A Dictionary of Music and Musicians/Proportion
PROPORTION (Lat. Proportio; Ital. Proporzione). A term used in Arithmetic to express certain harmonious relations existing between the several elements of a series of numbers; and transferred from the terminology of Mathematics to that of Music, in which it plays a very prominent part. In Music, however, the word is not always employed in its strict mathematical sense: for, a true Proportion can only exist in the presence of three terms; in which point it differs from the Ratio, which is naturally expressed by two. Now, the so-called 'Proportions' of Musical Science are almost always expressible by two terms only, and should therefore be more correctly called Ratios; but we shall find it convenient to assume, that, in musical phraseology, the two words may be lawfully treated as synonymous—as, in fact, they actually have been treated, by almost all who have written on the subject, from Joannes Tinctor, who published the first Musical Dictionary, in the year 1474,[1] to the Theorists of the 18th and 19th centuries.
Of the three principal kinds of Proportion known to Mathematicians, two only—the Arithmetical and Geometrical species—are extensively used in Music: the former in connection with differences of Pitch and Rhythm; the latter, in the construction of the Time-table, the Scale of Organ Pipes, and other matters of importance.
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[App. p.752 "in the diagram, above the figure 8 in the top row of figures, the sign should be a semicircle, not a circle. The note below the sign is correct."]
Thomas Morley, in his 'Plaine and easie Introduction to Practicall Musicke' (London 1597), gives a Table, which exhibits, at one view, all the different kinds of Proportion then iu general use; thereby saving so much time and trouble, in the way of reference, that we have thought it well to reproduce his Diagram, before proceeding to the practical application of our subject.
To use this Table, (1) When the name of the Proportion is known, but not its constituents, find the name in the upper part of the Diagram; follow down the lines of the lozenge in which it is enclosed, as far as the first horizontal line of figures; and the two required numbers will be found under the points to which these diagonal lines lead. Thus, Tripla Sesquialtera lies near the left-hand side of the Diagram, about midway between the top and bottom; and the diagonal lines leading down from it conduct us to the numbers 2 and 7, which express the required Proportion in its lowest terms. (2) When the constituents of the Proportion are known, but not its name, find the two known numbers in the same horizontal line; follow the lines which enclose them, upwards, into the diagonal portion of the Diagram; and, at the apex of the triangle thus formed will be found the required name. Thus the lines leading from 2 and 8 conduct us to Quadrupla.
The uppermost of the horizontal lines comprises all the Proportions possible, between the series of numbers from 1 to 10 inclusive, reduced to their lowest terms. The subsequent lines give their multiples, as far as 100; and, as these multiples always bear the same names as their lowest representatives, the lines drawn from them lead always to the apex of the same triangle.
By means of the Proportions here indicated, the Theorist is enabled to define the difference of pitch between two given sounds with mathematical exactness. Thus, the Octave, sounded by the half of an Open String, is represented by the Proportion called Dupla; the Perfect Fifth, sounded by 2-3 of the String, by that called Sesquialtera; the Perfect Fourth, sounded by 3-4, by Sesquitertia. These Ratios are simple enough, and scarcely need a diagram for their elucidation; but, as we proceed to more complex Intervals, and especially to those of a dissonant character, the Proportions grow far more intricate, and Morley's Table becomes really valuable.
A certain number of these Proportions are also used for the purpose of defining differences of Rhythm; and, in Mediæval Music, the latter class of differences involves even greater complications than the former.
The nature of Mode, Time, and Prolation will be found fully explained under their own special headings; and the reader who has carefully studied these antient rhythmic systems will be quite prepared to appreciate the confusion which could scarcely fail to arise from their unrestrained commixture. [See Notation.] Time was, when this commixture was looked upon as the cachet of a refined and classical style. The early Flemish Composers delighted in it. Josquin constantly made one Voice sing in one kind of Rhythm, while another sang in another. Hobrecht, in his 'Missa Je ne demande,' uses no less than five different Time-signatures at the beginning of a single Stave—an expedient which became quite characteristic of the Music of the 15th and earlier years of the 16th centuries. It was chiefly for the sake of elucidating the mysteries of this style of writing that Morley gave his Table to the world; and, by way of making the matter clearer, he followed it up by a setting of 'Christes Crosse be my speed,' for Three Voices, containing examples of Dupla, Tripla, Quadrupla, Sesquialtera, Sesquiquarta, Quadrupla-Sesquiquarta, Quintupla, Sextupla, Septupla, Nonupla, Decupla, and Supertripartiens quartas, giving it to his pupil, Philomathes, with the encouraging direction—'Take this Song, peruse it, and sing it perfectly; and I doubt not but you may sing any reasonable hard wrote Song that may come to your sight.'
Nevertheless, Morley himself confesses that these curious combinations had fallen quite into disuse long before the close of the 16th century.
Ornithoparcus, writing in 1517,[2] mentions eight combinations of Proportion only, all of which have their analogues in modern Music, though, the Large and Long being no longer in use, they cannot all be conveniently expressed in modern Notation, (1) The Greater Mode Perfect, with Perfect Time; (2) the Greater Mode Imperfect, with Perfect Time; (3) the Lesser Mode Perfect, with Imperfect Time; (4) the Lesser Mode Imperfect, with Imperfect Time; (5) the Greater Prolation, with Perfect Time; (6) the Greater Prolation, with Imperfect Time; (7) Perfect Time, with the Lesser Prolation; (8) Imperfect Time, with the Lesser Prolation.
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Adam de Fulda, Sebald Heyden, and Hermann Finck, use a different form of Signature; distinguishing the Perfect, or Imperfect Modes, by a large Circle, or Semicircle; Perfect, or Imperfect Time, by a smaller one, enclosed within it; and the Greater, or Lesser Prolation, by the presence, or absence, of a Point of Perfection in the centre of the whole; thus—
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In his First Book of Masses, published in 1554, Palestrina has employed Perfect and Imperfect Time, and the Greater and Lesser Prolation, simultaneously, in highly complex Proportions, more especially in the 'Missa Virtute magna,' the second Osanna of which presents difficulties with which few modern Choirs could cope; while, in his learned 'Missa L'homme armé,' he has produced a rhythmic labyrinth which even Josquin might have envied. But, after the production of the 'Missa Papæ Marcelli,' in the year 1565, he confined himself almost exclusively to the use of Imperfect Time, with the Lesser Prolation, equivalent to our Alla Breve, with four Minims in the Measure; the Lesser Prolation, alone, answering to our Common Time, with four Crotchets in the Measure; Perfect Time, with the Lesser Prolation, containing three Semibreves the Measure; and the Greater Prolation, alone represented by our 3-2. A very little consideration will suffice to shew that all these combinations are reducible to simple Dupla, and Tripla.
Our modern Proportions are equally unpretentious, and far more clearly expressed; all Simple Times being either Duple, or Triple, with Duple subdivisions; and Compound Times, Duple, or Triple, with Triple subdivisions. Modern Composers sometimes intermix these different species of Rhythm, just as the Greater and Lesser Prolation were intermixed, in the Middle Ages; but, the simplicity of our Time-signatures deprives the process of almost all its complication. No one, for instance, finds any difficulty in reading the Third and Fourth Doubles in the last Movement of Handel's Fifth Suite (the 'Harmonious Blacksmith'), though one hand plays in Common Time, and the other in 24-16. Equally clear in its intention, and intelligible in the appearance it presents to the eye, is the celebrated Scene in 'Don Giovanni,' in which the First Orchestra plays a Minuet, in 3-4; the Second, a Gavotte, 2-4; and the Third, a Valse, in 3-8; all blending together in one harmonious whole—a triumph of ingenious Proportion worthy of a Netherlander of the 15th century, which could only have been conceived by a Musician as remarkable for the depth of his learning as for the geniality of his style. Spohr has used the same expedient, with striking effect, in the Slow Movement of his Symphony 'Die Weihe der Töne'; and other still later Composers have adopted it, with very fair success, and with a very moderate degree of difficulty—for our Rhythmic Signs are too clear to admit the possibility of misapprehension. Our Time-table, too, is simplicity itself, though in strict Geometrical Proportion—the Breve being twice as long as the Semibreve, the Semibreve twice as long as the Minim, and so with the rest. We have, in fact, done all in our power to render the rudiments of the Art intelligible to the meanest capacity: and only in a very few cases—such as those which concern the 'Section of the Canon,' as demonstrated by Euclid, and other writers on the origin and constitution of the Scale; the regulation of Temperament; the Scale of Organ Pipes; and others of like nature are we concerned with Proportions sufficiently intricate to demand the aid of the Mathematician for their elucidation.