CHAP. I. Of Sound in General.

IN General (to pass by what is not pertinent to this Design) Sence and Experience confirm these following Properties of Sound.

1. All Sound is made by Motion, viz. by Percussion with Collision of the Air.

2. That Sound may be propaga­ted, and carried to Distance, it re­quires a Mdium by which to pass.

3. This Medium (to our Purpose) is Air.

4. As far as Sound is propagated a­long the Medium; so far also the Mo­tion passeth. For (if we may not say that the Motion and Sound are one and the same thing, yet at least) it is [Page 2]necessarily consequent, that if the Motion cease, the Sound must also cease.

5. Sound, where it meets with no Obstacle, passeth in a Sphere of the Medium, greater or less, according to the Force and Greatness of the Sound: Of which Sphere the sonorous Body is as the Centre.

6. Sound, so far as it reacheth, pas­sech the Medium, not in an Instant, but in a certain uniform Degree of Ve­locity, calculated by Gassendus, to be about the Rate of 276 Paces, in the space of a second Minute of an Hour. And where it meets with any Obstacle, it is subject to the Laws of Reflexi­on, which is the Cause of Eccho's, Meliorations, and Augmentations of Sound.

7. Sound, i. e. the Motion of Sound, or sounding Motion, is carried through the Medium or Sphere of Activity, with an Impetus or Force which shakes the [Page 3]free Medium, and strikes and shakes every Obstacle it meets with, more or less, according to the vehemency of the Sound, and Nature of the Obsta­cle, and Nearness of it to the Centre, or sonorous Body. Thus the impetu­ous Motions of the Sound of Thunder, or of a Cannon, shake all before it, even to the breaking of Glass Win­dows, &c.

8. The Parts of the sounding Bo­dy are moved with a Motion of Trem­bling, or Vibration, as is evident in a Bell or Pipe, and most manifest in the string of a musical Instrument.

9. This Trembling, or Vibration, is either equal and uniform, or else un­equal and irregular; and again, swif­ter or slower, according to the Con­stitution of the sonorous Body, and Quality and Manner of Percussion; and from hence arise Differences of Sounds.

10. The Trembling, or Vibration [Page 4]of the sonorous Body, by which the particular Sound is constituted and dis­criminated, is impressed upon, and carried along the Medium in the same Figure and Measure, otherwise it would not be the same Sound, when it ar­rives at a more distant Ear, i. e. the Tremblings and Vibrations (which may be called Undulations) of the Air or Medium, are all along of the same Velocity and Figure, with those of the sonorous Body, by which they are caused.

The Differences of Sounds, as of one Voice from another, &c. (besides the Difference of Tune, which is cau­sed by the Difference of Vibrations) arise from the Constitution and Figure, and other Accidents of the sonorous Body.

11. If the sonorous Body be requi­sitely constituted, i. e. of Parts solid, or tense, and regular, fit, being struck, to receive and express the Tremulous [Page 5]Motions of Sound, equally and swift­ly, then it will render a certain and even Harmonical Tone or Tune, re­ceived with Pleasure, and judged and measured by the Ear: Otherwise it will produce an obtuse or uneven Sound, not giving any certain or discernable Tune.

Now this Tune, or Tuneable sound, [...], i. e. [...], An agreeable Cadence of Voice, at one Pitch or Tension. This tune­able Sound (I say) as it is capable of other Tensions towards Acuteness, or Gravity, i. e. the Tensions greater or less, the Tune graver or more acute, i. e. lower or higher, is the first Matter or Element of Musick. And this Harmonick Sound comes next to be considered.

CHAP. II. Of Sound Harmonick.

THE first and great Principle upon which the Nature of Harmonical Sounds is to be found out and discovered, is this: That the Tune of a Note (to speak in our vulgar Phrase) is constituted by the Measure and Proportion of Vibrations of the sonorous Body; I mean, of the Velocity of those Vibrations in their Recourses.

For, the frequenter the Vibrations are, the more acute is the Tune; the slower and fewer they are in the same Space of Time, by so much the more grave is the Tune. So that any given Note of a Tune, is made by one cer­tain Measure of Velocity of Vibrati­ons; viz. Such a certain Number of [Page 7]Courses and Recourses. e. g. of a Chord or String, in such a certain Space of Time, doth constitute such a certain determinate Tune. And all such Sounds as are Unisons, or of the same Tune with that given Note, though made upon whatsoever different Bodies, (as String, Bell, Pipe, Larynx, &c.) are made with Vibrations or Trem­blings of those Bodies, all equal each to other. And whatsoever Tuneable Sound is more acute, is made with Vibrations more swift; and whatsoe­ver is more grave, is made with more slow Vibrations: And this is universal­ly agreed upon, as most evident to Ex­perience, and will be more manifest through the whole Theory.

And, That the Continuance of the Sound in the same Tune, to the last, (as may be perceived in Wire-strings, which being once struck will hold their Sound long) depends upon the Equa­lity of Time of the Vibrations, from [Page 8]the greatest Range till they come to cease: And this perfectly makes out the following Theory of Consonancy, and Dissonancy.

Some of the Ancient Greek Authors of Musick, took notice of Vibrations: And that the swifter Vibrations caused acuter, and the slower, graver Tones. And that the Mixture, or not Mixture of Motions creating several Intervals of Tune, was the Reason of their being concord or discord. And likewise, they found out the several Lengths of a Mo­nochord, proportioned to the several Intervals of Harmonick Sounds: But they did not make out the Equality of Measure of Time of the Vibrations last spoken of, neither could be prepa­red to answer such Objections, as might be made against the Continuity of the sameness of Tune, during the Conti­nuance of the Sound of a String, or a Bell, after it is struck. Neither did any of them offer any Reasons for the [Page 9]Proportions assigned, only it is said, that Pythagoras found them out by Chance.

But now, These (since the Acute Galileo hath observed, and discovered the Nature of Pendulums) are easie to be explained, which I shall do, pre­mising some Consideration of the Properties of the Motions of a Pen­dulum.

Hang a Plumbet C on a String or Wire, fixed at O. Bear C to A: Then let it range freely, and it will move toward B, and from thence swing back towards A. The Moti­on from A to B, I call the Course, and [Page 10]back from B to A, the Recourse of the Pendulum, making almost a Semi-Circle, of which O is the Centre. Then suffering the Pendulum to move of it self forwards and backwards, the Range of it will at every Course and Recourse abate, and diminish by degrees, till it come to rest perpendicu­lar at OC.

Now that which Galileo first obser­ved, was, that all the Courses and Re­courses of the Pendulum, from the greatest Range through all Degrees till it came to rest, were made in E­qual Spaces of Time. That is, e. g. The Range between A and B, is made in the same Space of Time, with the Range between D and E, the Plumbet moving swifter between A and B, the greater Space; and slower between D and E, the lesser; in such Proportions, that the Motions between the Terms AB and DE, are performed in Equal space of time.

And here it is to be noted, that where-ever in this Treatise, the swift­ness or slowness of Vibrations is spoke of, it must be always understood of the Frequency of their Courses and Recourses, and not of the Motion by which it passeth from one side to an­other. For it is true, that the same Pendulum under the same Velocity of Returns, moves from one side to the other, with greater or less Velocity, according as the Range is greater or less.

And hence it is, that the Librations of a Pendulum are become so excellent, and usefull a Measure of Time; espe­cially when a second Observation is ad­ded, that, as you shorten the Pendulum, by bringing C nearer to its Centre O, so the Librations will be made propor­tionably in a shorter Measure of Time, and the Contrary if you lengthen it. And this is found to hold in a Duplicate Proportion of Length to Velocity. [Page 12]That is, the Length quadrupled, will subduple the Velocity of Vibrations: And the Length subquadrupled, will duple the Vibrations, for the Proporti­on holds reciprocally. As you add to the Length of the Pendulum, so you diminish the Frequency of Vibrations, and increase them by shortning it.

Now therefore to make the Courses of a Pendulum doubly swift, i. e. to move twice in the same Space of Time, in which it did before move once; you must subquadruple the Length of it, i. e. make the Pendulum but a Quarter so long as it was before. And to make the Librations doubly slow, to pass once in the Time they did pass twice; you must quadruple the Length; make the Pendulum four Times as long as it was before, and so on in what Proportion you please.

Now to apply this to Musick, make two Pendulums, AB and CD, fasten together the Plumbets B and D, and [Page 13]Stretch them at Length, (fixing the Cen­ters A and C.) Then, being struck, and put into Motion; the Vibrations, which before were Distinct, made by AB, and CD, will now be United (as of one Entire String) both backward and forward, between E and F. Which Vibrations (retaining the aforesaid Ana­logy to a Pendulum) will be made in e­qual Spaces of Time, from the first to the last; i. e. from the greatest Range to the least, until they cease. Now, this being a double Pendulum, to Subduple the Swiftness of the Vibrations, you do but Double the Length from A to C, which will be Quadruple to AB. The lower Figure is the same with that above, only the Plummets taken off.

And here you have the Nature of the String of a Musical Instrument, resembling a double Pendulum moving upon two Centers, the Nut and the Bridge, and vibrating with the greatest Range in the Middle of its Length; and the Vibrations equal even to the last, which must make it keep the same Tune so long as it sounds. And be­cause it doth manifestly keep the same Tune to the last, it follows that the Vibrations are equal; Confirming one another by two of our Senses: in that we see the Vibrations of a Pendulum move equally; and we hear the Tune of a String, when it is struck, continue the same.

The Measure of Swiftness of Vibra­tions of the String or Chord, as hath been said, constitutes and determines the Tune, as to Acuteness and Gravi­ty of the Note which it sounds: And the Lengthning or Shortning of the String, under the same Tension, deter­mines [Page 15]the Measure of the Vibrations which it makes. And thus, Harmony comes under Mathematical Calculati­ons of Proportions, of the Length of Chords; of the Measure of Time in Vibrations; of the Intervals of tuned Sounds. As the Length of one Chord to another, Caeteris paribus, I mean, be­ing of the same Matter, Thickness and Tension; so is the Measure of the Time of their Vibrations. As the Time of Vibrations of one String to another, so is the Interval or Space of Acuteness or Gravity of the Tune of that one, to the Tune of the other: And conse­quently, as the Length is (Caeteris pa­ribus) so is the determinate Tune.

And upon these Proportions in the Differences, of Lengths, of Vibrations, and of Acuteness and Gravity; I shall insist all along this Treatise, very large­ly and particularly, for the full Infor­mation of all such ingenious Lovers of Musick, as shall have the Curiosity to [Page 16]inquire into the Natural Causes of Har­mony, and of the Phaenomena which occurr therein, though otherwise, to the more learned in Musick and Mathe­matical Proportions, all might be ex­pressed very much shorter, and still be more shortned by the help of Sym­bols.

And here we may fix our Foot: Concluding, that what is evident to Sence, of these Phaenomena, in a Chord, is equally (though not so discernably) true of the Motions of all other Bodies which render a tuneable Sound, as the Trembling of a Bell or Trumpet, the forming of the Larynx in our selves, and other Animals, the Throat of Pipes and of those of an Organ, &c. All of them in several Proportions sensibly trembling and impressing the like Un­dulations of the Medium, as is done by the several (more manifest) Vibrati­ons of Strings or Chords.

In these other Bodies, last spoken of, [Page 17]we manifestly see Reason of the Diffe­rence of the Swiftness of their Vibrati­ons (though we cannot so well measure them) from their Shape, and other Ac­cidents in their Constitution; and chiefly from the Proportions of their Magnitudes; the Greater generally vi­brating slower, and the Less more swiftly, which give the Tunes accor­dingly. We see it in the Greatness of a String. A greater and thicker Chord will give a graver and lower Tone than one that is more slender, of the same Tension and Length; but they may be made Unison by altering their Length and Tension.

Tension is proper to Chords or Strings (except you will account a Drumm for a Musical Instrument, which hath a Tension not in Length, but in the whole Surface) as when we wind up, or let down the Strings, i. e. give them a greater or less Tension, in tuning a Viol, Lute, or Harpsichord, [Page 18]and is of great Concern, and may be measured by hanging Weights on the String to give it Tension; but not ea­sily, nor so certainly.

But the Lengths of Chords (because of their Analogy to a Pendulum) is chiefly considered, in the Discovery of the Proportions which belong to Har­mony, it being most easie to measure and design the Parts of a Monochord, in relation to the whole String; and therefore all Intervals in Harmony may first be described, and understood, by the Proportions of the Length of Strings, and consequently of their Vi­brations. And it is for that reason, that in this Treatise of the Grounds of Harmony, Chords come so much to be considered, rather than other sounding Bodies, & those, chiefly in their Propor­tions of Length. It is true, that in Wind-Instruments, there is a regard to the Length of Pipes, but they are not so well accommodated (as are Chords) to be [Page 19]examined, neither are their Vibrations, nor the Measure of themso manifest.

There are some Musical Sounds, which seem to be made, not by Vi­brations but by Pulses, as by whisking swiftly over some Silk or Camblet­stuffs, or over the Teeth of a Comb, which render a kind of Tune more Acute or Grave, according to the Swift­ness of the Motion. Here the Sound is made, not by Vibrations of the same Body, but by Percussion of seve­ral Equal, and Equidistant Bodies; as Threads of the Stuff, Teeth of the Comb, passing over them with the same Velocity as Vibrations are made. It gives the same Modification to the Tune, and to the Undulations of the Ayr, as is done by Vibrations of the same Measure; the Multiplicity of Pul­ses or Percussions, answering the Multi­plicity of Vibrations. I take this notice of it, because others have done so; but I think it to be of no use in Musick.

CHAP. III. Of Consonancy and Dissonancy.

COnsonancy and Dissonancy are the Result of the Agreement, mixture or uniting (or the contrary) of the undulated Motions of the Ayr or Medium, caused by the Vibrations by which the Sounds of distinct Tunes are made. And those are more or less capable of such Mixture or Co-in­cidence according to the Proportion of the Measures of Velocity in which they are made, i.e. according as they are more or less commensurate. This I might well set down as a Postulatum. But I shall by several Instances indea­vour to illustrate the undulating Mo­tions or Undulations of the Ayr; and confirm what is said of their Agree­ment and Disagreements. And first [Page 41]the Undulations, by somewhat we see in other Liquids.

Let a Stone drop into the Middle of a small Pond of standing Water when it is quiet, you shall see a Motion forthwith impressed upon the Water, passing and dilating from that Center where the Stone fell, in circular Waves one within another, still propagated from the Center, spreading till they reach and dash against the Banks, and then returning, if the force of the Mo­tion be sufficient, and meeting those inner Circles which pursue the same Course, without giving them any Check.

And if you drop a Stone in another place, from that Centre will likewise spread round Waves; which meeting the other, will quietly pass them, each moving forwards in its own proper Figure.

The like is better experimented in Quick-silver, which being a more [Page 42]dense Body, continues its Motions longer, and may be placed nearer your Eye. If you try it in a pretty large round Vessel, suppose of a foot Diameter, the Waves will keep their own Motion forward and backward, and quietly pass by one another as they meet. Something of this may be seen in a long narrow Passage, where there is not room to advance in Circles.

Make a wooden Trough or long Box, suppose of two Inches broad, and two deep, and twenty long. Fill it three Quarters or half full of Quick-silver, and place it Horizontally, when it is at quiet, give it with your Finger a little patt at one End, and it will impress a Motion of a ridgid Wave across, which will pass on to the other End, and dashing against it, return in the same Manner, and dash against the hether End, and go back again, and thus backward and forward, till [Page 43]the Motion cease. Now if after you have set this Motion on foot, you cause such another, you shall see each Wave keep its regular Course; and when they meet one another, pass on without any Reluctancy.

I do not say these Experiments are full to my purpose, because these be­ing upon single Bodies, are not suffi­cient to express the Disagreements of Disproportionate Motions caused by different Vibrations of different sound­ing Bodies; but these may serve to illustrate those invisible Undulations of Ayr. And how a Voice reflected by the Walls of a Room, or by Ec­cho being of adequate Vibrations, re­turns from the Wall, and meets the commensurate Undulations passing forwards, without hindring one ano­ther.

But there are Instances which fur­ther confirm the Reasons of Conso­nancy and Dissonancy, by the Mani­fest [Page 44]agreeing or disagreeing Measures of Motions already spoken of.

It hath been a common Practice to imitate a Tabour and Pipe upon an Organ. Sound together two discord­ing Keys (the base Keys will shew it best, because their Vibrations are slow­er) let them, for Example, be Ga­mut with Gamut sharp, or F Faut sharp, or all three together. Though these of themselves should be exceeding smooth and well voyced Pipes; yet, when struck together, there will be such a Battel in the Ayr between their disproportioned Motions, such a Clat­ter and Thumping, that it will be like the beating of a Drum, while a Jigg is played to it with the other hand. If you cease this, and sound a full Close of Concords, it will appear surprizingly smooth and sweet, which shews how Discords well placed, set off Concords in Composition. But I bring this Instance to shew, how strong [Page 45]and vehement these undulating Moti­ons are, and how they correspond with the Vibrations by which they are made.

It may be worth the while, to re­late an Experiment upon which I hap­pened. Being in an Arched sound­ing Room near a shrill Bell of a House Clock, when the Alarm struck, I whistled to it, which I did with ease in the same Tune with the Bell, but, indeavouring to whistle a Note higher or lower, the Sound of the Bell and its cross Motions were so predominant, that my Breath and Lips were check'd so, that I could not whistle at all, nor make any Sound of it in that discor­ding Tune. After, I sounded a shrill whistling Pipe, which was out of tune to the Bell, and their Motions so clashed, that they seemed to sound like switching one another in the Ayr.

Galileo, from this Doctrine of Pen­dulums, easily and naturally explains [Page 46]the so much admired sympathy of Consonant strings; one (though un­touch'd) moving when the other is struck. It is perceptible in Strings of the same, or another Instrument, by trembling so as to shake off a Straw laid upon the other String: But in the same Instrument, it may be made ve­ry visible, as in a Bass-viol. Strike one of the lower Strings with the Bow, hard and strong, and if any of the other Strings be Unison or Octave to it, you shall plainly see it vibrate, and continue to doe so, as long as you continue the Stroke of your Bow, and, all the while, the other Strings which are dissonant, rest quiet.

The Reason hereof is this. When you strike your String, the Progressive sound of it strikes and starts all the other Strings, and every of them makes a Movement in its own pro­per Vibration. The Consonant string, keeping measure in its Vibrations with [Page 47]those of the sounding String hath its Motion continued, and propagated by continual agreeing Pulses or Strokes of the other. Whereas the Remainder of the Dissonant strings having no help, but being checked by the cross Mo­tions of the sounding String, are con­strained to remain still and quiet. Like as, if you stand before a Pendulum, and blow gently upon it as it passeth from you, and so again in its next Courses keeping exact time with it, it is most easily continued in its Motion. But if you blow irregularly in Measures different from the Measure of the Mo­tion of the Pendulum, and so most frequently blow against it, the Moti­on of it will be so checked, that it must quickly cease.

And here we may take notice, (as hath been hinted before) that this also confirms the aforesaid Equality of the Time of Vibrations to the last, for that the small and weak Vibrations of [Page 48]the sympathizing String are regulated and continued by the Pulses of the greater and stronger Vibrations of the sounding String, which proves, that notwithstanding that Disparity of Range, they are commensurate in the Time of their Motion.

This Experiment is ancient: I find it in Aristides Quintilianus a Greek Au­thour, who is supposed to have been contemporary with Plutarch. But the Reason of it deduced from the Pendu­lum, is new, and first discovered by Galileo.

It is an ordinary Trial, to find out the Tune of a Beer-glass without stri­king it, by holding it near your Mouth, and humming loud to it, in several single Tunes, and when you at last hitt upon the Tune of the Glass, it will tremble and Eccho to you. Which shews the Consent and Uniformity of Vibrations of the same Tune, though in several Bodies.

To close this Chapter. I may con­clude that Consonancy is the Passage of several Tuneable sounds through the Medium, frequently mixing and uniting in their undulated Motions, caused by the well proportioned com­mensurate Vibrations of the sonorous Bodies, and consequently arriving smooth, and sweet, and pleasant to the Ear. On the contrary, Dissonan­cy is from disproportionate Motions of Sounds, not mixing, but jarring and clashing as they pass, and arriving to the Ear Harsh, and Grating, and Of­fensive. And this, in the next Chap­ter shall be more amply explained.

Now, what Concords and Discords are thus produced, and in use, in or­der to Harmony, I shall next consi­der.

CHAP. IV. Of Concords.

COncords are Harmonic sounds, which being joyned please and delight the Ear; and Discords the Contrary. So that it is indeed the Judgment of the Ear that determines which are Concords and which are Discords. And to that we must first resort to find out their Number. And then we may after search and examine how the natural Production of those Sounds, disposeth them to be pleasing or unpleasant. Like as the Palate is absolute Judge of Tasts, what is sweet, and what is bitter, or sowr, &c. though there may be also found out some natural Causes of those Quali­ties. But the Ear being entertained with Motions which fall under exact Demonstrations of their Measures, the [Page 51]Doctrine hereof is capable of being more accurately discovered.

First then, (setting aside the Unison Concord, which is no Space or Inter­val, but an Identity of Tune) the Ear allows and approves these following Intervals, and only these for Concords to any given Note, viz. the Octave or Eighth, the Fifth, then the Fourth, (though by later Masters of Musick de­graded from his Place) then the Third Major, the Third Minor, the Sixth Ma­jor, and the Sixth Minor. And also such, as in the Compass of any Voice or In­strument beyond the Octave, may be compounded of these, for such those are, I mean compounded, and only the for­mer Seven are simple Concords; not but that they may seem to be com­pounded, viz. the greater of the less with­in an Octave, and therefore may be called Systems, but they are Originals. Whereas beyond an Octave, all is but Repetition of these in Compound with [Page 52]the Eighth, as a Tenth is an Eighth and a Third; a Twelfth is an Eighth and a Fifth; a Fifteenth is Disdiapa­son, i.e. two Octaves, &c.

But notwithstanding this Distinction of Original and Compound Concords; and, tho' these compounded Concords are found, and discerned by their Ha­bitude to the Original Concords com­prehended in the System of Diapason; (as a Tenth ascending is an Octave above the Third, or a Third above the Octave; a Twelfth is an Octave to the Fifth, or a Fifth to the Eighth, a Fifteenth is an Eighth above the Oc­tave, i.e. Disdiapason two Eighths, &c.) yet they must be own'd, and are to be esteemed good and true Concords, and equally usefull in Melody, espe­cially in that of Consort.

The System of an Eighth, contain­ing seven Intervals, or Spaces, or De­grees, and eight Notes reckoned inclu­sively, as expressed by eight Chords, [Page 53]is called Diapason, i. e. a System of all intermediate Concords, which were anciently reputed to be only the Fifth and the Fourth, and it comprehends them both, as being compounded of them both: And now, that the Thirds and Sixths are admitted for Concords, the Eighth contains them also: Viz. a Third Major and Sixth Minor, and a­gain a Third Minor and Sixth Major. The Octave being but a Replication of the Unison, or given Note below it, and the same, as it were in Minuture, it closeth and terminates the first perfect System, and the next Octave above [...] ascends by the same Intervals, and i [...] in like manner compounded of them, and so on, as far as you can proceed upwards or downwards with Voices or Instruments, as may be seen in an Organ, or Harpsichord. It is there­fore most justly judged by the Ear, to be the Chief of all Concords, and is the only Consonant System, which [Page 54]being added to it self, still makes Concords.

And to it all other Concords agree, and are Consonant, though they do not all agree to each other; nor any of them make a Concord if added to it self, and the Complement or Residue of any Concord to Diapason, is also Concord.

The next in Dignity is the Fifth, then the Fourth, Third Major, Third Minor, Sixth Major, and lastly Sixth Minor; all taken by Ascent from the Unison or given Note.

By Unison is meant, sometimes the Habitude or Ration of Equality of two Notes compared together, being of the very same Tune. Sometimes (as here) for the given single Note to which the Distance, or the Rations of other Intervals are compared. As, if we consider the Relations to Gamut, to which A re is a Tone or Second, B mi a Third, C a Fourth, D a Fifth, &c. [Page 55]We call Gamut the Unison, for want of a more proper Word. Thus C fa ut, or any other Note to which other Intervals are taken, may be called the Unison.

And the Reader may easily discern, in which Sense it is taken all along by the Coherence of the Discourse.

I come now to consider the natu­ral Reasons, why Concords please the Ear, by examining the Motions by which all Concords are made, which having been generally alledged in the beginning of the third Chapter, shall now more particularly be discussed.

And here I hope the Reader will pardon some Repetition in a Subject, that stands in need of all Light that may be, if, for his easie and more steady Progress, before I proceed, I call him back to a Review and brief Summary of some of those Notions, which have been premis'd and consi­dered more at large. I have shewed,

[Page 56] 1. That Harmonick Sound or Tune is made by equal Vibrations or Tremblings of a Body fitly constitu­ted.

2. That those Vibrations make their Courses and Recourses in the same Measure of Time; from the greatest Range to the lesser, till they come to rest.

3. That those Vibrations are under a certain Measure of Frequency of Courses and Recourses in a given Space of Time.

4. That if the Vibrations be more frequent, the Tune will be proportio­nably more Acute: if less frequent, more Grave.

5. That the Librations of a Pendu­lum become doubly frequent, if the Pendulum be made four times shorter; and twice flower, if the Pendulum be four times longer.

6. That a Chord, or String of a Musical Instrument, is, as a double [Page 57] Pendulum, or two Pendulums tacked together at length, and therefore hath the same Effects by dupling; as a Pen­dulum by quadrupling, i. e. by du­pling the Length of the Chord, the Vi­brations will be subdupled, i. e. be half so many in a given Time. And by subdupling the Length of the Chord, the Vibrations will be dupled, and proportionably so in all other Mea­sures of Length, the Vibrations bear­ing a Reciprocal proportion to the Length.

7. That these Vibrations impress a Motion of Undulation or Trembling in the Medium (as far as the Motion extends) of the same Measure with the Vibrations.

8. That if the Motions made by different Chords be so commensurate, that they mix and unite; bear the same Course either altogether, or alternate­ly, or frequently: Then the Sounds of those different Chords, thus mixing, [Page 58]will calmly pass the Medium, and ar­rive at the Ear as one Sound, or near the same, and so will smoothly and evenly strike the Ear with Pleasure, and this is Consonancy, and from the want of such Mixture is Dissonancy. I may add, that as the more frequent Mixture or Coincidence of Vibrations, render the Concords generally so much the more perfect: So, the less there is of Mixture, the greater and more harsh will be the Discord.

From the Premisses, it will be easie to comprehend the natural Reason, why the Ear is delighted with those forenamed Concords: and that is, be­cause they all unite in their Motions often, and at the least at every sixth Course of Vibration, which appears from the Rations by which they are constituted, which are all contained within that Number, and all Rations contained within that Space of Six, make Concords, because the Mixture [Page 59]of their Motions is answerable to the Ration of them, and are made at or before every Sixth Course. This will appear if we examine their Motions. First, how and why the Unisons agree so perfectly; and then finding the rea­son of an Octave, and fixing that, all the rest will follow.

To this purpose, strike a Chord of a sounding Instrument, and at the same Time, another Chord supposed to be in all respects Equal, i. e. in Length, Matter, Thickness, and Ten­sion. Here then, both the Strings give their Sound; each Sound is a certain Tune; each Tune is made by a cer­tain Measure of Vibrations: the same Vibrations are impressed upon, and carried every way along the Medium, in Undulations of the same Measure with them, until the Sounds arrive at the Ear. Now the Chords being sup­posed to be equal in all respects; it follows, that their Vibrations must be [Page 60]also equal, and consequently move in the same Measure, joyning and uni­ting in every Course and Recourse, and keeping still the same Equality, and Mixture of Motions of the String, and in the Medium. Therefore the Habitude of these two Strings is called Unison, and is so perfectly Conso­nant, that it is an Identity of Tune, there being no Interval or Space be­tween them. And the Ear can hard­ly judge, whether the Sound be made by two Strings, or by one.

But Consonancy is more properly considered, as an Interval, or Space between Tones of different Acuteness or Gravity. And amongst them, the most perfect is that which comes near­est to Unison, (I do not mean betwixt which there is the least Difference of Interval: but, in whose Motions there is the greatest Mixture and Agreement next to Unison.) The Motions of two Unisons are in Ration of 1 to 1, or [Page 61]of Equality. The next Ration in whole Numbers is 2 to 1, Duple. Divide a Monochord in two Equal parts, half the Length compared to the whole, being in Subduple Ration, will make double Vibrations, making two Recourses in the same time that the other makes one, and so uni­ting and mixing alternately, i. e. eve­ry other Motion. Then comparing the Sounds of these two, and the half will be found to sound an Octave to the whole Chord. Now the Octave (ascending from the Unison) being thus found and fixed to be in duple Proportion of Vibrations, and subdu­ple of Length; consequently the Pro­portions of all other Intervals are easi­ly found out.

They are found out by resolving or dividing the Octave into the mean Rations which are contained in it. Euclid, in his Sectio Canonis, Theorem. 6. gives two Demonstrations to prove, [Page 62]that Duple Ration contains, and is composed of the two next Rations, viz. Sesquialtera and Sesquitertia. There­fore an Octave which is in Duple Ra­tion 2 to 1 is divided into, and com­posed of a Fifth, whose Ration is found to be Sesquialtera 3 to 2; and a Fourth, whose Ration is Sesquitertia 4 to 3. In like manner, Sesquialtera is composed of Sesquiquarta and Ses­quiquinta. That is, a Fifth 3 to 2 may be divided into a Third Major 5 to 4, and a Third Minor 6 to 5; &c.

There is an easie way to take a view of the Mean Rations, which may be contained in any Ration gi­ven, by transferring the Prime or Ra­dical Numbers of the given Ration in­to greater Numbers of the same Ra­tion, as 2 to 1 into 4 to 2, or 6 to 3, &c. which have the same Ration of Duple. Again, 3 to 2 into 6 to 4, which is still Sesquialtera. Now in 4 to 2. the Mediety is 3. So that [Page 63]4 to 3 and 3 to 2 are comprehended in 4 to 2; that is, a Fourth and a Fifth are comprehended in an Eighth. In 6 to 4 the Mediety is 5, so 6 to 4 contains 6 to 5 and 5 to 4; i. e. a Fifth contains the 2 Thirds. Let 6 to 3 be the Octave, and it contains 6 to 5 Third less, 5 to 4 Third Ma­jor, and 4 to 3, a Fourth, and hath two Medieties, 5 and 4. Of this I shall say more in the next Chapter.

These Rations express the Difference of Length in several Strings which make the Concords; and consequent­ly the Difference of their Vibrations. Take two Strings A B, in all other Respects equal, and compare their Lengths, which if equal, make Uni­son or the same Tune. If A be dou­ble in Length to B, i. e. 2 to 1, the Vibrations of B will be duple to those of A, and unite alternately, viz. at every Course, crossing at the Recourse, and give the Sound of an Octave to A.

If the Length of A be to that of B as 3 to 2, and consequently the Vi­brations as 2 to 3, their Sounds will consort in a Fifth, and their Motions unite after every second Recourse, i. e. at every other or third Course.

If A to B, be as 4 to 3, they sound a Fourth, their Motions uniting after every third Recourse, viz. at every fourth Course.

If A to B, be as 5 to 4, they sound a Ditone, or third Major, and unite after every fourth Recourse, i. e. eve­ry fifth Recourse.

If A to B, be as 6 to 5, they sound a Trihemitone, or Third Minor, uni­ting after every fifth Recourse, at e­very sixth Course.

Thus by the frequency of their be­ing mixed and united, the Harmony of joyned Concords is found so very sweet and pleasing; the Remoter be­ing also combined by their Relation to other Concords besides the Unison. [Page 65]The greater Sixth, 5 to 3, is within the Compass of Rations between 1 and 6; but, I confess, the lesser Sixth, 8 to 5, is beyond it: but is the Complement of 6 to 5 to an Octave, and makes a better Concord by its Combinations with the Octave, and Fourth from the Unison; having the Relation of a Third Minor to One, and of a Third Major to the Other, and their Motions uniting accordingly. And the Sixth Major hath the same Advantage. Of these Com­binations I shall have occasion to say somewhat more, after I have made the Subject in hand as plain as I can.

I proposed the Collating of two se­veral Strings, to express the Consort which is made by them; but other­wise, these Rations are more certain­ly found upon the Measures of a Mo­nochord, taken, by being applied to the Section of a Canon, or a Rule of the Strings length divided into parts, as oc­casion requires: because there is no need [Page 66]so often to repeat Caeteris paribus, as is when several Strings are collated. And if you take the Rations as Fractions, it will be more easie to measure out the given Parts of a Monochord, or sin­gle String extended on an Instrument: Those parts of the String divided by a Moveable Bridge or Fret put under, and made to sound; That Sound, re­lated to the Sound of the Whole, will give the Interval sought after. Ex. gr. ½ of the Chord gives an Eighth, ⅔ give a Fifth, ¾ sound a Fourth, ⅘ sound a Third Major, ⅚ a Third Minor, ⅗ a Sixth Major, ⅝ a Sixth Minor: Now we thus express these Concords.

I said, that all Concords are in Ra­tions within the Number Six; and I may add, that all Rations within the Number Six, are Concords: Of which, take the following Scheme.

All that are Concords to the Uni­son, are also Concords to the Octave, And all that are Discords to the Uni­son, are Discords to the Octave. And some of the Intermediate Concords, are Concords one to another; as the two Thirds to the Fifth, and the Fourth to the two Sixths. So that the Unison, Third, Fifth, and Octave; or the U­nison, Fourth, Sixth, and Octave, may [Page 68]be sounded together to make a com­pleat Close of Harmony: I do not mean a Close to Conclude with, for the Plagal is not such; but a compleat Close, as it includes all Concords within the Compass of Diapason. A Scheme of which I have set down at the End of the foregoing Staff of five Lines, which containeth the Notes by which the aforesaid Concords are expressed. The former two which as­cend from the Unison, Gamut, by Third Major (or Minor) and Fifth, up to the Octave; are usually called Authentick, as relating principally to the Unison, and best satisfying the Ear to rest upon: The other two, which ascend by the Fonrth and Sixth Minor, (or Major) up to the same Octave, are called Plagal, as more combining with the Octave, seeming to require a more proper base Note, vzi. an Eighth below the Fourth, and therefore not making a good con­cluding Close: And on the continual [Page 69]shifting these, or often changing them, depends the Variety of Harmony (as far as Consonancy reacheth, which is but as the Body of Musick) in all Con­trapunct chiefly, but indeed in all Kinds of Composition. I do not ex­clude a Sprinkling of Discords; nor here medle with Ayr, Measure, and Rythmus, which are the Soul and Spi­rit of Musick, and give it so great a commanding Power. The Plagal Moods descend by the same Intervals, by which the Authentick ascend; which is by Thirds and Fifths; and the Au­thentick descend the same by which the Plagal ascend, viz. by Fourths and Sixths; one chiefly relating to the Uni­son, the other to the Octave.

But that, for which I described these full Closes, was chiefly, to give (as I promis'd) a larger account of the be­fore-mentioned Combinations of Con­cords, which increase the Consonan­cies of each Note, and make a won­derfull [Page 70]Variegation and Delightfulness of the Harmony.

Cast your Eye upon the First of them in the Authentick Scale; you will see that B mi hath 3 Relations of Conso­nnacy, viz. To the Unison, or given Note G; to the Fifth, and to the Octave: To the Unison as a Third Minor; to the Fifth as a Third Major; to the Octave a Sixth Major; so that its Motions joyn after every fifth Recourse, i. e. at every sixth Course, with the Unison; every fifth with the Diapente or Fifth; every sixth Course with the Octave. Then consider the Diapente, D sol re; as a Fifth to the Unison, it joyns with it every third Course; and as a Fourth to the Octave, they joyn every Fourth Course. Then, the Octave with the Unison, joyns after every second Vi­bration, i. e. at every Course.

Now take a Review of the Variety of Consonancies in these four Notes. Here are mixed together in one Con­sort [Page 71]the Rations of 2 to 1, 3 to 2, 4 to 3, 5 to 4, 6 to 5, 5 to 3. And just so it is in the other Closes, only changing alternately the Sixths.

You may see here, within the Space of three Intervals from the Unison, viz. 3d, 5th, and 8th; what a Con­course there is of Consonant Rations, to Variegate and give (as it were) a pleasant Purling to the Harmony with­in that Space. For now, all this Va­riety is formed within one System of Diapason, justly bearing that Name. But then, think what it will be, when the remote Compounded Concords are joyned to them; as when we make a full Close with both Hands upon an Organ, or Harpsichord, or when the higher Part of a Consort of Musick is reconciled to the lower, by the middle Parts; viz. the Treble to the Base, by the Mean and Tenor: And all this, refreshed by the Interchangings made between the Plagal and Authen­tick [Page 72]Moods. Add to all this, the In­finite Variety of Movement of some Parts, through all Spaces, while some Part moves slowly: And (as in Fuges) one part chasing and pursuing another.

The whole Reason of Consonancy, being founded upon the Mixture, and Uniting of the Vibrating Motions of se­veral Chords or sounding Bodies; it is fit, it should here be better explained and confirm'd. That their Mixtures accord to their Rations, it is easie to be computed: But it may be represented to your Eye.

Let VV be a Chord, and stand for the Unison: Let O O be a Chord half so long, which will be an Octave to the Unison, and the Vibrations double: Then I say, they will alternately, i. e. at every other Vibration unite: Let from A to B, be the Course of the Vi­bration, and from B to A the Recourse. Observing by the way, that (in rela­tion to the Figures mentioned in this Paragraph and the next, as also in the former Diagram of the Pendulum, Cap. 2. pag. 9.) When I say, [from B to A] and, [overtakes V, in A, &c.] I do there indeavour to express the mat­ter brief and perspicuous, without per­plexing the Figures with many Lines: and avoiding the Incumbrance of so many Cautions, whereby to distract the Reader: Yet I must always be un­derstood to acknowledge the continual Decrease of the Range of Vibrations be­tween A and B, while the Motion con­tinues; [Page 75]and by A and B, mean only the Extremities of the Range of all those Vibrations, both the First greatest, and also the Successive lessened, and gra­dually contracted Extremities of their Range. And the following Demon­stration proceeds and holds equally in both, being applied to the Velocity of Recourses, and not to the Compass of their Range, which is not at all here considered. Such a kind of Equity, I must sometimes in other parts of this Discourse, beg of the Candid Reader. To proceed therefore, I say, whilst V being struck, makes his Course from A to B; O (struck likewise) will have his Course from A to B, and Recourse from B to A. Next, whilst V makes Recourse from B to A; O is making its Course contrary, from A to B, but recourseth and overtakes V in A, and then they are united in A, and begin their Course together. So you see, that the Vibrations of Diapason unite [Page 76]alternately, joyning at every Course of the Unison, and crossing at the Re­course.

Thus also Diapente or Fifth having the Ration of 3 to 2, unites in like manner at every third Course of the Unison. Let the Chord DD be Dia­pente to the Unison V; whilst V cour­seth from A to B, the Chord D courseth from A to B, and makes half his Recourse as far as C; i. e. 3 to 2. Whilst V recourseth from B to A, D passeth from C to A, and back from A to B. Whilst V courseth again from A to B, D passeth from B to A, and back to C. Whilst V recourseth from B to A, D passeth from C to B, and back to A: And then they unite in A, beginning their Courses together, at every third Course of V. In like man­ner the rest of the Concords unite, at the 4th, 5th, 6th Course, accord­ing to their Rations, as might this same way be shewn; but it would take [Page 77]up too much room, and is needless; being made evident enough from these Examples already given.

Thus far the Rates and Measures of Consonance lead us on, and give us the true and demonstrable grounds of Harmony: But still it is not com­pleat without Discords and Degrees (of which I shall treat in another Chapter) intermixed with the Concords, to give them a Foyl, and set them off the better. For, (to use a homely resem­blance) That our Food, taken alone, though proper, and wholsome, and natural, may not cloy the Palate, and abate the Appetite; the Cook finds such kinds and varieties of Sawce, as quicken and please the Palate, and sharpen the Appetite, though not feed the Stomach: As Vinegar, Mustard, Pepper, &c. which nourish not, nor are taken alone, but carry down the Nourishment with better Relish, and assist it in Digestion. So the Practical [Page 78]Masters and Skilful Composers make use of Discords, judiciously taken, to relish the Consort, and make the Concords arrive much sweeter at the Ear, in all sorts of Descant; but most frequently in Cadence to a Close. In all which, the chief regard is to be had to what the Ear may expect in the Conduct of the Composition, and must be performed with Moderation and Judgment; which I now only mention, not intending to treat of Composing, which is out of my De­sign and Sphear, and would be too large; but my design is, to make these Grounds as plain as I can, thereby to gratifie those, whose Philosophical Learning, without previous Skill in Musick, will easily render them capa­ble of this Theory: And also, those Masters in Practick Musick, and Lo­vers of it, who, though wanting Phi­losophy, and the Latin and other For­reign Tongues, to read better Au­thors; [Page 79]yet, by the help of their know­ledge in Musick, may attain to under­stand the depth of the Grounds and Reasons of Harmony, for whose sakes it is done in this Language.

I shall conclude this Chapter with some Remarks, concerning the Names given to the several Concords: We call them Third, Fourth, Fifth, Sixth, and Eighth. Of these, the Third's being Two, and Sixth's being also Two, want better distinguishing Names. To call them Flat and Sharp Thirds, and Flat and Sharp Sixths is not enough, and lies under a mistake; I mean, it is not a sufficient Distinction, to call the greater Third and Sixth, Sharp Third, and Sharp Sixth; and the les­ser, Flat. They are so, indeed, in a­scending from the Unison; but in de­scending they are contrary; for to the Octave, that greater Sixth is a lesser Third, and the greater Third is a les­ser Sixth; which lesser Third and Sixth [Page 80]cannot well be called Flat, being in a Sharp Key; Flat and Sharp therefore do not well distinguish them in Gene­ral. The lesser Third from the Octave being sharp, and the greater Sixth flat. So, from the Fifth descending by Thirds, if the First be a Minor Third, it is Sharp, and the other be­ing a Major Third, cannot be said to be Flat.

The other Distinction of them, viz. by Major and Minor, is more pro­per, and does well express which of them we mean. But still, the common and confused name of Third, if the Distinction of Major and Minor be not always well remembred, is apt to draw young Practitioners, who do not well consider, into another Errour. I would therefore call the greater Third (as the Greeks do) Ditone, i. e. of two whole Tones; and the Third Minor, Trihemitone, or Sesquitone, as consisting of three half Tones, (or ra­ther [Page 81]of a Tone and half a Tone) And this would avoid the mentioned Errour which I am going to describe.

It is a Rule in composing Consort Musick, that it is not lawful to make a Movement of two Unisons, or two Eights, or two Fifths together; nor of two Fourths, unless made good by the addition of Thirds in another Part: But we may move as many Thirds or Sixths together as we please. Which last is false, if we keep to the same sort of Thirds and Sixths; for the two Thirds differ one from ano­ther in like manner as the Fourth dif­fers from the Fifth. For in the same manner as the Eighth is divided into a Fifth and Fourth: so is a Fifth into a 3d Major and 3d Minor. Now call them by their right names, and, I say, it is not lawful to make a Movement of as many Ditones, or of as many Sesquitones as you please; and there­fore when you take the liberty spoken [Page 82]of, under the general names of Thirds, it will be found, that you mix Ditones and Trihemitones, and so are not con­cerned in the aforesaid Rule; and so the Movements of Sixths will be made with mixture and interchanges of 6th Major and 6th Minor, which is safe e­nough.

Yet, I confess, there is a little more liberty in moving Trihemitones and Di­tones, as likewise, either of the Sixths, than there is in moving Fourths or Fifths; and the Ear will bear it better. Nay, there is necessity, in a gradual Movement of Thirds, to make one Movement by two Trihemitones toge­ther in every Fourth, and Fifth, or Fourth disjunct: That is, twice in Dia­pason, or, at least, in two Fifths; as in Gamut Key proper. The natural Ascent will be, Ut Re Mi Fa Sol La: Now, to these join Thirds in Natural Ascent, and they will be, Mi Fa Sol La Fa Sol. Mi Fa Sol La Fa Sol Ʋs Re Mi Fa Sci La And thus it will [Page 83]be in other Cliffs, but with some va­riation, according to the place of the Hemitone. Here Fi [...] and [...] are two Trihemitones succeeding one ano­ther, and you cannot well alter them without disordering the Ascent, and disturbing the Harmony; because, where there is a Hemitone, the Tone below joined to it, makes a Trihemi­tone, and the next Tone above it, join­ed to it, makes the same. Thus you see the necessity of moving two Tribe­mitones together, twice in Diapason, or a 9th, in progression of Thirds, in Diatonic Harmony, but you cannot well go further.

Now, there is Reason, why two Tri­hemitones will better bear it, because of their different Relations, by which one Trihemitone is better distinguished from another, than one Octave, or one Fifth, or one Fourth from ano­ther.

In a third Minor, which hath two Degrees or Intervals, consisting of a Tone and Hemitone, the Hemitone may be placed either in the lower Space, and then generally is united to his 3d Major (which makes the Complement of it to a Fifth) downward, and makes a sharp Key; or else it may be placed in the upper Space, and then generally takes his 3d Major above, to make up the 5th upward, and con­stitute a Flat Key. And thus a Tritone is avoided both ways. I say, if the Hemitone, in the 3d Minor be below, then the 3d Major lies below it, and the Air is sharp. If the Hemitone be above, then the 3d Major lies above, and the Air is Flat. And thus the two Mi­nor Thirds joined in consequence of Movement, are differenced in their Re­lations, consequent to the place of the Hemitone; which variety takes off all Nauseousness from the Movement, and renders it sweet and pleasant.

You cannot so well and regularly make a Movement of Ditones, though it may be done sometimes, once or twice, or more, in a Bearing Passage (in like manner as you may sometimes use Discords) to give, after a little grating, a better Relish. The Skil­ful Artist may go farther in the use of Thirds and Discords, than is ordi­narily allowed.

I might enlarge this Chapter, by setting down Examples of the Lawful and Unlaw­ful Movements of Thirds Major and Minor, and of the Use of Discords; but, as I said before, my design is not to treat of Com­position: However you may cast your Eye upon these following Instances; and your own Observation from the best Masters will furnish you with the rest.

Lawful Movement of Thirds, Mix'd.

Unlawful Movement of Thirds Major.

That the Reader may not incurr any Mi­stake or Confusion, by several Names of the same Intervals, I have here set them down together, with their Rations.

Note, Whenever I mention Diesis without Distinction; I mean Diesis Minor, or Enharmonic: and when I to mention Comma; I mean Comma Majus, or Schism.

I should next treat of Discords, but because there will intervene so much use of Calculation, it is needful that (before I go further) I premise some account of Proportion in General, and apply it to Harmony.

CHAP. V. Of Proportion; and Applyed to Harmony.

WHereas it hath been said be­fore, That Harmonick Bo­dies and Motions fall under Nu­merical Calculations, and the Rati­ons of Concords have been already assign'd: It may seem necessary here (before we proceed to speak of Dis­cords) to shew the manner how to calculate the Proportions appertaining to Harmonick Sounds: And for this, I shall better prepare the Reader, by premising something concerning Pro­portion in General.

We may compare (i. e. amongst themselves) either (1.) Magnitudes, (so they be of the same kind;) Or (2.) the Gravitations, Motions, Velocities, [Page 89]Durations, Sounds, &c. from thence a­rising; or further, if you please, the Numbers themselves, by which the things Compared, are Explicated. And if these shall be Unequal, we may then consider, either, First, How much one of them Exceeds the other; or Se­condly, After what manner one of them stands related to the other, as to the Quotient of the Antecedent (or former Term) divided by the Consequent (or latter Term:) Which Quotient doth Expound, Denominate, or shew, how many times, or how much of a time, or times, one of them doth contain the other. And this by the Greeks is called [...], Ratio; as they are wont to call the Similitude, or Equality of Ratio's, [...], Analogic, Proportion, or Proportionality. But Custome, and the Sense assisting, will render any over-curious Application of these Terms unnecessary.

From these two Considerations last [...] there are wont to be de­ [...] sorts of Proportion, Arith­ [...]l, Geometrical, and a mixt Pro­ [...]tion, resulting from these two, cal­led Harmonical.

1. Arithmetical, When three or more Numbers in Progression, have the same Difference; as, 2, 4, 6, 8, &c. or discontinued, as 2, 4, 6; 14, 16, 18.

2. Geometrical, When three or more Numbers have the same Ration; as 2, 4, 8, 16, 32; or Discontinued; 2, 4; 64, 128.

Lastly, Harmonical, (partaking of both the other) When three Numbers are so ordered, that there be the same Ration of the Greatest to the Least; as there is of the Difference of the two Greater, to the Difference of the two less Numbers. As in these three Terms; 3, 4, 6; the Ration of 6 to 3 (being the greatest and least [Page 91]Terms) is Duple. So is 2, the Diffe­rence of 6 and 4 (the two greater Numbers) to 1. the Difference of 4 and 3 (the two less Numbers) Duple also. This is Proportion Harmoni­cal, which Diapason 6 to 3, bears to Diapente 6 to 4, and Diatessaron 4 to 3; as its mean Proportionals.

Now for the kinds of Rations most properly so called; i. e. Geometrical; first observe, that in all Rations, the former Term or Number (whether greater or less) is always called the An­tecedent; and the other following Number, is called the Consequent. If therefore the Antecedent be the greater Term; then the Ration is ei­ther Multiplex, Superparticular, Super­partient, or (what is compounded of these) Multiplex Superparticular, or Multiplex Superpartient.

1. Multiplex; as Duple, 4 to 2; Triple, 6 to 2; Quadruple, 8 to 2.

[Page 92] 2. Superparticular; as 3 to 2, 4 to 3, 5 to 4; Exceeding but by one aliquot part, and in their Radical, or least Numbers, always but by one; and these Rations are termed Sesquial­tera, Sesquitertia, (or Supertertia) Ses­quiquarta (or Superquarta) &c. Note, that Numbers exceeding more than by one, and but by one aliquot part, may yet be Superparticular, if they be not expressed in their Radical, i. e. least Numbers; as 12 to 8 hath the same Ration as 3 to 2; i. e. Superpar­ticular; though it seem not so, till it be reduced by the greatest Common Divi­sor to its Radical Numbers 3 to 2. And the Common Divisor (i. e. the Num­ber by which both the Terms may se­verally be divided) is often the Diffe­rence between the two Numbers; as in 12 to 8, the Difference is 4, which is the Common Divisor. Divide 12 by 4, the Quotient is 3; Divide 8 by 4, the Quotient 2; so the Radical [Page 93]is 3 to 2. Thus also 15 to 10, di­vided by the difference 5, gives 3 to 2; yet in 16 to 10, 2 is the com­mon Divisor, and gives 8 to 5; be­ing Superpartient. But in all Superpar­ticular Rations, whose Terms are thus made larger by being Multiplied: the Difference between the Terms is always the greatest common Divisor; as in the foregoing Examples.

The Third kind of Ration, is Su­perpartient, exceeding by more than One; as 5 to 3, which is called, Su­perbipartiens Tertias (or Tria) contain­ing 3 and ⅔; 8 to 5, Supertripartiens Quintas, 5 and ⅗.

The Fourth is Multiplex Superparti­cular, as 9 to 4, which is Duple, and Sesquiquarta; 13 to 4, which is Tri­ple, and Sesquiquarta.

The Fifth and last is Multiplex Su­perpartient, as 11 to 4; Duple, and Supertripartiens Quartas.

When the Antecedent is less than the Consequent; viz. when a less is compared to a greater; then the same Terms serve to express the Rations, only prefixing Sub to them, as Sub­multiplex, Subsuperparticular (or Sub­particular) Subsuperpartient (or Sub­partient) &c. 4 to 2 is Duple: 2 to 4 is Subduple. 4 to 3 is Sesquitertia; 3 to 4 is Subsesquitertia; 5 to 3 is Superbipartiens Tertias; 3 to 5 is Sub­superbipartiens Tertias, &c.

This short account of Proportion was necessary, because almost all the Philosophy of Harmony consists in Rations, Of the Bodies; Of the Mo­tions; and of the Intervals of Sound; by which Harmony is made.

And in searching, stating, and comparing the Rations of these, there is found so much Variety, and Cer­tainty, and Facility of Calculation, that the Contemplation of them may seem not much less delightful, than [Page 95]the very hearing the good Musick it self, which springs from this Foun­tain. And those who have already an affection for Musick, cannot but find it improv'd and much inhansed by this pleasant recreating Chase (as I may call it) in the Large Field of Harmonic Rations and Proportions. where they will find, to their great Pleasure and Satisfaction, the hidden causes of Harmony (hidden to most, even to Practitioners themselves) so amply discovered and laid plain be­fore them.

All the Habitudes of Rations to each other, are sound by Multiplica­tion or Division of their Terms; by which any Ration is Added to, or Substracted from another. And there may be use of Progression of Rations; or Proportions; and of finding a Medium, or Mediety between the Terms of any Ration. But the main work is done by Addition, and Sub­straction [Page 96]of Rations; which, though they are not performed like Addition and Substraction of Simple Numbers in Arithmetick; but upon Algebraic Grounds; yet the Praxis is most easie.

One Ration is added to another Ration, by Multiplying the two An­tecedent Terms together, i. e. the An­tecedent of one of the Rations, by the Antecedent of the other (for the more ease, they should be reduc'd into their least Numbers or Terms) And then the two Consequent Terms in like manner. The Ration of the Product of the Antecedents, to that of the Pro­duct of the Consequents, is equal to the other Two added or joined together. Thus (for Example) Add the Ration of 8 to 6; i. e. (in Radical Numbers) 4 to 3, to the Ration of 12 to 10; i. e. 6. to 5; the Product will be 24 and 15; i. e. 8 to 5; You may set them thus, 436524.15. and multiply 4 by 6, they [Page 97]make 24, which set at the Bottom; then multiply 3 by 5, they make 15; which likewise set under, and you have 24 to 15; which is a Ration com­pounded of the other two, and Equal to them both. Reduce these Products, 24 and 15, to their least Radical Numbers, which is, by dividing as far as you can find a Common Divi­sor to them both (which is here done by 3) and that brings them to the Ration of 8 to 5. By this you see, that a Third Minor, 6 to 5; added to a Fourth, 4 to 3; makes a Sixth Mi­nor, 8 to 5. If more Rations are to be added, set them all under each other, and multiply the first Antece­dent by the Second, and that Product by the Third; and again, that Pro­duct by the Fourth, and so on; and in like manner the Consequents.

This Operation depends upon the Fifth Proposition of the Eighth Book of Euclid; where He shews, That the [Page 98]Ration of plain Numbers is compound­ed of their sides. See these Diagrams.

Now compound these Sides. Take for the Antecedents, 4 the greater Side of the greater Plane, and 3 the great­er Side of the less Plane, and they multiply'd give 12; then take the remaining two Numbers 3 and 2, being the less Sides of the Planes (for Consequents) and they give 6. So, the Sides of 4 and 3, and of 3 and 2 compounded (by multiplying the Antecedent Terms by themselves, and the Consequents by themselves) make 12 to 6; i. e. 2 to 1. Which being [Page 99]apply'd, amounts to this; Ratio Ses­quialtera, 3 to 2, added to Ratio Ses­quitertia 4 to 3; makes Duple Ration, 2 to 1. Therefore Diapente added to Diatessaron, makes Diapason.

Substraction of One Ration from a­nother greater, is performed in like manner, by Multiplying the Terms; but this is done not Laterally, as in Ad­dition, but Crosswise; by Multiplying the Antecedent of the Former (i. e. of the Greater) by the Consequent of the Latter, which produceth a new Antece­dent; and the Consequent of the For­mer by the Antecedent of the Latter; which gives a new Consequent. And therefore it is usually done by an Ob­lique Decussation of the Lines. For Ex­ample, If you would take 6 to 5 out of 4 to 3, you may set them down thus. Then 4 mul­tiply'd by 5 makes 20; and 3 by 6 gives 18. So 20 to 18; i. e. 10 to 9, 436520.28.10.9. is the Remain­der. [Page 100]That is, Substract a Third Mi­nor out of a Fourth, and there will re­main a Tone Minor.

Multiplication of Rations is the same with their Addition; only it is not wont to be of divers Rations, but of the same, being taken twice, thrice, or oftener, as you please. And as before, in Addition, you added di­vers Rations by Multiplying them: So here, in Multiplication, you add the same Ration to it self, after the same manner, viz. by Multiplying the Terms of the same Ratio by them­selves; i. e. the Antecedent by it self, and the Consequent by it self (which in other words is to Multiply the same by 2) and will, in the Operation, be to Square the Ration first propounded (or give the Second Ordinal Power; the Ration first given being the First Power or Side) And to this Product, if the Simple Ration shall again be ad­ded (after the same manner as before) [Page 101]the Aggregate will be the Triple of the Ration first given; or the Pro­duct of that Ration Multiply'd by 3; viz. the Cube, or Third Ordinal Power. Its Biquadrate, or Fourth Power, proceeds from Multiplying it by 4; and so successively in order as far as you please you may advance the Powers. For instance, The Du­ple Ration 2 to 1, being added to it self, Dupled, or Multiply'd by 2, produceth 4 to 1 (the Ration Qua­druple) and if to this, the first again be added (which is equivalent to Mul­tiplying that said first by 3) there will arise the Ration Octuple, or 8 to 1. Whence the Ration 2 to 1, be­ing taken for a Root, its Duple 4 to 1, will be the Square; its Triple 8 to 1, the Cube thereof, &c. as hath been said above. And to use another instance; To Duple the Ration of 3 to 2; it must be thus Squar'd; 3 by 3 gives 9; 2 by 2 gives 4. [Page 102]So the Duple or Square of 3 to 2, is 9 to 4. Again, 9 by 3 is 27; and 4 by 2 is 8: So the Cubic Rati­on of 3 to 2 is 27 to 8. Again, to find the Fourth Power, or Biquadrate; (i. e. Squar'd Square) 27 by 3 is 81; 8 by 2 is 16: So 81 to 16 is the Ration of 3 to 2 Quadrupled; as 'tis Dupled by the Square, Tripled by by the Cube, &c. To apply this In­stance to our present purpose; 3 to 2 is the Ration of Diapente, or a Fifth in Harmony; 9 to 4 is the Ration of twice Diapente, or a Ninth (viz. Diapason with Tone Major) 27 to 8 is the Ration of thrice Diapente, or three Fifths; which is Diapason with Six Major (viz. 13th Major) The Ration of 81 to 16 makes four Fifths, i. e. Dis-diapason, with two Tones Major; i. e. a Seventeenth Major, and a Comma of 81 to 80.

To Divide any Ration, you must take the contrary way; And by Ex­tracting of these Roots Respectively, Division by their Indices will be per­formed. E. gr. To Divide it by 2, is to take the Square Root of it; by 3, the Cubic Root; by 4, the Biqua­dratick, &c. Thus to divide 9 to 4, by 2; The Square Root of 9, is 3; the Square Root of 4, is 2: Then 3 to 2 is a Ration just half so much as 9 to 4.

From hence it will be obvious to any to make this Inference; That Ad­dition and Multiplication of Rations are (in this Case) one and the same thing. And these Hints will be suffi­cient to such as bend their Thoughts to these kinds of Speculations, and no great Trespass upon those that do not.

The Advantage of proceeding by the Ordinal Powers, Square, Cube, &c. (as is before mentioned) may be ve­ry usefull where there is occasion of [Page 104]large Progressions. As to find (for Example) how many Comma's are contained in a Tone Major, or other Interval. Let it be, How many are in Diapason; Which must be done by Multiplying Comma's; i. e. Ad­ding them, till you arrive at a Ration Equal to Octave (if that be sought) viz. Duple. Or else by Dividing the Ration of Diapason, by that of a Comma, and finding the Quotient; which may be done by Logarithms. And herein I meet with some Diffe­rences of Calculations.

Mersennus finds, by his Calculation, 58 1/ [...] Comma's, and somewhat more in an Octave. But the late Nicholas Mercator, a Modest Person, and a Learned and Judicious Mathematici­an, in a Manuscript of his, of which I have had a Sight; makes this Re­mark upon it. In solvendo hoc Proble­mate aberrat Mersennus. And He, working by the Logarithms, finds out [Page 105]but 55, and a little more. And from thence has deduced an Ingenious Inven­tion of finding and applying a least Common Measure to all Harmonic Intervals; not precisely perfect, but very near it.

Supposing a Comma to be 1/53 part of Diapason; for better Accommo­dation rather than according to the true Partition 1/55; which 1/53 he calls an Artificial Comma, not exact, but dif­fering from the true Natural Comma about 1/20 part of a Comma, and 1/1000 of Diapason (which is a Difference imperceptible) Then the Intervals within Diapason will be measured by Comma's according to the following Table. Which you may prove by adding two, or three, or more of these Numbers of Comma's, to see how they agree to constitute those Intervals, which they ought to make; and the like by substracting.

This I thought fit, on this occasi­on, to impart to the Reader, having leave so to doe from Mr. Mercator's Friend, to whom He presented the said Manuscript.

Here I may advertise the Reader; that it is indifferent whether you com­pare the greater Term of an Harmo­nic Ration to the less, or the less to the [Page 107]greater; i. e. whether of them you place as Antecedent. E.gr. 3 to 2, or 2 to 3. Because in Harmonics, the proporti­ons of Lengths of Chords, and of their Vibrations are reciprocal or Counter-changed. As the Length is increased, so the Vibrations are in the same proportion decreased; & è con­tra. If therefore (as in Diapente) the length of the Unison String be 3, then the length (caeteris paribus) of the String which in ascent makes Diapente to that Unison must be 2, or 2ds / 3. Thus the Ration of Diapente is 2 to 3 in respect of the length of it, compared to the length of the Unison String.

Again, the String 2 vibrates thrice, in the same time that the String 3 vi­brates twice. And thus the Ration of Diapente in respect of Vibrations is 3 to 2. So that where you find in Au­thors, sometimes the greater Number in the Rations set before and made the Antecedent, sometimes set after [Page 108]and made the Consequent; You must understand in the former, the Ration of their Vibrations; and in the latter, the Ration of their Lengths; which comes all to one.

Or you may understand the Uni­son to be compared to Diapente above it, and the Ration of Lengths is 3 to 2; of Vibrations 2 to 3: or else Di­apente to be compared to the Unison, and then the Ration of Lengths is 2 to 3; of Vibrations 3 to 2. This is true in single Rations, or if one Ration be compared to another; then the two Greater Terms must be ranked as Antecedents: or otherwise, the two Less Terms.

The Difference between Arithme­tical and Geometrical Proportion is to be well heeded. An Arithmetical mean Proportion is that which has Equal difference to the Antecedent and Consequent Terms of those Num­bers to which it is the Mediety; and [Page 109]is found by adding the Terms and ta­king half the Sum. Thus between 9 and 1, which added together make 10, the Mediety is 5; being Equidif­ferent from 9 and from 1; which Dif­ference is 4: As 5 exceeds 1 by 4; so likewise 9 exceeds 5 by 4. And thus in Arithmetical Progression 2, 4, 6, 8; where the Difference is onely consider­ed, there is the same Arithmetical Pro­portion between 2 and 4, 4 and 6, 6 and 8; and between 2 and 6, and 4 and 8. But in Geometrical Proporti­on where is considered, not the Nu­merical Difference, but another Habi­tude of the Terms, viz. How many times, or how much of a time, or times, one of them doth contain the other (as hath been explained at large in the beginning of this Chapter.) There the Mean Proportional is not the same with Arithmetical, but found another way; and Equidifferent Progressions make different Rations. The Rations [Page 110](taking them all in their least Terms) expressed by less Numbers, being grea­ter than those of greater Numbers, I mean in Proportions Super-particu­lar, &c. Where the Antecedents are Greater than the Consequents: (as on the Contrary, where the Antecedents are Less than the Consequents, the Ra­tio's of Less Numbers are Less than the Ratio's of Greater.) The Mediety of 9 to 1, is not now 5, but 3; 3 having the same Ration to 1, as 9 has to 3 (as 9 to 3, so 3 to 1) viz. Triple. And so in Progression Arith­metical, of Terms having the same Differences; if considered Geometrical­ly, the Terms will all be comprehen­ded by unequal Rations. The Diffe­rences of 2 to 4, 4 to 6, 6 to 8, are Equal; but the Rations are unequal: 2 to 4 is less than 4 to 6, and 4 to 6 less than 6 to 8. As on the Contrary; 4 to 2, is greater than 6 to 4; and [...] to 4 greater than 8 to 6. For 4 to 2 [Page 111]is Duple; 6 to 4 but Sesquialtera (one and a half onely, or 2/2) and 8 to 6 is no more than Sesquitertia, (one and a Third part, or 4/3) which shews a con­siderable Inequality of their Rations. In like manner, 6 to 2 is Triple; 8 to 4 is but Duple; and yet their Differen­ces Equal. Thus the mean Rations comprehended in any greater Ration divided Arichmetically; i. e. by Equal Differences; are unequal to one ano­ther considered Geometrically. Thus 2, 3, 4, 5, 6; if you consider the Numbers, make an Arithmetical Pro­gression: But if you consider the Rati­ons of those Numbers, as is done in Harmony, then they are Unequal; eve­ry one being greater or less (according as you proceed by Ascent or Descent) than the next to it. Thus in this pro­gression, 2 to 3 is the greatest, being Diapente; 3 to 4 the next, Diatessaron; 4 to 5 still less, viz. Ditone; 5 to 6 the least, being Sesquitone. Or, if you des­cend, [Page 112]6 to 5 least; 5 to 4 next, &c. These are the mean Rations compre­hended in the Ration of 6 to 2, by which Diapason cum Diapente, or a 12th, is divided into the aforesaid Intervals, and measured by them: viz. as is 6 to 2, (viz. Triple.) So is the Aggregate of all the mean Rations within that Num­ber; 6 to 5, 5 to 4, 4 to 3, and 3 to 2. Or 6 to 5, 5 to 2; or 6 to 4, 4 to 2; or 6 to 3, 3 to 2. The Ag­gregates of these are Equal to 6 to 2, viz. Triple.

This is premised in order to pro­ceed to what was intimated in the foregoing Chapter.

Taking notice first of this procedure, peculiar to Harmonics; viz. To make Progression or Division in Arithmetical Proportion in respect of the Numbers; but to consider the things Numbred ac­cording to their Rations Geometrical. And thus Harmonic proportion, is said to be compounded of Arithmetical and Geometrical.

You may find them all in the Divi­sion of the Systeme of Diapason, into Diapente and Diatessaron, i. e. 5th and 4th; ascending from the Unison.

If by Diapente first, Then by 2, 3, 4, Arithmetically: If first by Diatessa­ron, Then by 3, 4, 6, Harmonically. And these Rations considered Geome­trically, in Relation to Sound; There is likewise found Geometrical Proporti­ons between the Numbers 6, 3, to 4, 2; and 6, 4, to 3, 2.

The Antients therefore owning one­ly 8th, 5th, and 4th, for Simple Con­sonant Intervals; concluded them all within the Numbers of 12, 9, 8, 6, which contained them all: viz. 12 to 6, Diapason; 12 to 8, Diapente; 12 to 9, Diatessaron; 9 to 8, Tone. And which served to express the three Kinds of Pro­portion; viz. Harmonical, between 12 to 8, and 8 to 6; Arithmetical, between 12 to 9, and 9 to 6; and Geometri­cal, between 12 to 9, and 8 to 6; [Page 114]and between 12 to 8, and 9 to 6. It was said therefore, That Mercu­rius his Lyre was strung with four Chords, having those Proportions, 6, 8, 9, 12. Gassend.

I intimated that I would here more largely explain that ready and ea­sie way of finding and measuring the mean Rations contained in any of those Harmonick Rations given, by transfer­ring them out of their Prime or Radi­cal Numbers, into greater Numbers of the same Ration. By Dupling (not the Ration, but the Terms of it: still conti­nuing the same Ration) you will have one Mediety: as 2 to 1 Dupled is 4 to 2; and you have 3 the Mediety. By Tripling you will have two Means; 2 to 1 Tripled is 6 to 3, containing 3 Rations; 6 to 5, 5 to 4, 4 to 3; and so still more, the more you multiply it.

Now observe, First, That any Ra­tion Multiplex or Superpartient (or by [Page 115]transferring it out of its Radical Num­bers made like Superpartient) contains so many Superparticular Rations, as there are Units in the Difference be­tween the Antecedent and the Conse­quent. Thus in 8 to 4 (being 2 to 1 transferred by Quadrupling) the Difference is 4, and it contains 4 Su­perparticular Rations; viz. 8 to 7, 7 to 6, 6 to 5, and 5 to 4. Where though the Progression of Numbers is Arith­metical, yet the Proportions of excess are Geometrical and Unequal. The Superparticular Rations expressed by less Numbers, being Greater, as hath been said, than those that consist of Greater Numbers; 5 to 4 is a Greater Ration than 6 to 5, and 6 to 5 Greater than 7 to 6, and 7 to 6 than 8 to 7; as a Fourth part is Greater than a Fifth, and a Fifth Greater than a Sixth, &c. But in this Instance, there are two Ra­tions not appertaining to Harmonics; viz. 8 to 7, and 7 to 6.

Secondly therefore, you may make unequal Steps, and take none but Har­monick Rations, by Selecting Greater and fewer intermediate Rations, tho' some of them composed of several Su­perparticulars; provided you do not discontinue the Rational Progression, but that you repeat still the last Conse­quent, making it the next Antecedent. As if you measure the Ration of 8 to 4, by 8 to 6, and 6 to 4; or by 8 to 5, and 5 to 4; or by 8 to 6, and 6 to 5, and 5 to 4. In these three ways the Rations are all Harmonical, and are respectively contained in, and make up the Ration of 8 to 4. Thus you may measure, and divide, and compound most Harmonick Rations without you Pen.

To that End, I would have my Reader to be very perfect in the Radical Numbers, which express the Rations of the Seven first (or uncompounded) Consonants: viz. Diapason, 2 to 1; [Page 117] Diapente, 3 to 2; Diatessaron, 4 to 3; Ditone, 5 to 4; Trthemitone, 6 to 5; Hexachordon Majus, 5 to 3; Hexa­chordon Minus, 8 to 5. And likewise of the Degrees in Diatonick Harmony, viz. Tone Major, 9 to 8; Tone Minor, 10 to 9; Hemitone Major, 16 to 15. And the Differences of those Degrees; Hemitone Greatest, 27 to 25; and He­mitone Minor, 25 to 24; Comma, or Schism, 81 to 80; Diesis Enharmonic, 128 to 125.

Of other Hemitones, I shall treat in the Eighth Chapter.

Now if you would divide any of the Consonants into two Parts, you may do it by the Mean, or Mediety of the two Radical Numbers; if they have a Mean: And where they have not (as when their Ratio's are Super­particular) you need but Duple those Numbers, and you will have a Mean (one or more.) Thus Duple the Num­bers of the Ration of Diapason, 2 to 1; [Page 118]and you have 4 to 2; and then 3 is the Mean by which it is divided into two Unequal, but Proper and Harmonical parts; viz. 4 to 3, and 3 to 2. After this manner Diapason, 4 to 2; com­prehends 4 to 3, and 3 to 2. So Dia­pente, 6 to 4; is 6 to 5, and 5 to 4. Ditone, 10 to 8; is 10 to 9, and 9 to 8. So Sixth Major, 5 to 3; is 5 to 4, and 4 to 3.

Though, from what was now ob­serv'd, you may divile any of the Consonants into intermediate Parts; yet when you divide these three fol­lowing, viz. Sixth Minor, Diatessaron, and Trihemitone; you will find that those Parts into which they are divided, are not all such Intervals as are Harmoni­cal. The Sixth Minor, whose Ration is 8 to 5, contains in it three Means; viz. 8 to 7, 7 to 6, and 6 to 5; the last whereof onely is one of the Har­monick Intervals, of which the Sixth Minor consists; viz. Tribemitone: and [Page 119]to make up the other Interval, viz. Diatessaron; you must take the other two, 8 to 7, and 7 to 6; which be­ing added (or, which is the same thing, taking the Ratio of their two Extream Terms, That being the Sum of all the Intermediate ones added) you have 8 to 6, or (in the least Terms) 4 to 3. Again Diatessaron, in Radical Numbers, 4 to 3; being (if those Numbers are dupled) 8 to 6, gives for his Parts, 8 to 7, and 7 to 6; which Ratio's agree with no Inter­vals that are Harmonick. Therefore you must take the Ration of Diatessa­ron in other Terms, which may afford such Harmonick Parts. And to do this, you must proceed farther than dupling (or adding it once to it self) for to this Duple you must add the first Radical Numbers once again (which in effect is the same with Tri­pling it at first) viz. 4 and 3, to 8 and 6; and the Aggregate will be a [Page 120]new, but Equivalent, Ration of Dia­tessaron; viz. 12 to 9. And this gives you three Means, 12 to 11, and 11 to 10; both Unharmonical; but, which together are, as was shewed be­fore, the same with 12 to 10 (or 6 to 5) Trihemitone; and 10 to 9, Tone Minor: and are the two Harmonical Intervals of which Diatessaron consists, and which divide it into the two near­est Equal Harmonick Parts. Lastly Trihemitone, or Third Minor, 6 to 5; or (those Numbers being dupled) 12 to 10, gives 12 to 11, and 11 to 10; which are Unharmonical Rations: but Tripled (after the former manner) 6 to 5 gives 18 to 15; which divides it self (as before) into 18 to 16, Tone Major; and 16 to 15, Hemitone Ma­jor.

Thus by a little Practice all Har­monick Intervals will be most easily measured, by the lesser Intervals com­prized in them. Now, (for exercise [Page 121]sake) take the Measures of a greater Ration: Suppose that of 16 to 3 be gi­ven as an Harmonick Systeme. To find what it is, and of what Parts it con­sists: First find the gross Parts, and then the more Minute. You will pre­sently judge, that 16 to 8 (being a Part of this Ration) is Diapason; and 8 to 4 is likewise Diapason: then 16 to 4 is Disdiapason, or a Fifteenth; and the remaining 4 to 3, is a Fourth. So then, 16 to 3, is Disdiapason, and Dia­tessaron; i. e. an Eighteenth: 16 to 8, 8 to 4, and 4 to 3.

But to find all the Harmonick Inter­vals within that Ration (for we now consider Rations as relating to Harmo­ny) take this Scheme.

16 to 3 contains,

Or Thus,

All these Invervals thus put toge­ther are comprehended in, and make up the Ration of 16 to 3; being ta­ken in a Conjunct Series of Rations.

But otherwise, within this Compass of Numbers are contained many more Expressions of Harmonick Ra­tion. Ex. gr.

And now I suppose the Reader bet­ter prepared to proceed in the remain­der of this Discourse, where we shall treat of Discords.

CHAP. VI. Of Discords and Degrees.

ALL Habitudes of one Chord to another, that are not Concords (such as are before described) are Discords; which are, or may be in­numerable, as are the minute Tensi­ons by which one Chord may be made to vary from it self, or from a­nother. But here we are to consider onely such Discords as are useful (and in truth necessary) to Harmony, or at least relating to it, as are the Diffe­rences found between Harmonick In­tervals.

And these apt and useful Discords, are either Simple uncompounded In­tervals, such as immediately follow one another, ascending or descending [Page 125]in the Scale of Music: As Ut Re Mi. Fa Sol La Fa Sol, and are called De­grees: Or else, greater Spaces or In­tervals compounded of Degrees inclu­ding or skipping over some of them, as all the Concords do, Ut Mi, Ut Fa, Ut Sol, &c. And such are the Discords of which we now treat, as principally the Tritone, False Fifth, and the two Se­venths; Major, and Minor, if they be not rather among the Degrees, &c. For more perspicuity I shall treat of them severally; viz. of Degrees, of Discords, and of Differences.

CHAP. VII. Of Discords.

BEsides the Degrees, which, though they constitute and compound all Concords, yet are reckoned amongst Discords; because every Degree is Dis­cord to each Chord, to, or from which it is a Degree, either Ascending, or De­scending, as being a Second to it: Be­sides these I say, there are other Dis­cords, some greater and some less. The less will be found amongst the Differ­ences in the next Chapter; and are fit, rather to be known as Differences, than to be used as Intervals.

The greater Discords are generally made of such Concords, as, by reason of misplaced Degrees, happen to have a Comma, or Diesis, or sometimes a He­mitone [Page 107]too much, or too little: And so become Discords, most of them be­ing of little use; only to know them, for the better Measuring, and Rectify­ing the Systems. Yet they are found a­mongst the Scales of our Music.

Sometimes a Tone Major being where a Tone Minor should have been placed, or a Tone Minor instead of a Tone Ma­jor; sometime other Hemitones, getting the place of the Diatonic Hemitone Major, and serving for a Degree, create unapt Discording Intervals: amongst which may be found at least two more Se­conds, two more Thirds, two more Sixths, and two more Sevenths. In each of which, one is less, and the o­ther greater, than the true Legitimate In­tervals, or Spaces of those Denominati­ons; as will be more explained in the ensuing Discourse.

But besides these (or rather amongst them, for I here treat of Degrees as Discords) there are two Discords emi­nently [Page 168]considerable, viz. Tritone, and Semidiapente. The Tritone, (or False Fourth) whose Ration is 45 to 32, consists of 3 whole Notes; viz. 2 Tones Major, and 1 Minor. The Semidiapente, (or False Fifth) 64 to 45; is com­pounded of a Fourth, and Hemitone Mi­jor.

And these two divide Diapason, 64 to 32, by the Mediety of 45; And they divide it so near to Equality, that in Practice, they are hardly to be di­stinguished, and may almost pass for one and the same: but in Nature, they are sufficiently distinguished, as may be seen, both by their several Rations, and several Compounding parts.

I think we may reckon Sevenths for Degrees, as well as among the greater Discording Intervals; because they are but Seconds from the Octave, and are as truly Degrees Descending, as the Seconds are in Ascent: though they be great Inter­vals in respect of the Unison, and as such may be here regarded.

These Discords, the Tritone, and Semi­diapente; as also, the Seconds, and Se­venths, are of very great use in Music, and add a wonderfull Ornament and Pleasure to it, if they be judiciously ma­naged. Without them, Music would be much less gratefull; like as Meat would be to the Palate without Salt or Sawce. But, the further Consideration of this, and to give Directions when, and how to use them; is not my Task, but must be left to the Masters of Com­position.

Discords then, such as are more apt and usefull, (Intervalla Concinna) are these which follow.

• 2d. Minor; or, Hemitone Major, 16 to 15.
• 2d. Major; Tone Minor, 10 to 9.
• 2d. Greatest; Tone Major, 9 to 8.
• 7th. Minor; 5th. & 3d. Minor, 9 to 5.
• 7th. Major; 5th. & 3d. Major, 15 to 8.
• Tritone; 3d. Maj. & Tone Maj. 45 to 32.
• Semidiapente; 4th. & Hemit. Maj. 64 to 45.

These are the Simple Dissonant apt Intervals within Diapason; if you go a further compass, you do but repeat the same Intervals added to Diapason, or Dis-diapason, or Tris diapason, &c. as, Ex. gra. a

• 9th. is Diapason with a 2d.
• 10th. Diapason with a 3d.
• 11th. Diapason with a 4th. or,
• 11th. Diapason cum Diatessaron.
• 12th. Diapason with a 5th. or,
• 12th. Diapason cum Diapente.
• 15th. Dis-diapason.
• 19th. Dis-diapason cum Diapente.
• 22th. Tris-diapason. &c.

Here, by the way, the Reader may take a little Diversion, in practising to measure the Rations of some of those Intervals, in the foregoing Catalogue of Discords, by comparing them with [Page 171] Diapason; as those of the Sevenths, which I select, because they are the most di­stant Rations under Diapason: Viz. Se­venth Minor, 9 to 5; and Seventh Ma­jor, 15 to 8. Now to find what De­gree or Interval lies between these and Diapason.

First, 9 to 5 is 10 to 5, wanting 10 to 9 (Tone Minor.) Next 15 to 8 is 16 to 8, wanting 16 to 15 (Hemi­tone Major.) So the Degree between Se­venth Minor and Diapason, is Tone Minor; and between Seventh Major and Diapa­son, is Hemitone Major.

Then he may exercise himself in a Survey of what Intervals are comprized in those several Sevenths, and of which they are compounded.

First, 9 to 5 comprizeth 9 to 8, and 8 to 5: Or 9 to 8, 8 to 6, and 6 to 5. Next, 15 to 8 contain 15 to 12, 12 to 10, 10 to 9, and 9 to 8: Or 15 to 12, and 12 to 8; Or 15 to 10, and 10 to 8, &c. I suppose, that the [Page 172]Reader, before this, is so perfect in these Rations, that I need not lose time to name the Intervals expressed by the Mean Rations, contained in the fore­going Rations of the Sevenths, which shew of what Intervals the several Se­venths are compounded.

Besides these, (by reason of Degrees wrong placed) there are two more 7ths. [false 7ths] one, less than the true ones; and another greater. The least com­pounded of two Fourths; whose Rati­on is 16 to 9, and wants a Comma of 7th. Minor, and a Tone Major of Diapa­son: The other is the greatest, called Semidiapason, whose Ration is 48 to 25; being a Diesis more than 7th. Major; and wanting Hemitone Minor of Diapa­son.

Now first, 16 to 9, is 16 to 8 (2 to 1) wanting 9 to 8; i. e. wanting Tone Ma­jor of Diapason; and contains 16 to 10 (8 to 5) and 10 to 9. Or, 16 to 15, 15 to 12 (5 to 4) 12 to 10 (6 to 5) [Page 173]and 10 to 9. Next, Semi-diapason 48 to 25, is 50 to 25, wanting 50 to 48; i. e. 25 to 24 (viz. Hemitone Minor) of Diapason.

And the like happens, as hath been said, to the other Intervals, which ad­mit of Major and Minor; viz. 2ds, 3ds, and 6ths. The 4th, and 5th, and 8th, ought always to remain immutable; though they may suffer too sometimes, and incline to Discord, if we ascend to them by very wrong Degrees; as you may see in the II Scale in the forego­ing Chapter: where the Fourth having 2 Tones Major, is a Comma too much.

All these Intervals may be subject to more Mutations, by more absurd pla­cing of Degrees, or of Differences of Degrees; but it is not worth the Curio­sity to search farther into them. The Reader may take pleasure, and suffici­ently Exercise himself in comparing and measuring these which are already laid before him.

But to return from this Digression. There are many unapt Discords, which may arise by continual Progression of the same Concords; i. e. by Adding (for Example) a 4th. to a 4th, a 5th. to a 5th, &c. For it is observable, That, onely Diapason added (as oft as you please) to Diapason, still makes Concord: But any other Concord, added to it self, makes Discord.

You will see the Reason of it, when you have considered well the Anatomy (as I may call it) of the Constitutive parts of Diapason; which contains, and is composed of 7 Spaces of Degrees, or of 4th. and 5th, or of 3ds. and 6ths, or of 2ds. and 7ths; which must all keep their true Measures and Rations belong­ing to them, and otherwise are easily and often disordered.

Then, consider Diapason as constitu­ted of two Fourths Disjunct, and a Tone Major between them. And this last is most needfull to be very well consi­dered; [Page 175]as most plainly shewing the Reasons of those Anomalies, or Irregu­lar Intervals, which are produced by Changing the Key; and consequently giving a new and wrong place to this Odd Tone Major, which stands in the midst of Diapason, between the two Fourths Disjunct.

Every 4th. must consist of one Tone Major, one Tone Minor, and one Hemitone Major, as its Degrees, placing them in what Order you please; whose Rations, added together, make the Ration of Dia­tessaron. And of these same Degrees contained in the 4th, are made the two 3ds, which constitute the 5th. Tone Major and Hemitone Major make the less 3d, or Trihemitone; Tone Major and Tone Minor make the Greater 3d, or Ditone; Trihemitone and Ditone make Diapente; Trihemitone and Tone Minor (as likewise Ditone and Hemitone Major) make Dia­tessaron.

Now this Tone Major, that stands in the middle of Diapason, between the two 4ths, which it disjoins; and the Degrees required to the 4ths, will not in a fixed Scale stand right, when you alter your Key, and begin your Scale of Diapason from another Note. For that which was the 5th, will now be the 4th, or 6th, &c. And then the Degrees will be disordered, and create some discording Intervals. If you continue conjunct Fourths, there will be a Defect of Tones Major; if you continue conjunct Fifths, there will be too many Tones Major in the Systems produced. And if a Tone Major be found, where it ought to have been a Tone Mi­nor; or a Minor instead of a Major; that Interval will have a Comma too much, or too little. And so likewise will from a wrong Hemitone be found the Diffe­rence of a Diesis. And these two, Com­ma and Diesis, are so often Redundant, or Deficient, according as the Degrees happen to be disordered or mis-placed; [Page 177]that thereby, the Difficulties of fixing half Notes of an Organ in tune for all Keys, or giving the true Tune by Fretts, become so Insuperable.

You see, that in every Space of an Eighth, there are to be 3 Tones Major, and 2 Tones Minor, and 2 Hemitones Ma­jor: One Tone Major between the Dia­tessaron and Diapente, and a Tone Major, a Tone Minor, and Hemitone Major in each of the Disjunct Fourths.

These are the Proper Degrees by which you should always Ascend or Descend through Diapason, in the Diatonic Kind; which Diapason, being the compleat Sy­stem, containing all primary Simple Harmonic Intervals that are; (and for that reason called Diapason;) You may multiply it, or add it to its self as oft as you please, as far as Voice or Instru­ment can reach; and it will still be Concord, and cannot be disordered by such addition: because every of them will contain (however placed) just [Page 178]3 Tones Major, 2 Tones Minor, and 2 He­mitones Major.

Whereas, if you add any other Inter­val to it self, the Degrees will not fall right, and it will be Discord, because all Concords are compounded of une­qual Parts, as hath been shew'n before; and if you carry them in Equal Pro­gression, they will mix with other In­tervals by incongruous Degrees, and those Disordered Degrees will create a Dissonant Interval. See the following Scheme of it.

• 2 3ds. Minor make 5th, wanting Hem. Min.
• 2 3ds. Major make 5th, and Hemit. Minor.
• 2 4ths. make 8th, wanting Tone Maj.
• 2 5ths. make 8th, and Tone Major.
• 2 6ths. Minor make 8th, and Ditone, & Diesis.
• 2 6ths. Major make 8th, & 4th, & Hem. Min.

To which may be added, That

• 2 Tones Min. make Ditone, wanting a Comma.
• 2 Tones Maj. make Ditone, and a Comma.

It was said above, that Diapason may be added to it self as oft as you please, and there will be no disorder, because every one of them will still retain the same Degrees of which the first was Composed: But it is not so in other Concords; of which I will add one more Example, because of the use which may be made of it.

Make a Progression of 4 Diapente's, and, as was shewed in the 5th. Chapter, (Pag. 102.) it will produce Dis-diapason, and 2 Tones Major, which is a 17th, with a Comma too much. Because in that Space there ought to be just 7 Tones Ma­jor, and 5 Tones Minor; Whereas in 4 Fifths continued, there will be found 8 Tones Major, and but 4 Tones Minor: So that a Tone Major, getting the place of a Tone Minor, there will be in the whole System a Comma too much. One of these Major Tones should have been a Tone Minor, to make the Excess above Dis-diapason a just Ditone.

On the other side, if you [...] the Ration of 4 Diatessarons, t [...] be a Tone Minor, instead of a [...] jor; and consequently a Comma [...] in constituting Diapason and 6th [...] For since every Fourth must [...] the Degrees of Tone Minor, [...] Major, one Hemitone Major; i [...] that if you continue 4 Fourth [...] will be 4 Tones Minor, 4 Tone [...] and 4 Hemitones Major. Where [...] Interval of Diapason with 6th [...] there ought to be 5 Tones Major, [...] 3 Minor.

By this you may see the Reas [...] to put an Organ or Harpsichc [...] more general usefull Tune, y [...] tune by 8ths, and 5ths; mak [...] 8ths. perfect, and the 5ths. a litt [...]ing downward; i. e. as much as [...]ter of a Comma, which the Ear w [...] with in a 5th, though not in [...] For Example, begin at C Fa ut [...] C Sol fa ut a perfect 8th. to i [...]

[Page] [Page 181] re ut a bearing 5th; Then tune a per­fect 8th. to G, and a bearing 5th. at D La sol re; and from thence down­wards (that you may keep towards the middle of the Instrument) a perfect 8th. at D Sol re: and from thence a bearing 5th. up at A; and from A, a perfect 8th. upwards, and bearing 5th. at E La mi. From E an 8th. downwards; and so go on, as far as you are led by this Method, to tune all the Mid­dle part of the Instrument: and at last fill up all above, and below, by 8ths. from those which are settled in Tune; according to the Scheme an­nexed. Observing (as was said) to Tune the Eighths perfect, and the Fifths a lit­tle bearing Flat; except in the three last Barrs of Fifths, where the Fifths begin to be taken downward from C, as they were upwards in all before: Therefore, as before, the Fifth above bore down­ward; so here, the Fifth below must bear upward, to make a Bearing Fifth; but [Page 182]That being not so easie to be judged, Al­ter the Note below, till you judge the Note above to be a Bearing Fifth to it. This will rectifie both those Anomalies of 5ths and 4ths. For the 5th. to the Unison, is a 4th. to the Octave; and what the 5th. looseth by Abatement, the 4th. will gain: which doth in a good De­gree rectifie the Scale of the Instrument. Taking care withall, that what Anoma­lies will still be found in this Hemitonic Scale, may, by the Judgment of your Ear, in Tuning, be thrown upon such Chords as are least used for the Key: as ♯G, ♭E, &c. Even which the Ear will bear with, as it doth with other Dis­cords in binding passages; if so, you close not upon them. But the other Dis­cords, so used, are most Elegant; these only more Tolerable.

CHAP. VIII. Of Differences.

ALL Rations and Proportions of inequality, have a Difference be­tween them, when compared to one another; and consequently the Intervals, expressed by those Rations, differ likewise. A Fifth is Different from a Fourth, by a Tone Major; from a 3d. Minor, by a 3d. Major; so an 8th. from a 5th, by a 4th. Of the Compounding parts of any In­terval, one of them is the Difference be­tween the other part and the whole In­terval.

But I treat now of such Differences as are generally less than a Tone, and create the Difficulties, and Anomalies occurring in the two foregoing Chap­ters. I have the less to say of them [Page 184]a part, because I could not avoid touch­ing upon them all along. It will onely therefore be needfull, to set before you an orderly view of them. And first, taking an account of the true Harmonic Intervals with their Differences, and the Degrees by which they arise; it will be easier to judge of the false Intervals, and of what concern they are to Har­mony.

Table of true Diatonic Intervals within Diapason, with the Differences be­tween them.

Those which arise from the Differences of Conso­nant Intervals, are called Intervalla Concinna, and pro­perly appertain to Harmony: The rest are necessary to be known, for making and understanding the Scales of Musick.

Table of false Diatonic Intervals, caused by Improper Degrees; with their Ra­tions and Differences from the true Intervals.

This Mark + stands for More; − for Less.

Here in this account may be seen, how frequently the Comma, and the Diesis Abounding, or Deficient, by reason of Mis-placed Degrees, occasion Discord in Harmonic Intervals.

The Comma, by reason of a wrong Tone, i.e. too much; when a Tone Ma­jor happens where there ought to be a Tone Minor: or too little, when the Tone Minor is placed instead of the Major. And the Diesis is Redundant, or Deficient, by reason of a wrong Hemitone; when the Major happens instead of the Minor, or the Contrary: the Diesis being the Dif­ference between them. And if Hemito­nium Maximum get in the place of Hemi­tonium Majus, the Excess will be a Comma; if in the place of Hemitone Minor, the Excess will be Comma and Diesis.

And these Anomalies are not Imagi­nary, or only Possible, but are Real in an Instrument fixed in Tune by Hemi­tones; as, Organ, Harpsichord, &c. And the Reader may find some of them a­mongst [Page 188]those 4 Scales of Diapason, in the 6th. Chapter; to which also more may be added: Out of the First of which, I have Selected some Examples; using the common Marks, as before: viz. +, for More; and − for Less, or Wanting.

From

• ♯C, to ♭E; Tone Major, + Diesis; or, 3d. Min. Hemit. Min.
• ♯C, to F; 3d. Major, + Diesis; or, 4th, − Hemit. Minor.
• D, to G; 4th, + Comma.
• ♭E, to ♯F; 3^d Min. − Dies. & Com. or Tone Min. + Hem. Min.
• ♭E, to ♯G; 4th, − Diesis; or, 3d. Maj. + Hem. Min.
• ♯F, to ♭B; 3d. Maj. + Dies. & Com. or 4th, − Hem. Subminim.
• ♯F, to B; 4th, + Comma.
• ♯G, to ♭B; Tone Maj. + Diesis; or, 3d. Min. − Hemit. Min.
• ♯G, to C; 3d. Maj. + Diesis; or, 4th, − Hemit. Min.
• B, to D; 3d. Min. − Comma.

Next, take account of some Diffe­rences which constitute several Hemi­tones.

Difference between

• Tone Major, and Hemit. Minor. Hem. Maxim. 27 to 25
• 3d. Major, and 4th. Hemit. Majus: 16 to 15
• Tone Major, and Hemit. Major. Hemitone Medium. 135 to 128
• 3d. Minor, and 3d. Maj. Hem Min. Di­esis Chromat. 25 to 24
• 2 Tones Major, and 4th. Hemiton. (or Limma) Pythagor. 256 to 243
• Tone Maj. and Limma. Apotome 2187 to 2048; or Hemit. Med. with Comma.

To which may be added, out of Mersennus.

Differ. between

• Hemit. Maxim. and Hemit Minor. Hemitonium. Minimum. 648 to 625
• Tone Minor, and Hemitone Maxim. or, Hemitone Minor, and Comma. Hemitonium. Subminimum. 250 to 243

Next, take a farther View of Diffe­rences, most of which arise out of the preceding Differences, by which you [Page 190]will better see how all Intervals are Compounded, and Differenced; and more easily judge of their Measures.

Table of more Differences.

Difference between

• Tone Maj. and Tone Min. Comma.
• Tone Maj. and Hem. Greatest. Hemitone Minor.
• Tone Maj & Hem. Medium. Hemitone Major.
• Tone Maj. and Hem. Pythag. Apotome.
• Hem. Greatest, and Hem Maj Comma
• Hem. Greatest & Hem. Min. Comma, and Diesis; viz. Hem. Minimum.
• Hemit. Major, and Minor. Diesis.
• Hemit. Major, and Medium. 2048 to 2025. viz. Comma Minus.
• Hemit. Major, and Pythag. Comma.
• Apotome, and Hemit. Majus. Diff. betw. Comma Majus, and Min.
• Apotome, and Hemit. Med. Comma.
• Apotome, and Hem. Pythag. Comma, and the aforesaid Differ.
• Apotome, and Hemit. Minus. 2 Comma's.
• Hemit. Medium, and Phythag. Differ. of Comma Majus, and Minus.
• Hemit. Medium, and Minus. Comma.
• Hemit. Pythag. and Minus. Comma Minus.
• Hemit. Minus, and Diesis. Somewhat more than Comma, viz. 3125 to 3072.
• Hemit. Minus, and Comma. Hem Subminimum.
• Diesis, and Comma. Comma Minus, viz. 2048 to 2025.
• Com. Majus, and Com. Minus. 32805 to 32768.

These Differences (with some more) are found between several other Inter­vals; of which more Tables might be drawn, but I shall not trouble the Rea­der with them. Having here shewn what they are, he may (if he please) exercise himself to examine These by Numbers, and also find out Them; and to some it may be pleasant and de­lightfull; And for that reason, I have the more largely insisted on this part of my Subject, which concerns the Mea­sures, Habitudes, and Differences of Harmonic Intervals.

I shall add one Table more; of the Parts, of which these lesser Intervals are Compounded; which will still give more Light to the former; and is, in Effect, the same.

• [Page 192]Tone Major con­tains, & is com­pounded of Tone Minor, and Comma. Hemitone Maxim. and Hemitone Min.
• Tone Major con­tains, & is com­pounded of Hemitone Maj. Hemitone Med. Limma, Apotome. 2 Hem. Min.
• Tone Major con­tains, & is com­pounded of Hemitone Maj. Hemitone Med. Limma, Apotome. 1 Diesis,
• Tone Major con­tains, & is com­pounded of Hemitone Maj. Hemitone Med. Limma, Apotome. 1 Comma.
• Tone Min. Hem. Maxim. Hem. Submin. Hemit. Major, Hemit. Min. 2 Hemit. Min.
• Tone Min. Hem. Maxim. Hem. Submin. Hemit. Major, Hemit. Min. 1 Diesis.
• Hem. Max. Hem. Maj. Comma. Hem. Med. Diesis. Hem. Pyth. 2 Comma's. Hem. Min. Dies & Com.
• Hem. Maj. Hem. Med. Com. Min. Hem. Pyth. Comma. Hem. Min. Diesis. Hem. Submi. Dies & Com.
• Hem. Med. Hem. Min. Comma. Hemitone Pythagoricum. Difference between Comma Majus, and Minus, viz. 32805 to 32768.
• Hem. Pyth. Hemitone Minus. Comma Minus.
• Hem. Min. Hemit. Submin. Comma. Diesis, and 3125 to 3072.
• Diesis. Comma, Comma Minus.
• Comma. Comma Minus, 32805 to 32768.

I think there scarce needs an Apology for some of these Appellations, in respect of Gram­mar. That I call Hemitonium, and Hexachor­don, Majus, and Minus; sometimes Hemitone, and Hexachord, Major, and Minor. These two last Words are so well adapted to our Language, that there is no English-man, but knows them. Therefore when I make Hemitone an English word, I take Major, and Minor, to be so too; and fittest to be joyned with it, without re­spect of Gender.

CHAP. IX. Conclusion.

TO Conclude all. Bodies by Mo­tion make Sound; Sound, of fitly Constituted bodies, makes Tune: Tune, by Swiftness of Motion is ren­dred more Acute; by Slowness more Grave: in Proportion to the Measure of Courses and Recourses, of Trem­blings or Vibrations of Sonorous Bo­dies. Those Proportions are found out by the Quantity and Affections of Sounding Bodies. Ex. gr. by the Length of Chords. If the Proportion of Length (Caeteris paribus,) and consequently of Vibrations of several Chords, be com­mensurate within the Number 6; then those Intervals of Tune are Consonant, and make Concord, the Motions mix­ing [Page 194]and uniting as they pass: If incom­mensurate, they make Discord by the jarring and clashing of the Motions. Concords are within a limited Number, Discords innumerable. But of them, those only here considered, which are (as the Greeks termed them) [...], Con­cinnous, apt and usefull in Harmony: Or which, at least, are necessary to be known, as being the Differences and Measures of the other; and helping to discover the reason of Anomalies, found in the Degrees of Instruments tuned by Hemitones.

All these I have endeavoured to ex­plain, with the manifest Reasons of Consonancy and Dissonancy, (the Pro­perties of a Pendulum giving much light to it,) so as to render them easie to be understood by almost all sorts of Rea­ders; and to that end have enlarged, and repeated, where I might, to the more intelligent Reader, have compri­zed it very much shorter. But I hope [Page 195]the Reader will pardon that, which could not well be avoided, in order to a full, and clear Explanation of that, which was my design, viz. the Philoso­phy of the Natural Grounds of Har­mony.

Upon the Whole; You see how Ra­tionally, and Naturally, all the Simple Concords, and the Two Tones, are found and demonstrated, by Sub-divisions of Diapason.

• 2 to 1, i. e. 4 to 2; into 4 to 3, and 3 to 2.
• 2 to 1, i. e. 6 to 3; into 6 to 5, and 5 to 3.
• 2 to 1, i. e. 8 to 4; into 8 to 5, and 5 to 4.
• 2 to 1, i.e. 10 to 5; into 10 to 9, 9 to 8, and 8 to 5.

In which are the Rations (in Radical, or Least Numbers) of the Octave, Fifth, Fourth, Third Major, Third Minor, Sixth Major, Sixth Minor, and Tone Ma­jor, and Tone Minor.

And then, All the Hemitones, and Diesis, and Comma, are found by the Differences of these, and of one ano­ther; as hath been shewn at large.

Now, certainly, this is much to be preferred before any Irrational Contri­vance of expressing the several Intervals. The Aristoxenian way of dividing a Tone [Major] into 12 Parts, of which 3 made a Diesis, 6 made Hemitone, 30 made Diatessaron; (as hath been said) might be usefull, as being easier for Apprehensi­on of the Intervals belonging to the three Kinds of Musick; and might serve for a least common Measure of all Intervals, (like Mr. Mercator's artificial Comma) 72 of them being contained in Diapason.

But this way, and some other Me­thods of dividing Intervals equally, by Surd Numbers and Fractions, attempt­ed by some Modern Authors; could never constitute true Intervals upon the Strings of an Instrument, nor afford any Reason for the Causes of Harmo­ny, as is done by the Rational Way, explaining Consonancy by united Mo­tions, and Coincidence of Vibrations. And though they supposed such Divisi­ons [Page 197]of Intervals; yet we may well be­lieve, that they could not make them, nor apply them in tuning a Musical In­strument; and if they could, the Inter­vals would not be true nor exact. But yet, the Voice offering at those, might more easily fall into the true Natural Intervals. Ex. gr. The Voice could hardly express the Antient Ditone of 2 Tones Major; but aiming at it, would readily fall into the Rational Consonant Ditone of 5 to 4, consisting of Tone Ma­jor, and Tone Minor. It may well be rejected as unreasonable, to measure In­tervals by Irrational Numbers, when we can so easily discover and assign their true Rations in Numbers, that are Mi­nute enough, and easie to be understood.

I did not intend to meddle with the Artificial part of Musick: The Art of Composing, and the Metric and Rhyth­mical parts, which give the infinite Va­riety of Air and Humor, and indeed the very Life to Harmony; and which [Page 198]can make Musick, without Intervals of Acuteness and Gravity, even upon a Drum; and by which chiefly the won­derfull Effects of Musick are performed, and the Kinds of Air distingushed; As, Almand, Corant, Jigg, &c. which vari­ously attack the Fancy of the Hearers; some with Sprightfulness, some with Sadness, and others a middle way. Which is also improved by the Diffe­rences of those we call Flat, or Sharp Keys; The Sharp, which take the Great­er Intervals within Diapason, as 3ds, 6ths, and 7ths. Major; are more Brisk and Airy; and being assisted with Choice of Measures last spoken of, do Dilate the Spirits, and Rouze them up to Gal­lantry, and Magnanimity. The Flat, consisting of all the less Intervals, con­tract and damp the Spirits, and produce Sadness and Melancholy. Lastly, A mixture of these, with a suitable Rhyth­mus, gently fix the Spirits, and compose them in a middle Way: Wherefore the [Page 199]First of these is called by the Greeks Diastaltic, Dilating the second, Systaltic, Contracting; the last, Hesychiastic, Ap­p [...]sing.

I have done what I designed, search­ed into the Natural Reasons and Grounds, the Materials of Harmony; not pre­tending to teach the Art and Skill of Musick, but to discover to the Reader the Foundations of it, and the Reasons of the Anomolous Phoenomena, which oc­curr in the Scales of Degrees and Inter­vals: Which though it be enough to my Purpose, yet is but a small (though indeed the most certain, and, conse­quently most delightfull) Part of the Phi­losophy of Musick; in which there remain Infinite Curious Disquisitions that may be made about it: As what it is, that makes Humane Voices, even of the same Pitch, so much to differ one from ano­ther? (For though the Differences of Hu­mane Countenances are visible; yet we cannot see the Differences of Instruments [Page 200]of Voice, nor consequently of the Moti­ons and Collisions of Air, by which the Sound is made.) What it is that con­stitutes the different Sounds of the Sorts of Musical Instruments, and even single Instruments? How the Trumpet, on­ly by the Impulse of Breath, falls into such variety of Notes, and in the Low­er Scale makes such Natural Leaps into Consonant Intervals of 3d, 4th, 5th, and 8th. But this I find is very ingeniously explicated by an Honourable Member of the R. S. and published in the Philo­sophical Transactions, N^o 195. Also how the Tube-Marine, or Sea-Trumpet (a Monochord) so fully expresseth the Trumpet; and is also made to render other Varieties of Sounds; as, of a Vio­lin, and Flageolet, whereof I have been an Ear-witness? How the Sounds of Harmony are received by the Ear; and why some persons do not love Mu­sick? &c.

As to this last; the incomparable Dr. Willis mentions a certain Nerve in Brain, which some Persons have, and some have not. But further, it may be considered, that all Nerves are composed of small Fibres; Of such in the Gutts of of Sheep, Cats, &c. are made Lute-Strings: And of such are all the Nerves, and amongst them, those of the Ear, composed. And, as such, the latter are affected with the Regular Tremblings of Harmonic Sounds. If a false String (such as I have before described) trans­mit its Sound to the best Ear; it displeas­eth. Now, if there be found falseness in those Fibres, of which Strings are made; Why not the like in those of the Audi­tory Nerve in some Persons? And then it is no wonder if such an Ear be not pleased with Musick, whose Nerves are not fitted to correspond with it, in com­mensurate Impressions and Motions. I gave an Instance in Chap. 3d. how a Bell-Glass will Tremble and Eccho to its own [Page 202]Tune, if you hit upon it: And I may add, That if the Glass should be irre­gularly framed, and give an uncertain Tune, it would not answer your Trial. In fine, Bodies must be Regularly named to make Harmonic Sounds, and the Ear Regularly constituted to receive them. But, this by the by; and only for a Hint of Inquiry.

I was saying, that there remain Infi­nite Curiosities relating to the Nature of Harmony, which may give the most A­cute Philosopher business, more than e­nough, to find out; and which perhaps will not appear so easie to demonstrate and explain, as are the Natural Grounds of Consonancy and Dissonancy.

After all therefore, and above all, by what is already discovered, and by what yet remains to be found out; we cannot but see sufficient cause to Rouze up our best Thoughts, to Admire and Adore the Infinite Wisdom and Good­ness of Almighty God. His Wisdom, [Page 203]in ordering the Nature of Harmony in so wonderfull a manner, that it surpas­seth our Understanding to make a through Search into it, though, (as I said) we find so much by Searching, as does recompence our Pains with Plea­sure, and Admiration.

And his Goodness, in giving Musick for the Refreshings and Rejoycings of Mankind; so that it ought, even as it relates to common Use, to be an Instru­ment of our great Creator's Praise, as he is the Founder and Donor of it.

But much more, as it is advanced and ordained to relate immediately to his Holy Worship, when we Sing to the Honour and Praise of God. It is so Essential a part of our Homage to the Divine Majesty, that there was never a­ny Religion in the World, Pagan, Jew­ish, Christian, or Mehumetan, that did not mix some kind of Musick with their Devotions; and with Divine Hymns, and Instruments of Musick, set forth the [Page 204]Honour of God, and celebrate his Praise. Not only, Te decet Hymnus Deus in Sion. (Psal. 65.) but also —Sing unto the Lord all the whole Earth. (Psal. 96.)

And it is that, which is incessantly performed in Heaven before the Throne of God, by a General Consort of all the Holy Angels and the Blessed.

In short, we are in Duty and Gra­titude bound to bless God, for our De­lightfull Refreshments by the use of Mu­sick; But especially in our publick De­votions, we are obliged by our Religi­on, with Sacred Hymns and Anthems, to magnifie his Holy Name; that we may at last find Admittance above, to bear a Part in that Blessed Consort, and Eternally Sing Allelujahs, and Trisagions in Heaven.