Thomas Morley, in the preface to his A Plain and Easy Introduction to Practical Music of 1597, says the following:

But as concerning the book itself, if I had before I began it imagined half the pains and labour which it cost me, I would sooner have been persuaded to anything than to have taken in hand such a tedious piece of work, like unto a great sea, which the further I entered into the more I saw before me unpassed, so that at length, despairing ever to make an end (seeing that grow so big in mine hands which I thought to have shut up in two or three sheets of paper) I laid it aside in full determination to have proceeded no further, but to have left it off as shamefully as it was foolishly begun. But then, being admonished by some of my friends that it were a pity to lose the fruits of the employment of so many good hours, and how justly I should be condemned of ignorant presumption in taking that in hand which I could not perform if I did not go forward, I resolved to endure whatsoever pain, labour, loss of time and expense and what not, rather than to leave that unbrought to an end in the which I was so far engulfed.

Taking, therefore, those precepts which being a child I learned, and laying them together in order, I began to compare them with some other of the same kind set down by some late writers. But then was I in a worse case than before, for I found such diversity betwixt them that I knew not which part said truest or whom I might best believe. Then was I forced to run to the works of many, both strangers and Englishmen (whose labours, together with their names, had been buried with me in perpetual oblivion if it had not been for this occasion) for a solution and clearing of my doubt. But to my great grief then did I see the most part of mine own precepts false and easy to be confuted by the works of Taverner, Fayrfax, Cooper, and infinite more, whose names it would be too tedious to set down in this place; but what labour it was to tumble, toss, and search so many books, and with what toil and weariness I was enforced to compare the parts for trying out the value of some notes (spending whole days, yea and many times weeks for the demonstration of one example which one would have thought might in a moment have been set down), I leave to thy discretion to consider, and none can fully understand but he who hath had or shall have occasion to do the like.^[1]

If Morley found the situation of tactus, rhythm, and mensuration confused in his day, so much the greater is the confusion in our day, both because of the remoteness of the times and the additional contradictions of modern authorities with their problems of transcription. For there is probably no other aspect of Renaissance music in which there is more disagreement among modern scholars than in the matter of tactus and mensuration and their effect on meter, rhythm, and proper methods of transcription. Some scholars seem to ignore the theory of tactus completely in their transcriptions, while others have become slaves to a method of transcription which is tied to a single unvarying concept of the tactus. Some regard the tactus as the equivalent of one modern beat, some as two. Some scholars insist on what they regard as an objective, even mechanical, method of transferring mensural systems into specific meters, whereas others insist that mensuration has nothing to do with meter, but only with relative note values. Some employ bar lines regularly according to a strict interpretation of the meter according to the mensuration sign; others employ bar lines irregularly according to their own interpretation of the rhythmic structure of the music. Some use solid bar lines, some use dotted, some use both solid and dotted bar lines. Some put the bar lines through the staff, others between the staves, and some do a mixture. Some regard the use of ties as a gross misinterpretation of the original rhythmic concept, whereas others regard them as one of the blessings of modern notation. Some scholars have transcribed Renaissance music without reduction in note values. Most modern scholars agree that a reduction of note values is desirable for modern use, but they disagree as to the scale of reduction.

It is obviously impossible even to attempt a resolution of these many conflicting opinions within the confines of an introductory paper such as this. The purpose of this paper is, rather, to give a highly condensed preview of a more complete and thorough study in which the writer is presently engaged, with special attention to the practical function of the tactus in the conducting of Renaissance music. Almost four years have passed since this paper was presented in its original form at the Valparaiso University Church Music Seminar. Since that time the progress of these studies^[2] has made it desirable to revise some portions considerably, but this presentation should still be regarded as a preliminary report, not as a final study. A full explanation of the methods of transcription employed and the full documentation of many complicated and controversial issues will have to await the completed study.

The tactus is a method of conducting by simple down-and-up movements of the hand or finger (or even the foot, in the case of a player who needs a method of keeping time while his hands are otherwise occupied). The concept is often referred to in English writings by the word stroke, in German by the word Schlag (or sometimes Taktschlag), in Italian by battuta. The Latin word tactus is often found shortened in other languages to simply tact. The term tactus always refers to the complete down-and-up movement (). Thus the tactus is a compound unit which may be separated into two components or half-tactus units.

The tactus is, of course, inextricably bound up with the notation and with the mensural theory of the 15th and 16th centuries. One aspect of mensural theory must be reviewed here as a necessary preliminary to understanding the interrelationships among tactus, rhythm, and meter—namely, the mensuration schemes that form the basis of the rhythmic systems of the 14th to the 16th century.

A mensuration scheme is a hierarchical arrangement of note values involving five different levels of notes: the maxima, the longa, the breve, the semibreve, and the minima. These five levels of note values are organized into four levels of rhythm (“rhythm” here referring to the relationship between two adjacent levels of note values). The levels of rhythm are expressed in the four terms (1) maximodus, which refers to the relation between maxima and longa; (2) modus, which refers to the relation between longa and breve; (3) tempus, which refers to the relation between breve and semibreve; and (4) prolatio (or prolation), which refers to the relation between semibreve and minima. Each level of rhythm may be either perfect (that is, by threes), or imperfect (that is, by twos).

Altogether there are 16 possible combinations of perfection and imperfection at the four levels of rhythm, producing 16 possible mensuration schemes. These 16 mensuration schemes are listed by many theorists of the 15th and 16th centuries. Tinctoris even refers to them as “species of composition” and deals with each one separately and with an accompanying musical example.^[3]

Renaissance theorists have elaborate and often conflicting terminology for these 16 mensuration schemes. For the sake of brevity it is convenient today to refer to them by means of a system of two Roman and two Arabic numerals.^[4] For example, II-III-3-2 means imperfect maximodus, perfect modus, perfect tempus, and imperfect prolation. Reference to any two or three of these levels can be made by the use of the appropriate numerals, the arrangement of Roman or Arabic numerals indicating the levels intended (for example, II-III, or III-3, or 2-3). A single level will be identified by its proper term without recourse to the system of abbreviations.

Unfortunately, despite the importance attached to these “species of composition” by the older theorists, modern scholars have made no effort to reproduce these mensuration schemes in modern transcriptions, and worse, they have almost invariably obliterated in their transcriptions many of the bits of evidence in the original notation that would enable the reader to reconstruct the mensuration scheme for himself. I shall here propose a system of transcription that will enable the modern reader to identify the original mensuration schemes by means of the “modern” time signatures.

Table I shows a list of time signatures that express unequivocally (1) the rhythmic hierarchy analogous to the hierarchy found in the original mensuration scheme, and (2) the ratio of reduction in note values used in the transcription.^[5] A measure in the transcriptions corresponds to a maxima in the original notation. The time signatures on the staff show the constitution of the measure corresponding to the constitution of the maxima of the original notation. Since these measures are often rather long for ready comprehension, they are usually subdivided by dotted bar lines into smaller groupings. The time signatures for these dotted subdivisions are given above the staff in smaller figures.

Click here for Table I.

How regularly do these mensuration schemes work out in the actual musical works of the period? As might be supposed, a high degree of variation is found between complete regularity in some pieces and much irregularity in others. However, despite the irregularities that arise in the practical application of the mensuration schemes (and the corresponding modern measures), there is ample evidence to support the contention that these mensuration schemes are fundamental to the rhythmic system of the 14th to the 16th century and should be the point of departure for a rhythmic analysis and transcription of the music.

What is the nature of the irregularities that do occur? These can be briefly outlined here as follows:

1. Syncopation—the displacement of any note value from its normal position in the rhythmic structure. A syncopation may affect any part of a mensuration scheme or the entire mensuration scheme (syncopation of the maxima). The former may cause a displacement of dotted bar lines in the transcription; the latter will cause a displacement of an entire measure in the transcription. (Such a concept of syncopation as a displacement entails the idea that the displaced note or rhythmic figure retains its identity or “integrity.” Since modern ties tend to destroy this identity, they are avoided in the transcriptions as much as possible.)

2. Redistribution of rhythmic values and/or figures. The most common of these is the hemiolia, which may be described as a redistribution of 2 x 3 into 3 x 2. Like syncopation, the hemiolia may take place at any rhythmic level, and if it affects the maxima, it will create a different-sized measure in the transcription. The hemiolia may be effected in the original notation by means of coloration (white or red notes instead of black, black notes instead of white). Coloration is indicated in the transcription by a bracket above colored notes ( or ). Other types of redistribution may be effected by means of syncopation. These again may be internal (such as a redistribution of a 12/2 measure into 3 x 4, or of a (3 x 4/2) measure into a 4 x 3 grouping, or the oft-remarked redistribution of 8 into 3+3+2, etc.),^[6]or they may affect the measure structure. Thus a period of five maximae that would ordinarily be transcribed as 5 measures of 4/2 (5 x 4/2) may actually show a rhythmic structure of 4 x 5/2. Syncopation often combines with coloration to produce regular or irregular redistributions in the rhythm.

3. Fractional mensuration schemes. There are times when a mensuration scheme is incomplete. It may consist of only a single longa or perhaps only a single breve, or it may be truncated at any time before it has run its full course. The most obvious indication of such fractional schemes in the original notation is the appearance of a new mensuration sign that forces the curtailment of the previous mensuration scheme and begins a new one.

There are likewise problems that arise in regard to an interpretation of the original notation and its significance. Sometimes there are discrepancies between the mensuration sign of the original notation and the actual mensuration scheme operating in the music. Discrepancies may also exist between mensuration and meter, so that, for example, a triple meter may be found expressed in a duple mensuration. (Cf., for example, Fig. 3, below.)

How is the tactus related to these mensuration schemes? In the development of mensural notation from the early 14th century to the end of the 16th century, three basic systems can be discerned in the application of the tactus. In the ensuing discussion these three systems are designated as Systems I, II, and III, or the “preclassical system,” the “classical system,” and the “postclassical system.” The three systems may be roughly distinguished from one another by the kind of note value that receives the tactus in integer valor (the normal or “whole” time value of the notes): the imperfect breve in System I, the imperfect semibreve in System II, and the minima in System III. The “preclassical” system is so designated because it is not clearly described by the theorists, but must be reconstructed by deduction and by means of retrospective references by later writers. It is found primarily from 1300 to approximately 1450, with remnants carrying over to 1600 and later. The “classical system” is so named because it is clearly defined, precise, and predictable in its application, and it is the system in vogue at the time theorists begin describing the tactus. Examples may be found from the early 15th century to the early 16th century, with the basic principles retaining some validity throughout the 16th century and to a lesser extent even in the 17th century and later. The “postclassical system” is so named because it represents a breakdown and disintegration of the classical principles. It is not so much a unified system as it is a convenient grouping of different practices which can be brought into logical relation with one another. The unifying principle is simply that in integer valor the minima receives the tactus. Its earliest manifestation is the use (in conjunction with the classical system) of perfect prolation as augmentation, forming in effect a new integer valor which was itself subject to diminution and augmentation.

Some scholars insist that the value of the tactus should be made a whole note in the modern transcription; most modern transcribers make the half note the tactus; a few modern scholars prefer the quarter note. I believe that all three of these modern note values can be justified under differing circumstances. It would seem logical to transcribe music in the preclassical system of the tactus in a 4:1 ratio and music in the classical system in a 2:1 ratio, producing in both cases a half-note tactus in the modern notation. The problem encountered in following this procedure is that ambiguities exist between the two systems that cannot always be resolved with certainty. Furthermore, both the preclassical and the classical system were used in both black and white mensural notation, though it can be said that the preclassical system was more “at home” in black notation and that the classical system was more “at home” in white notation. Inasmuch as both of these systems of tactus could be notated in either black or white notation, it would seem desirable to have the difference in notation reflected in the modern transcription. Considering, then, the ambiguities between the two systems of the tactus and considering the desirability of reflecting the difference between black and white notation in the modern transcription, I here propose the following system of ratios for the transcription of mensural notation:

1. Transcribe black notation in a 4:1 ratio ( or ) for integer valor. In the transcription, then, the preclassical system of tactus will give the half note the tactus (dotted half note if the prolation is perfect); the classical system will give the quarter note the tactus.

2. Transcribe white notation in a 2:1 ratio for integer valor in Systems I and II ( or ). In the transcription, the preclassical system of tactus will give the whole note the tactus (dotted whole note if the prolation is perfect); the classical system will give the half note the tactus.

3. Transcribe white notation without reduction in note values (1:1 ratio) when it can be ascertained that System III is in effect, most notably in perfect prolation as augmentation in conjunction with the classical system. The half note will receive the tactus.

Admittedly some problems will be encountered in carrying out this scheme of ratios for transcription, but such a scheme has the advantage of giving a logical unity to the transcription procedure for the entire body of music in mensural notation from the 14th to the 16th century. Table II summarizes this proposal, showing the ratios of transcription for integer valor, duple diminution, and duple augmentation in these various situations and showing the resultant note values that will receive the tactus.

One of the most significant features of the tactus method of conducting is that it makes no difference to the manner of conducting whether the rhythm is duple or triple on any level. All meters are conducted alike. This feature is illustrated in Table III, which shows the classical system of tactus applied to the four possible combinations of tempus and prolation, A in mensural notation and B in modern notation (2:1 ratio of reduction). (It is unnecessary to illustrate this application of the tactus with the various combinations of modus and maximodus, since these involve only varying extensions of the partial schemes shown in Table III.) It will be noted that in the classical system the constant unit in all mensuration schemes is the minima, which in integer valor always receives the half tactus. In transcription from white notation, then, the quarter note gets the half tactus in all the time signatures found in Column C of Table I.

As strange as this “rhythmic neutrality” may seem to the modern conductor, it is nevertheless a basic and integral part of the system of tactus and is the only method which permits the many counterrhythms and simultaneously conflicting mensuration schemes that characterize the music of the 15th century. A little experience with the system also shows that the conducting of triple meters, such as in perfect prolation, with a binary tactus is not actually as disadvantageous as it may at first appear. For example, the oft-encountered hemiolia fits easily into the binary tactus as follows:

Click for example of hemiolia in binary tactus.

Click here for Table II.

Click here for Table III.

The Kyrie I of the Missa Prolationum of Ockeghem^[7] may serve as an illustration of the simultaneous use of four different mensuration schemes in the classical system of tactus. It may be found in Fig. 1 in the method of transcription here proposed, that transfers the original mensuration schemes into modern time signatures in accordance with the relationships given in Table I.

Click here for Figure 1.

In diminution all the note values are halved. Fig. 2, an excerpt from the Choralis Constantinus by Heinrich Isaac,^[8] illustrates the combining of integer valor with duple and quadruple diminution. Fig. 3 illustrates the simultaneous use of duple and triple diminution in a famous mensuration canon by Josquin.^[9] Note that here the tactus is subdivided by twos in the lowest voice and by threes in the highest voice.

Click here for Figure 2.

Click here for Figure 3.

Some indication of the degree of variability in the application of the tactus in the different systems may be had by a consideration of the ways in which perfect prolation can be related to the tactus. Four such different relationships are indicated in Table IV. In the 14th century there is little doubt that in perfect prolation the semibreve was worth a half tactus (Table IV, A). In some sources of the 15th and 16th centuries, perfect prolation is shown with the semibreve equal to the full tactus (Table IV, B). In the classical system, as we have seen, the minima is worth a half tactus (Table IV, C) And finally, in the closing years of the 15th and throughout the 16th century it became common practice to treat perfect prolation as a form of augmentation, in which the minima is equal to a tactus (Table IV, D). This tactus alla minima interpretation becomes a new integer valor for perfect prolation, which was then again subject to diminution and augmentation.

Click here for Table IV.

In the 16th century the problem of the tactus becomes further complicated by the presence of three kinds of tactus described by the theorists. But before the relationship between these three kinds of tactus and the rhythm of the music can be adequately analyzed, it is necessary to define some terms.

The concepts of “beat,” “pulse,” and “rhythmic unit” are not always clearly defined in modern discussions of rhythm. For the purpose of the ensuing analysis the following definitions are established:

A “rhythmic unit” is a note value or its equivalent in any level in the rhythmic hierarchy. It is, therefore, a flexible term, which can be applied to long or short durations or metric structures.

A “beat” is a rhythmic unit which is capable of a consistent twofold division in the rhythmic hierarchy.

A “pulse” is the first division of the beat. It is therefore capable of consistent division into a still lower level of note values.

The division of the pulse is referred to as an “elementary unit.” It is not capable of further consistent division in the musical styles under consideration in this paper, but isolated elementary units may be subdivided into two smaller note values. Since then, these smaller note values occur only in pairs, they are not regarded as an essential part of the rhythmic hierarchy. They are in the nature of tiny melodic flourishes and make no appreciable contribution to the metric structure.

When these concepts are related to the mensuration schemes as employed in the musical sources from the 14th to the 16th centuries, it will be found that any of the note values may represent the beat in the rhythmic organization: the maxima and longa may represent the beat, but only in diminution; the breve represents the beat in 14th-century music and in much of the music of the 15th century; the semibreve represents the beat in much 15th- and 16th-century music; and the minima represents the beat in perfect prolation and other forms of augmentation of the 15th and 16th centuries and in the so-called “black notation” of madrigals in the latter half of the 16th century.

It should be emphasized that the above definitions relate to the rhythmic organization of the music and are independent of the method of conducting. Such a distinction is a necessary preliminary to an adequate understanding of the relation of the tactus to the rhythmic structure of the music. In the simplest and most obvious relation between tactus and beat, the tactus equals the beat. However, this is by no means always true, and other relationships are possible. For example, in the classical system of the tactus, the tactus equals the beat only when the prolation is imperfect. In perfect prolation, since the pulses (represented by the minimae) are grouped by threes, the beat equals 1½ tactus, as illustrated in Table V, A. A similar relation occurs in tempus perfectum diminutio. (See Table V, B)

Click here for Table V.

In Table V, A and B, the relation between beat and tactus is the same in spite of the fact that in A the notation is in integer valor and in B it is in diminution. Here the diminution in note values is compensated for by the use of correspondingly larger note values. In the latter part of the 15th century, however, it became common practice to use diminution as a device to obtain a faster movement, so that there were now as many notes within a half tactus as had previously appeared in the entire tactus (see Table V, C). It will be noted that now the half tactus is the beat. To the practical musician this situation soon suggested the possibility of doubling the speed of the tactus to make it easier to accommodate the greater number of notes. This possibility led to the employment of two kinds of tactus—the greater and the lesser, a situation first clearly documented by Ornithoparcus in 1519:

Tact is three-fold, the greater, the lesser, and the proportionate. The greater is a Measure made by a slow, and as it were reciprocall motion. The writers call this Tact the whole, or totall Tact. And, because it is the true tact of all Songs, it comprehends in his motion a Semibreefe not diminished: or a Breefe diminished in a duple.

The lesser Tact, is the halfe of the greater, which they call a Semitact. Because it measures by it[s] motion a Semibreefe, diminished in a duple: this is allowed of onely by the vnlearned.^[10]

(Ornithoparcus’ third kind of tact will be discussed later.)

Table VI shows the two kinds of tactus applied to diminutions of the four partial mensuration schemes previously shown in Table III in integer valor. The decision as to whether the greater or the lesser tactus should be employed is a decision that lies in the hands of the performer. The choice exists, however, only in those cases where the half tactus of the greater tact is a full beat. It is obvious that the lesser tactus is easier for beginners (see Ornithoparcus’ reference, above, to the “unlearned”). It is also clear that there is more danger of too slow and stodgy a tempo when the lesser tactus is used.

The greater and the lesser tactus, then, theoretically stand in a ratio of 2:1. However, if the greater tactus is maintained in accordance with the strict classical rules, despite the shift in the level of the beat to the semibreve through the introduction of smaller note values in diminution, there is almost inevitably going to be a slowing down of the tactus to accommodate the greater number of notes. Thus will mean a faster tempo (that is, beat) than C, but a slower tactus. In any given piece where the choice between the greater tactus (with half-tactus beats) and the lesser tactus (with full-tactus beats) presents itself, the relation between the two kinds of tactus will indeed be 2:1. If, on the other hand, the lesser tactus under or is compared with the greater tactus under C or O (assuming the full tactus to be just one beat) the speed of the lesser tactus will be found to be faster, but it probably will not be twice as fast. Thus  and came to represent a faster tempo and a faster tactus, but not necessarily in a 2:1 proportion. This procedure is documented by Glareanus when he says:

But whenever musicians wish to accelerate the tactus, which they consider should be done when they believe the hearing is fatigued, namely, in order to remove weariness, they draw a line downwards through the circle or semicircle, as ,, and they then call this contrary quality diminutio, not because either the value or number of notes is lessened, but because the tactus becomes faster.^[11]

According to these directions, the proportional relation between C and seems to have lost its significance, and the crossed semicircle simply stands for a faster tempo.

By the second half of the 16th century the lesser tactus under seems to have been a widely accepted norm. But meanwhile a similar development was also taking place in undimished signatures. Before the middle of the century examples may be found where even in integer valor (C or O) the half tactus becomes the beat. This opens the way for the same choice between the greater and lesser tactus under C and O as under and . About the middle of the century the lesser tactus under C, producing a tactus alla minima, becomes a normal procedure in the so-called “black notation” for madrigals. This tactus alla minima under C now stands in a logical relation to the above-mentioned tactus alla semibreve under , the former used mostly in secular music, the latter in both sacred and secular music.

Obviously, when first introduced, the lesser tact restores the simple equation of one tactus equals one beat. But late in the 16th century again the pressure of an increased number of small notes (introduced especially through ornamentation) once more forced the half tactus into representing a full beat, and from this development stems the present-day concept that stands for a measure of two half-note beats, one down and one up.

There is still another type of tactus which is mentioned by Ornithoparcus in 1519 and described more completely by Agricola in 1532.[12] This is the tactus proportionatus, or what Praetorius later called the tactus inaequalis.^[13] According to this method of beating the tactus, any arrangement of three pulses forming a triple beat, which under the system previously described resulted in one beat to each 1½ tactus, could be beat in one tactus in which the down strokes and the up strokes were uneven, as follows:

Click here for an example of down strokes and up strokes.

It should be noted, however, that this method was not appropriate for arrangements of three beats, for then the second beat would be left completely unarticulated in the tactus. Agricola makes this restriction of the tactus proportionatus clear when he says that each of the minimae in a tactus proportionatus is equal to a minima in the lesser tact under ,^[14] a situation where, at this time, the minima could be only a pulse and not a beat.

A statement of Glareanus suggests that this tactus proportionatus is what was used in the so-called tripla sections of works of his day under the signs O3, C3, or O₃. He objects to the common or popular use of the word “tripla” for such sections, which term should be reserved for a true tripla proportion between voices. But he does speak of the tactus in these sections as an “admirable and majestic tactus.”^[15] This suggests that the tactus was slower than the common lesser tactus employed under , though perhaps not as slow as the 1:1½ ratio that Agricola’s explanation would imply. Thus here again we are confronted with at least the possibility that a nonproportional change in tempo is involved in these

“tripla” sections, so that a tactus inaequalis or proportionatus is employed in which the full tactus is somewhat slower than the lesser tactus that preceded.

From this study of the development of the tactus concept during the 16th century it may readily be seen that the supposed “absolute” tempo of the tactis is largely a myth. The tempo of the tactus was variable, after all; in fact, the developments traced here could not have taken place if the speed of the tactus had been invariable. Just as in any period, the actual musical situation confronting the performer will have to be the main consideration in determining tempo.

Various attempts were made by 16th-century theorists to define the speed of the tactus, the most reliable indications being those that relate the tactus to the human pulse, about 72 per minute. Unfortunately it is not always clear whether the writers are speaking of the greater or the lesser tactus, or whether they are referring to the entire tactus or to the individual down and up movements. A little experimentation and experience suggests that 72 is actually a good average speed for the beat in those situations where either the lesser tact or the greater tact could be applied. This speed may sometimes be reduced to about 60 per minute, which means that the greater tact would be as slow as 30 full tactus per minute. In situations where the full tactus is a beat, it seems reasonable to suppose that it may go as fast as 80 or 90 per minute. So that, even though the tactus may not be a means to fix a precise tempo, it may serve to determine fairly adequately certain limits to a suitable tempo and may serve to prevent gross misconceptions.

With all of the complication of the three kinds of tactus, and situations where one or the other should be given preference, it is easy to understand why in 1547 Glareanus gives up—almost in despair, it seems—trying to explain the tactus and sends the reader back to the authority of the late 15th century, Franchinus Gafurius. Glareanus then says: “Perhaps it would be better to warn the reader in passing that the tactus or measuring is understood principally through the solution which has to be made by an examination of modus, tempus, and prolatio.”^[16]

This is still good advice. The conductor of today, when he approaches music of the 15th and 16th centuries, should make himself familiar with the basic principles of its rhythmic structure and with the various possibilities in the application of the tactus, and then determine the method of conducting that best solves the rhythmic problems at hand. Likewise, the editors of this music should establish a method of transcription faithful to the original notation and to the basic rhythmic concepts underlying the musical styles of the period.

Cited References and Notes

1. Modern ed., ed. Alec Harman (New York, 1952), pp. 5, 6.
2. These studies have been made possible in part by grants from The Newberry Library (Fellowship, Summer 1960) and Valparaiso University (Research grants, 1960 and 1962, sabbatical leave, 1962).
p
3. Cf. E. de Coussemaker, Scriptorum de musica medii aevi (Paris, 1869), IV, 50–53.
4. This system represents a slight modification of that presented by Willi Apel in The Notation of Polyphonic Music, 900–1600, 4th ed. (Cambridge, 1949), pp. 97–100.
5. The rhythmic hierarchy implicit in these time signatures may be determined by factoring the numerator into twos and threes and then continuing to divide the note value represented by the denominator by twos until one has the four levels of rhythm. If a number is encountered in the numerator which contains both two and three as factors, factor out the twos first, then the threes (going from the higher rhythmic levels to the lower), thus:
   6 = 2 x 3 (not 3 x 2!)
   12 = 2 x 2 x 3
   18 = 2 x 3 x 3
   24 = 2 x 2 x 2 x 3
   36 = 2 x 2 x 3 x 3
   54 = 2 x 3 x 3 x 3

The number in the denominator merely refers to a note value and carries no connotation of beat or tactus; thus 3/1 does not necessarily mean that the whole note gets the beat or the tactus, but merely that there are three whole notes in the measure.
6. Cf. Curt Sachs, Rhythm and Tempo (New York, 1953), p. 65 et passim, and Willi Apel, “Drei plus Drei plus Zwei = Vier plus Vier,” Acta Musicologica, XXXII (1960), 29–33.
7. Cf. the facsimile of the chief source for this work (Roma, Vat. Chigi, Cod. C. VIII. 234) in Johannes Ockeghem, Collected Works, ed. Dragan Plamenac, Vol. II (New York, 1947), Plate II.
8. Cf. facsimile in Apel, Notation, p.169.
9. Cf. facsimile, ibid., p.181.
10. This translation from Andreas Ornithoparcus, Micrologus; or, Introduction: Containing the Art of Singing, trans. John Dowland (London, 1609), p. 46.
11. This translation from “The Dodecachordon of Heinrich Glarean,” trans. and ed. Clement Albin Miller (University of Michigan Ph.D. dissertation, University Microfilms, No. 2424), pp. 390–91.
12. Cf. Martin Agricola, Musica figuralis deudsch (Wittenberg, 1532).
13. Cf. Michael Praetorius, Terpsichore (1612), Bd. XV, Gesamtausgabe der musikalischen Werke von Michael Praetorius (Wolfenbüttel, 1929), pp. xi and xii.
14. Cf. Agricola, chap. vi.
15. Cf. translation, pp. 391 and 392.
16. Ibid., p. 388.

Valparaiso University

From The Musical Heritage of the Church, Volume VI (St. Louis, Mo.: Concordia Publishing House, 1963). Copyright Concordia Publishing House. Printed by permission. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of Concordia Publishing House.