A treatise of the natural grounds and principles of harmony by William Holder ...

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 Quali∣ties. But the Ear being entertained with Motions which fall under exact Demonstrations of their Measures, the

Doctrine hereof is capable of being more accurately discovered.

First then, (setting aside the Unison Concord, which is no Space or Inter∣val, 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 de∣graded from his Place) then the Third Major, the Third Minor, the Sixth Ma∣jor, and the Sixth Minor. And also such, as in the Compass of any Voice or In∣strument beyond the Octave, may be compounded of these, for such those are, I mean compounded, and only the for∣mer Seven are simple Concords; not but that they may seem to be com∣pounded, viz. the greater of the less with∣in 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

the Eighth, as a Tenth is an Eighth and a Third; a Twelfth is an Eighth and a Fifth; a Fifteenth is Disdiapa∣son, 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 Ha∣bitude to the Original Concords com∣prehended 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 Oc∣tave, 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, espe∣cially in that of Consort.

The System of an Eighth, contain∣ing seven Intervals, or Spaces, or De∣grees, and eight Notes reckoned inclu∣sively, as expressed by eight Chords,

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 a∣gain 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 there∣fore most justly judged by the Ear, to be the Chief of all Concords, and is the only Consonant System, which

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.

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 natu∣ral 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 consi∣dered more at large. I have shewed,

1. That Harmonick Sound or Tune is made by equal Vibrations or Tremblings of a Body fitly constitu∣ted.

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 proportio∣nably more Acute: if less frequent, more Grave.

5. That the Librations of a Pendu∣lum 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

Pendulum, or two Pendulums tacked together at length, and therefore hath the same Effects by dupling; as a Pen∣dulum by quadrupling, i. e. by du∣pling the Length of the Chord, the Vi∣brations 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 Mea∣sures of Length, the Vibrations bear∣ing 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 alternate∣ly, or frequently: Then the Sounds of those different Chords, thus mixing,

will calmly pass the Medium, and ar∣rive 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, be∣cause 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

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 rea∣son 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 Ten∣sion. Here then, both the Strings give their Sound; each Sound is a certain Tune; each Tune is made by a cer∣tain 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 sup∣posed to be equal in all respects; it follows, that their Vibrations must be

also equal, and consequently move in the same Measure, joyning and uni∣ting 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 Conso∣nant, that it is an Identity of Tune, there being no Interval or Space be∣tween them. And the Ear can hard∣ly 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 near∣est 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

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 uni∣ting and mixing alternately, i. e. eve∣ry 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 subdu∣ple of Length; consequently the Pro∣portions of all other Intervals are easi∣ly 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,

that Duple Ration contains, and is composed of the two next Rations, viz. Sesquialtera and Sesquitertia. There∣fore an Octave which is in Duple Ra∣tion 2 to 1 is divided into, and com∣posed 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 Ses∣quiquinta. 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 gi∣ven, by transferring the Prime or Ra∣dical Numbers of the given Ration in∣to greater Numbers of the same Ra∣tion, 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

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 Ma∣jor, 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 consequent∣ly the Difference of their Vibrations. Take two Strings A B, in all other Respects equal, and compare their Lengths, which if equal, make Uni∣son or the same Tune. If A be dou∣ble 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 Vi∣brations 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. eve∣ry fifth Recourse.

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

Thus by the frequency of their be∣ing mixed and united, the Harmony of joyned Concords is found so very sweet and pleasing; the Remoter be∣ing also combined by their Relation to other Concords besides the Unison.

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 Com∣binations 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 se∣veral Strings, to express the Consort which is made by them; but other∣wise, these Rations are more certain∣ly found upon the Measures of a Mo∣nochord, taken, by being applied to the Section of a Canon, or a Rule of the Strings length divided into parts, as oc∣casion requires: because there is no need

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 sin∣gle 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, re∣lated 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.

[illustration] musical chords

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

6	to	5	3d	Minor.
	to	4	5th
	to	3	8th
	to	2	12th
	to	1	19th
5	to	4	3d	Major.
	to	3	6th	Major.
	to	2	10th	Major.
	to	1	17th	Major.
4	to	3	4th
	to	2	8th
	to	1	15th
3	to	2	5th
	to	1	12th
2	to	1	8th
6	to	5	3d	Minor.
5	to	4	3d	Major.
4	to	3	4th
3	to	2	5th
2	to	1	8th

All that are Concords to the Uni∣son, are also Concords to the Octave, And all that are Discords to the Uni∣son, 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 U∣nison, Fourth, Sixth, and Octave, may

be sounded together to make a com∣pleat 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 as∣cend 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 con∣cluding Close: And on the continual

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 Con∣trapunct chiefly, but indeed in all Kinds of Composition. I do not ex∣clude a Sprinkling of Discords; nor here medle with Ayr, Measure, and Rythmus, which are the Soul and Spi∣rit 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 Au∣thentick descend the same by which the Plagal ascend, viz. by Fourths and Sixths; one chiefly relating to the Uni∣son, 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 be∣fore-mentioned Combinations of Con∣cords, which increase the Consonan∣cies of each Note, and make a won∣derfull

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 Conso∣nnacy, 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 Vi∣bration, i. e. at every Course.

Now take a Review of the Variety of Consonancies in these four Notes. Here are mixed together in one Con∣sort

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 Con∣course there is of Consonant Rations, to Variegate and give (as it were) a pleasant Purling to the Harmony with∣in that Space. For now, all this Va∣riety 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 Authen∣tick

Moods. Add to all this, the In∣finite 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 se∣veral 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.

[illustration] vibrating string patterns

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 Vi∣bration, and from B to A the Recourse. Observing by the way, that (in rela∣tion 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 mat∣ter brief and perspicuous, without per∣plexing the Figures with many Lines: and avoiding the Incumbrance of so many Cautions, whereby to distract the Reader: Yet I must always be un∣derstood to acknowledge the continual Decrease of the Range of Vibrations be∣tween A and B, while the Motion con∣tinues;

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 gra∣dually contracted Extremities of their Range. And the following Demon∣stration 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

alternately, joyning at every Course of the Unison, and crossing at the Re∣course.

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 Dia∣pente to the Unison V; whilst V cour∣seth 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 man∣ner the rest of the Concords unite, at the 4th, 5th, 6th Course, accord∣ing to their Rations, as might this same way be shewn; but it would take

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 com∣pleat 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 resem∣blance) 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

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 De∣sign 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 capa∣ble of this Theory: And also, those Masters in Practick Musick, and Lo∣vers of it, who, though wanting Phi∣losophy, and the Latin and other For∣reign Tongues, to read better Au∣thors;

yet, by the help of their know∣ledge in Musick, may attain to under∣stand 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 les∣ser, Flat. They are so, indeed, in a∣scending from the Unison; but in de∣scending they are contrary; for to the Octave, that greater Sixth is a lesser Third, and the greater Third is a les∣ser Sixth; which lesser Third and Sixth

cannot well be called Flat, being in a Sharp Key; Flat and Sharp therefore do not well distinguish them in Gene∣ral. 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 be∣ing a Major Third, cannot be said to be Flat.

The other Distinction of them, viz. by Major and Minor, is more pro∣per, 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 ra∣ther

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 ano∣ther in like manner as the Fourth dif∣fers 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 there∣fore when you take the liberty spoken

of, under the general names of Thirds, it will be found, that you mix Ditones and Trihemitones, and so are not con∣cerned 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 e∣nough.

Yet, I confess, there is a little more liberty in moving Trihemitones and Di∣tones, 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 toge∣ther in every Fourth, and Fifth, or Fourth disjunct: That is, twice in Dia∣pason, 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

be in other Cliffs, but with some va∣riation, according to the place of the Hemitone. Here Fi 〈◊〉〈◊〉 and 〈◊◊〉〈◊◊〉 are two Trihemitones succeeding one ano∣ther, 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 Trihemi∣tone, and the next Tone above it, join∣ed to it, makes the same. Thus you see the necessity of moving two Tribe∣mitones 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 Tri∣hemitones 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 ano∣ther.

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 con∣stitute 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 Mi∣nor Thirds joined in consequence of Movement, are differenced in their Re∣lations, 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 Skil∣ful Artist may go farther in the use of Thirds and Discords, than is ordi∣narily allowed.

I might enlarge this Chapter, by setting down Examples of the Lawful and Unlaw∣ful Movements of Thirds Major and Minor, and of the Use of Discords; but, as I said before, my design is not to treat of Com∣position: 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.

[illustration] musical intervals

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

8th.	Octave, Diapason.	2	to	1
7th. Major.	Heptachord Major.	15		8
7th Minor.	Heptachord Minor.	9		5
6th. Major.	Hexachord Major.	5		3
6th. Minor.	Hexachord Minor.	8		5
5th.	Diapente. Pentachord.	3		2
5th. False (in de∣fect)	Semidiapente.	64		45
4th. False (in ex∣cess)	Tritone.	45		32
4th.	Diatessaron. Tetrachord.	4		3
3d. Mayor.	Ditone.	5		4
3d. Minor.	Sesquitone	6		5
3d. Minor.	Trihemitone.	6		5
3d. Minor.	Semiditone.	6		5
2d. Maj or Whole note Major.	Tone Major Degrees Diatorin. Max.	9		8
2d. Min. or Whole note Minor.	Tone Minor. Degrees Diatorin. Min.	10		9
2d. Least. or Half∣note Greater.	Hemi∣tone May. Degrees Diatorin. Minim.	16		15
2d. Least. or Half∣note Greater.	Semi∣tone May. Degrees Diatorin. Minim.	16		15
Half note Les••.	Hemi∣tone Minor	25		24
Half note Les••.	Semi- tone Minor	25		24
Half note Les••.	Diesis Chromatic.	25		24
Half note Les••.	Diesis Major.	25		24
Quarter Note.	Diesis Enharmonic.	128		125
Quarter Note.	Diesis Minor.	128		125
Difference be∣tween Tone Major & Tone Minor.	Comma Comma Maj•…•…. Schism.	81		80

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.