IV

Quantitas Intrinseca:
The Perception of Meter

TIME SIGNATURES EVOLVED in order to specify what note value was to be equated with the beat and how many beats were to be included in a measure. But what is it that enables us to distinguish the beat, or groups of beats, in a measure? This question arose when the measure of the tactus no longer coincided with the beat.

Theorists described meter using words nearly as elusive as the perception of meter itself. Although meter is clear to our senses, it is difficult to describe. This may be as much a factor of the mysterious way our minds work as it is of musical organization.

John Holden says that notes that come on the beat are notes to which we give a “superior regard.” German theorists call them “intrinsically long” notes, Italian theorists call them “good” notes, and notation places these “good” notes after bar lines and on odd-numbered beats of the measure. It is not necessary to perform these notes in any special manner in order to associate them with the beat, although musicians may articulate them in such a way as to enhance this perception.

To clarify this point, John Holden asks the reader to consider the steady, equal, unstressed sounds of a watch:

We find them proceeding by pairs . . . which is owing to the pulses being alternately a little stronger and weaker: 1 2, 1 2, 1 2; each single pulse . . . may represent the time of a semiquaver. We can also . . . place our regard on the alternate stronger pulses and disregard the weaker ones, so as to apply the same way of counting 1 ; 2 : 1 ; 2 : in a slower manner to the successive pairs . . . considering each pair as constituting one pulse . . . answering the time of a quaver.

This perception of pairs, and of the coupling of pairs together, is a model of the meter of , or common time, according to Holden.

He continues,

In the performance of music, there is a certain emphasis or accent laid on the beginning of every measure, which plainly distinguishes one species of time from another; so that a hearer is naturally led to distribute a tune into its proper measure, though he should take no notice of the manner of beating time; nay though he should know nothing at all of the rules. . . . The emphasis always falls upon the number 1, in the method of counting a watch . . . and accompanies the putting down of the hand, or toe, in beating time.

There is no occasion to make the beginning or emphatic part, of the measure always stronger, or louder than the rest, though it is sometimes best to do so; for it is not so much the superior loudness of the sound, as the superior regard which a hearer is led to bestow upon it that distinguishes one part of the measure from another. This is a truth of great importance as will hereafter appear, and deserves to be well fixed in mind, before we proceed.¹

Holden’s interpretation of accent or emphasis as distinct from the perception of rhythmic order is indeed a “truth of great importance.”

Joshua Steele’s definition of measure organization attempts to explain speech rhythms in terms borrowed from music.

As all speech, prose as well as poetry, falls naturally under emphatical divisions, which I will call cadences: let the thesis or pulsation, which points out those divisions, be marked by bars, as in ordinary music. Modern musicians, very improperly, use the words accented and unaccented in the place of thesis and arsis. . . . Of modes of time there are only two genera; the one, where the whole time of a bar or cadence, is divided by 2, and its sub-duples or sub-triples; the other, where the whole time of a bar or cadence is divided by three, and its sub-duples or sub triples. But here let it be observed, that this emphasis of cadence and the expression of loudness, are not to be considered as equivalent terms or affections of the same kind; for the arsis or remiss, may be loud, or forte; and the thesis, or emphatic, piano or soft occasionally. The thesis and arsis being periodically alternate, whether expressed or supposed: whereas the application of the forte and piano are ad libitum or apropos. . . .

Here then are the two general modes or measures of time. The first, wherein each step makes a cadence, and is divided equally by the even number 2, and the pace, or double cadence, by 4; and is in music called common time, andante, or the measure of a march. The second, where the whole pace, making only one cadence, may be equally divided by the number 6, as the double of 3; and is called triple time or the measure of the menuet and jigs.

The affections of heavy and light were always felt in music, though erroneously called by some moderns accented and unaccented; however the accented or heavy note, was never understood to be necessarily loud, and the other necessarily soft; because if it were so, there could be no occasion for separate directions, where to apply the forte and piano, in as much as the affections of heavy and light are continued in every cadence of every air, . . . whereas the forte and the piano are often applied directly contrary to heavy and light. ²

John Holden’s “superior regard” and Steele’s “pulsation and remission” indicate a perception of the essence of metrical order. This concept was also defined, although with less precision, in the discussions of quantitas intrinseca, or of “good” and “bad” notes that were offered by German, Italian, and French writers in the seventeenth and eighteenth centuries.

Quantitas intrinseca is discussed in the Satyrischer Componist of Wolfgang Caspar Printz:

Ferner ist zu wissen/ dass die Zahl eine sonderbare Krafft und Tugend habe/ welche verursacht/ dass unter etlichen/ der Zeit nach/ gleich-langen Noten oder Klängen/ etliche länger/ ediche kürzer zu seyn scheinen: Welches sonderlich wohl zu mercken/ so wohl wegen des Textes/ als auch wegen der Consonantien und Dissonantien.

Diese unterschiedliche Länge etlicher/ der Zeit oder Wahrung nach/ gleichlange Noten/ wird genennet Quantitas Temporalis Intrinseca, die innerliche Zeit- Länge.

Dass dieses seinen Grund in re ipsâ, in der That selbst habe/ kan ein jeder leicht sehen und hören/ wenn er einen Text unter gleich Noten setzet. Als zum Exempel [Ex. 4.1].

In diesem Wort ist die erste und dritte Sylbe lang/ die andere und vierdte kurz. Wird nun solches Wort mit denen Sonis ABCD gesungen/ so wird es fein ins Gehöre kommen/ und jede Sylbe seine rechte Länge haben. Verwechsele ich aber die Zahl der Noten/ ob ich schon die Noten selbst behalte [Ex. 4.2].

So wird es gar unanmuthig und Wiederwärtig ins Gehöre fallen/ weil die langen Sylben chri und a mit kurzen Sonis E & G hergegen die kurzen Sylben sti & nus mit langen Sonis F & H hervor gebracht und gesungen werden: Welches nicht seyn könte/ wenn die Soni EFGH keinen Unterschied an innerlicher Zeitlänge hätten.

Diese Quantität recht zu erkennen/ muss man wissen/ dass jede Note entweder in zwey oder drey gleiche Theile getheilet werde.

Ist der Theiler der Noten die zweyte Zahl/ so seyn alle Noten/ so mit einer ungeraden Zahl/ als 1, 3, 5, 7, &c. gezehlet werden/ lang; Hergegen/ die mit einer geraden Zahl, als 2, 4, 6, 8, &c. gezehlet werden/ kurz.

Hieraus erhellet/ dass eine jede Semibrevis oder ganzer Tact auch der innerlichen Quantität nach/ lang sey/ weil sie mit einer ungeraden Zahl, Nehmlich 1 gezahlet wird: Sintemahl allezeit, im Niederschlagen des Tactes, das Zehlen sich anfangen muss.

Ingleichen seyn alle und jede syncopirte Noten lang/ weil in denselben die gerade und ungerade Zahl zusammen kommen und sich vereinbaren.

Ist die dritte Zahl der Theiler/ so ist die erste lang/ die andere und dritte kurz.

Wenn aber die erste schweiget/ so ist die andere lang die dritte kurz.³

Further, the position in the measure has a peculiar power and virtue which cause notes equal to one another, according to the time signature, to seem longer or shorter. This should be especially noted as much because of the text as because of consonance and dissonance.

The apparent different length of notes that are equal according to their time or value, is called Quantitas Temporalis Intrinseca, or the inner duration. It can easily be seen that this has its basis in the notes themselves when a text is set to notes of like duration [Ex. 4.1].

EX. 4.1

In this word, the first and third syllables are long and the second and fourth are short. If they are sung to notes ABCD above, they sound right to the ear and each syllable is the correct length.

If the position of the notes in the measure is changed, whether or not the notes themselves are changed [Ex. 4.2], the phrase falls disagreeably and contrarily on the ear because the long syllables Chri and a are used and sung with the short sounds E and G, and the short syllables sti and nus are used and sung with the long sounds F and H. This would not occur if the sounds EFGH were not different in their inner duration.

EX. 4.2

To know these quantities correctly, one must know that every note is divided into either two or three like parts.

If the subdivisions of notes are duple, all odd numbered notes 1, 3, 5, 7, etc. are considered long and all even numbered notes 2, 4, 6, 8, etc. are short. . . .

I explain here that every semibrevis or entire Tactus, according to the inner quantity, is also long because it is figured with an odd number, one, since this number always begins on the downbeat of a measure.

Also each and every syncopated note is long because the odd and even numbers are mingled together and mixed in it.

If the subdivisions (of a note) are three in number, the first is long and the second and third are short.

When the first part is silent, the second is long and the third is short.

Printz does not advocate lengthening or shortening the duration of notes in the Satyrischer Componist: he implies that it is only the listener’s perception of their “peculiar power and virtue” that distinguishes long from short.

Heinichen used the terms notae virtualiter longae and notae virtualiter breves for “long and short” when explaining the harmonic passing tone in figured bass.⁴ J. A. Scheibe used anschlagende (struck) for “long” and durchgehende (passing) for “short.”⁵ These terms seem to have been applied in relation to harmonic rather than rhythmic considerations.

In Italian treatises beginning with Girolamo Diruta,⁶ nota buona and nota cattiva, “good” and “bad” notes, are used to refer to metrical order. Diruta explained that when a two-note group comes on the beat, the good note is the first and the bad note is the second. His fingering rules for keyboard performance place “good” fingers, the second and fourth, on good notes, and “bad” fingers, the first, third, and fifth, on bad notes.⁷ Lorenzo Penna⁸ and subsequently other writers, both German and Italian,⁹ used these terms in broader contexts than that of determining fingerings for the keyboard.

“Good” and “bad” notes and their musical consequences are carefully explained by Georg Muffat in Latin, German, Italian, and French in the preface to Florilegium secundum (1698). Muffat was trained in both the Italian and French styles of composition and performance; he studied in Paris from 1663 to 1669, and with Pasquini in Rome in 1681—82, where he became acquainted with Corelli’s music.¹⁰ According to him, bar lines precede good notes and mark recurring patterns of organization, most of which reflect dance steps. Good and bad notes are identified by their position in the measure.

Of all the notes found in any composition to be played, there are those that are good (nobiliores; edle; buone e principals, bonne, noble ou principales), and others that are bad (ignobiliores, seu viliores; schlechte; cattive, ò vili; chetives ou viles). Good notes are those that seem naturally to give the ear a little repose. Such notes are longer, those that come on the beat or essential subdivisions of measures, those that have a dot after them, and (among equal small notes) those that are odd numbered and are ordinarily played down-bow. The bad notes are all the others, which like passing notes, do not satisfy the ear so well, and leave after them a desire to go on. ¹¹

Muffat uses the Italian idea of good and bad notes to clarify his presentation of French performance directions to German musicians, a truly international undertaking. Walther’s Lexikon defines all these terms, German as well as Italian, as befits a dictionary:

Quantitas Notarum extrinseca, & intrinseca [lat.] die äusserliche und innerliche Geltung der Noten; nach jener art ist jede note mit ihres gleichen in der execution von gleicher; nach dieser aber, von ungleicher länge: da nemlich der ungerade Tact-Theil lang, und der gerade Tact-Theil kurz ist. ¹²

Quantitas notarum extrinseca, and intrinseca [lat.] is the apparent (or outward) and the inner value of the notes. According to the former, every note is performed equal to other notes of the same value, but according to the latter the notes are of unequal length: since, to be specific, the uneven-numbered parts of the beat are long and the even-numbered ones short.

Walther defined good and bad notes, tempo di buona and tempo di cattiva, as follows:

Tempo di buona [ital.] der gute Tact-Theil, ist in Tactu aequali, unter 2 Minimis die erste Minima, oder die erste helffte des Tacts; unter 4 Vierteln, das lste und 3te Viertel; unter 8 Achteln, das lste, 3te, 5te, und 7te Achtel u.s.w. Weil erwehnte tempi oder ungerade Tact-Theile bequem sind, dass auf ihnen eine Caesur, eine Cadenz, eine lange Sylbe, eine syncopirte Dissonanz, und vor allen eine Con- sonanz (als von welcher eben der Bey-Nahme: di buona entstanden) angebracht werde.

Tempo di buona [ital.] is the good part of the beat. Under the equal tactus, the first of two minims, or the first half of the beat is good; also the first and third of four quarter notes, the first, third, fifth and seventh of eight eighth notes and so forth, because these tempi, or odd-numbered parts of the beat are suitable for the placement of a caesura, a cadence, a long syllable, a syncopated dissonance, and above all a consonance (from which comes its name—di buona).

Tempo di cattiva, oder di mala [ital.] der schlimme Tact-Theil, ist in Tactu aequali, unter 2 Minimis die zweyte Minima, oder die zweyte Helffte des Tactes; unter 4 Vierteln, das 2te und 4te Viertel; unter 8 Achteln, das 2te, 4te, 6te und 8te Achtel; weil Nurbesagte tempi oder gerade Tact-Theile einige von ober-zehlten Stücken Nicht, wohl aber desen contrarium leiden. ¹³

Tempo di cattiva, or di mala [ital.] is the bad part of the beat. In the Tactu aequali or beat with two equal strokes, the second of two minims or the second half of the beat is bad; also the second and fourth of four quarters, sixth and eighth of eight eighth notes, because these tempi or even-numbered parts of the beat are all different from the above-mentioned parts, and are their opposites.

In the discussions of the terms quantitas intrinseca, “good and bad” notes, or Holden’s “superior regard,” no method of enhancing or articulating these differences is specified.

Hiller’s Anweisung defines the elusive concept of meter more clearly than any other German treatise of the eighteenth century. Hiller’s concept of musical meter follows the tradition of the seventeenth- and earlier eighteenth-century theorists, even though his compositional style reflects his admiration of more modern composers such as Hasse, J. C. Bach, and Quantz, rather than Sebastian Bach. His good sense and pedagogical skill are evident in the clarity of his explanation of meter:

Unter zwo neben einander stehenden, der Gestalt und dem Werthe nach gleichen Noten, ist, bey gleicher oder gerader Abtheilung des Tacts, der innerlichen Quantität nach (Die ausserlich Quantität verändert sowohl die Gestalt als den Werthe der Noten; wir haben es als dann mit ganzen und halben Tactnoten, mit Vierteln, Achteln u.s.f. zu thun . . .) immer die eine lang, und die andere kurz. Dieser Umstand hat seinen Grund in dem Natürlichen Gefühle der Menschen, und äussert sich sogar in der Sprache; indem man nicht zwo Sylben nach einander aus- prechen kann, dass nicht die eine kürzer scheine, als die andere, die Prosodie mag auch dagegen einwenden, was sie will. Welche nun unter zwo gleichgeltenden Noten lang, und welche kurz sey, das muss die Tactabtheilung bestimmen. Auf dem Papiere geschieht diess vermittelst eines Strichs, welcher durch alle fünf Linien gezogen wird, und daher den Namen eines Tactstrichs (Was zwischen zween solchen Strichen stehet, er mögen nun eine, zwey, drey oder zwanzig Noten seyn, wird ein Tact genennt) fürt, bey der Ausführung eines Stücks aber geschieht es durch das Niederschlagen und Aufheben der Hand. (Ein paar Kuntswörter, die wir den Griechen abgeborgt haben, und welche in diesem Werke sich wohl bisweilen mit eindrängen möchten, so gern ich auch sonst alles deutsch sage, was sich deutsch sagen lässt, mag man sich im voraus bekannt machen. Das Niederschlagen der hand heisst Thesis, und das Aufgeben derselben, oder wie man insgemein sagt, der Aufschlag, Arsis. In einigen musikalischen Büchern findet man diese beyden Wörter, in der Anwendung auf die Tacttheile, anders verdeutscht und das erste durch guter Tacttheile, das andere durch schlimmer Tacttheile, nach dem itali- änischen nota buona, nota cattiva, übersetzt. . . . So erlaube man mir immer statt guter Tacttheile; langer Tacttheile, und statt schlimmer, kurzer Tacttheile zu sagen. Schlimm im Gegensatze von gut, ist so viel als böse; das ist nun wohl von dem armen Tacttheile zu lieblos gesprochen. . . .) Das letze gilt für die kurze, das erste für die lange Note, wenn nicht mehr als zwo Noten in einem Tacte beysammen sind. Der Tactstrich, welcher zugleich den Niederschlag andeutet, steht allemal unmittelbar vor der langen Note. ¹⁴

Of two notes, side by side, of the same kind and value, and in a duple or equal division of the beat, one will always be long and the other short, according to their inner quantity (the outward or apparent quantity deals with the change in the kind and value of notes; we are concerned with whole and half notes, with quarters, eighths, etc.).. . . This fact has its basis in man’s natural feelings, and also in speech. Two syllables cannot be spoken together without it appearing that one is shorter than the other, although prosody sometimes wishes to take exception to this. The division of the measure determines which of the two is long and which is short. This is made apparent on paper by means of a line drawn through all five lines, called a bar line (what is included between two bar lines, whether one, two, three or twenty notes is called a measure); and in performance by a down-and-up motion of the hand. (Note: These two technical terms, which we have borrowed from the Greek, must be explained, for although I like to say everything in German that German can say, in this work I will be forced sometimes to use them. The downstroke of the hand is called thesis and the upstroke, or as is sometimes said, the upbeat, is called arsis. These two words are found in several books to describe the parts of the measure, others translate this to “good part” and “bad part” of the measure, after the Italian words nota buona, nota cattiva.. . . Therefore allow me to say “longer part” instead of “good part” and “shorter part” instead of “bad part. . . .”) The latter [Aufheben] refers to the short, and the former to the long note if there are no more than two notes in a measure. The bar line, which indicates the downbeat, always comes directly before the long note.

According to Hiller, various terms used by earlier writers to describe metrical perceptions are identical; his explanation is easily reconciled with the definitions of Holden and Steele. It seems to be the belief of most seventeenth- and eighteenth-century theorists that musical meter is naturally and adequately perceived by the listener and only secondarily heightened through performance techniques. Most performers today are aware of how crude it is to suggest that the measure is identified by a regular accent or dynamic stress based on bar lines and time signatures. This confidence in the listener’s basic metrical perception as defined by quantitas intrinseca gives rise to a remarkable variety of subtle articulation techniques that delicately enhance and shape that perception.