# ASK FATHER: Permutations of the Novus Ordo according to options

QUAERITUR:

Your recent post on ways to enrich the Novus Ordo has brought to mind something I’ve wondered for a while, how many different ways can Mass be said validly in the OF given all of the different options? e.g. just going off of 4 Eucharistic Prayers and yes/no on the sign of peace you’re already at 8 options.

Interesting question.  The number of permutations might be hard to calculate.

First, you would need to know all the variables/options.  Then you would have to figure which options negatively exclude other, subsequent options.  Then you would do the math.

However, in a simple series of options, without lots of exclusions, etc., the number can get big really fast. You mention 8 options. Let’s double that. Let’s say that there are 16 options.  There are way more than 16, but let’s say there are 16. So, 16 × 15 × 14 × 13 × … = ?

The total permutations are: 20,922,789,888,000

You decide.

Fr. Z is the guy who runs this blog. o{]:¬)
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### 34 Responses to ASK FATHER: Permutations of the Novus Ordo according to options

1. teomatteo says:

factoring in the brevity constant (b), this would determine the likelihood the mass (m), would be said by the priest (p) at the large suburban (s) parish anywhere.

2. jfk03 says:

You might want to subtract one option on account of the (unwritten) rubric that the first Eucharistic prayer (Roman canon) should never be used because it takes too long.

3. Gripen says:

Now I want to do the math. Anybody have a missal I could borrow? Probably isn’t worth the effort, but it might be interesting (shocking?) to compare the permutations in the Ordinary and Extraordinary Forms.

4. nchen says:

Father,

It is much higher than 16.

There are:
– 3 options at the introductory rite
– 3 options at the penitential act,
– 2 options at the Gospel (short form)
– 2 options at the Creed
– 2 options at the prayers of the faithful (to do or not to do)
– 8 options for the preface (Sunday in OT)
– 10 options for the Eucharistic Prayer (4 + 2 reconciliation + 4 for special needs)
– 2 options at the sign of peace (to do or not to do)
– 4 options at the conclusion

Those are just the ones I can think of off the top of my head. There are many, many more than 16.

5. nchen says:

Also, there are some exclusions, because you can’t do more than one of a part, so it would just be multiplying across for each option: 2*2*3*8*10… etc. I’ll leave the math to someone else.

6. Gabriel Syme says:

I had no idea it would be such a massive number, although I did expect it to be in the millions.

An American author, Dan Graham, wrote a great book called “Lex Orandi” comparing the modern and traditional mass and sacraments. He considered this kind of thing.

I learned that there are over one million possible ways for someone to be baptised via the modern form of sacrament. It always sticks in my head because I read that around the time we were getting our daughter baptised (we went for a traditional baptism).

He’s also done an analysis of the two forms of mass compared, in a article called “Words Count” which shows how the mass was gutted by the change from traditional to modern, all in the name of making the mass acceptable to protestants.

Of course, its well known that half a dozen protestant ministers of varying denominations worked together with ++Bugnini to design the new mass – but even still it is shocking to see the large changes and deliberate ambiguities quantified.

Here’s a copy of the “Words Count” article (are we allowed to post links? – apologies if not!):

http://christus.cx/DanGrahamMassdifferences.pdf

(I expect the book mentioned above and this article overlap to a degree.)

7. JustaSinner says:

Telephone game starts with Suzy had a red dress and ends with Johnny had a blue bicycle.

8. Father:

You are right on.

Just to expand the list of examples…

> There are, in fact, 13 Eucharistic Prayers, not ten as indicated above. Recall that there are 3 prayers intended for children. Although they weren’t retranslated, as I recall, they are still able to be used.

> At least once a year — Easter Sunday — there is a third option for the Creed; that is to recall the baptismal promises.

> For Sundays in Ordinary Time, there are times when a Mass other than the assigned Mass may be used; for Mission Sunday, I think. I bet there are others.

> The Roman Canon has two places where the lists can be abbreviated. In addition, there are concluding phrases, “Through Christ our Lord, Amen” that are in brackets, meaning the celebrant can include those, or omit them. Right there gives multiple variations of the Roman Canon.

> The Ordinary Form can be celebrated in any language, so long as there is an approved translation. Moreover, several languages can be used in the same Mass. The Mass can include both Latin and the vernacular. I have been at “big Masses” where the Petitions featured seven or eight languages.

> The offertory prayers can be prayed aloud or quietly.

> If Eucharistic Exposition is to take place following Mass, the Monstrance is placed on the altar near the conclusion of Mass, and the Mass ends differently.

> Oh, and how many know this? Did you know that Morning or Evening Prayer in the Liturgy of the Hours can be combined with Mass? Maybe the other offices too, I can’t recall.

If one considers the three fateful words that destroyed Catholic music – “alius cantus aptus” – then the number of permutations goes into the realm of large powers of ten very quickly.

10. mburn16 says:

I remember seeing an answer to this question years ago where the total combinations was at least several thousand. Obviously, the vast majority of parishes use a fairly consistent form, but the options are out there. I will try and find the source tonight.

11. Chris Garton-Zavesky says:

Hmmm. That number looks to be about the size of the National Debt.

Wow! That’s a lot of options. Isn’t the expressions “unstable liturgy” an oxymoron?

12. asciiduck says:

Father, you’ve mixed up your math a bit. Permutations like are used when ordering matters, when A, B, and C is different than B, A, C. While there are many options in the Ordinary Form, they don’t include the ability swap parts of the mass into a different part of the mass.

13. Kerry says:

Among all the possible variations, (validity notwithstanding), which NO Mass is the proof set?

14. Someone please be the Garrigue says:

I was never any good at maths, but I think you should be looking at *combinations* rather than permutations. [I’m not really looking for either. However, perhaps a mathematician can jump in. I think permutations is proper when the order of the items makes a difference.]

If each of sixteen options has a yes/no answer, then the total number of combinations is 2 times 2 times 2 times… or 2 to the power 16 – a ‘mere’ 65536. Once you factor in three or more Eucharistic Prayers and various LGBTSJ kiss of peace preferences, then some of those twos will be a little larger than 2, but not exponentially so.

Of course, nobody really needs a 64-bit Mass.

15. Lepidus says:

Only because I probably should have been a lawyer or a member of the grammar police….

The actual answer the specific question is that there is an infinite number of ways that Mass could be said validly. I’ve been to my share valid Masses that were clearly illicit.

16. Moro says:

nchen – Thanks for the comprehensive list. However, let’s not forget that with all these options there are other variables: ad orientam or no? What music? Chant or CCM? When? Some Latin or no Latin? The local vernacular or the vernacular of an immigrant group? Or both? Or both plus something else? Piano, guitar, harp, or nothing? Incense? At which of the allowable times? Fiddleback, Gothic, or ugly vestments? I could go on and on. This thread alone proves why the extraordinary form is the better choice.

17. AmjdhA says:

Not just options of incense or no, but when incense is used, when and who/what is incensed? There is an option to include a sprinkling rite at each Mass, particularly during the Easter Season. NO also offers the option to distribute the Eucharist under one or both species. Have the Deacon assist or leave him in the pew? In this case, the LESS of fewer options is MORE beneficial to the faithful.

18. The Egyptian says:

With my poor math skills I will defer to the Rev Sagan
https://youtu.be/HZmafy_v8g8

There are about 10 distinct portions of the Mass. To use the list that nehen provided, above: Introductory Rite, Penitential Rite, Gospel, Creed, Prayers of the Faithful, Preface, Eucharistic Prayer, Sign of Peace, and Conclusion. In addition, there is the choice of Language and Chanted.

Each of these is a distinct set and has its own number of possible variations. Let us represent the portions of the Mass as S, so S1 corresponds to the set of possible Introductory Rites, S2 is the set of Penitential Rites, etc. Set S1, the Introductory Rite set has three members. Let’s call them S11, S12, and S13. Likewise, set S2, the Penitential Rite has three members or options. Let’s call them S21, S22, S23 (the first number is the set number and the second number is the element or option within that set). Altogether, from the list provided, there are ten sets of portions of the Mass with options, S1…S10, however, one must, also, consider the ordinary portions of the mass, such as the Kyrie, Gloria, Santus, Pater Noster, Agnus Dei, etc. Because these are sets with one member that do not admit of options and because they have cardinality of 1 (i.e., one member in the set), do not affect the overall number of possibilities, but since except Language and Chant are valid options that must be included, since, then, even the Ordinary prayers have cardinality of at least 4 (vernacular, Latin, chanted, unchanted). To make life simple, we will include Language and Chant as an option within the various sets and not count them as separate sets, themselves, which (note to mathematicians) might be called compactification of the dimensionality.

Now, let us call the number of options for each part of the Mass, i.e., the number of set elements for each part of the Mass, Sn, its cardinality, cn. Thus, from above, the cardinality of the Introductory Rite, S1 is 3, so, c1 = 3 (technically, including chant or no chant for each option, there are 6). Likewise, S2, the Penitential Rite, S2 has a cardinality, c2 = 3, S3, the Gospel, has a cardinality, c3 = 2, etc. Since each of these admit, additionally, to Language and Chant, one must increase the total cardinality of each portion of the Mass by the number of Languages possible and the number of Chants possible (more on that, later). Let us specify a total of Ln possible valid Languages (which is finite and countable and not more than 1000) and a total of CHn possible valid chants (finite and countable and less than 100).

Okay, now we can answer the question of how many possible types of valid Masses there can be, irrespective of time – since some options depend on the time of year, the total number of types of valid Masses will vary throughout the year. By assuming time-independence, we simply assume that all options are possible at any time, which is a simplification.

What you are looking for, here, are not permutations, but the variations within fixed subsets of a total set. For example, let’s say that set A has two members, {a1, a2}, and set B has three members, {b1,b2,b3}. To enumerate all possibilities, realizing that one must always have an a and b, although they can be one of the different options, one arrives at the following list: {a1,b1}, {a1,b2}, {a1,b3}, {a2,b1}, {a2,b2}, {a3,b3}. This is the exhaustive list. Since one must have an Introductory Rite followed by some form of Penitential Rite, followed by etc., with possible variations of each, what one wants is called, the Cartesian Product, which is represented as A x B (which can be pronounced as, “The Cartesian Product of set A and set B.”). In the example, above, A x B is the list we wrote out. How does one figure out the Cartesian Product? It is easy. List one set horizontally and one vertically and just combine them like ordinary math:
……..b1………b2………b3
a2 {a2,b1} {a2,b2} {a2,b3}

a1 {a1,b1} {a1,b1} {a1,b3}

One gets a grid of all possibilities. If this looks familiar, it should. In high school math you studied the x-y plane, which is nothing more than numbers going horizontally and vertically, so the point, (1,2) is a point on the 2-dimensiopnal number line, which can be graphed. Since the x-axis contains all of the real numbers and so does the y axis, mathematically, we say the 2-d axis is R x R or R^2 space.

In chess, one denotes the position on the board by using a through h for horizontal and 1 through 8 for vertical (or vice-versa, I forget). This one can speak of moving a bishop from a4 to b5. The chess board is an 8 x 8 listing of the Cartesian Product of {1…8} x {a…h}.

So, for the Mass, since we have a total of 15 sets, including the Ordinary parts (because they admit variations in Language and Chant), then the Cartesian Product would be S1 x S2 x S3…S15, which would reside in a 15-dimensional space and each point in the space would represent a different Mass. Thus, one can specify a particular Mass by its coordinates in the space. A Mass using the first Introductory Rite, the chanted Kyrie, the second Penitential Rite, etc., woould be represented as {1,2,2…}. If someone wants to make such a 15 x 15 grid of all Mass possibilities, be my guest. Notice, that if I have forgotten any options or parts of the Mass, the theory is quite general. It simply would increase the number of sets or options, which would increase the Cartesian Product grid size.

How many distinct types of Masses are there? Well. one must not think that the 15-dimensional space in this case is infinite, since there are only finitely-many options, unlike on a number line where numbers go from – infinity to + infinity. What we have, here, is a finite subspace of the full 15-dimensional space of R^15. The question is, what is the size of the subspace?

It is known from linear algebra that if each set in a Cartesian Product has a cardinality of c, so,
S1 = c1
S2 = c2
S3 = c3, etc.
then to find the total cardinality of the Cartesian Product, S1 x S2 x…Sn, one simply multiplies the cardinalities of the sets, so the total size of the cardinality is just c1 x c2 x…cn.

In the example that we worked, above, for set A and B, set A had a cardinality of 2 and set B had a cardinality of 3, so we expect the Cartesian Product to have 2 x 3 = 6 distinct members, which is what we found.

So, since the Introductory Rite, S1, has 3 options (that is its cardinality), the Kyrie has 2 options (spoken or chanted), etc., in themselves, and since each of these sets can, theoretically be spoken in any permissible language, thus increasing the number of variations to however many languages are permitted in the world plus the number of variations within the Rite itself, the Introductory Rite really has 3 + 100 possible variations, if there are 100 permissible languages (although why one would want to do the Introductory Rite in Russian at an American Mass is beyond me), the total number of possible Mass types would be, based on the information provided above by the various posters and assuming, say, 200 permissible languages and 7 standard chant types – one for each mode be approximated by
210 x 209 x 205… which is approximately 200^15 = 3,28 x 10^34 possible Mass types.

Now, if one only wants to consider only vernacular Masses with one chant possibility (so, it doubles the number for each), then the total number of Mass types would be:
6 Introductory Rite (3 x 2 for chant)
x 2 Kyrie (chanted or not)
x 6 Penitential Rite (3 options x 2 chant)
x 4 Gospel (2 x chant)
x 4 Creed (2 x chant)
x 4 Sanctus (vernacular, Latin, chanted, unchanted)
x 16 Preface (8 x 2 chant)
x 20 Eucharistic Prayer (10 x 2 chant)
x 4 Pater Noster (vernacular, Latin, chanted, unchanted)
x 4 Agnus Dei (vernacular, Latin, chanted, unchanted)
x 4 Sign of Peace (2 x 2 chanted)
x 8 Conclusion (4 x 2 chanted)

So, the theoretical minimum cardinality for the time-independent Mass is 754,974,720 or about 7.5 x 10^8 and the theoretical maximum is about 3.28 x 10^34 depending on how many languages there are for the Mass. Obviously, the true theoretical minimum is 1, because a priest could say the same Mass the same way, all the time (seasons excluded), but we are talking about the total number if all options are used. Now, if a priest had been saying a Mass at the speed of one Mass per second since the Big Bang, he would have only said 4 x 10^17 Masses. If he wanted to say all of the theoretically possible Masses in all of the languages possible starting at the Big Bang and just be finishing, roughly, today, he would have to say a Mass at the speed of one Mass per 7.5 x 10^-16 seconds or .75 femtoseconds, which is the speed of the fastest known laser.

Realistically, since most priests only use about two or three variants, the number of variations most priests normally use in a lifetime is about 2048.

Hope that helps.

The Chicken

P. S. A priest once gave a homily that mentioned the number of Masses being said at any given time during the day in the world. I sent him a long e-mail correcting the number. That was interesting. Did you know that there is a, “Mass wave,” that propagates around the earth at a particular speed? If anyone is interested, I might dig up the calculations.

Oopps. The A x B Cartesian Product table should read:

……..b1………b2………b3
a2 {a2,b1} {a2,b2} {a2,b3}

a1 {a1,b1} {a1,b2} {a1,b3}

Anyway, the theory is general.

The Chicken

Sorry, for all of the typos. I sound a little like English is not my native language (is it, hmm…).

To correct, yet again, if A = {a1, a2} and B = {b1, b2, b3), then A x B = {a1,b1}, {a1,b2}, {a1,b3}, {a2,b1}, {a2,b2}, {a2,b3}.

The Chicken

22. Richard A says:

Then there’s balloons, or no balloons? Barney costume or not? Giant masks, or not? Is wearing a Darth Vader costume the same as wearing a Barney costume, or is it different?

23. kurtmasur says:

Also, don’t forget that the NO rubrics have 3 ways in how communion can be distributed, including via intinction, and via a “pipet”? The latter is made out of metal, but not exactly sure if it is more like a straw or a spoon.

And to add to Richard A’s comment, don’t forget to factor in if you’ll have EMHCs including whether it will be lay people or the “devil” distributing, as was the case of a lady wearing devil ears :-o

24. Atra Dicenda, Rubra Agenda says:

Old man Peabody…had this crazy idea of breeding pine trees…

25. Rob83 says:

The most important stat is really that every parish and every priest can, if they so choose, say Mass differently than every other parish or priest out there. It’s not like Mass wasn’t variable before (the Missale Romanum is a very large book due to the variable parts), it’s just that the variables followed a well-defined script that determined what was supposed to be said any given day in a given place. Variation with a purpose.

26. Simon_GNR says:

jfk03:
“You might want to subtract one option on account of the (unwritten) rubric that the first Eucharistic prayer (Roman canon) should never be used because it takes too long.”

Quite. Our parish priest has been with us over two years now and, as far as I can remember, we’ve had the first Eucharistic Prayer once and the fourth one once. It’s usually EP2 or sometimes EP3. Why do we have to have the shortest one? Is everybody in a hurry to get out of church? Mass is only once a week for most of us, and I think many of us would like the Roman Canon – it is the Roman Catholic Church after all. It’s so redolent of history and tradition. Actually, that probably answers the question of why so many priests don’t like to use it.
Even on the rare occasions when EP1 is used it’s often the abbreviated version which leaves out Linus, Cletus, Clement…etc and Felicity, Perpetua, Agatha, Lucy etc. I love to hear these names of ancient saints and martyrs: it reminds us of the long history of the Church and those that have gone before us on the path to Heaven.
One diocesan priest of my acquaintance says he never uses anything BUT the Roman Canon – it’s the only one he regards as properly historical. He also says Mass in the Extraordinary Form once a week too – on a weekday morning. I wish I lived in his parish!

27. Suburbanbanshee says:

To be fair, there were also plenty of options in the Extraordinary Form, and tons more in the pre-Trent world. [So?]

In the last month, I ran across a little article about the meaning of how many times the priest fractionated the Eucharist. He only went through the numbers of pieces from 2 – 12, so obviously he wasn’t doing Mozarabic Rite!

28. Suburbanbanshee says:

A little Latin article. Filling up some space in Jammy’s edition of St. Albert the Great’s stuff. So we’re talking the 1650’s, not the 1960’s.

29. NoraLee9 says:

Oh, Chicken.

30. un-ionized says:

NoraLee, yes, Ow!

31. amsjj1002 says:

Dear Masked Chicken, you have blown my mind away, but I am still in awe of your gifts. Thank you!