The Sounds of Strings

Axiom 6: When 2 strings emit identical sounds, a third note which is consonant with one of them will also be consonant with the other; and one which is dissonant from one will also be dissonant from the other, and so with various different kinds of consonances or discords also.

Note that identity of sound is put in the subordinate position as a species, and consonance in the antecedent position as genus.

Hence the two following points should be understood. First, it does not follow that if two strings are consonant in any way, then a third is also consonant with both of them, or dissonant with both of them. For that is false of the genus, but true of the species of identity of sound.

Secondly, it does not follow that if the third is consonant with one of the sounds which are identical in some particular consonance, it will be consonant with the other with the same species of consonance; for that would not always be true, which I shall demonstrate with an example, though in anticipation. Let there be two notes, making the diapason G and g. Let there be a third note d. It makes a fifth with G, and is therefore also consonant with g; yet not by a fifth, but by a fourth.

The chief application of this axiom is in Proposition IV.

Axiom 7: The cause of the harmonies are metaphysical

When 2 strings or voices emit identical sounds, a third note which is identical in sound with one of them, will also be identical in sound with the other.

What could not be affirmed in the previous axiom of the genus, is true in the species as to identity of sound.

The application is in Proposition 3.

Then contemplation of these axioms, especially of the first five, is lofty, Platonic, and analogous to the Christian faith, looking towards metaphysics and the theory of the soul.

For geometry, the part of which that looks in this direction was embraced in the two previous books, is coeternal with God, and by shining forth in the divine mind supplied patterns to God, as was said in the preamble to this Book, for the furnishing of the world, so that it should become best and most beautiful and above all most like to the Creator. Indeed all spirits, souls, and minds are images of God the Creator if they have been put in command each of their own bodies, to govern, move, increase, preserve, and also particularly to propagate them.

Then since they have embraced a certain pattern of the creation in their functions, they also observe the same laws along with the Creator in their operations, having derived them from geometry.

Also they rejoice in the same proportions which God used, wherever they have found them, whether by bare contemplation, whether by the interposition of the senses, in things which are subject to sensation, whether even without reflection by the mind, by an instinct which is concealed and was created with them, or whether God Himself has expressed

these proportions in bodies and in motions invariably, or whether by some geometrical necessity of infinitely divisible material, and of motions through a quantity of material, among an infinity of proportions which are not harmonic, those harmonic proportions also occur at their own time, and thus subsist not in BEING but in BECOMING.

Nor do minds, the images of God, merely rejoice in these proportions; but they also use the very same as laws for performing their functions and for expressing the same proportions in the motions of their bodies, where they may.

The following Books will offer two splendid examples.

One is that of God the Creator Himself, who assigned the motions of the heavens in harmonic proportions. The second is that of the soul which we generally call Sublunary Nature, which actuates objects in the atmosphere in accordance with the rules of the proportions which occur in the radiations of the stars.

So let the third example, and the one which is proper to this Book, be that of the human soul, and also that of animals to a certain extent. For they take joy in the harmonic proportions in musical notes which they perceive, and grieve at those which are not harmonic. From these feelings of the soul the former (the harmonic) are entitled consonances, and the latter (those which are not harmonic) discords.

But if we also take into account another harmonic proportion, that of notes and sounds which are long or short, in respect of time, then they move their bodies in dancing, their tongues in speaking, in accordance with the same laws.

Workmen adjust the blows of their hammers to it, soldiers their pace. Everything is lively while the harmonies persist, and drowsy when they are disrupted.

Whether these and the like are intentional or involuntary, that is the work of the mind; and whether it is by the necessity of the nature of the elements and of matter that no tuning can suit the senses but that which is based on the harmonic proportions of the figures, has been argued in various ways by the philosophers. All ask the source of that pleasure which glides into the ears from the proportion of notes, pleasure by which we define consonances.

Those who incline towards matter and the motion of the elements, adduce as an example the fact, in itself indeed certainly remarkable, that a string which is set in motion draws another string which has been set in motion into sounding with it, if it has been tightened into consonance with itself, but if it has been tightened into dissonance leaves it motionless.

Since that cannot come about by the intervention of any mind, because the sound, the supposed cause of it, does not have mind or understanding, it follows that we can say it comes about by the adjustment of the motions to each other.

For the sound of the string has higher or lower pitch, from the speed or slowness of the vibration with which the whole free length of the string vibrates.

These differences in the sounds do not arise primarily and immediately in the actual length or shortness, but secondarily, that is to say because when the length is diminished the slowness of the vibration is diminished, and its speed is increased.

The reason is that, if the free length of the string remains the same, the actual tightening of it raises the pitch of the sound, because by leaving the string less slack, it also diminishes the space through which it can vibrate in its reciprocating motion.

Then if the tension of two strings is equal, so that they can sound in unison, in that case the sound of one, that is the immaterial emanation of the body of the string, which is set in vibration, gliding from its string, strikes the other string, just as when someone shouts at a lute, or something else hollow. With that shout he strikes the hollow object and makes all its strings resonate.

Now that emanation of the vibration strikes the other string with the same rhythm of speed and the latter also moves in that rhythm because it is equally tight; so that individual beats (into which the vibration is understood to be divided) continually come upon individual stationary points of the other string as they strike it. So it comes about that the string which is tightened to unison with the first moves most of all. Yet the string which is of twice or half the speed also moves, because two beats of the vibration are completed for one stationary point of the string, and thus every third beat after the previous one always coincides with the extreme of one stationary point. Lastly the string which is of one and a half times the speed also moves to some extent, because 3 little beats occur for two stationary points of the former string.

But now the beats on the one hand and the stationary points on the other begin to meet each other more frequently and to impede each other.

While 2 beats of the former string miss the end of a stationary point of the latter, only one coincides; and when they meet in that way the motion of the other strings is halted, exactly as if someone had applied a finger to the one which was vibrating.

This seems to me the remarkable cause of this discovery; and if anyone is more fortunate than I in his intellectual search, I shall yield him the palm.

What follows, then? If the speed of one string has the power to move another which is in proportion to it, but which, as far as can be seen, remains untouched, will not the fact that two strings have the same speeds as each other have the power to titillate the hearing pleasantly, on account of the fact that in a way it is moved uniformly by both strings, and that two beats from two sounds or vibrations cooperate in the same impulsion? It is vain, say I, to dispose of this matter so easily; and I wonder that Porphyry the commentator on the Harmonics of Ptolemy could have been satisfied with something like that for the cause of this phenomenon, although he is a philosopher of the most profound insight. Unless, as is probable, he was constrained by the difficulty of seeking out the cause from penetrating as far as he wished, and thought it better to make some statement than to be completely silent, which they always say is a disgrace to a philosopher.

For what, I ask, is the proportion of titillation of the hearing, a corporeal thing, to that unbelievable pleasure, which we feel totally within the mind from harmonic consonances? Surely if any pleasure does come from the titillation, the chief participant in that pleasure is the organ which undergoes the titillation?

For it seemed to me that every sense should be defined in this way, in the D io p tric s because the particular sensation is complete, generating pleasure or pain, when the emanation of the organ which is ordained for that sensation, as it is affected by the external circumstance, comes within the tribunal of the common sense, by the passage of the spirits.

Yet in fact in the hearing of consonant notes or sounds, what parts of the pleasure, I ask, are attached to the ears? Surely we are pained sometimes by our ears, when we gape at what we hear, and put a hand in the way of excessive noises; yet we are no less eager to perceive consonances, and our hearts leap within us? Add the fact that this explanation deduced from the motion applies particularly to unison, whereas it is not unison which is especially pleasurable, but other consonances, and their combination.

Much can be adduced to overthrow this explanation which has been adduced for the pleasure of consonances, which I refrain for the present from setting out in too much detail. I emphasize a single point, which I have already touched on above, and which can represent the whole: that the operations and motions of bodies, which imitate the harmonic proportions, are on the side of the soul and the mind, assigning them a cause for their delight in consonance.

Nor is the authority of the ancients against them. When they defined soul now as motion, now as harmony, it was not so much that they spoke absurdly as that they were interpreted inappropriately, since in difficult matters there often lurk mystical senses concealed beneath the husks of the words.

Indeed the philosophy of Timaeus the Locrian on the composition of the soul from harmonic proportions, mentioned in the preamble, was refuted by Aristotle in the sense conveyed by the actual words; but I should not dare to affirm that there is nothing lurking in those writings but what the actual words convey.

On the contrary I think no-one will deny that the author at least holds what I here ascribe to him, that it is Mind or the human intellect by the judgement or instinct of which the sense of hearing discriminates pleasant, that is consonant proportions from the unpleasant and dissonant, especially if he ponders carefully that proportions are entities of Reason, perceptible by reason alone, not by sense, and that to distinguish proportions, as form, from that which is proportioned, as matter, is the work of Mind,

From the Now since we have expounded two properties of the regular knowledge-finrures,^” the knowledge-producing constructibility of the sides in producing ^ 0 1 0 /

construction each case, and the congruence of those which are wholly linked to of the figures,

each Other, which clearly do not both apply over the same range, our axioms refer chiefly to constructions, because that is more closely associated with the proportions of the motions, from which sounds are also derived.

For congruence belongs to figures as wholes; whereas motions (in which harmonic proportions occur) extend in a straight line the sides of the figure from which they are derived (since generally all of them are considered as rectilinear) and thus undo and destroy their own figure, as serpents do their mother.

A figure, insofar as it is congruent, divides a complete circle into parts: the harmonic proportions extend the divided circle into a straight line, and cancel the effect of the division made by the figure.

Thus consonances along with constructible figures reach to infinity: congruent figures are limited by the twelvefold number.

Lastly any figure makes a single division of a circle; but the parts established in a circle always make two consonances with the whole.

Although in fact the argument in this third Book will be more And from concerned with the knowledge-producing construction of sides than congruence, with the congruence of complete figures, nevertheless on account of their close relationship the latter will not be neglected in its place.

For first the Latin meaning of the word congruence, if you make a thorough investigation, is the same as that of the Greek word harmony—the words with which we shall deal in this Book—except that usage has quickly made a distinction between these words from the subjects to which they refer.

Secondly, the congruence of figures imparts a certain congruence to motions (with which this and the fifth Book will be concerned).

Thirdly, although we are examining not so much the whole figure as one side of it, and it is the part which that side cuts off which is consonant, yet at the same time it is also true that we are not so much considering the size of the part of the circle which is intercepted as the nature of the figure by which that is done, whether it is constructible and congruent, or the contrary. For any figure has, from its angles through which it was allotted congruence in Book II, also acquired a construction in Book I.

The examination of the congruence of figures is therefore not to be dissociated from harmonies.