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

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 in∣numerable, as are the minute Tensi∣ons by which one Chord may be made to vary from it self, or from a∣nother. 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 Diffe∣rences found between Harmonick In∣tervals.

And these apt and useful Discords, are either Simple uncompounded In∣tervals, such as immediately follow one another, ascending or descending

in the Scale of Music: As Ut Re Mi. Fa Sol La Fa Sol, and are called De∣grees: Or else, greater Spaces or In∣tervals compounded of Degrees inclu∣ding 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 Se∣venths; 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.

And First of Degrees.

Degrees, are uncompounded Inter∣vals, which are found upon 8 Chords and in 7 Spaces, by which an imme∣diate Ascent or Descent is made from the Unison to the Octave or Diapason; and by the same progression to as ma∣ny Octaves as there may be occasion.

These are different, according to the different Kinds of Music; viz. Enhar∣monic, Chromatic, and Diatonic, and the several Colours of the two Lat∣ter: (All which I shall more conveni∣ently explain by and by.) But of these now mentioned, the Diatonic is the most Proper and Natural Way: The other two, if for Curiosities sake we consider them only by running the Notes of an Octave up or down in these Scales, seem rather a force up∣on Nature; yet herein probably might lye the Excellency of the Anci∣ent Greeks. But we now use only the Diatonic kind, intermixing here and there some of the Chromatic, (and more rarely some of the Enharmonic:) And our Excellency seems to lye in most artificial Composing, and joyning se∣veral parts in Symphony or Consort; which they cannot be supposed to have effected, at least in so many Parts as we ordinarily make; because (as is gene∣rally

affirmed of them) they owned no Concords, besides Eighth, Fifth, and Fourth, and the Compounds of these.

F. Kircher (cited also by Gassendus without any Mark of Dissent) is of Opinion, That the Anclent Greeks ne∣ver 〈◊〉〈◊〉 Consort Music, i. e. of diffe∣rent parts at once; but only Solitary, for one single Voice, on Instrument. And that Guido Aretinus first invented and throught in Music of Symphony or Consort both for the one and the other. They applyed Instruments to Voice, but how they were managed, He must be wiser than I, that can tell.

This way of theirs seems to be more proper (by the Elaborate Cu∣riosity and Nicety of Contrivance of Degrees, and by Measures, rather than by Harmonious Consonancy, and by long studied performance) to make great Impressions upon the Fancy, and operate accordingly, as some Histories

relate: Ours, more Sedately affects the Understanding and Judgment from the judicious Contrivance, and happy Composition of Melodious Consort. The One quietly, but powerfully af∣fects the Intellect by true Harmony: The Other, chiefly by the Rythmus, violently attacks and hurries the Imagi∣nation. In fine, upon the Natural Grounds of Harmony (of which I have hitherto been treating) is found∣ed the Diatonic Music; but not so, or not so regularly, the Chromatic and En∣harmonic kinds. Take this following view of them.

The Ancients ascended from the Unison to an Octave by two Systemes of Tetrachords or Fourths. These were either Conjunct, when they began the Second Tetrachord at the Fourth Chord; viz. with the last Note of the First Tetrachord; and which being so joyned, constituted but a Seventh: And therefore they assumed a Tone be∣neath

the Unison (which they there∣fore called Proslambanomenos) to make a full Eighth.

Or else the two Tetrachords were disjunct; the Second taking its begin∣ning at the Fifth Chord; there being always a Tone Major between the Fourth and Fifth Chords. So, the Degrees were immediately applyed to the 4ths, and by them to the Octave; and were different according to the different Kinds of Music. In the common Dia∣tonic Genus, the Degrees were Tone and Semitone; Intervals more Equal and Easy, and Natural. In the com∣mon Chromatic, where the Degrees were Hemitones and Trihemitones; the Dif∣ference of some of the Intervals was Greater. But the Greatest Difference, and consequently Difficulty, was in the Enharmonic Kind, using only Diesis, or Quarter of a Tone, and Ditone; as the Degrees whereby they made the Tetrachord.

And to constitute these Degrees, some of them, viz. the Followers of Aristoxenus, divided a Tone Major into 12 Equal Parts; i.e. Supposed it so di∣vided: Six of which being the Hemi∣tone, (viz. half of it,) made a Degree of Chromatic Toniaeum. And Three of them, or a Quarter called Diesis; a Degree Enharmonic. The Chromatic Fourth rose thus, viz. from the First Chord to the Second was a Hemitone; from the Second to the Third, a He∣mitone; from the Third to the Fourth, a Trihemitone; or as much as would make up a just Fourth. And this last Space (in this case) was accounted as well as either of the other, but one Degree or undivided Interval. And they called them Spiss Intervals (〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) when two of those other Degrees put together, made not so great an Inter∣val as one of these; as, in the Enhar∣monic Tetrachord, two Dieses were less than the remaining Ditone, and in the

common Chromatic, two Hemitone De∣grees were less than the remaining Trihemitone Degree.

Then for the Enharmonic Fourth, the first Degree was a Diesis, or Quarter of a Tone; the Second also 3 of those 12 parts, viz. a Diesis; the Third a Ditone; such as made up a just Fourth. And this Ditone, (though so large a Degree) being considered as thus pla∣ced (in the Enharmonic Tetrachord) was likewise to them but as one un∣compounded or entire Interval.

These were the Degrees Chromatic, and Enharmonic. Though they also might be placed otherwise, i. e. The greater Degree in these may change its place, as the Hemitone, (the less De∣gree) doth in the Diatonic Genus. And from this change chiefly arose the seve∣ral Moods, Dorian, Lydian, &c. From all which, their Music no doubt (though it be hard to us to conceive) must af∣ford extraordinary delight and pleasure;

if it did bear but a reasonable Propor∣tion to their infinite Curiosity and La∣bour. And as we may suppose ie to have differed very much from that which now is, and for several Ages hath been used: So consequently we may look upon it as in a manner lost to us.

In prosecution of my Design I am only, or chiefly to insist on the o∣ther Kind of Degrees; which are most proper to the Natural Explanation of Harmony; viz. the Degrees Dia∣tonic; which are so called; not be∣cause they are all Tones; but because most of them, as many as can be, are such; viz. in every Diapason, 5 Tones, and two Hemitones. Upon these I say I am to insist, as being, of those before mentioned, the most Natural and Ra∣tional.

Digression.

But before we proceed, it may perhaps be a satisfaction to the Reader, after what has been said, to have a little better Prospect of the An∣cient Greek Music, by some general Account; not of their whole Doctrine, but of that which relates to our present Subject, viz. their Degrees, and Scales of Harmony, and Notes.

First then, take out of Euclid the Degrees according to the three Genera; which were, Enharmonic, Chromatic, and Diatonic; which Kinds have six Co∣lours (as they call'd them.) Euclid, Introd. Harm. Pag. 10.

The Enharmonic Kind had but one Colour; which made up its Tetra∣chord by these Intervals; a Diesis (or Quarter of a Tone,) then such another Diesis; and also a Ditone incomposit.

The Chromatic had three Colours; by which it was divided into Molle, Sescu∣plum, and Toniaeum.

1st. Molle, in which the Tetrachord rose by a Triental Diesis (four of those 12 parts mentioned before) or third part of a Tone; and another such Die∣sis; and an Incomposit Interval, con∣taining a Tone, and half, and third part of a Tone: and it was called Molle, be∣cause it hath the least, and consequent∣ly most Enervated Spiss Intervals with∣in the Chromatic Genus.

2d. Sescuplum; by a Diesis which is Sesquialtera to the Enharmonic Diesis, and another such Diesis, and an Incompo∣sit Interval of 7 Dieses Quadrantal; viz. Each being 3 Duodecimals of a Tone.

3d. Toniaeum; by a Hemitone, and Hemitone, and Trihemitone; and is cal∣led Toniaeum, because the two Spiss In∣tervals make a Tone. And this is the ordinary Chromatic.

The Diatonick had 2 Colours; it was Molle, and Syntonum.

1st. Molle; by a Hemitone, and an incomposit Interval of 3 Quadrantal Dieses, and an Interval of 5 such Dieses.

2d. Syntonum, by a Hemitone, and a Tone, and a Tone. And this is the com∣mon Diatonic.

To understand this better, I must re-assume somewhat which I mention∣ed, but not fully enough before. A Tone is supposed to be divided into 12 least parts, and therefore a Hemitone contains 6 of those Duodecimal (or Twelfth) parts of a Tone; a Diesis Trientalis 4; Diesis Quadrantalis 3; The whole Diatessaron 30. And the Dia∣tessaron in each of the 3 Kinds, was made and perform'd upon 4 Chords, having 3 mean Intervals of Degrees, according to the following Numbers and Proportions of those 30 Duodeci∣mal parts.

• Enharmonic, by 3, and 3, and 24.
• Chromatic, Molle, by 4, and 4, and 22.
• Chromatic, Hemiolion, or Sescuplum, by 4½, and 4½, and 21.
• Chromatic, Toniaeum. by 6, and 6, and 18.
• Diatonic, Molle, by 6, and 9, and 15.
• Diatonic, Syntonum, by 6, and 12, and 12.

To each of these Kinds, and the Moods of them, they fitted a perfect Systeme, or Scale of Degrees to Disdia∣pason; as in the following Example ta∣ken out of Nichomachus: To which I have prefixed our Modern Letters.

E. Nichomacho; Pag. 22.

A	Nete Hyperbolaeon.
G	Paranete Hyper∣bolaeon.	Enh. Chro. Diat.
F	Trite Hyperbolaeon.	Enh. Chro. Diat.
E	Nete Diezeugme∣non.
D	Paranete Diezeug∣menon.	Enh. Chro. Diat.

C	Trite Diezeugme∣non.	Enh. Chro. Diat.
B	Paramese.
D	Nete Synemmenon.
C	Paranete Synem∣menon.	Enh. Chro. Diat.
B	Trite Synemmenon.	Enh. Chro. Diat.
A	Mese.
G	Lichanos Meson.	Enh. Chro. Diat.
F	Parypate Meson.	Enh. Chro. Diat.
E	Hypate Meson.
D	Lichanos Hypaton.	Enh. Chro. Diat.
C	Parypate Hypaton.	Enh. Chro. Diat.
B	Hypate Hypaton.
A	Proslambanomenos.

In this Scale of Disdiapason, you see the Mese is an Octave below the Nete Hyperbolaeon, and an Octave above the Proslambanomenos: And the Lichanos, Parypate, Paranete, and Trite, are chan∣geable; as upon our Instruments are the Seconds, and Thirds, and Sixths,

and Sevenths: the Proslambanomenos, Hypate, Mese, Paramese, and Nete, are Immutable; as are the Unison, Fourths, Fifths, and Octaves.

Now from the several changes of these Mutable Chords, chiefly arise the several Moods (some call'd them Tones) of Music, of which Euclid sets down Thirteen; to which were joyn∣ed two more, viz. Hyperaeolian and Hy∣perlydian; and afterwards Six more were added.

I shall give you for a Tast Euclid's Thirteen Moods. Euclid. Pag. 19.

• Hypermixolydius, five Hyperphrygius.
• Mixolydius acutior, five Hyperiastius.
• Mixolydius gravior, five Hyperdorius.
• ...Lydius acutior.
• Lydius gravior, five Aeolius.
• ...Phrygius acutior.
• Phrygius gravior, five Iastius.
• ...Dorius.
• ...Hypolydius acutior.

• ... Hypolydius gravior, five Hypoaeolius.
• ...Hypophrygius acutior.
• Hypophrygius gravior, five Hypoiastius.
• ...Hypodorius.

Of these the most Grave, or Lowest, was the Hypodorian Mood; the Pros∣lambanomenos whereof was fixed upon the lowest clear and firm Note, of the Voice or Instrument that was to ex∣press it; And then all along from Grave to Acute the Moods took their Ascent by Hemitones, each Mood being a Hemi∣tone higher or more acute than the next under it. So that the Prostamba∣nomenos of the Hypermixolydian Mood, was just an Eighth higher than that of the Hypodorian, and the rest accord∣ingly.

Now each particular Chord in the preceding Scale had two Signs or Notes [σημεια] by which it was chara∣cterized or described in every one of these Moods respectively, and also for

all the Moods in the several Kinds of Music; Enharmonic, Chromatic, and Di∣atonic; of which two Notes, the upper was for reading [λέξις] the lower for percussion [κροῦσις] One for the Voice, the other for the Hand. Consider then how many Notes they used; 18 Chords severally for 13 Moods (or rather 15, taking in the Hyperaeolian, and Hyperlydian, which are all descri∣bed by Alypius) and these suited to the three Kinds of Music. So many Notes, and so appropriated, had the Scholar then to learn and conn, who studied Music. Of these I will give you in part a View out of Alypius.

Notes of the Lydian Mood in the Diatonic Genus.

〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Φ.	Ϲ.	Ρ.	Μ.	Ι.	Θ.
〈☐〉〈☐〉.	Γ.	〈☐〉〈☐〉.	Ϝ.	Ϲ.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.
1	2	3	4	5	6	7	8	9

Γ	〈☐〉〈☐〉.	Ζ.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Μ.	Ι.
Ν.	Ζ.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Ζ.	η.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.
10	11	12	13	14	15	16	17	18

• 1 Proslambanomenos. Zeta imperfect, and Tau jacent.
• 2 Hypate Hypaton. Gamma averted, and Gamma right.
• 3 Parypate Hypaton. Beta imperfect, and Gamma inverted.
• 4 Hypaton Diatonos. Phi, and Digamma.
• 5 Hypate Meson. Sigma, and Sigma.
• 6 Parypate Meson. Rho, and Sigma inverted.
• 7 Meson Diatonos. My, and Pi drawn out.
• 8 Mese. Iota, and Lambda jacent.

• 9 Trite Synemmenon. Theta, and Lambda in∣verted.
• 10 Synemmenon Diato∣nos. Gamma, and Ny.
• 11 Nete Synemmenon. Ο Squared, lying Su∣pine upwards; and Zeta.
• 12 Paramese. Zeta, and Pi jacent.
• 13 Trite Diezeugmenon. E Squared, and Pi in∣verted.
• 14 Diezeugmenon Dia∣tonos. Ο Squared, Supine, and Zeta.
• 15 Nete Diezeugmenon. Phi jacent, and a care∣less Eta (η) drawn out.
• 16 Trite Hyperbolaeon. Υ looking downwards, and Alpha, left half, looking upwards.
• 17 Hyperbolaeon Dia∣tonos. My, and Pi lengthened, with an Acute above.
• 18 Nete Hyperbolaeon. Iota, and Lambda ja∣cent, with an Acute above.

The Numeral Figures I have added under the Signs (or Marks) only for Reference to the Names of the Notes signified by them, to save describing them twice.

Notes of the Aeolian Mood in the Diatonic Genus.

〈☐〉〈☐〉.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Χ.	Τ.	Ϲ.	Ο.	Κ.	Ι.
〈☐〉〈☐〉.	〈☐〉〈☐〉.	Γ.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Ϲ.	Κ.	λ.	〈☐〉〈☐〉
1	2	3	4	5	6	7	8	9

Ζ.	Α.	Η.	Ζ.	Α.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	Ο.	Κ
〈☐〉〈☐〉.	\.	〈☐〉〈☐〉.	〈☐〉〈☐〉.	\.	〈☐〉〈☐〉.	η.	Κ́.	λ́.
10	11	12	13	14	15	16	17	18

• 1 Proslambanomenos. Eta (H) imperfect a∣verted, and E Qua∣drate averted.
• 2 Hypate Hypaton, &c. Delta inverted, and Tau jacent, averted, &c.

Aristides (Pag. 91.) enumerates and describes all the Variations of every Let∣ter in the Greek Alphabet; by which the Signs or Notes above mentioned,

and those of the other Moods, were contrived out of them. They are in all 91; including the Proper Letters; I shall not describe, but only number them.

Out of

Α	were made	7	Ν	were made	2
Β		2	Ξ		2
Γ		7	Ο		2
Δ		4	Π		7
Ε		3	Ρ		2
Ζ		2	Σ		6
Η		5	Τ		4
Θ		2	Υ		3
Ι		4	Φ		4
Κ		3	Χ		4
Λ		5	Ψ		2
Μ		5	Ω		4
		49			42
91

I shall only add a word or two con∣cerning their Antient use of the Words Diastem and System. Diastem signifies an Interval or Space; System a Con∣junction

or Composition of Intervals. So that generally speaking, an Octave, or any other System, might be truly called a Diastem, and very frequently used to be so called, where there was no occasion of Distinction. Though a Tone, or Hemitone, could not be called a System: For when they spoke strictly, by a Diastem they understood only an Incomposit Degree, whether Diesis, He∣mitone, Tone, Sesquitone, or Ditone; for the two last were sometimes but Degrees, one Enharmonic, the other Chromatic. By System they meant, a Comprehensive Interval, compounded of Degrees, or of less Systems, or of both. Thus a Tone was a Diastem, and Diatessaron was a System, compounded of Degrees, or of a 3d. and a Degree. Diapason was a System, compounded of the lesser Systems, 4th, and 5th; or 3d. and 6th; or of a Scale of Degrees: and the Scale of Notes which they used, was their Greatest, or Perfect System. Thus with

them, a 3d. Major, and a 3d. Minor, in the Diatonic Genus, were (properly speak∣ing) Systems; the former being com∣pounded of two Tones, and the latter of three Hemitones, or a Tone and Hemi∣tone: But in the Enharmonic Kind, a Ditone was not a System, but an Incom∣posit Degree; which, added to two Die∣ses, made up the Diatessaron: And in the Chromatic Kind, a Trihemitone was the like; being only an Incomposit Dia∣stem, and not a System.

But to return from this Digression (which is not so much to my purpose, as to gratify the Reader's Curiosity) and continue our Discourse according to Nature's Guidance, upon the Diatonic Degrees. It was said that there are 5 Tones and 2 Hemitones in every Dia∣pason. Now the reason why there must be 2 Hemitones, is, because an Eighth is Naturally composed of, and divided in∣to 5th. and 4th; and a Fifth is 3 Tones and a half; a Fourth 2 Tones and a half;

and the Ascent, by Degrees, must pass by Fourth and Fifth; which are always unchangeable, and keep the same Di∣stance from Unison; and a just Tone Major of 9 to 8 always between them. Therefore the Diapason has not an Ascent of 6 Tones; but of 5 Tones and 2 He∣mitones, One Hemitone being placed in each Fourth Disjunct; in either of which Fourths, the Degrees may be altered by placing the Hemitone in the First, or Se∣cond, or Third Degree of either. As, MI, FA, Sol, La. La, MI, FA, Sol. Sol, La, MI, FA. If this be done in the former Tetrachord, then is changed the Second, or Third Chord; If in the other Disjunct Tetrachord, then the Sixth, or Seventh is changed: The Fourth and Fifth being Stable and Immutable. By them we Naturally divide the Diapason: The Second, Third, Sixth, and Seventh, are alterable, as Minor, and Major, ac∣cording to the place of the Hemitone.

〈2 pages missing〉〈2 pages missing〉

In Diatonic Music, there is but one Sort of Hemitone amongst the Degrees, called Hemitone Major; whose Ration is 16 to 15: being the Difference, and making a Degree between a Tone Major, and Third Minor; or between a Third Major, and a Fourth.

There are two Sorts of Tones; viz. Major, and Minor. Tone Major (9 to 8) being the Difference between a Fourth and Fifth: And Tone Minor (10 to 9) which is the Difference between Third Minor and Fourth. But both the Tones arising (as hath been said) out of the Partion of a Third Major, in like man∣ner as 5th. and 4th. do by the Partition of an Eighth: I may (with submission) make the following Remark; wherein, if I be too bold, or be mistaken, I shall beg the Reader's pardon.

The Ancient Greek Masters found out the Tone by the Difference of a Fourth and Fifth, substracting one from the other. But had they found it also

(and that more Naturally) by the Divi∣sion of a Fifth; first into a Ditone and Sesquitone, and then by the like proper Division of a true Ditone (or Third Ma∣jor) into its proper parts; they must have found both Tone Major, and Tone Minor. Euclid rests satisfied, That, In∣ter super-particulare non cadit Medium. A super-particular Ration cannot have a Mediety; viz. in whole Number: which is true in its Radical Numbers. But had he doubled the Radical Terms of a Super-particular, he might have found Mediums most Naturally and Uniform∣ly dividing the Systems of Harmony. Ex. gr. The Duple Ration 2 to 1, as the Excess is but by an Unity; has the Nature of Super-particular: but 2 to 1, the Terms being dupled, is 4 to 2; where 3 is a Medium, which divides it into 4 to 3 (4th.) and 3 to 2 (5th.) Again, 3 to 2, dupling each Term, is 6 to 4; and in the same Manner gives the 2 Thirds, viz. 6 to 5, (3d. Minor)

and 5 to 4, (3d. Major.) Likewise the 3d. Major, 5 to 4, dupled as before, 10 to 8, give the 2 Tones; i. e. 10 to 9, Tone Minor, and 9 to 8, Tone Major.

And it seems to be a reason why the Antients did not discover and use the Tone Minor, and consequently not own the Ditone for a Concord; because They did not pursue this way of dividing the Systems. Although Euclid had a fair Hint to search further, when he mea∣sured the Diapason by 6 Tones [Major] and found them to exceed the Interval of Diapason.

The Pythagoreans, not using Tone Mi∣nor, but two Equal Tones Major, in a Fourth, were forced to take a lesser In∣terval for the Hemitone; which is called their Limma, or Pythagorean He∣mitone; and, which added to those two Tones, makes up the Fourth: it is a Comma less than Hemitone Major, (16 to 15;) and the Ration of it, is 256 to 243.

Yet we find the later Greek Masters, Ptolemy, to take Notice of Tone Minor; and Aristides Quintilianus, to divide a Sesquioctave Tone (9 to 8) by dupling the Terms of the Ration thereof, into 2 Hemitones; 18 to 17, and 17 to 16. And those again, by the same way, each into two Dieses; 36 to 35, 35 to 34; the Division of 18 to 17, the less He∣mitone: And 34 to 33, and 33 to 32; the parts of 17 to 16, the greater Hemi∣tone. But yet, none of these were the Complement of two Sesquioctave Tones to Diatessaron: but another Hemitone, whose Ratio is about 20 to 19; not exactly, but so near it, that the Differ∣ence is only 1216 to 1215; both which together make the Limma Pytha∣goricum.

But I no where find, that they thus divided the 5th, and 3d. Major, but ra∣ther seemed to dislike this way, because of the Inequality of the Hemitones and Dieses thus found out; and chose rather

[illustration]

[illustration] organ tuning chart

to Constitute their Degrees by the Ses∣quioctave Tone, and those Duodecimal supposed Equal Divisions of it. But to return.

There are, you see, 3 Degrees Diato∣nic; viz. Hemitone Major, Tone Minor, and Tone Major. The First of these, some call Degree Minor; the Second, Degree Major; and the Third, Degree Maxim. Now these three Sorts of De∣grees are properly to be intermixed, and ordered, in every Ascent to an Eighth, in relation to the Key, or Uni∣son given, and to the Affections of that Key, as to Flat, and Sharp, in our Scale of Music; so, that the Concords may be all true, and stand in their own set∣tled Ration. Wherefore if you change the Key, they must be changed too; which is the Reason why a Harpsichord, whose Degrees are fixed; or a fretted Instrument, the fretts remaining fixt, cannot at once be set in Tune for all Keys: For, if you change the Key, you

withall change the place of Tone Minor, and Tone Major, and fall into other He∣mitones that are not proper Diatonic De∣grees, and consequently into false Inter∣vals.

You may fully see this, if you draw Scales of Ascent fitted to several Keys (as are here inserted) and compare them. For an Example of this, Take the First Scale of Ascent to Diapason (I) viz. up∣on C Key Proper, by Diatonic Degrees; (making the first to be Tone Minor, as convenient for this Instance) intermixing the Chromatic and other Hemitones, as they are usually placed in the Keys of an Or∣gan; i. e. run up an Eighth upon an Or∣gan (tuned as well as you can) by half Notes, beginning at C Sol fa ut, and you will find these Measures. The Proper Degrees standing right, as they ought to be, being described by Breves; the o∣ther by Semibreves: The Breves repre∣senting the Tones of the broad Gradual Keys of an Organ; the Semibreves re∣presenting

the Narrow Upper Keys, which are usually called Musics. And let this be the first Scale, and a Stan∣dard to the rest.

Then draw a Second Scale (II) run∣ning up an Eighth in like manner; but let the Key, or First Note, be D. Sol re Proper, on the same Organ standing tuned as before; which Key is set a Note (or Tone Minor) higher than the former.

Draw also a Third Scale (III) for D Sol re Key with Sharps, viz. Third, and Seventh, Major; i. e. F, and C, Sharp.

In the First of these Scales, the De∣grees (expressed by Breves) are set in good and natural Order.

In the Second Scale (changing the Key from C to D). you will find the Fourth, and Sixth, a Comma (81 to 80) too much; but between the Fourth, and Fifth, a Tone Minor, which should be al∣ways a Tone Major. So from the Fourth

to the Eighth, is a Comma short of Dia∣pente; and from the Sixth, a Comma short of 3d. Minor. And this, because in this Scale, the Degrees are misplaced.

The Third Scale makes the Second, Third, Fourth, and Sixth, from the Uni∣son, each a Comma too much; and from the Octave, as much too little. In it, the 3d. Degree, between ♯F and G, is not the Proper Hemitone, but the Great∣est Hemitone, 27 to 25. And all this, be∣cause in this Scale also, the Degrees are misplaced; and there happen (as you may see) three Tones Minor, and but two Major: the Deficient Comma being added to the Hemitone.

I have added one Example more, of a Fourth Scale; (IV) viz. beginning at the Key ♯C; with the like Order of Degrees, as in the First Scale (from the Note C♯) upon the same Instrument, as it stands tuned after the First Scale. And this will raise the First Scale half a Note higher.

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In this Scale, all the Hemitones are of the same Measure with those of the First Scale respectively.

And the Intervals should be the same with those of the First Scale; which has Third, Sixth, Seventh, Major.

But in this Fourth Scale, the 1st. De∣gree, from ♯C to ♭E, is Tone Major, and Diesis; as being compounded of 16 to 15, and 27 to 25.

The 2d. Degree from ♭E to F, is Tone Minor; therefore the Ditone, made by these two Degrees, is too much by a Diesis, (128 to 125) and as much too little the Trihemitone, from the Ditone to the Fifth.

The 3d. Degree, from F to ♯F, is a Minor Hemitone, 25 to 24; which, (though a wrong Degree) sets the Dia∣tessaron right.

The 4th. Degree, from ♯F to ♯G, is Tone Major, and makes a true Fifth.

The 5th. Degree, from ♯G to ♭B, is Tone Major, and Diesis; setting the Hexa∣chord

(or Sixth) a Diesis and Comma too much, or too High. It ought to have been Tone Minor.

The 6th. from ♭B to C, is Tone Minor; too little in that place by a Comma.

The 7th. from C to ♯C, is Hemitone Minor; too little by a Diesis. And so, these two last Degrees are deficient by a Diesis and Comma; which Diesis and Comma, being Redundant (as before) in the 5th. Degree, are balanced by the Deficiency of a Comma in the 6th. De∣gree; and of a Diesis in the 7th: And so the Octave is set right.

These Disagreements may be better viewed, if we set together, and com∣pare the Degrees of this IV Scale, and those of the I: where we shall find, no one of all the 7 Degrees, to be the same in both Scales.

Scale I.	Scale IV.
Degrees.
1st. Tone Minor.	Tone Maj. & Diesis.
2d. Tone Major.	Tone Minor.
3d. Hemit. Major.	Hemitone Minor.
4th. Tone Major.	Tone Minor.
5th. Tone Major.	Tone Maj. & Diesis.
6th. Tone Major.	Tone Minor.
7th. Hemit. Major.	Hemitone Minor.

And thus it will succeed in all Instru∣ments, Tuned in order by Hemitones, which are fixed upon Strings; as Harp, &c. Or Strings with Keys; as Organ, Harpsichord, &c. Or distinguished by Fretts; as Lute, Viol; &c. For which there is no Remedy, but by some alte∣rations of the Tune of the Strings, in the Two former; and of the Space of the Fretts in the latter; as your present Key will require, when you change from one Key to another, in perform∣ing Musical Compositions.

Though the Voice, in Singing, be∣ing Free, is naturally Guided to avoid and correct those before described Ano∣malies, and to move in the true and proper Intervals: It being much easier with the Voice to hit upon the Right, than upon the Anomalous or Wrong Spa∣ces.

Much more of this Nature may be found, if you make and compare more Scales from other Keys. You will still find, that, by changing the Key, you do withall change and displace the De∣grees, and make use of Improper De∣grees, and produce Incongruous Inter∣vals.

For instead of the Proper Hemitone, some of the Degrees will be made of other sort of Hemitones; amongst which chiefly are these two: viz. Hemitone Maxim. 27 to 25; and Hemitone Minor, or Chromatic, 25 to 24. Which Hemi∣tones constitute and divide the two Tones; viz. Tone Major, 9 to 8: the

Terms whereof Tripled, are 27 to 24; and give 27 to 25, and 25 to 24. The Tone Minor likewise is divided into two Hemitones: viz. Major, 16 to 15; and Minor, 25 to 24.

These two serve to measure the Tones, and are used also, when you Divert in∣to the Chromatic kind. But the Hemi∣tone-Degree in the Diatonic Genus, ought always to be Hemitone Major, 16 to 15; as being the Proper Degree and Differ∣ence between Tone Major and Trihemi∣tone, between Ditone and a Fourth, be∣tween Fifth and Sixth Minor, between Sixth Major and Seventh Minor, and also between Seventh Major and Oc∣tave.

Music would have seem'd much Ea∣sier, if the Progression of Dividing had reached the Hemitones: I mean, If, as by Dupling the Terms of Diapason, 4 to 2; it Divides in 4 to 3, and 3 to 2; Diates∣saron, and Diapente: And the Terms of Diapente dupled, 6 to 4; fall into 6 to 5,

and 5 to 4, Third Minor, and Third Major; And Ditone, or Third Major, so Dupled, 10 to 8, falls into 10 to 9, and 9 to 8, Tone Minor, and Tone Major: If, I say, in like manner, the dupled Terms of Tone Major 18 to 16, thus divided, had given Usefull and Proper Hemitones 18 to 17, and 17 to 16. But there are no such Hemitones found in Harmony, and we are put to seek the Hemitones out of the Differences of Other Intervals; as we shall have more Occa∣sion to see, when I come to treat of Differences, in Chap. 8.

I may conclude this Chapter, by shewing, how All Consonants, and o∣ther Concinnous Intervals, are Com∣pounded of these three Degrees: Tone Major, Tone Minor, and Hemitone Ma∣jor; being severally placed, as the Key shall require.

• Tone Major, and Hemitone Major, joyn'd, make 3d. Minor.
• Tone Major, and Tone Minor, joyn'd, make 3d. Major.
• Tone Major, and Tone Minor, and Hemitone Major, joyn'd, make 4th.
• 2 Tones Major, joyn'd, make 5th.
• 1 Tone Minor, joyn'd, make 5th.
• 1 Hemitone Maj. joyn'd, make 5th.
• 2 Tones Major, joyn'd, make 6th. Minor.
• 1 Tone Minor, joyn'd, make 6th. Minor.
• 2 Hemitones Maj. joyn'd, make 6th. Minor.
• 2 Tones Major, joyn'd, make 6th. Major.
• 2 Tones Minor, joyn'd, make 6th. Major.
• 1 Hemitone Maj. joyn'd, make 6th. Major.
• 3 Tones Major, joyn'd, make 7th. Minor.
• 1 Tone Minor, & joyn'd, make 7th. Minor.
• 2 Hemitones Maj. joyn'd, make 7th. Minor.
• 3 Tones Major, joyn'd, make 7th. Major.
• 2 Tones Minor, joyn'd, make 7th. Major,
• 1 Hemitone Maj. joyn'd, make 7th. Major,

• 3 Tones Major, joyn'd, make Diapason.
• 2 Tones Minor, joyn'd, make Diapason.
• 2 Hemit. Major, joyn'd, make Diapason.
• 2 Tones Major, joyn'd, make Tritone, or false 4th.
• 1 Tone Minor, joyn'd, make Tritone, or false 4th.
• 1 Tone Major, joyn'd, make Semidiapente, or false 5th.
• 1 Tone Minor, joyn'd, make Semidiapente, or false 5th.
• 2 Hemit. Major, joyn'd, make Semidiapente, or false 5th.