would be nearly on top of one another, and we couldn’t use the same x axis).  As before, the amplitude of this sine wave thermal cycle varies with latitude, with the maximum change in daily temperature (maximum amplitude) at either pole of our Earth.  If you’d like to try your hand at working out the formula, go for it – the answer I worked out for my location here in Michigan is y_T (in °F) = 22.2 + 25.3*Sin(2π*(x)/365.25-3π/4+0.22).

The cycles of temperature and daylight for the year.  The y axis is unlabelled because this graph is normalized to 1, and because these values (daylight minutes and average temperature) are a function of location.  The actual data curves (for daylight and temperature) are based on data from mid-Michigan.

The same data, if plotted on a polar graph instead of a Cartesian graph, gives this diagram for the Wheel of the Year in mid-Michigan:

3.  “Thermstice” and “Equitherm” are hybrid words (words containing roots from different languages – in this case Latin and Greek).  Hybrid words pretty common, including well known words like automobile, bioluminescence, hetero & homosexual, hexavalent, Homo sapiens, neuroscience, television, and many more.

4.  So which cycle (thermal or daylight) is used to define our four seasons?  Well, both have been used.

The traditional celebration of the Cross-Quarters may be due to the marking the seasons by the cycle of daylight.  For instance, if “summer” is defined as the brightest quarter of the year, then summer would start at Beltane and end at Lunasa, and Litha would be called “Midsummer” – which indeed is Litha’s traditional name.  The seasons, when based on the daylight cycle, give the remaining Cross-Quarters as “boundaries”.  The minor point that the Cross-Quarter Sabbats are traditionally a little earlier than the exact halfway point isn’t a big deal.  In fact, that little shift usually moves them closer to the actual thermstice or equitherm.

However, here in America, most calendars mark the seasonal boundaries using the Quarter Sabbats (so Litha is “the first day of summer”, not “midsummer”.  Here at least, defining the seasons based on the thermal cycle is more common – which makes the Cross-Quarters the center of each season.   Regardless of which cycle is looked at, both cycles give us the full 8 Sabbats of the Wheel of the Year when the midpoints and boundaries of all four seasons are celebrated.

5.  An example is given here: https://atheopaganism.wordpress.com/2015/01/29/an-atheopagan-life-celebrating-riverain-and-adapting-the-wheel-of-the-year/  same essay also here:  http://humanisticpaganism.com/2015/01/26/an-atheopagan-life-riverain-and-adapting-the-wheel-of-the-year-by-mark-green/

 The Authors

In addition to writing the Starstuff, Contemplating column here at HumanisticPaganism, Dr. Jon Cleland Host is a scientist who earned his PhD in materials science at Northwestern University & has conducted research at Hemlock Semiconductor and Dow Corning since 1997. He holds eight patents and has authored over three dozen internal scientific papers and eleven papers for peer-reviewed scientific journals, including the journal Nature. He has taught classes on biology, math, chemistry, physics and general science at Delta College and Saginaw Valley State University. Jon grew up near Pontiac, and has been building a reality-based spirituality for over 30 years, first as a Catholic and now as a Unitarian Universalist, including collaborating with Michael Dowd and Connie Barlow to spread the awe and wonder of the Great Story of our Universe (see www.thegreatstory.org, and the blog at evolutionarytimes.org). Jon and his wife have four sons, whom they embrace within a Universe-centered, Pagan, family spirituality. He currently moderates the yahoo group Naturalistic Paganism.

Heather is a parent and a scientist raising her four children to explore the world through scientific understanding and with spiritual appreciation of the Universe. She has a Master of Science degree in Physics from Michigan State University, a Bachelor of Science degree in Aerospace Engineering from the University of Michigan, and a Bachelor of the Arts degree in English Literature, also from the University of Michigan. She teaches physics as an adjunct instructor at Delta College, runs the Math Mania program at a local elementary school, has worked at Dow Corning as an engineer and at NASA as an intern, and she has led science outreach workshops for K-12 students through joint programs between NASA and the University of Michigan. She is a naturalistic non-theist, whose faith has been shaped by her childhood within the Episcopal Church, her adult membership in the Unitarian Universalist church, and through Buddhist meditation.  She has a passion for bringing science and spirituality to everyone in a fun way, both for her own family and for the wider community of the Earth.  She is a co-author with Jon Cleland-Host of Elemental Birthdays: How to Bring Science into Every Party.