Mathematicians and Music 2.2

We give below a version of the end of the second section of Raymond Clare Archibald's Presidential Address to the Mathematical Association of America on Mathematicians and Music.

The previous part of Archibald's talk is here

Mathematicians and Music

by R C Archibald, Brown University.


The next outstanding figure in our survey is Boetius who flourished in the early part of the sixth century of our Christian era. He was a Roman senator and a philosopher, - "the last of the Romans whom Cato or Tully could have acknowledged for their countryman," as Gibbon expresses it. Not only did Boetius exert great influence in his own time through his summaries of logical and scientific works of the ancients, but for six centuries after his death they were the leading authorities. He wrote works on arithmetic, geometry and music. While the first printed edition of the arithmetic appeared at Venice in 1488, all three united seem to have first been published in 1492. Details of the mathematical works have been given by Moritz Cantor. His extensive treatise on music is a valuable repertory of the knowledge of the ancients in this art. It was long used as a text at the Universities of Oxford and Cambridge. Boetius sets forth the details of the accomplishments of the Pythagoreans and the teachings of such writers as Aristoxenus that were opposed to those of Pythagoras. He also surveys the Ptolemaic musical scheme in connection with those of Pythagoras and Aristoxenus. Since the doctrine of Boetius was mainly Pythagorean, this was the system that prevailed for centuries later.

As the Roman absorbed the Greek, so the Christians accepted the Roman organization of learning. In the medieval curriculum the scope of this learning on the secular side was comprised within the seven liberal arts and philosophy. The seven liberal arts, divided into the Trivium (grammar, dialectic, rhetoric), and the more advanced Quadrivium (geometry, arithmetic, music and astronomy), were an inheritance from a period at least as early as the second century before Christ; indeed the Quadrivium division of mathematical studies is Pythagorean. Some explanation of the nature of the subjects of the Trivium is necessary in order to make their scope clear; but we are only concerned with the mathematical sciences of the Quadrivium in which, early in the middle ages, the course in geometry was more a course in geography and surveying than in the subject matter of Euclid's Elements which later became a text. The study of music consisted mainly in becoming acquainted with the mathematics of the subject, and with the mystic properties of its numbers, - much as taught by the Pythagoreans. As a liberal art it concerned itself neither with singing (apart from its rules), nor with playing on an instrument. Astronomy, with its practical applications to the calendar and sundial, was the most popular of the Quadrivium subjects but there was probably more of astrology in it than astronomy as we now understand the term.

Some four hundred years after the time of Boetius, the polyphonic period in the development of music had its inception in the composition of certain two-part song-forms. During the six hundred years which followed, that is, till towards the close of the sixteenth century, polyphonic music adorned with canon, fugue, and counterpoint was developed to a notable degree.

Of mathematicians who flourished in this period I shall refer to only one, Girolamo Cardano. Among those of the sixteenth century achieving a reputation in mathematics and medicine none was better known than he, whose greatest mathematical work, Ars Magna (1545), contains the first solution of the general cubic equation in print.

Cardano was an ardent lover of music and while living in Milan his house was constantly filled with men and boys of somewhat sinister reputation but capable of joining with him in part-singing so popular in the polyphonic period. During the last twenty-five years of his life he spent considerable time in writing a work on music, which was in many respects original and must have been welcomed by all musical students as a valuable contribution to the literature of the subject. This work begins by laying down at length the general rules and principles of the art, and then goes on to treat of ancient music in all its forms; of music as Cardano knew and enjoyed it; of the system of counterpoint and composition, and of the construction of musical instruments.

An interesting glimpse of Cardano's personality may be gleaned in another place from his listing of the joys of home and children. Incidentally he suggests:-

Let the young child . . . be shut out from the sight or hearing of all ill. When he is about seven years old let him be taught elements of geometry to cultivate his memory and imagination. With syllogisms cultivate his reason. Let him be taught music, and especially to play upon stringed instruments; let him be instructed in arithmetic and painting, so that he may acquire taste for them, but not be led to immerse himself in such pursuits. He should be taught also a good hand-writing, astrology, and when he is older, Greek and Latin.
In the early part of the third period in the development of music, namely, the period of Harmonic or Modern Music, we have the first opera and the first oratorio, and, as I have already said, the discovery by Galileo that the simple ratios of the lengths of strings existed also for the pitch numbers of the tones they produced, an observation later generalized by Newton. By the time of Rameau, the most eminent French composer and writer on the theory of music in the eighteenth century, the harmonics or upper partial tones of the human voice had been recognized and made the basis for more satisfying harmonic development. A string, for example, vibrates not only as a whole but also, at the same time, in each of its aliquot parts 1/2, 1/3, 1/4, 1/5, 1/6, and so on. Thus the first upper partial tone is the upper octave of the prime tone, the second is the fifth of this octave, the third upper partial is the second higher octave, the fourth is the major third of this second higher octave, the fifth is the fifth of the second higher octave, making six times as many vibrations as the prime in the same time; and so on, each successive upper partial tone being fainter than the preceding. It may be shown that beginning with the twenty-fourth upper partial all the notes of a major scale may be obtained from the dominant, that is, the fifth. The dominant and not the tonic is thus the root, of the whole scale. In the bugle, trumpet, French horn, and other instruments only the fundamental tone of the instrument and some of its harmonics can be sounded. On a horn about four feet long the notes are c, c', g', c", e", and g", the primes denoting tones in higher octaves.

Not all upper partials need exist in connection with a fundamental musical tone. Certain tuning forks have no upper partials. In 1800, the noted physicist, Thomas Young, who first furnished the key to decipher Egyptian hieroglyphics, was also the first to show that:-

... when a string is plucked or struck, or, as we may add 'bowed' at any point in its length which is the node of any of its so-called harmonics those simple vibrational forms of the string which have a node in that point are not contained in the compound vibrational form. Hence if we attack at its middle point, all the simple vibrations due to the even numbered partials, each of which has a node at that point, will be absent. This gives the sound of the string a peculiarly hollow or nasal twang.
Because of this law piano makers eliminate certain undesirable upper partials by striking the middle strings of their instruments at a point 1/7 to 1/9 of their lengths from their extremities. So too in making other instruments it is possible to eliminate, or reinforce, certain partials.

But we have got ahead of our story. Returning to the beginning of the Harmonic period let us consider the musical writings which were issued in the seventeenth century by such mathematicians as Kepler, Wallis, Mersenne, Desargues, Descartes and Christian Huygens.

Pythagorean ideas on the ratios of numbers and of proportions applied to the constitution of the universe seem to have been the point of departure of Kepler in his famous work Harmonices Mundi published in 1619. It is in the fifth book of this work that one first finds the third fundamental law of modern astronomy, "The squares of the periodic times of the several planets are proportional to the cubes of their mean distances from the sun," demonstration of which furnished Newton with the basis for his theory of gravitation. The third book of the work is especially devoted to music and it may be characterized as mainly a work on the philosophy of music. The fifth book to which I have referred is somewhat allied to the third, since in it the author endeavoured to establish curious analogies between the harmonic proportions of music and astronomy.

Markedly contrasted to Kepler in abilities and habits of thought was John Wallis, the notably able Savilian professor at Oxford University, where a brilliant mathematical school was developed under his direction. He is well known as mathematician and cryptographer, but few have observed his extensive writings on musical mattes filling more than 500 folio pages in the third volume of his collected works. The first of these is a Greek and Latin edition of Ptolemy's Harmony, and Porphyry's third century commentary on the same, with an extensive appendix by Wallis on ancient and modern music. Then comes the only published text, with Latin translation, of a musical work by Manuel Bryenne, a fourteenth century Greek, four manuscripts of whose work are to be found at the Bodleian. Among other writings of Wallis on acoustics and music may be mentioned four memoirs published in the Philosophical Transactions, and bearing the following titles: "On the trembling of consonant strings," "On the division of the monochord, or section of the musical canon," "On the imperfections of an organ," and "On the strange effects of music in former times."

The Franciscan friar Marin Mersenne, Wallis's senior by nearly 30 years, is known to the general run of mathematicians through the numbers with which his name is associated and which arise in discussion of perfect numbers. He was widely acquainted with French and foreign contemporary mathematicians and actively corresponded with them. His work in physics dealt chiefly with questions in acoustics. He determined ratios of the vibration numbers of strings varying in thickness and tension, results included in those of Brook Taylor derived mathematically about 70 years later. I have not been able to verify the statement that Mersenne noticed, but attached no importance to the observation, that a vibrating string gave forth not only the fundamental tone but also higher sounds. We have already remarked that Rameau made much of the fact in the following century. Mersenne wrote half a dozen works on harmony and musical instruments but his most notable one is L'Harmonie Universelle, a great work of 1500 pages with an immense quantity of engraved plates and musical examples. This was published in 1636-7. It is really a combination of several treatises, for example, On the Nature of Sounds and Movements of All Sorts of Bodies, On Voice and Songs, and On Instruments. There is also a treatise on mechanics, by Roberval, which no one but a Mersenne could regard as appropriately placed in his work on harmony. While no sections of the work are of transcendent merit, one finds a great amount of information, especially regarding Frenchmen, which is no longer to be found elsewhere. It is only here, for example, that we learn that the geometer Desargues was the author of a method of singing.

Among Mersenne's friends was one, some eight years his junior, René Descartes. That he was interested in music is attested by the fact that a score of his published letters treat of motions of vibrating strings and various musical topics. Moreover in 1618, when 22 years of age, he wrote a Compendium Musicae, but this was first published as a little tract of 58 pages in 1650, the year of his death. The material is arranged under about a dozen headings such as: the object of music is the sound; number and time that one should observe in the sounds; concerning the diversity of sounds; consonances; the octave; the fifth; the fourth; the second, minor third, and sixth; the degrees or tones of music; dissonances; and the manner of composing - in connection with which five principles are laid down in an interesting manner.

A copy of the manuscript of the Compendium found its way to one afterwards to become a particular friend of Descartes. This was Constantin Huygens, a many-sided genius possibly best known as a poet and a musician; he was a competent performer on several instruments and author of several musical works. His second son was Christian Huygens the great Dutch mathematician, mechanician, astronomer and physicist. Two publications dealing with musical matters were written by Christian Huygens. The first of these is a brief sketch of 1691, entitled Novus Cyclus Harmonicus, and occupying only 8 quarto pages. In them he suggests another solution of the problem of how suitably to arrive at a tempered scale. If we divide the octave into twelve equal parts or degrees, we have a cycle in which a fifth of 7 and a major third of 4 degrees approximates to Pythagorean intonation. A cycle of 53, with a fifth of 31 and a major third of 18, had also been proposed, and led to similar results. The new harmonic cycle of Huygens contained 31 degrees, with a fifth of 23 and a major third of 10, and closely imitates mean tone temperament. He refers to the writings of Mersenne, and of Zarlino, "one of the most learned and enlightened music theorists of the sixteenth century." I have already drawn attention to the natural way in which logarithms enter into the discussion of musical intervals. So far as I have been able to determine, this little publication of Huygens is the first to illustrate this fact.

The second work of Huygens containing musical material was finished for the press just before his death. Three years later it appeared simultaneously in Latin and English and is an exceedingly entertaining work. It is entitled The Celestial Worlds discover'd: or Conjectures Concerning the Inhabitants Plants and Productions of the Worlds in the Planets. In order adequately to present an idea of a section on mathematics and music I shall quote somewhat extensively.

The author surmises that if the surfaces of Jupiter and Saturn are divided like ours into sea and land it is reasonable to suppose that the inhabitants must know of the art of navigation. He then infers that they must have the:-

... Mechanical Arts and Astronomy, without which Navigation can no more subsist, than they can without Geometry.
Huygens then continues:-
But Geometry stands in no need of being proved after this manner. Nor doth it want assistance from other Arts that depend upon it, but we may have a nearer and shorter assurance of their not being without it in those Earths. For that Science is of such singular Worth and Dignity, so peculiarly imploys the Understanding, and gives it such a full Comprehension, and infallible certainty of Truth, as no other Knowledge can pretend to: it is moreover of such a Nature, that its Principles and Foundations must be so immutably the same in all Times and Places, that we cannot without Injustice pretend to monopolize it and rob the rest of the Universe of such an incomparable Study. Nay Nature itself invites us to be Geometricians: it presents us with Geometrical Figures, with Circles and Squares, with Triangles, Polygons, and Spheres, and proposes them as it were to our Consideration and Study which abstracting from its usefulness is most delightful and ravishing. Who can read Euclid or Apollonius, about the Circle, without Admiration? Or Archimedes of the Surface of the Sphere, and Quadrature of the Parabola without Amazement? Or consider the late ingenious Discoveries of the Moderns, with Boldness and Unconcernedness? And all these Truths are as naked and open, and depend upon the same plain Principles and Axioms in Jupiter and Saturn as here, which makes it not improbable that there are in the Planets some who partake with us in these delightful and pleasant studies
Then a little later the author continues:-
It's the same with Music as with Geometry, it's everywhere immutably the same, and always will be so. For all Harmony consists in Concord, and Concord is all the World over fixed according to the same invariable Measure and Proportion so that in all Nations the Difference and Distance of Notes is the same, whether they be in a continued gradual Progression, or the Voice skips over one to the next. Nay, very credible Authors report, that there's a sort of Bird in America, that can plainly sing in order six musical Notes: Whence it follows, that the Laws of Music are unchangeably fixed by Nature, and therefore the same Reason holds for their Music, as we even now showed for their Geometry.
Discussing the probability of other planets' being inhabited and of the inhabitants' possible interest in music and invention of musical instruments, he continues:-
What if they should excel us in the Theory and practical part of Music, and outdo us in consorts of vocal and instrumental Music, so artificially composed, that they show their skill by the Mixtures of Discords and Concords and of this last sort 'tis very likely the 5th and 3rd are in use with them.

This is a very bold Assertion, but it may be true for aught we know, and the Inhabitants of the Planets may possibly have a greater insight- into the Theory of Music than has yet been discovered among us. For if you ask any of our Musicians why two or more perfect Fifths cannot be used regularly in Composition; some say 'tis to avoid that Sweetness and Lushiousness which arises from the repetition of this pleasing Chord. Others say, this must be avoided for the sake of that Variety of Chords that are requisite to make a good Composition; and these Reasons are brought by Descartes and others. But an Inhabitant of Jupiter or Venus will perhaps give you a better Reason for this, viz. because when you pass from one perfect Fifth to another, there is such a Change made as immediately alters your Key, you are got into a new key before the Ear is prepared for it, and the more perfect Chords you use of the same kind in Consecution, by so much the more you offend the Ear by these abrupt Changes.

It may interest harmony students of our day to learn that the prohibition of consecutive fifths was not something recently invented for their undoing, but was a matter of fundamental importance adequately explained over two hundred years ago. And this is not the only passage of interest for such students in the last work of Christian Huygens.

I have already referred to Thomas Young's memoir of 1800 and his explanations of varied qualities of tone through agitation of a string at different points. The mere fact of such differences of quality had been already noted by Huygens in connection with harpsichords - those precursors of pianos in the sixteenth, seventeenth, and eighteenth centuries.

The final part of Archibald's talk is here

JOC/EFR August 2006

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