Integer and Noninteger Multiples

Sometimes when you want to understand...you've got to unpack first.

 In his article "Menstrual-Cycle Synchrony:  Problems and New Directions for Research," Jeffrey C. Schank wrote: "Only rhythms that have periods of integer multiples of each other can, in a strict mathematical sense, be in a synchronous state over time (Winfree, 1980, 1987).

When one looks at rhythms of noninteger multiple periods of each other, one sees that cycle onsets repeatedly converge and diverge from each other..."

Rhythms?” “Periods of integer multiples of each other?”  “Synchronous state?”  Since when did menstrual synchrony get to be so complicated?

The above paragraph is from an article appearing in the Journal of Comparative Psychology, vol. 115, and like most paragraphs in publications aimed at researchers, it packs a whole lot of information in a very small space (see suitcase above).  Let’s try unpacking it, with a few definitions:

Rhythm

Movement with a regular succession of strong and weak elements.”  This just means thinking of the days a woman is menstruating as the strong element, and the days when she is not menstruating, the weak element

Viewing a woman’s menstrual cycle over many months, we see that a rhythm arises, strong element-weak element-strong element-weak element-strong element-weak element

Period

An interval between recurrences of an astronomical or other phenomenon.”  In menstrual cycle research, the “recurring phenomenon” is the first day of the strong element, or a woman menstruating; thus an example of the “interval between recurrences” would be the portion of the RHYTHM between slash marks [/] below:

/strong element-weak element-s/trong element-weak element-strong element-weak element

Of course, as months go by, the pair of slash marks [/] would keep shifting forward, enclosing the first day of menstruation of the earlier month and the first day of menstruation of the later month.  In this way, menstrual synchrony researchers accumulate data, sometimes on hundreds of women, sometimes over decades!

It is important to note that the above definition is NOT what is usually meant by ‘period’ (as in, “I just got my period,” or “I’m on my period”) – what is usually meant looks like this instead:

/strong element/-weak element-strong element-weak element-strong element-weak element

Can you see the difference?

Integer

A whole number.”

For example, 24, 27, 28…etc.

Multiple

A number that may be divided by another a certain number of times without a remainder (28 is a multiple of 7).

36 is a multiple of 6; 100 is a multiple of 5; etc.

Integer Multiple…and Noninteger Multiple

An integer multiple is a whole number that may be divided by another a certain number of times without a remainder.  Using some period lengths as examples, we see that:

28/28 = 1; there is no remainder, so 28 is an integer multiple of 28;

40/20 = 2; there is no remainder, so 40 is an integer multiple of 20;

28/27 = 1.037; there is a remainder, so 28 is a noninteger multiple of 27.

Synchronous State

Synchronous:  “Existing or occurring at the same time.”

State:  “The existing condition or position of a person or thing.”

* * * * *

Interpreting the above in the context of menstrual synchrony research, a synchronous state can only exist between two (or more) menstrual rhythms when the periods of those rhythms are integer multiples of each other.

That is, a woman with a menstrual rhythm whose period is 28 days can be in a synchronous state with a woman whose menstrual rhythm has a period of 28 days also; or, in an example given by researchers Weller and Weller (and later commented upon by Jeffrey Schank,) a woman whose menstrual rhythm has a period of 40 days can be in a synchronous state with a woman whose menstrual rhythm has a period of 20 days.

If the periods of two (or more) menstrual rhythms are noninteger multiples of each other, then it is impossible for them to be in a synchronous state, because “cycle onsets,” that is, the first day of the strong element, “repeatedly converge and diverge from each other.”

Which brings us back to the experience that Terry Farrah had with her roommate.  Why not go back and read her story again, and see if you can pick out the period of each woman’s rhythm, and determine whether indeed the periods are noninteger multiples of each other.

* * * * *

But, before we go on to the first MOLTXPERIMENT, there is something else to consider:  Some have suggested that in the case of periods that are noninteger multiples of each other, synchrony can be defined as occurring when cycle onsets diverge less than would be expected by chance.  There are a few problems with this approach:

1.  No studies have SPECIFICALLY looked at women with, for example, 27 and 28-day periods respectively, over a sufficiently long period of time, to determine whether their cycle onsets are actually diverging less than would be expected by chance.

2.  If it were found that these women’s periods WERE diverging less than would be expected by chance, this would be more properly referred to as “hypodivergence,” rather than synchrony.  (‘Hypo’ means “under” or “less than.”)

3.  As neither synchrony nor hypodivergence have been conclusively proven to occur, perhaps it’s best to stick with the concept of “menstrual overlap.”

4.  Of course, “we must be hypodiverging!” doesn’t have the same ring to it that “we must be synchronizing!” does; nor does “we must be overlapping!”  At the end of this MOLTXIBIT, we will explore some of the reasons why “we must be synchronizing!” is so much more satisfying a way of putting things.

Click here for a simple experiment demonstrating the repeated convergence and divergence of cycle onsets when periods (as defined above) are noninteger multiples of each other; AND the “synchronous” state menstrual rhythms fall into, when their periods ARE integer multiples of each other.

X MOLTXPERIMENT:  A Tale of Two Tape Measures