Mensuration in renaissance music

Mensuration - the music

Mensuration signs
Groups of smaller values
Two kinds of dots!

Partial imperfection
More about alteration

Modes and their notation

Musical excerpts

Previous page: The rules

The rhythmical range of most pieces exceeds a single division! That's why I had to wait till now to introduce mensuration signs: all of them specify simultaniously the type of the two very important divisions tempus and prolatio.

Mensuration signs
Like nowadays, they appear at the beginning of each part to initialize its mensuration, or later on to change it. It wasn't so unfrequent that the various parts of the same piece showed different mensurations.

sign: The perfect circle (not the letter O) shows tempus perfectum, that is the ternary division of breves into semibreves, while the inner dot shows prolatio major, that is ternary division of semibreves into minims.
A modern transcription without reduction of the values would lead to a 9/2 time signature; the modern dot used in this transcription might be confusing, but it's unavoidable in our modern binary-only notation system: here a perfect breve is transcribed to three tied dotted whole notes, a perfect semibreve to a dotted whole note weighting three half notes.

sign: The perfect circle shows tempus perfectum, but the absence of an inner dot here shows prolatio minor, that is binary division of semibreves.
A modern transcription without reduction of the values would lead to a 3/1 time signature: a perfect breve is transcribed to three tied whole notes, each of them weighting two half notes.

sign: The imperfect circle (which is not the letter C) shows tempus imperfectum (binary breves), and the dot shows prolatio major (ternary semibreves).
A modern transcription without reduction leads to a 6/2 time signature: a perfect semibreve is translated to a dotted whole note.

sign: shows tempus imperfectum and prolatio minor (binary semibreves); indeed, would have there been this only sign, we wouldn't have to write much :-)
A modern transcription with no reduction leads to a 2/1 time signature.

I'm sorry myself because I couldn't draw inner dots larger than a pixel; at least this will perhaps lead you to give still more attention to my text :-)

Groups of smaller values
I'll honour my promise to drop the N n notation used on the previous page. However we'll still have to write note values: now that we work with two divisions simultaniously, we will take as new unit the smaller note of the prolation, that is the minim.
Therefore, here are, for each mensuration, the note values expressed in terms of this new unit:


1	1	1	1
3	2	3	2
9	6	6	4

(We'll avoid, as long as possible, to write the non integer values of shorter notes)

From now we'll have the great pleasure to hear more sophisticated rythms:

For obvious typographical reasons, I'll show note values only, not their pitch which is still not relevant presently.

Well, but... what about our rules, expressed up to now in terms of note figures, if we replace these ones by several values of same total weight? Do these rules still apply, and, if they do, how?

Rule G2 : The total value of a group is never modified by an imperfection or an alteration rule.

And these are good news: for, if we had had to split the change of value to every smaller note of the group, the system would have turned to a real nightmare!

example 2bis (to be compared with example 2) :

= 2 2 2 | 2 4 according to I2.
Thus, as said just above, I1 is not applied to the group of the three first semibreves as a whole!

Rule I4 : A group may cause an imperfection, as might be the case for the note it was replacing.

example 2ter : a)

= 4 1 1 | 6 ; b)

= 4 (1) 1 | 6 ; c)

Here we return to the same evaluation as for the initial example 2: a) the group of two minims is causing imperfection according to rule I1 which overrides I2; b) minim rests act the same way; c) here, values are just halfed (

means perfect semibreves).

example 4bis :

= 4 1 1 | 2 2 1 1 | 1 1 4
The first group of two minims triggers rule I1, the last one rule I2 (see ex 4).

Conversely, when a note or a group does not trigger the application of a rule, then, obviously, it does not either when replaced by smaller values:

example 21 :

taken from Apel page 108.
The groups wheighting three semibreves do not allow to apply rules I1 or I2, since they already did not when written with semibreve figures, thus: 6 | 2 2 2 | 6 | 2 2 1 1 | 6 | 1 2 2 1 | 6 with a nice small syncope near the end of the sequence!

Two kinds of dots!
Working with two different divisons simultaniously, we'll come upon the two sorts of dots; let's summarize about them:

The addition dot can only occur after a binary note, the one just on its left (it's so obvious for us today that we might forget about it!); most often than not, the smaller complementary value appears soon after the dotted note;
the division or perfection dot (same meanings) is only possible in a ternary divison.

example 5bis :

= 4 2 | 1 1 3 1 | 6
A straightforward case: the dot is applied to the binary semibreve of minor prolation, and the complementary value is found immediately after it.

example 22 :

= 2 3 2 2 2 1 | 6 taken from Apel page 117.
Same mensuration, same comment, only the minim which makes the syncope end is found a little further.

Sooner or later, one comes upon the two kinds of dots close to one another in the same passage! Here is an example:

example 23 :

= 6 | 3 1 2 | 2 4 taken from Apel page 117.
The first dot can't be an addition dot: it follows a perfect breve; as a matter of fact, it prevents it from beeing imperfected by the group weighting four semibreves on its right side. Then comes an addtion dot, and then the last breve is imperfected for the reason just mentionned above, but on its left side.

Sometimes the scribe has been kind enough to use two different signs for the two kinds of dots, writing one of them a little higher on the staff, or using writing it as a v-like symbol.
Why hasn't this obvious notation not been used more often? Maybe because it cannot be used whenever the two kinds of dots should be written at the same place... Apel thinks he's found such an example:

example 24 :

= 4 3 1 2 2 | 6 taken from Apel page 117.
To be frank, I'm wondering whether Mr Apel doesn't feel too happy to have found a very subtle case. Let's have a look at it:
In a first step, leaving aside any detail at prolation level, we imagine the second and third notes as semibreves, and we notice that the first breve, followed by a group summing up to four semibreves, is imperfected on the right by one semibreve. This can also be seen from our example 23 taken from... Apel himself, a little lower on the same page! For the division dot here prevented imperfection of the first breve.
In the second step, we take in account the details omitted up to now, and then the additive function of the dot following the semibreve is enough to get the result; thus I can't see any need to say that "the dot functions as a punctus divisionis with respect to the tempus by marking off a group of perfection, and as a punctus additionis with respect to the prolatio by adding a half to the value of the S".
Indeed, there is something strange in this sequence: it seems that the value of the breve is diminished by a quantity smaller than the cause of imperfection! But this just means - against Apel's opinion - that division does not occur at the same place as the addition dot: for, this latter one defines 3 and 1 values, while their total 4 is splitted into 2 and 2 belonging to the left and right perfections!

Transcribed to modern notation, this example shows a tie crossing the bar - here compared to a perfection border.

Later we'll come upon other cases in which this two step procedure will be fruitful.

Partial imperfection
This is a slightly new event, which may happen only within two or more simultanious division levels! To understand it, let's first come back for a while to the imperfectio ad totum process studied up to now, that we can now make more explicit:

The whole breve can be divided into three parts; one of these parts is replaced by a note or a group weighting the third of a perfection, resulting in 4 2.

Here is what Blockland de Monfort told about that on his page 35:

"When, before a note being perfect according to the division, one finds a neighbouring smaller note, or an equivalent value, equal to its third part, and exceding the number three: as if, in perfect major mode, a superfluous long or its value is found to be superfluous before or after a maxima (like are two or three breves, or a long rest), then one must substract from the longer note a value equal to the smaller one, so that nothing remains superfluous over the mentionned number [three]".
"The same in perfect minor mode when, before or after some long one finds a superfluous breve or its value: and in perfect time before or after some breve, [one finds] a semibreve or its value: and in perfect prolation before or after some semibreve, a minim or its value."

Unfortunately, I feel quite unable to translate the early flavor of this text. Some of you might like to taste it on the French version of the page.
Next part

Lets now consider a long followed by a semibreve, in perfect time. The long might be binary or ternary - I choose it to be binary in this example:

Let's imagine we divide the long in two halves, each of them being equal to a perfect breve, and let's apply imperfection rule to the second part: the third semibreve of this breve is replaced by the semibreve following the long (or by an equivalent group), thus a result of 10 2.

Of course this sort of imperfection can also happen on the left side:

= 2 10.

Here is the explanation given by Blockland, page 36:

"If the smaller note, or its value, can't be equal to a third of the longer one, but rather to some other perfect part of it, then it will make imperfect, not the longer note (it can't in no way, since it's not equal to its third) - as if in major and minor perfect mode one finds a breve or its value before or after a maxima - but only the next long included in it, to which it can be equal. It happens so when semibreves are found after longs in perfect [minor] mode and time: and also when minims are found after breves in perfect time and prolation."

example 25 : a)

= 8 2 2 ; b)

= 2 8 2 (Apel page 111)
a) here one removes from the long the last two semibreves of its second half (it couldn't get smaller because of rule C0) ;
b) imperfection on left and right sides for each half of the long.

Let's notice that the same process remains valid when the long is ternary, provided the divison at lower level is also ternary - since this division is the one in which the imperfection mechanism will happen. Therefore, as stated by Blockland, things happen the same way in such mensurations like:

(the longer note is here binary, like it was just above)

(here it is ternary)

Let's not conclude that the system is foolishly complicated: we shouldn't forget that, in the absence of ties, we need some way to build complex values, and this substractive process is, after all, quite natural... Here is what it makes possible from now on:

example 26 :

(Apel page 120)
First two notes: imperfectio ad partem propinquam on the right side; two next ones: imperfectio ad totum on the right; eigth, nineth, tenth notes : imperfectio ad partem propinquam on left and right sides; four next to last notes : imperfectio ad totum on the right, and then, after the division dot, on the left (without a dot the second minim would in principle be altered at prolatio level).
Result: 5 1 | 2 1 | 1 1 1 | 1 4 1 | 2 1 | 1 2 | 6 (under this mensuration, our borders show perfections at prolatio level, each of them weighting three minims).

example 27 :

= 1 7 1 | 9 (Apel page 122)
AFAIK, the division dot is here more a gift from the scribe, than really necessary (for the imperfection of the last breve on its left side has a low precedence, according to rule I3).

In such cases, the value of the breve can go down to four minims, but never less, since it could then be written as a perfect semibreve (and this was, after all, the meaning of our rule C0 - taking in account the change of units which happened in between).

example 28 :

= 1 (1) 1 1 4 1/2 1/2 | 9 (Apel page 122)
A new example showing that a rest may cause imperfection exactly like does the corresponding note. In the absence of a dot, one must prefer imperfection (here partial) of the first breve by the two semiminims on the right, rather than imperfection of the second breve by the same notes on the left.

example 29 :

= 4 1/2 1/2 3 1 | 9 (Apel page 122)
A rather subtle example! Let's live dangerously, here is my understanding:
The first division dot prevents the semibreve to be imperfected by the following minim at prolatio level (Under ternary division, this dot can in no way be an addition dot - always applied to... binary notes! This approach, though leading sometimes - like here - to the right result, has been pointed out as quite wrong already by Yssandon [folio 20 verso] in 1582!).
The second dot makes things more straightforward at tempus level: musically speaking, it would be quite possible that only a part of the group in the middle would imperfect the first breve, and the other part the second one.

Remote part

I can't omit to tell you about some sort of imperfection involving a still shorter part of the longer note, as shown in the following sequences:

, or

etc... ;

(the first note is a maxima); etc....

To be honest, these cases of imperfectio ad partem remotam are quite unfrequent: I haven't personnally found any, Blockland de Montfort doesn't mention them, and even Apel fails to give an example, for himself cleverly demonstrates that the sequence given on his page 112 is, indeed, a case of imperfectio ad partem propinquam.
However, one must remember that this may happen, that's necessary in order to get a correct transcription in such a case...

A question of vocabulary

We've told that the divison in which partial imperfection occurs must be ternary. However, Machaut, already, seems to have been the first one violating this principle:

(transcribed to white notation from an example in Apel page 345).

Please notice the mensuration (no dot) : here imperfectio ad partem propinquam happens in the division prolatio minor, hence one half of a binary semibreve is substracted from the breve! Indeed one understands that the process works as well in this case, however one should rather name it by such a phrase like "partial replacement" rather than imperfection - since the latter word has no meaning outside a ternary division.

More about alteration
In principle, alteration must be used only when rule C2 forbids to write the rhythm 1 2 in the standard way. Now, let's notice that, obviously, the similis ante similem perfecta principle is only relevant in the presence of note or rest figures, but not before groups of smaller values of the same total. Therefore a group cannot cause alteration of the previous note:

...

= ... 2 2 3 1 2 to be compared with

= 2 4 | 6

I was quite amazed to read this passage on Blockland de Monfort's page 38:
"When you'll see two semibreve rests hanging from the same staff line, with two semibreves and a breve, then the second one is altered, and weights two semibreves : for the above mentionned rests under the same line are counted for a breve [sic], and to avoid alteration one must write them under different staff lines.

The more I read this passage, the more obscure it seems to me!
First, how can one say that two semibreve rests are equal to a breve under... perfect mensuration??? Maybe Blockland, taking this sequence from a real example, had in mind some other semibreve coming on the right, that he might have included when counting, but forgotten to write in his book? We have to find an explanation for this absurd statement...
Besides that, I notice that it seems incorrect to talk of alteration here, since the rhythm that it is supposed to show could as well be obtained in a natural way

�

I do think that nobody would say here that the second breve must be perfect because of the similis ante similem perfecta principle, since no similar figure can be found on its right side...
In case you would understand any part of this passage, you would be kind to

!
Historical evolution

As mentionned on the previous page, imperfection must often be preferred to alteration even in the absence of a division dot, in such sequences like the following ones:

a)

; b)

According to Apel, it's still more often true in the case b), with the argument that the rhythm arising from imperfection is "musically more natural" in binary time. I'm not quite happy with this explanation: first, this author rejects this sort of argument when used by somebody else (see a footnote on page 130), secondly I don't see any reason why the argument couldn't be applied as well to case a) when the long is binary.
Perhaps could we better understand this situation if we remember that the minim was introduced later, that is when alteration had already begun to be used less frequently? One more evidence: such a sequence occurring in a ligature must be transcribed by alteration, but we know that no minim has ever be included in a ligature...

The mensuration

Not that it's really a special theoretical case, but the presence of two ternary divisions has interesting consequences:

Division dots are quite frequent, since they may happen at both levels: prolatio and tempus ;
Imperfection and alteration may also happen at both levels, hence a note can be altered at tempus level, and then imperfected at prolatio level.

On the other hand, addition dots are unfrequent, for they can be applied only to minims or shorter notes.

I suggest that the following examples will be better analysed in two steps: considering first the tempus level, omitting notes shorter than a semibreve, and, then only, considering the prolatio level while adding them back:

example 30 :

= 9 | 3 5 1 | 9 (Apel page 122)
The second semibreve is altered at tempus level, and then imperfected by the minim on its right side at prolatio level.

example 31 :

= 6 3 | 2 1 5 1/2 1/2 | 9 (Apel page 122)
The semibreve after the division dot is imperfected on the right at prolatio level. The following one is doubled by alteration at tempus level, and then imperfected at prolatio level by the two semiminims on its right side.

And now comes the crowning piece, involving division dots and alteration at both levels simultaniously!

example 32 :

= 8 1 | 3 1 4 1 | 9 (Apel page 122)
On the first breve is applied an imperfectio ad partem propinquam ; next comes a division dot at tempus level. The first semibreve remains perfect despite the minim on its right side because of a second division dot, this one acting at prolatio level! The second semibreve is doubled by alteration at tempus level, then imperfected at prolatio level, on its left and right sides.

example 33 :

= ? ? ? | 1 2 1 4 1/2 1/2 | 9 (Apel page 122)
The first dot acts at tempus level: followed by a group weighting three semibreves, the first breve could have stayed perfect; but now it's the last semibreve of the group (before the semiminims) which is doubled by alteration. The second dot, acting at prolatio level, makes the last minim to lay in the same group as the next semibreve, and this has two consequences: the second minim of the group in the middle (between the dots) is altered at prolatio level; and the last semibreve, already altered at tempus level, is imperfected at prolatio level on its left and right sides.
I don't find obvious the transcription of the beginning of the sequence! Apel here argues that the second minim should be altered at prolatio level, leading to the result 6 1 2. However, if such was the rhythm that the composer had in mind, it could have been written

since it's not followed by any semibreve figure. In such a case it's in principle not correct to use alteration! Nothing helps here to choose between the scribe's or the musicologist's points of view, since the latter doesn't give any argument to support his view: often one can take in account the difference in value to choose between two such options; but that's not true here, since this difference obtained at prolatio level will in both cases be substracted from the breve because of imperfection at tempus level, so that the transcription 7 1 1, summing up to the same total as 6 1 2, seems to me quite possible as well...

This 7 1 1 rhythm is not as strange as it might appear at first sight: let's imagine, in modern 9/2 time signature, a dotted square tied with a half note, then a breathe, and then an upbeat made of two other half notes...

Modes and their notation
Division of longs into breves is called modus longarum, or modus minor, or just modus.
Division of maximas into longs, which appeared a little later, is called modus maximorum, or modus major, or maximodus.

When modes are marked perfect (see below), they are to be understood exactly like other ternary divisions; this must be taken in account in order to get an exact transcription: longs and/or maximas may be imperfected, and longs may be altered before a maxima (Apel gives on his page 124 a striking example involving two embedded alterations!).

On the other hand, in the absence of any indication, modes are supposed to be binary. Happily enough, this happens quite often, and that's why - beeing quite naive - I've been able to transcribe without problem several documents dated from around 1500: for, reading again my first ideas about mode signs, I realize that they were optimistic, that is, simpler that the real story:

the notation of modes has been changing with time passing;
this change has been smooth enough to let two systems live together;
even at a given time, serious fights occured between such systems of signs (not an exageration!) ;
the winning system could nevertheless be read in various ways, and had some relationship with the notation of proportions, this latter one being itself quite complex and highly controversial.

We'll only try to give the mainline of this complex adventure, the details of which you'll find for instance in Busse Berger.

Notation by mode rests

This notation, born in the middle of the 1400s, takes its roots in a simple idea: modes are shown by a few rests written at the beginning of the staff - before the mensuration sign, their meaning depending on their length and on the way they're grouped. However the idea has been implemented in various ways:

An overwhelming majority of people agreed with Tinctoris, specially in Italy: the height of the rests - filling two or three staff lines - shows how many breves are included in a long (that is, the nature of minor mode), and the number of such rests written one aside the other shows how many longs are included in a maxima (hence major mode). This is the system mentionned by Apel on his page 124:


Perfect major mode Perfect minor mode	Perfect major mode Imperfect minor mode	Imperfect major mode Perfect minor mode	Imperfect major mode Imperfect minor mode
written as 3-3 inside text below	written as 3-2 inside text below	written as 2-3 inside text below	written as 2-2 inside text below

(The last line shows the notation that we'll be using inside the current paragraph, for typographical reasons)
Another method was advocated by Gaffurius and a few authors, mostly German ones. This notation, explained by Blockland de Monfort on his pages 26 and 27, tries to keep minimum the amount of information written on the staff:

shows perfect minor mode; in the absence of this sign, the minor mode is binary.

shows perfect major mode; in the absence of this sign, the major mode is binary.

This is an illogical notation, as Gaffurius himself eventually admitted, when trying - clumsily - to reconciliate it with Tinctoris's notation. Indeed, it only has a misleading appearance of simplicity: obviously, a single group of rest(s) is not enough to notate the four combinations 3-3, 2-2, 3-2 and 2-3...

Whichever of its variants beeing used, notation by mode rests had enough drawbacks to make it disappear near the end of the 1400s already. First, it didn't tell anything about time and prolation; thus a mensuration sign had to be added, leading to a verbose notation. Secondly, and above all, for obvious reasons it couldn't be used anywhere inside a piece of music to show a change of mode.

The Modus cum tempore notation

It gradually replaced the previous notation, since it was able to show the nature of each division level everytime it was necessary. Here are a few variants:

full version

32 for instance shows the perfect major mode (whole circle), perfect minor mode, imperfect time, and major prolation.

23 : perfect major mode, imperfect minor mode, perfect time, minor prolation (no dot).
etc...

abbreviated version (frequent)

2 for instance shows perfect minor mode (whole circle), imperfect time, and major prolation.

3 : imperfect minor mode, perfect time, minor prolation.
etc...

reversed version

32 shows imperfect time (half circle), perfect minor mode, imperfect major mode, major prolation.
etc...
Those supporting this last version - a few ones indeed - were arguing that it was a better idea to keep the circle always show the tempus (and, personnally, I do find illogical the first versions which write the prolation dot inside the circle attached to the mode).

Only the first two versions have been explained - quite clearly - by Blockland (pages 31-32), since they seem to him "quite enough for beginners". Yssandon (folios 14 verso to 16 recto) says more about this subject, but his comments are rather confused to my taste.

To make the subject still spicier, this system shared several signs with the notation of proportions; BTW, this is why the meaning of the circle shifted from indication of time to indication of mode: when some passage had to be played twice faster (diminution two), it wasn't notated with smaller values, but rather with the same ones preceded by such a symbol like

2, for instance, if perfect time was to be diminished. But then semibreves were played as fast as were minims under the original mensuration, breves like were semibreves, longs like were breves (thus the shift time => minor mode), etc. Furthermore, double diminution of perfect time or triple diminution of imperfect time brought up delicate questions about the nature of the target mensuration and about to which note one should now attach the tactus...

Believe me, the situation was quite confused, here is a comment by Yssandon (folio 18 verso) : "Diminution can be done in four ways, the first one shown by a binary number, shown like this

2, thus, though these signs are those of minor mode they can also show double proportion. I've already written in a separate chapter how reserved a musician must be when using these confused signs. [...]". And Blockland wasn't more encouraging, who so ended his chapter about proportions (folio 47): "There are some other sorts of proportions, about which we won't tell, partly because they are difficult, and partly because they're not so frequently used by modern authors".

Though useful to my present subject, these quotations from the end of the 1500s shouldn't be taken too seriously: obviously, nearly one century after Ockeghem's death, these authors are looking back at an obsolete past.

Thus, we are now facing a paradoxical situation: music nowadays helps scholars to understand mode and proportions signs, even though these signs were formerly designed to command its performance!

Long and maxima rests

Their notation is quite simple and natural, though in relationship with the previous one: a long rest is made of a vertical stack of breve rests, and a maxima rest is an horizontal sequence of long rests; the number of rests in the stack or in the sequence tells about the perfect or imperfect nature of the note:

perfect or imperfect long rests.

imperfect maxima rest under perfect minor mode (that is, perfect long).

imperfect maxima rest under imperfect minor mode.

Why aren't we talking about perfect maxima rests (three rests written aside one another)? Because it is quite unfrequent to come upon an isolated perfect maxima; it only occurs inside groups the total value of which is equal to a maxima, and the same is true for the maxima rest:

;

(imperfect minor mode) ;

(perfect minor mode) ; etc...

Last but not least: rests are written aside one another in such a way they don't contradict the mode. For instance, under imperfect major mode, one will come upon:

etc...

For, three rests at same height would have suggested perfect major mode. Just as, under imperfect minor mode, one will write:

but not

which would show a perfect long.

Inference of the mode

Often the music itself allows to deduce the nature of a division when mensuration or mode signs are absent or ambiguous.

This paragraph is a partial synthesis of pages 346 to 349 in Apel, dealing with a previous period of time in which mensuration signs were quite often omitted.
Let's take minor mode as an example:

Proofs of a perfect long
coming upon

; a dot not followed by any smaller complementary value (which may lay rather far sometimes!)

Hints of a perfect long
every

is preceded or followed by a

; many

groups

Proofs of an imperfect long
coming upon

Hints of an imperfect long
every

group is sooner or later followed by a

(syncope)

You'll find on this separate page some explanations by early authors, Blockland et Yssandon (about black notes mentionned by these authors, see the page on coloration).

Musical excerpts
At last, let's taste music and... simplicity! For, you'll see that I've told about much more details than those I've personnally encountered up to now...

This page being already bigger than it should be, the examples will appear in separate windows:

Ockeghem :	Kyrie of the mass Ecce ancilla Domini, Cantus.
Festa :	Hymn Conditor alme siderum, Tenor.
Isaac :	Song Palle, palle, Cantus & Bassus.
Busnois :	Kyrie of the mass O Crux lignum, Tenor.
Barbireau :	Kyrie of the mass Virgo parens Christi, Cantus.

Had you omitted to study ligatures, it might be the right moment to do so by clicking here...
What... you have been studying them, but you don't remember my little diagram? Ok, here it is !

And, please don't refrain from the pleasure to listen to some of these pieces which have been transcribed!

Last, if you've walked with me till here: thank your for your patience!