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Mensuration - the rules

Our notation standard (1/2)

We will first study the rules for a division without specifying its level (tempus: breves => semibreves or prolatio: semibreves => minims) or its type (binary or ternary).

Note symbols

We'll write N for the longer note of the division (a breve when considering time, a semibreve for prolation), and n for the shorter one (thats is, a semibreve or a minim).

Note values

The value of the larger note will be written V, that of the shorter one will be written v. As a convention, values will be expressed in terms of this shorter one v taken as a unit. Interpretation rules will then allow us to evaluate V depending on mensuration and context.

Notation of rests

Following Apel, we will notate rest symbols and values between a pair of parenthesis: (N) et (n) for symbols seen on the document, (V) et (v) for their value once they've been interpretated.

This notation standard may be summarized as follows:

 larger note N larger rest (N) shorter note n shorter rest (n) tempus prolatio

Some readers might dislike this abstract notation used here for a while, but it nevertheless brings three benefits, allowing us to:

• talk about time and prolation simultaniously, as explained just before;
• ignore here the reduction ratio mentioned in the introductory page;
• avoid the confusion which would result if we'd use modern note symbols, since in a ternary division we'd have to introduce dotted notes.

Rules for a binary division

They are quite simple: V = 2v = 2 without exception!
That is, in tempus imperfectum a breve always weights two semibreves, and in prolatio minor a semibreve weights always two minims.

Exactly like our modern dot, it increases by half the value of whichever note on its left side. However, double dot didn't exist.

I didn't know whether I should use dot or point - I don't mean Apel didn't know himself, but he's using punctus all the time :-)
After such a short mention of binary division, most authors seem to feel guilty, and take a chance to add some digresssions (about musica ficta, for instance). I won't celebrate this ritual, if you don't mind!

The ternary sentence

A group of symbols, the value of which is equal to 3, is named a perfection. It's convenient to consider a sentence as a sequence of perfections.

Nominal values

The nominal value of the N symbol is: V = 3 ; it is: v = 1 for the n symbol.
One will come upon such symbols in musical patterns like this one, for instance:

(to be read from left to right like a small musical sentence)
 symbol N n n n N N value 3 1 1 1 3 3 tempus perfectum prolatio major

A ternary note weighting its nominal value 3 is said to be perfect.

Actual values

The nominal value 3 of the N symbol often decreases to 2 (or even less), depending on the context, that is, depending on the neighbouring notes on its right or left side. This substractive imperfection process will be explained in more details further on.

The nominal value 1 of the symbol n is, in a special circumstance, doubled to 2; this is the so called alteration process (see below).

In this still unsettled situation, a first rule can nevertheless be held as certain:

Rule C0 : N will never be small enough to weight only 1, and n will never be greater than 2.

As a matter of fact, an N decreased to 1 would be simply written n ! And the alteration is applied on a yes-no basis, therefore n can weight 1 (nominal) or 2 (altered), nothing else.

In such a context dependant notation system, one must have some symbols keeping always the same value, and acting as milestones in the sentence, thus preventing context dependance to propagate too far away. Thus, here are two other certainties :

Rule C1 : Rests keep always their nominal value (V) = (3) and (v) = (1), whichever note or rest may they have as neighbours.

Rule C2 : The first note in a sequence N N or N (N) is always equal to 3 (similis ante similem perfecta), in other words it is never imperfected by any context on its left.

The meaning of the dot (1/2)

In ternary mensuration, the dot may have two quite different meanings, depending on where it appears:

• After a binary note (that is, not an N), and followed sooner or later by a note of half its value, it's an addition dot, like our modern one, as was already explained for binary mensuration;
• Otherwise, specially after an N, it's a perfection dot (explained below) ; a dot following a ternary note is never an addition dot.

Our notation standard (2/2)

Rules C1 et C2 allow us to find milestones in the sentence, which are indeed perfection borders that we will notate by this symbol |

For instance, in a sequence ... N N ... the first note is perfect (from C2), thus we'll write ... | 3 | N ... ; in so doing we decrease the number of possible interpretations for the preceding and following notes.
As another example, C1 allows to put borders on both sides of a rest ... (N) ... and thus such a sequence can be notated ... | (3) | ...

The beginning of a staff is in itself a perfection border. But one has to be more careful with its end, since a final longua, often seen at this place, has no definite metric meaning, as already mentionned on the page about ligatures.

The meaning of the dot (2/2)

The perfection dot (or its synonym division dot) is in itself a border. We'll notate it ° to avoid any confusion with punctuation.

For instance, a sequence ... | N ° n ... will be understood as ... | 3 | n ... since the N keeps here its nominal value 3 and thus makes up a perfection (as we shall see very soon, the dot here prevents the larger note to be imperfected by the smaller one on its right side - and that's of course the reason for the name of the perfection dot).

Our goal is to translate all the symbols seen on the original document into their values: N => 3 or less, n => 1 or more, etc.

In order to do so, we have first to search for the various perfection borders mentionned above: beginning of the staff, perfection dots, rules C1 and C2.

These perfection borders will allow us to give their values to a first set of symbols, and will help us to evaluate the remaining ones.

Last we will scan each part of staff between these borders, and apply to them the rules explained further (imperfection, alteration) in order to evaluate the still untranslated symbols.

Rule C3 : A sequence located between two sure borders like the ones we've defined must sum up to an integer number of perfections, that is, its value must be a multiple of 3.
Or, to state it in a little more musical way: a syncope cannot be so long that it would go across such a border.

The imperfection process

The main wish of a large note N is to keep its nominal value, that is, to fill alone a whole perfection. In such a case we will write: | N | = | 3 |
However, fulfilling this wish won't be so easy, since many small n notes will wish to stay in the same perfection - the total value remaining equal to 3.

Rule I1 : N followed by a single n weights 2 ; that is: N n | = 2 1 |
The same is true when an N is followed by 4 n or more, and even if one or several of these n is replaced by an (n) rest.
Early musicians called that imperfectio a parte post (abbreviation app), that we could replace today by the simpler: imperfection on the right side.

Rule I2 : N following a single n, or 4 n or more weights 2: | n N = | 1 2
The same holds when one or several n is/are replaced by rests (n).
(imperfection on the left, formerly named imperfectio a parte ante, abbreviation apa)

Rule I3 : when both rules might apply, the imperfection on the right (I1) is to be applied.

Probably because imperfection on the left side didn't exist in former times - thanks to Jason Stoessel for this information!
Notice that these rules don't say anything about sequences made of three or two small notes n. In the first case, it's understandable that a sequence n n n in ternary mensuration tends to make up a perfection when possible (see example 1 just below). The second case, n n before an N will be explained later (alteration process).

example 1 : | N n n n N |
None of the rules can be applied here, because there are three n, thus the result is: | 3 | 1 1 1 | 3 |
Let's notice the perfection borders on both sides of the example (on the document, they could be rests, for instance). If they'd be lacking, the N notes might be imperfected by some context on the left or right side; for example | N n n n N n | would result in | 3 | 1 1 1 | 2 1 | by application of I1.

example 2 : | N n N | = | 2 1 | 3 | by application of I1 and I3.
For the erroneous application of I2 would have resulted in: | 3 | 1 2 |

example 3 : | n N n N |
The correct result is obtained by applying I2 twice: | 1 2 | 1 2 | (it would have no musical meaning to apply I1 on the N n sequence in the middle: this would give | 1 2 1 3 | for a total of 7, thus a syncope crossing a perfection border, in violation of our certainty rule C3).

example 4 : | N n n n n n N |
We apply I1 on the two first notes, and I2 on the two last ones: | 2 1 | 1 1 1 | 1 2 |

example 5 : | N n n n n N |
A sequence n n n makes up a perfection, the remaining n imperfects the neighbouring N. According to I3 we have to apply I1 rather than I2, thus: | 2 1 | 1 1 1 | 3 | (applying I1 and I2 on the two first and the two last notes would give | 2 1 1 1 1 2 | summing up to 8 in violation of C3)

example 6 : | n n n n N n n n N ...
No imperfection happens at the end of the sequence since there are three n ; but I2 is to be applied on the left of the first N resulting in | 1 1 1 | 1 2 | 1 1 1 | N ...

example 7 : | N ° n n n n N |
The first large note remains perfect: the dot after it cannot be anything else than a perfection dot. On the other hand, the last one of the four n triggers the application of I2 on the second N, resulting in | 3 | 1 1 1 | 1 2 | to be compared with the example 5 in which the dot was lacking.

Rule states that a rest keeps always the same value. On the other hand, I1 and I2 say that a rest may (n) cause an imperfection exactly like the n note itself. Therefore, the previous examples are still valid if we replace an n note by its rest (n).

Howver, one has to be more careful when coming upon a sequence of several such rests.

example 8 : | N "two small rests" n n N |

The ambiguous notation of this example will be detailed as follows:

• two rests written at the same height belong to the same perfection;
• when written at different heights, they stay in different perfections.

Let's notice how quickly the notation system is wishing to use a free parameter: changing notes in rests, height becomes irrelevant, and thus can get a new meaning!
example 9 : | N (n) (n) n n N |  (rests of example 8 written at different heights).
The first rest, and only this one, stays in the first perfection, thus the result | 2 (1) | (1) 1 1 | 3 |

example 10 : | N (n) (n) n n N |  (rests of example 8 written at same heights).
The rests stay in the same perfection, which cannot be the first one: for the N value would decrease to 1 and then would have been written n (rule C0). Thus the first N remains perfect, and the second one is imperfected on the left, resulting in | 3 | (1) (1) 1 | 1 2 |

That's all for facts. I'll now give - more carefully - my comments to this situation, which is less simple than one might think at first sight, if we compare it to examples 5 and 7 involving only notes.

First, in the example 9, different heights obviously act as a perfection dot. However - as a first paradox - this dot was not necessary in the example 5 (the result could be obtained from rule I1 alone)!

That notations N (n) (n) and N (n)°(n) are equivalent is evidenced by Blockland de Monfort himself on his page 37, though Apel doesn't say a word about that, and though I've personnally never come upon the second one (a possible reason is suggested just below).
Secondly - as another paradox, if we replace back the rests of the example 10 by notes, we do have to insert a perfection dot in order to get the example 7 - failing to do so we would get again the example 5!

I've tried to understand this notation practice. Having transcribed a few manuscripts, it seems to me that, among all possible traps, the one occurring more often is the confusion that can be made between dots and semibreve or minim rests (even discarding the bad condition of documents, which they couldn't foresee). Thus I imagine that they wanted to avoid as possible to mix these symbols in sequence, by replacing systematically N (n)°(n) by N (n) (n), and by assuming that in N (n) (n) ... one is allowed to omit the dot after N every time it would have been necessary.

Let's examine one more interesting sequence given by Apel on his page 110:

example 11 : | N (N) n N n n n n N n (n) (n) ... (last rests at same height)
The sequence begins with | 3 | (3) | 1 2 | 1 1 1 | according to rules C1 and I2. Parsing the remaining part | n N n (n) (n) ... comes down to finding whether the N is perfect or not. In the first case one would get a syncope | 1 3 1 (1) | (1) ... denied by the notation of the rests at same height, thus the correct result for the whole sequence: | 3 | (3) | 1 2 | 1 1 1 | 1 2 | 1 (1) (1) ...

 Last, let's come back for a while to the simpler situation of isolated rests, to bring attention on a notation practice slightly different than our modern one: while we write rests at a standard height, they preferred to include them in the melodic flow, that is to write them at a somewhat intermediate height with respect to neighbouring notes.

Anti-doping test

Dear reader, please don't use any suspicious medecine to overcome your discouragement, but rather do try this suggestion: stop for a while reading this page, and go to browse the beginning of the next one dedicated to music. For, the examples given in it, taken from real cases and written with common note symbols, will refresh your motivation and bring some more light on how are working the imperfection rules...

The alteration process

Rules I1 and I2 don't say anything about the case of two n in sequence, and that's what we'll study now.

example 12 : ... n N | N ...
We know that, according to rule C2, the first N is perfect, whatever may be the context on the left or right side of the sequence.
From this it appears that we now have a problem to notate a 1 2 rhythm on the left on an N: for the only mean at our disposal up to now, that is n N, cannot be used, since it would lead to the current example 12 resulting in ... 1 | 3 | N ...
The alteration rule given below solves this problem by exceptionnally changing the evaluation of n n:

Rule A : In a sequence | n n N ... the value of the second n is doubled, resulting in | 1 2 | N ...

It's essential to notice that alteration was introduced only because of the existence of rule C2. Or, to state it another way, the altered evaluation of an n n sequence is correct only when it appears on the left of an N or its rest (N) - Apel hilights the fact that many authors insisted on this point.

example 13 : | N n n N |
This is the very kind of sequence which led to introduce alteration, thus: | 3 | 1 2 | 3 |
Let's notice that imperfection rules I1 et I2 don't apply to a sequence of two n, just to allow for alteration in such a situation.

example 14 : | N n ° n N |
Here the perfection dot forbids an altered evaluation; one applies I1 in the first perfection on its left, and I2 on the right side, to get | 2 1 | 1 2 |

Let's memorize carefully the notation and the evaluation of the sequences of examples 13 and 14, we'll have to say more about them later on.

example 15 : | N n ° n n N |
Result: | 2 1 | 1 2 | 3 | (I1, perfection dot, alteration). For the point separates the sequence n n N from the n belonging to the first perfection on its left side, and thus calls for an altered evaluation before the last N.
Notice, once again, the difference with the sequence | N n n n N | which had no dot (example 1).

example 16 : | N n n N N |
In this sequence beginning like the example 13, many apparent reasons - all of them being one same reason indeed - act together to invalidate the evaluation | 2 1 | 1 2 | 3 | :  a) a sequence of two n doesn't call for imperfection ;  b) on the contrary, this sequence has to be altered ; c)  and, by the way, the next to last N, before a same note as itself, remains perfect. Thus the correct result: | 3 | 1 2 | 3 | 3 |

Alteration and rests

First, C2 can be applied also in the case N (N); therefore, by the same arguments as above, alteration happens in a sequence | n n (N) ... resulting in | 1 2 | (N) ...

Secondly, C1 tells us that in a sequence n (n) N the (n) rest is never altered.

example 17 : | N n (n) N |
No possible alteration for (n) thus | 2 1 | (1) 2 | according to I1 and then I2 ; to be compared with example 13!

Last, it's not surprising to learn that an (n) rest may cause alteration of its neighbouring n on the right side.

exemple 18 : | N (n) n N n (n) N | (taken from Apel page 115)
Alteration is applied on the first part of the sequence, on the left of the N in the middle, but not on the second part, since the (n) rest cannot be altered. Therefore the N in the middle is imperfected by the neighbouring n on its right, and so is the last N by the (n) rest on its left, resulting in: | 3 | (1) 2 | 2 1 | (1) 2 |

It's now the moment to state a general rule that we've in fact already used in this last example and many others before:

Rule G1 : N weights its nominal value 3 whenever none of the I or A rules can be applied.

Obviously I couldn't state such a rule before having given all the other ones; but you'll perhaps ask me to argue a little more because it seems to enounce a sort of paradox: in perfect mensuration, an N note is perfect only when there's no reason to make it imperfect!

That's, after all, what we said at the beginning of the paragraph about imperfection: N fullfills its wish to weight 3... whenever nothing forbids that. I like to imagine a perfection as a kind of container in which notes are negociating their values in order to stay together, while sharing the total available space.
Though full of respect for Apel who has learnt me what I know, I presently feel that my approach is at least as close to the philosophy of the mensuration system as the one he took himself: on his page 108, he states very soon a positive rule of perfection "[An N] is perfect if followed by two or three [n]" (brackets because his notation is not mine; by the way, his footnote on this page pertains to a different point that we shall talk about further on).
As a consequence, at the top of page 110, when examining a sequence from which we've built our example 6, he must admit that it "shows that [an N] followed by three [n] may occasionally be imperfect, namely, by imperfectio apa" (brackets for the same reason as above).
Such a striking exception, happening quite soon, has puzzled me very much, and that was my first reason for a different approach; the second one beeing that I need to state things inside my own logics in order to understand them. Hoping, of course, that the few examples that I'll come upon on my little amateur's way won't invalidate some rule of this beautiful building...

Imperfection or alteration?

The rules are in principle sufficient to make a choice between imperfection or alteration. More, the perfection dot - also called division dot, which was introduced to force a different evaluation (examples 14 and 15), may always be added to help solving an ambiguity.

However, this sort of gift is made by the scribe only when he wishes to do so!

A borderline case!

example 19 : | N n n n n n n N | (Apel page 115)
Rules make the evaluation straightforward : imperfection at the beginning, alteration at the end, thus the result | 2 1 | 1 1 1 | 1 2 | 3 |
However it seems that Apel had sometimes to make a different evaluation, taking in account that n n n groups tend to make up a perfection, thus leaving the initial and final N perfect, to give : | 3 | 1 1 1 | 1 1 1 | 3 |

In such cases, choice can be made easier by looking at the other voices...

Apel doesn't quote any source, the dates of which would have been interesting to know: either this tricky case illustrates the limits of an underspecified system, or, more probably, it has something to do with the historical evolution of alteration, the importance of which decreased when time passed. For, definite evidence of such an evolution does exist, even in much simpler cases than the previous one...

Historical evolution of alteration

The sequences shown in examples 13 and 14 are specially meaningful: as soon as the end of the 15th century, one may come upon some cases in which, despite the absence of a dot, | N n n N | must be evaluated to | 2 1 | 1 2 | (here, it's decisive to compare the different voices, since the two evaluations in question differ by one perfection). As years are passing, these cases happen so often in musical sources that one cannot explain them anymore by unfortunate omissions of the perfection dot by the scribes.

Meanwhile, theoreticians still teach the former evaluation (given in example 13)! And it remains valid indeed, according to Apel, everytime it appears in a ligature. Though this author states that without an explanation, I'm wishing to suggest one, deduced from the interesting juxtaposition of these well established facts:

• the altered evaluation is the former one; it became old-fashioned, being replaced by imperfection;
• ligatures, too, happened less and less often with time passing;
• when written in a ligature, the sequence we're talking about must be understood in the old way.

Thus it would be a sort of fossilisized interpretation, remaining alive only when written in the more and more obsolete way. By the way, Apel himself (and probably all documentalists with him) strongly advises not to confuse the date of a document with the date of its subject. To say it another way, provided the scribe hadn't got any particular directive, he might have minimized his effort by leaving unchanged the notation when copying music written dozens of years before, though habits had already changed at the time of his work...

A much more obscure question arises in the footnote on page 108. Apel quotes Glarean who, mentionning himself Gaffurius, reports that composers as famous as Josquin would have violated many times our rule C2! One might like to link this fact with the previous one, since alteration was only introduced because of the existence of rule C2 (see our comments under rule A). However, it would be a lot of work to evaluate all the consequences of violations of C2 (at least in some cases?), and since, on the other hand, Apel-Glarean-Gaffurius neither quote any source nor give any example to prove their statement, I feel suddendly unwishing to speculate further more... However, in case some expert, having lost his way, would arrive on this page, I'd be grateful for the gift he would make me by some hint about this point!

Let's play music!

I now invite you to browse the next page, where these dry rules are illustrated by more and more realistic examples involving both divisions tempus and prolatio simultaniously.