Notes on this transcription: August/September 2002.
This rendition has been transcribed from a photocopy of the original manuscript as supplied by the Library of Congress, Washington D.C.
The manuscript has many corrections, insertions, additions, crossings out, and underlines, and uses extremely long sentences and paragraphs. In transcribing it I have attempted to keep everything that Harrison wrote, and to position it in the order sequence that he intended. I have maintained his spelling, and Capitalisation, and attempted to discover and indicate which parts of the text he intended as footnotes and margin notes. In some cases it has not been clear exactly where certain passages were intended to appear, and there seem to be a few missing interlines, which are referred to in the manuscript. They may have been written on loose sheets which have become separated from the manuscript. I have included these references and inserted notes at the appropriate points in the text, where I have found difficulty in interpreting Harrison's intentions.
The body of the manuscript consists:
Preface - six sheets (Preface first page + II to VI)
Body of Book - 170 pages - Title and first sheet + 169 others which Harrison numbered in sequence from 2 to 170;
[Pages 142 to 170 are corrections, additions and amendments)
Corrections and amendments - Seven unnumbered pages of corrections.
One sheet of a receipt from the 1922 for 5/5/-, presumably for the sale of the manuscript.
To enable readers to more easily find subjects in the text, which would otherwise be continuous,
I have concluded that there four main subjects; and have labelled them accordingly as:
1) The logic and specifications of Tuning from Pi.
2) The shortcomings of Integer derived tuning systems.
3) Church Services, Psalms, Parsons, and Choirs
4) The Tuning of Bells using "Radical Numbers".
A true and full Account of the Foundation of Musick,
or ,
as principally therein, of the Existense of the Natural Notes of Melody:
Wherein is shown the Absurdity of such imaginary stuff as was first crowded into the Scale of Musick by the Ancients, and has still been retained therein by our more modern Writers, in order to make the matter seem more Reasonable or Philosophical, whenas in reality it is even as hence, or would be as thence, bereaved of all such Ornament, and as, when at the same time, its Foundation or basis as in another Sense tho' secret, is most strictly true or perfect in itself; and therefore, as being otherwise therefrom not only to a much higher degree than what we should think. or have thought to have been Mathematically Beautiful, or even as otherwise so Perfection itself, consequently, it can have nothing to do with any such Arbitrary Conjectures:
And herein are also, very necessary Observations exhibited and as with Instructions about the highest, the most Important, or the most Undeniable life of the Subject:
Where Does This Go???? --Start.
To the Book cover at the latter end of the book at E, the interline following.
E or as in other words, and as still implying amongst us, so much reason as when He desir'd that Labourers might be sent into His Vineyard, or Reapers to the Harvest, or as when St. Paul fought (as it were) with beasts at Ephesus &c. but certainly, as now otherwise, and even for the same Reason [I am far from saying in Opposition thereto] we ought for the better to make the Praising of God, and that according to David's way, the more or farther to shine; so therefore, was it to so be said that Christ taught Parsons [or Bishops] by Preaching to stifle the Psalms, viz. by not affording them their due, or as by
Cx on the Book cover.
c [not but that the parson might do it as he pleas'd, and could do it well, but I have not heard or seen a Clark likely so to do it in London was the Parson to permit it, but I say when so] and to be jointly
x [for the pride of the Parson]
Written in the left margin vertically and crossed out;
b nay, from late Experience my son has found that Forks can be apply'd even with greater success to the Tuning of A Harpsichord or Spinet than the Monochord i.e. with much more ease, and to greater exactness than the said Monochord itself by which they are Tuned, fine Mediators indeed! but in that point they have not yet been experienced as with respect to the Tuning of
At A on the Book cover.
Where Does This Go???? --End.
As Experimented and Illustrated by John Harrison, Inventor and maker of the Time-Keeper for the Longitude at Sea :
and who was for several years, Parish Clark [or Clark of the Psalmody] at Barrow near Barton upon Humber, in the county of Lincoln.
A true and full Account of the Foundation of Musick, or ,
as principally therein, of the Existense of the Natural Notes of Melody:
The Preface
The subject matter, or Foundation of Musick, as contained in the fore-end of this Book, comes to light in the World through my becoming acquainted with Mr. Benjamin Darlin of Hull, who was a very able Mathematician but no Musician, nor, as it happened, and that as in consequence much for the better, had he ever had any the least occasion to consider about, or at all to reflect upon what is called Theory, as touching any Ground-work concerning the Thing.
Now, It was upon occasion of my going to Hull [as desir'd] to try [as with respect to Tune] two Peals of bells which they had got cast, that thereby got acquainted with the above mentioned Gentleman, and to whom I communicated the Foundation of their Division, or imperfect Scale of Musick which was then upon the Monochord I had along with me, as was also [by fretts] upon my Viol which I had at Home; The place of my abode being then near to that Town, viz. at Barrow in Lincolnshire, across the River Humber opposite thereto. now that said Foundation [as from as it were Arbitrary Experiments, without any help] was upon the major 3rd being taken a little flat, but 'twas not so much as 1/20 Comma; and the whole or result of which, in the Logarithm way, I then communicated to him, but told him. as also the Organist, that it had never given me right content, in that my Viol would not afford a Tune to true and fine is a good Voice could sing it, but, as notwithstanding, I told 'em it was the best Division that I could, and with the greatest truth of Strings possible acquire from Experience, i.e. the best Division of the Octave that I, as by the means of playing a few Country Tunes could ascertain, as when under that Condition of truth in the strings, viz. by, or from thence altering or varying the Calculation for the placing of the fretts accordingly as when, or as I had then, at any time judg'd or imagin'd faults to lye, I say from thence, and as when at last from my acquiring, or resting upon the Ground of the Major 3rd, being taken almost 1/20 Comma flat [or as from my making that the ground] that I thought I had got the Division of the Octave as good as it was possible ever at all to be, having, as it is to be understood, a strong confinement upon me from Theory, viz. according to the common way of proceeding as of old; but now, as I had told Mr. Darlin that my Division had never pleas'd me right, he in a few days after [in as small a Compass as I had given mine to him] sent me the result of the Circumference and Diameter of a Circle seeing that that was not, as he thought, a great deal different from what I had communicated to him; now he sent this in order for me to try whether it would do better than mine or not, and withal that I might see how near I had come then to not thinking anything about it; or at all considering it in that light, he did not tell me, nor could he give me any the least assurance according as Dr. Smith relates in his Preface, that it would be better, nor as it seemed contradictory (as already intimated) to my reason in making the Major 3rd, as also, and as I thought farther to be regarded the 5th flatter than mine, did I think it would, and afterwards shew'd Mr. Darlin that reason; so I say as it was contradictory to my reason, or to such reasons as was ingrained in me from false Notions, or Prejudice of Theory, it laid by me for several Years, or as untill I had been some years in London, [for it is to be understood, that for some years after Mr. Darlin had sent me his bit of paper, as I was then otherwise engag'd had but seldom time to play] but at last upon (as it were accidentally} discovering that my major 6th was excessively too sharp, or was as thence no 6th at all, as being quite out of its Bounds or beyond such Latitude as I found it would only admit of, for [after which Discovery] by trying farther, or more strictly I found, that if the said major 6th was but 3/20 Comma too sharp [viz. as with respect to what is styl'd perfect] it would be good for nothing, as likewise the same, or but scare discernibly better, as with respect to flatness in its complement to the Octave the Minor 3rd, I say I did not juge that it would be so in the Major 6th, according to Dr. Smith's Preface, but I tried and found it so, and it is moreover to be observ'd, as still contrary to Dr. Smith, that it could be no judging or trial, as pertaining to Mr. Huygen's Division, the which I had then never heard of, and the which was afterwards [viz. after Mr.Graham had told me of it] so plain from its badness as not to need any trying, but was, as on the other hand from such Experiments, consequently bad, as were on my behalf truly genuine, Nay, And I am now further sure viz. from my strict Experiments than what I was able to make at that time the same Experiments as touching the truth, are much more to be regarded or depended on, than what, upon this subject can otherwise be demonstrated from the power of any theory whatever, nay as touching the matter, such Experiments are indeed only to be rely'd on. and from thence [viz. from my Experiments} I took occasion to try what Mr. darlin had sent me so long before, and the which to my great surprise produc'd the truth of the matter, or was the very Thing wanting;
For it is to be understood, that if any one Chord or Interval be not exactly what it ought to be, the whole Scale or Division must as thereby be rendered false or thrown out of order, but meaning principally as with respect to the natural Steps of melody; but in consequence thereof, as I shall shew, and tho' it may be in some measure contradictory to our reasoning, yet if otherwise it must then be thrown or varied from what must withal produce (as Providence or Nature has ordered the Matter) the most Lofty, or the most elegant degrees of Harmony.
Now after I had fully satisfy'd myself of the fact, I wrote to my brother to desire him to acquaint Mr. Darlin therewith, as also to know whether he did it of himself (as I have good reason to think he did) or whether he took the hint from any Book, but the Answer came that Mr Darlin was dead; and along with which my Brother sent me [still in the Logarithm way] another Division of the Octave [viz. as founded upon the Major 3rd being taken Perfect] the which Mr. Darlin had afterwards communicated to Mr Norwood of Hull, and the which I still have by me: bit as this was some years after my first communicating the subject to mr. Darlin, he might perhaps, by some means or other, as in that Interim have heard of Dr. Smith's Foundation and Division thereupon built. from whence it is more plain that Mr. Darlin had no assurance of either of them being what was wanted to be. nor even whether the one or the other of 'em, or any other whatever could be any better than what I at first had communicated to him [contrary to Dr. Smith's relation of the Matter] but still, 'tis nevertheless to his commendation to be remembered, that Mr. Darlin was a very ingenious worthy Man.
And now to conclude my preface, As I have spoken [in this Book] of the chief use or most undoubted Design of Musick, as subsequent to such its Administration as I have shewn or treated of, as from its Scale or secret Foundation, and tho' in this point of the Matter, viz. its chief use not perhaps so perfectly or properly as might in consequence from the same its surprizing Foundation have ensu'd by some other more able pen, yet still, or as notwithstanding, I hope my Labours therein, or as taken in the whole, may prove to some acceptable.
Of the Ground-work or Existence of the Natural Notes of Melody &c.
I have found from strict Experiments, as with suitable preparations or Apparatuses to the Purpose, that the natural Notes of Melody are certainly free from all the inconsistent nonsense arising from the Imagination of there being a Tone Major of 9 to 8, a Tone Minor of 10 to 9, and a nonsensical Comma (as being their difference) of 81 to 80, they being things according to my Experience, which Nature ever despised, yet always seemed to have their room wherein to create controversy, or inconsistency about the Scale of Musick; I say, as free from these &c. they [the said Notes] are, as I found on the contrary in a regular scale, consequently in such a one as cannot then, or upon the same account be said to a tempermental scale, but on the other hand a truly Melodious One, it being only the Notions of men, from false conjectures of old (as shewn in this Book) that imagined it a temperamental, in their adhering to Harmony as their Principal, and not as Melody; whereas on the contrary, melody ought certainly to have been, nay must certainly be the Motive, for even most of the Country Plow-men and Milk-maids can naturally sing a Tune [or do naturally sing Tunes] one ought not as therein to be contradicted, may rather [as touching the Scale] will not, or cannot as therein suffer to be obtruded upon -- by any Instrument or its Tuning, since (as already advanced) they, as well as any other fine, or all finer singers whatever do naturally sing in Tune, i.e. without any consideration, or idle thoughts in them about a Temperament; consequently a beginning with Harmony [as touching any Treatise of Musick] must be to begin at the wrong end of the work; and therefore, as granting melody so be the Principal, it ought most chiefly, nay I may say entirely as on the other hand to have been [by proper Methods or Experiments] examined to what degree or perfection Harmony might as thence be, or rather as thence was derived; for even without taking any farther notice of the inconsistency which would arise from such Tones major and Minor as have been imagined, the following, tho' short succession of 3rds and 4ths must be sufficient to show the absurdity of the Thing, for such perfect Intervals [viz. as falsely so styled] never was, nor can be used in any Melody whatever, because [and even without Notifying such unpleasantness as would consequently be heard in sounding the succession in such a Monstrous manner once over; the which could not once be done by the Voice, tho' always thought the contrary, neither upon any Instruments any more than once, viz. without, as in going on, a second Preparation both of Strings and Tuning for a second time &c.] because I say the Pitch, in, or after each time would be varied or flattened what has been called a Comma, the three descending 3rds being bigger by that quantity than the two ascending 4ths [the which in Nature cannot be], so in consequence of which, as with regard to Melody [the Voice, or as I may the Voice of Nature was adhering to, or admitting of these sorts of Mathematical 3rds and 4ths &c.] some other intervals therefore more truly perfect [or to a secret Perfection] are always used by the voice, I do not mean only in this succession (here below) but in all others for it is certain, that the same 3rds and 4ths which are used here, both must and will according to nature be used in all successions whatever, as it is certain that they are not, or can not be such Mathematical ones as intimated above; see the Notes or Proof of the Matter as followeth; where in the fifth place let a certain Pitch be given for the Note C, then for easiness or perspicuity, instead of working with the Ratios themselves take their Logarithms as in manner below,
That of the 4th ------------, 12494
That of the Sharp or Major 3rd -, 09691
That of the Flat, or Minor 3rd --, 07918
The Pitch C F D G E C
or
The Pitch C F D G E C
Now, these examples are only as instances, but the notes are to be imagined as taken upon the proper key.
++++++++++++
Lucy comments: This must be the log values for the small integer ratios of the intervals which he mentioned earlier
12494 is the log of 3:4 - The just or Pythagorean Fourth
(498.045 cents) (e.g. C to F) i.e. (2L+s)
09691 is the log of 4:5 - The just major third (5th harmonic) (407.820 cents) (e.g. C to E) i.e. (3L+s)
07918 is the log of 5:6 - The just minor third
(315.641 cents) (e.g. D to F) i.e. (L+s)
++++++++++++
Hence F, as here so be supposed
the first Note, would be above ,12494
the pitch [as such a 4th] viz.
Subtract
And D as such as Flat 3rd
descending from F ------ ,07918
________
The quantity or space there
fore that D would be above the ,04576
Pitch --------------------
Add
G a 4th ascending from D --- ,12494
_________
Therefore the Quantity that
G would be above the Pitch ,17070
would be -----------------
Subtract
The the descent of a Flat
3rd from G to E ------------ ,07918
_________
Hence the quantity that E
would be above the Pitch --- ,09152
Subtract
And then as lastly, the descent
of a Sharp 3rd from E
must or would [ according to the
absurdity of these ratios] go lower
than the Pitch, -- as first given to C,
the log of which being ------ ,09691
__________
So hence the whole, the Pitch
would be flattened the feigned
non-musical Comma [viz. the Ratio
of 80 to 81] the log of which being ,00539
So, as hence in the whole [viz. as in the once going over with these notes in succession] the Pitch would thus be flattened a Comma, or would be so much sharpened in the same Notes in succession be so varied as the 3rds.
++++++++++++
Lucy notes.
Harrison demonstrates the comma (used in just Intonation) mathematically using logs and shows how if we assume:
A IVth is 4:3
IIIrd is 5:4
and bIIIrd is 6:5
Ascending by IVth (C to F) = * (4/3)
Descending by bIIIrd (F to D)= / (6/5)
Ascending by IVth (D to G) = * (4.3)
Descending by bIIIrd (G to E)= / (6/5)
Descending by IIIrd (E to C)= / (5/4)
Would be expected to arrive at the same pitch for C,
Yet the result is that the C is flattened by a comma
(81/80)
This is a widely known and valid criticism of integer ratio tuning systems. i.e. the existence of the comma.
Any meantone tuning would overcome this problem.
What is unknown about Harrison's solution is why he chose pi as the basis of his meantone system, and how he can justify it.
++++++++++++
so ascend and the 4ths descend, as
,
so it must in consequence be infered that the Ratios corresponding to these logarithms can never be used fine, or perfect melody, must therefore from Voices &c. be from some other Intervals; the which others for readiness, as well as for the great exactness required in the affair [and the which as naturally arising from the Plan from whence the same results]
I shall illustrate by the Logarithms, viz. after I have spoken a little farther concerning the Logarithms of ratios; and in both which cases it is to be understood that such correspondent numbers are not in this affair to be looked upon only as logarithms, but as real quantities, or proportionable representations as used here above.
Now the log of any ratio is had by subtracting the log. of the lesser number from that of the greater. This the ratio of the 5th being as 2 to 3, so [from Table Of Logarithms] the log. of 2 subtract'd from the log of 3 gives the log or proportion for the ratio of the 5th &c. Hence the logs of the most remarkable ratios are as followeth.
The log. of the 5th viz. of 2 to 3 ----,17609
The log. of the 4th ----- of 5 to 4 ----,12494
The log. of the sharp 3rd of 4 to 5 ----,09691
That of the flat 3rd of 5 to 6 ----,07918
That of the sharp 6th of 3 to 5 ----,22185
That of the flat 6th of 5 to 8 ----,20412
Now the logarithms or numbers must indeed be looked upon, as if to be the very substance of the matter yet as notwithstanding without their proper seasonings respectively as hereafter, are as already intimated, they are or can be nothing there not being, as is in the first place already shewn, a proper field for their admittance or reception neither of these was, could it be any better, as will fairly appear from this plain treatise, but on the contrary much worse; but from strict experience I have found the logarithms as hereafter, not only, as with the greatest strictness to be the logarithms or exact quantities of the intervals of melody or true steps of tune, the most principal thing to be notified, but withal, as at the same time, and the which is very astonishing, in that to the very completion of the matter when an instrument is tuned exactly to them, I say exactly, without nay the least deviation flatter or sharper in each or every note respectively, as can only be verified by means of a curious monochord divided therefrom and with its string drawn to such a degree of accuracy as truly to correspond therewith, For it is to be understood that the common drawing of Wire for Musical Instruments will not produce a String for this Purpose. I say, I then as farther found 'ere to afford the consonances of harmony much more lively, bright or elegant than if nature had permitted true melody &c. and in my consequence thereof, a division of the octave from the logarithms of the others as above, or, what is all one, from their corresponding chords to have been hence to proceed,
The Vibrations of the Octave as a Standard being 2 to 1 [or as the Ratio of 2 to 1] in which all the rest [and even as supposed may be wanted upon any occasion] must have their Portions or as it were Stations assigned; I say as so with regard to the Octave, its Logarithm or Quantity must be the difference (as shewn above) betwixt the Log. of 1 and that of 2, consequently, as the log of 1 is 0, so the Log of the Octave must remain to the Log of the Number 2, viz. ,30103: and for the making of this noble division (as I have found) viz. for the Stations of all the Intervals or Chords to be contained therein, Providence or Nature has provided, or rather has settled the Affair on such a Foundation, as whereupon from the the most accurate of Experiments [nay indeed it is only such that can be of service in the matter] no fault in the least is to be found, but, for the better, every Chord, whether as strictly in their primary sense they be to be look'd upon as intervals of melody, or secondarily as Consonances of Harmony, they are thereupon rendered to the most elevated height of Perfection; whereas on the other hand, or without such Basis, as the Matter is so very difficult, it might perhaps never have been in the power of Man to have brought the Division to the right Decision or Establishment, tho' from faulty Experiments, or even at the best from small faults in Experiments [it being a hard matter to keep from such in coming up to so nice a point, or to such nice points as is or are therein required] they might perhaps [or notwithstanding] have sometimes imagin'd that they had done the right thing [viz. to the best that's possible according to the false Conception of such imperfectness as must have been thought to have attended the same] but in that case, when [with respect to the niceness of the Matter] so uncertain, and the which in consequence would have been the Case, even from as it were invisible accidents which attend the Experiments [was no wrong thought, as concerning imperfection so have interfered] where then must have been the certainty, or any of the least ground for the Matter? but that such a Division must have remained at least as Mathematically speaking Arbitrary! But now, as Providence or Nature has for this purpose afforded a Foundation, as mentioned in the Preface, and of which presently, we are, as I have found from Experience by a curious Monochord founded thereon [viz. by application to the Spinet] not only free'd from incertainty in the Matter, but withal as I shall show from the false Notion of Temperament [i.e. from any suffering thereby]; for tho' Experiments of this kind, as hath just been shewn, would hardly ever alone, or by means of Theory, have been sufficiently able to have established such a Division, yet I say, by means of a curious Monochord, I found sufficient cause to acquiesce in a Division resulting from what I from hence affirm to be the true Foundation of Musick, and that is (seem it ever strange or foreign to the Matter) the proportion which the Diameter of a Circle bears to its Circumference! But supposing it should be asked why it should be so? I answer I found it to be so from the most accurate Experiments that I not only think, but am sure is possible to be made; but if that Answer be not just here, or at present satisfactory, I may also ask a Question, and that is why [upon the Spinet] the 5th as a single Consonance will from what has been called its State of Perfection, viz. as 3 to 2 will from thence I say but hardly bide two thirds of the Quantity sharpening that it will flattening? as may also be verified by the Monochord; But it must be understood that the Monochord be exactly what it ought to be, viz. true to itself in every respect, or as near as possible to a mathematical degree of exactness: and it is also to be understood, as touching a single Consonance, as of the 5th As that there is a difference to be discern'd betwixt when, as only so, to what it or any other Chord will, as at the same time be for the better in Tunes or Lessons of Musick [nay still, as therein and thereby to such betterness as declar'd above], but 'tis not thus to be observ;d as touching the Steps of Melody as hath already been shewn. so I say, as our Reason cannot solve this, it must be debar'd from being a Judge in the Matter, and the division, which I have confirm'd from Experience must ever stand good as a compleat Structure to the purpose, viz. according to the results of the Numbers following notwithstanding whatever may [as without accurate Experiments, by a perfect Monochord] be thought to the contrary: whether the Question can be solv'd or not, or whether such a Question ought ever at all to be taken notice of or not, I am very sure the Scale of Musick as followeth; Nor but it might be allow'd that the solution of the question, of fact, would be of great Importance as with respect to Theory, was it possible that such a thing could take place in the Matter, viz. as touching the making this most truly Melodious, and as I found in my consequence of which,. altho' in some measure beyond my reach [nay even so far beyond as not to be esteem'd in a small degree.] the most truly Harmonious, or Musical Division; for in that each Interval or Chord from its respective Latitude as term'd hereafter, and as allotted by the said Division, and only as so, viz. neither more nor less in the least degree, and whether as in consequence flat or sharp, should thereby be rendered so truly good as shewn hereafter, is very astonishing; or rather in other words, that the Foundation of Musick should be upon what it really is, or whereupon I strictly found is to be, is quite stupendious, and not to be solv'd by any Mathematical reasoning or Theory whatever.
Let the octave, as above be repre- |
sented by the logarithm of 2 and |
let that same number be also esteemed | ,30103
as the circumference of a circle viz. |
Then [as I am shew or verify be- |
fore I have done] the diameter will | .09582 *
be the greater 3rd Viz. ------ |
Note, As well known, the proportion of the circumference of a Circle to the Diameter is about at 3,1416 to 1; true enough, as Mathematically speaking to this purpose, as well as to many, or most of all others, or as even to all the most useful Intents in the World; hence As 3.1416 is to 1; so is ,30103 to ,09582.
And the radius
or half of which the Larger note viz | ,04791
And as thence, Five times the larger |
note subtracted from the octave |
will leave ,06149, so the half of |
which must be the lesser note [since | .03074
five of the larger notes, and two of the |
lesser notes exactly compleat or make |
up the octave] Hence the lesser note is |
And from these by addition and subtraction,
all the others may be found,
As for instance the 5th, the which con- |
sists of three of the larger notes and one | .17447
of the lesser -------- |
The sharp 6th of four of the larger |
notes and one of the lesser ------ | .22238
Note , It is to be observ'd that by larger Note I do not mean the old imaginary Tone Major, nor by the lesser that same nonsense of the old Tone Minor, such not having any being at all in the Matter; but however the same Names may be very applicable here as with respect to the truth providing they be not misunderstood to mean the fictitousness of old: But the difference of these, the real notes, the which is .01717 ought not to be termed as hath non-sensically been, viz. Hemitone Minor, or Semitone Minor, but as properly what it is, and that is a flat or a sharp, or the difference of the Notes (so for shortness Diff. or some other Term) Nature not by speaking it to be any other in its being the Complement of the lesser Note to the larger; the lesser Note having the same Authority to be call'd a whole one that the other has, since it may as well as the other [as of which by and by] be esteem'd as a proper piece of Symmetry in the Scale of Musick, altho' its Ratio be as it were nothing, viz. 44 to 41 nearly.
+++++++++++++++++
Lucy Notes:
Here Harrison clearly lays out his methods of calculation, using logarithms
Number Log Interval
2. .30103 Octave VIIIth
Pi .09582 2 Large IIIrd
pi^(1/2) .04791 Large IInd
s .03074 small bIInd
3L+s .17447 3L + s Vth
4L+s .22238 4L + s VIth
Harrison compares the values of the fifth and sixth intervals to the integer ratios which he had mentioned earlier.
i.e. 3/2 log =.17609 701.955 cents for the fifth
and 5/3 log =.22185 813.687 cents for the sixth (or as Harrison calls it the " Sharp 6th ").
Harrison identifies the (L-s) as the difference and explains how this is the interval which sharpens or flattens a note.
The 44 to 41 ratio 1.073170731707317 log = 0.03066881976645196 cents = 122.255536823056 = close to lesser note = small (s) interval.
+++++++++++++++++++
I have here as in particular notify'd the 5th and Sharp 6th, as being the Chords upon which the stress of the Bearings [as hitherto term'd] are chiefly to be taken notice of and hence as may be observe'd that the 5th as with regard to the Log of 2 to 3 is flatten'd rather more than three times as much as the Sharp 6th is sharpen'd, as with regard to the Log of .3 to 5, but notwithstanding this seemingly inequality, I found them to be equally, or truly good as with respect to the principal Matter, the Steps of Tune or Intervals of Melody, yea even as therein to a Mathematical degree of Sweetness [if so it may properly be Denominated], altho in that respect as above [or as in so high a respect] not as thence to be solv'd or reason'd for; and as farther, I likewise found 'em through these their correspondent Seasonings [ for so I think they may be properly said to be] to come up to the most elevated height of perfection, as sound the finest, or most relishing degrees of Harmony and the same to be said of all the rest; for to be many of them (the Octave to be excepted) what has been thought to be perfect [could it be so] would, and even in this secondary respect, viz. Harmony, make but dull work in comparison of these, could again, as at the same time the principal matter Melody pass unregarded; but however, as to what they then would make or produce [in other respects] is of small signification, since Nature has otherwise taken care of the Matter, by its securely debaring such the Absurdity out of the Question. *
* But however it may not be unnecessary to observe that what has been thought would be perfect, could such have been admitted, would have had (as in consequence of what has been and will be farther advanced) not only a great occasion to have been called Temperament, but even worse than Temperament, since the whole scale of Musick must thence have been thrown quite out of the course of Nature, and that as chiefly * with regard to the Principal Matter Melody; as to which even the Violin, when played well gives a very wrong Proof, there not being room for it to do otherwise, tho' misunderstood to the contrary, viz. as if upon it the Notes are, or can be taken to greater Perfection [Difference in Voice unnotify'd] than can be done upon the Spinet or Harpsichord, supposing they would stand ever so well to what they may, and as most especially by a Monochord be set [or rather, as the most conveniently, by Forks strictly tun'd thereby]; whereas it might only be from the want of more Notes or Divisions in each Octave that it can only play, or that Tunes can only thereupon be played in more different Keys, or as if may say, at more different Pitches than can be done upon these other Instruments, but not more perfect in any one than may be had from the others, no, nor as the Case is now Suppos'd [and as without notifying any close Shake or Disguise] not so perfect by far, viz. so far as according to their correspondent Divisions, or number of different sounds in the Octave; but this has nothing to do, nay rather as I may ask, what has this to do with the perfectness of any Tune? or rather, as nearer the point to be ask'd, with the perfectness of every Note in any Tune? viz. any more than upon the other Instruments as under the Circumstance here supposed, or as yet more properly speaking, to do with the real, or natural Scale of Musick? supposing that Tunes could not be played at all, viz. neither upon It nor any Thing else, the Humane Voice only to be excepted, I say what? save only for the most part [viz. after, as is to be suppos'd in these other Instruments, three or four more Keys, as are sometimes wanted as necessary Divisions in each Octave] with what great differences in Pitch you may as thereon farther have, or as perhaps you may as thence still farther be occasion'd to want, or, as in the most unnecessary Musick, may unnecessarily or rather, as nonsensically thence delight to have: And agreeable hereto, or as according to the tenor of this it has been affirm'd that no Instrument with Fretts could ever possibly come near the truth, whereas the contrary is really true, and fairly to be asserted [since there is no Temperament; no nonsensical Tones Major and Minor in true Melody the Principal, or if you please in the Principal of true Melody]; but that such Instruments may indeed do so, or come up even to the very truth, or most exquisite degree of sweetness, their Fretts, with their strings true, must be plac'd in a much different manner to what they foolishly were upon the Viols and Lutes, and now still [or again] are upon the Guitars; nay a Viol, from the great difference in the thickness of its Strings, will require some alteration in the Nut, viz. the fourth String, third String, Second and fifth Strings to be dented somewhat longer therein, but the fourth String from its thickness [as uncover'd] the most, and that about 6/100 Inch, so the second but a very little, but the third more, and the reason for this is, that so much of String as may be rightly concerned in sounding or vibrating, is not from the thickness or stiffness of the String, and as occasioned by the pressure into the Notches of the Nut, is not I say terminated precisely at the point of the bearing, but at a little distance therefrom or before, and on which account a contrary effect happens withal, viz. from the said Stiffness, as with respect to the Fingers [or softness of the Fingers] at the Fretts, but as in some measure, implying both which in the denting, will, or if left may, as above, be in the fourth string about 6/100 Inch Now the most convenient thickness of the gut Strings whereon to make the Fretts for a Viol is about 3/100 Inch in Diameter, all of one and the same thickness; and in their placing with respect to a Lath or Gage [viz. from the Nut to the bridge] by which they must be set the Divisions thereupon, to the Perfection of the Instrument, being to be from the same sort of Calculation as spoken of above for a Monochord, I say as respect to That they must lie in little Slanting, the Notes upon the Bass Strings being more sharpen'd by the pressing down to the Finger-Board than them upon the Trebles; but that is not the sole matter in the Case, for withal, from the smallness of the Treble Strings, together with the Softness of the Fingers (as mention'd above) it must be, that the Fretts as with respect to them [or as under them] must (I say) be at a little Sharper [or even at a pretty bit sharper] than as from the Lath or Calculations, but not so for reason already given as touching the Basses; therefore, as thence t he Fretts must lie a little slanted, but still in the whole but little; for as with the heights of Bridge and Nut as low as properly may be, or as with the range of the Strings as thence with respect to the Finger-Board, the most general slant of the Fretts must be only about 1/20 Inch, save only b ut as farther [from the circumstances above] it is to be observ'd that from the pressing down to the said Finger-Board or the Nature of the bending thence required, i.e. as with regard to the distance prest down, and at what distance from the Nut, the Fretts [as under the small Strings] must be plac'd rather the most sharp about the 3rds and 4th from the Nut; Whenas not so from experience upon the Guitar [viz. from the nature of the Strings and Fretts thereon] but if anything the contrary in very small degree about the places here mention'd , neither must the Fretts be plac'd a-slant as in the Viol; but the length of the Treble String there, viz. from the Nut to the Bridge must be longer than from what the frets are Calculated, and still the more so as touching its Basses, [ save only out of which, as with respect to both or all, the Nut must be plac'd shorter by 26/100 of 1/4 Inch than where the calculation for the Fretts is from] as is easy to be verified with true strings; so this Bridge must stand a little a-slant: But tho' I speak about the Guitar, yet I cannot but think it as an Instrument hardly worthy notice and withal especially as with regard to the upper, or most distant Fretts from the Nut, a little curving or bowing [but in this Article very small] the middle of the Bridge being somewhat farther from the Nut than its extremities; but farther Instrument [the Viol] hereafter.
N ow from the Log of the Sharp 6th as last above, it may be observ'd, that as with regard to that of 3 to 5 it has, 00053 sharp Latitude [for so I think the most proper to term it, as containing so much more, or as having that difference sharp], and accordingly the 5th, 00162 flat latitude; and as the nature of the sharp 6th [upon which Experiments are the easiest to be verified] will allow of no more [the 5th bearing, as with respect to Harmony, the 162 flat, full as well, if not rather better than the sharp 6th, the 53 sharp, and as when at the same time, that could hardly be or be minded if Melody requir'd it, or was Melody to require it otherwise for, as by and by, when the larger Note is truly sweet as with respect to Melody, it is as bad as bad can be as with respect to Harmony, Page 29] hence the absurdity of tuning the sharp 3rd sharp, as hath been commonly done, may easily be seen as followeth.
Suppose the sharp 3rd was not to be tun'd sharp, but only so wide as to be what is unfitly, or poorly styl'd Perfect [as being without its correspondent quantity of seasoning, viz. of ,
, 00109 flat Latitude] that is, to |
be tun'd exactly to the Log. of | ,09691
4805,012 ---------------- |
Then, as Tones Major, Minor &c. |
as conjectur'd of old are not of the | ,04845.5
Question [as will be confirm'd below] |
half the Log. of 4 to 5 [as on course |
in tuning] would fall to be the |
larger Note, viz. ----------------
And in manner as above, five |
times the larger Note being, Sub- |
tracted from the Octave would | ,02937.75
leave ,05875.5 the half of which |
to be the lesser Note ------- |
And now the Sharp 6th consist- |
ing of four of the larger Notes and | ,22319.75
and one of the lesser would be ..... |
So its sharp Latitude would be - ,00134.75
w henas for its right seasoning, as I found from the small Latitude in which its Harmony is confined, as also, and as chiefly therein with regard to the true steps of Melody that it would not allow of any more than ,00053: * neither withal did I find any cause or reason from other ( ...... continues after "Monstrous usage Indeed!")
++++ Start of Footnote
I t being as some remark which I shall here make to the purpose, and to which, what I try'd upon the Viol as mentioned in the Preface is agreeable, viz. when so Sharp as 1/20 of the foolish Comma more, i.e. when about the 1/200 part of the larger note more, and the which according to the term of Latitude [the properest] when so much as ,00080 sharp or somewhat before it be so much, I say for a remark to the purpose it is then [upon the Spinet] turn'd from its being a fine Chord or Consonance to being no better than a meer Snarl; and the which is what all the Schemes, before my Experiments would, nay even far worse than so have produc'd or turn'd out, so consequently the nearer any Scheme comes up to mine, and as when still not quite so it, 'tis only a bad use made of mine, viz. if put in practice as expest; and this is what Dr. Smith's will do, providing what he says or describes be truly put in execution; for it is to be observ'd, that when from me more than 1/9 Comma flat in the Major 3rd, the Latitude of the Major 6th will be render'd ,00084,17 sharp; and when from no more than 1/9 Comma Flat in the Major 3rd, the Latitude of the said Major 6th will then be render'd ,00089,795 sharp; so here it is to be observ'd, that if Dr. Smith saves his Major 6th at all [viz. as only with respect to Harmony] he must tune according to the drift of my Scale, and not so that of his own, or at least it might be much nearer to mine than to what he calls his own, for let him make a Monochord, or let one be made, upon his Principle, or upon his Theoretical Principles, and tune to it, to try whether it will do better than mine or not [but as here by the by, not meaning for it to be judg'd or sentenc'd by a Board of Longitude] it not being, I am very sure, in the power of any Demonstration whatever, neither of that of any of my Experiment in the world, viz. whether at Cambridge or else where, before my Experiments were made, to shew, towards a Mathematical truth how far the major 6th would bear, and that as a Key to open the Lock to bring the whole Scale into Order, as intimated in the preface; I say, before I had completed my Apparatus to the purpose, and had acquir'd proper means of trying it, it was never in the power of Man (seem the Assertion ever so presumptuous) to know the Matter [or Foundation of Musick] &c. any tolerable degree by Numbers considering the niceness of the points requir'd therein, whenas the which are hence fairly illustrated, or to Mathematical truths de-fin'd; neither can it as yet be done or confirm'd by any save only by me or a proper use of my Apparatus, as being as already imply'd, too nice or secret a Matter even as primarily to be defin'd from any theory, neither as consequently afterwards, to be practicably distinguish'd by Beats; whenas to tune all the Strings of a good spinet, that's contain'd in one Octave by my Monochord, and as thereby from its nice execution upon its Foundation imply'd must be quite sufficient to satisfy all the reason in the World, was there no farther Proof to be had: and here I cannot help calling to Mind, that when I first told Dr. Smith of any Scale [i.e. of the real Scale of Musick] that he said he would soon try it upon the Reed Stop of the Organ; a famous Stop indeed whereupon to try Such a Matter or Nicety!
i.e. in good round Numbers, if such would but do! or rather as without any regard to Numbers at all, for it could not as in consequence, by that means or method, be any other sort of trial than as it were at random, or without regard to exactness, whereas the contrary was not only in a high, but even in the highest degree requir'd! But as farther to proceed. Where would Dr. Smith's Major 6th have got to had he not convers'd with me, accidentally as it were about the Matter, viz. when he had founded his Scale (and the which, as I afterwards heard by Mr. Graham had been two years in hand) and was a going to publish his Book upon the Major 3rd being taken where falsely styl'd, as with respect to Musick perfect? why, it would have got (as above) to ,00134,75 sharp, and so might consequently as well have been a Mile farther, Nay was it to have been two Miles farther, or only so far out of the Question as according to Mr. Huygen's poor imaginary low Division [viz. as when computed ,00149 Sharp] yet it must still, with veneration have been look'd upon as something, if not to have been taken as a Step towards right, providing it still was but adorn'd, besmear'd or bedawb'd with Greek and Latin &c. and still had then doubtless made his Mathematical Schemes with his Algebraical reasonings to serve to the purpose [tho' no more to do in the Matter, at either time, than they could have to do with the Pallats, Balance-Spring and Thermometer of my Longitude Watch, i.e. nothing at all] and at last [viz. after f or all his Rectification and Curiousity] to tune into a many Keys, viz. into more than usual, by, or through a Succession of 5ths, wherein all the Nicety, was there any [tho' to be nicer than mine] must very probably be lost, for when a Surveyor makes use of a many Stations his Map can seldom or never be so correct as when from fewer,
I n a marginal space this secondary Note against this place. "But indeed, was it here, or be it here by a perfect Monochord, it might or may then be said, or still be said to be as from one station."
consequently, such stuff as that (and whether with thought or without) must only be fitting to be put at the end, or after any the meanest handling of the Scale of Musick, unless it could be said, that through the Power of Theory, as jointly with that of Algebra, and as not withstanding any inaccuracy whatever that might attend or may happen in the putting in Practice, the nicety must still remain: [whenas from his tuning there can be no proof of what he writes to be the scale of Musick}: But even as thence, supposing as impossible to the greatest truth that the most prodigiously curious or natural Foundation of Musick would admit of, yet still as then not so be in the least better towards what ought as most chiefly to be regarded [and not what is treated of Page 16] and that is for the pitching of a Psalm-Tune in the Church; whenas in my conversing with him, more than once or twice, I had told him [from Experience] to what nicety the pitching of a Tune for a Consort of fine Psalm-Singers requir'd, so towards which he ought, as one would have thought to have made his chief, or as a Rev. nd Man his most becoming drift, as of which hereafter; hence surely in the whole such a Book as his, as touching the natural Scale of Music &c., and as considering withal the Means by which he came by it, or as above the Means whereby he took occasion so for to rectify it as he did, must be as Ignoble a Thing as ever was done or produc'd in the World, and more especial so as from such a Man, and who himself [viz. after my conversing with him about the Matter] had told me, and had also told Mr. Nettleton, that he was so far satisfy'd of the truth of mine, that he would not publish his, not that I greatly car'd whether he was so or not, or would p ublish his or not, but as thence I am in expectation that something else [as about the Nature of Ballances &c.] may, or will be issued out from Cambridge as if the knowledge of which was first discover'd there, whereas it was exhibit'd by me, and that from something which happen'd in my long course of Machinery to fall in my way; But it is otherwise certain that the University Men (no commendation to 'em sure) are generally (and besides as in Country Villages) pretty arbitrary, and think they may do whatever they please, as was (by them) wanted arbitrarily to be done in my other Affair, as of still more noted Longitude, and wherein Dr. Smith took indeed friendly care to advise me, or rather to persuade me to be quiet and rest easy, tho' I should not or could not [because of them] have Justice done; whenas he ought to have sent a Letter certifying to the Board [the which Favour I greatly requested but could not succeed] that nothing [in fact] as touching the Completion of Longitude could do so well as the true Time-Keeper; but Parsons will be Parsons, nothing contrary to their Partiality must take place, or even nothing at all but what they please if they can help it {University's Poison]; but I thank God for the better! and that then through the hands of an upright Judge, together with other Unprejudiced, Honest, or real Commissioners for the Longitude, along with His Lordship, I mean along with the Rt. Hon. He the Earl of Sandwich: But as nevertheless tho' the wrong headed, barbarous Inhumane Method, whereby to bring my Time-Keeper into the World [or rather to keep it out of the World] was at that time stav'd off, yet it had afterwards [viz. after another Voyage] through the same Inhumanity, Barbarity, or Disingenuity, and as notwithstanding such Virtue as ought to have been expected from the Act of Parliament, is remaining still in force, and as upon which my Son was also sent that second Voyage, yet I say as notwithstanding that, it had like to have been brought on again, and then as might still perhaps, for the residue of my life, have been made a meer Vassal to the most ignorant Men; for then accordingly as at the first, because I had contrived and made such a Thing, as the which for its case and certainty, as well as with its afforded opportunities for frequency in discovering the Longitude [and as was confirmed in each of the two Voyages] far surpast all that ever was or could be imagin'd by any Rev. nd Professor, or any Body else whatever; yet I say, because I had produc'd such a Thing, as was handled much the same, or at least no better than a poor Bull at the Bull-Ring bated by Bull-Dogs; Monstrous usage indeed!
End of Footnote ++++
other Chords or Intervals form its wanting any more, or for its being forc'd to have any more [viz. than .00053 Sharp] the 5th for Instance, and as in chief, not being render'd [as I can make bold to say, from so far as is to be, or can be determinable from Experience] to its finest Flavour by any less than ,00162 flat Latitude, therefore, not on the contrary suffering by it, as was suppos'd by a Temperament, But is thence acquiring its goodness, and that as most especially in its primary Case, viz. as an Interval of Melody, as being thereby, and only thereby rendere'd to be a true Step of Tune;
It being then and only then, according to the disposal of Nature or Providence what the Ear best likes and the Voice always takes, and the which must certainly be from its becoming, natural or true Proportion, whatever our reason may have suggested, pretended or imagin'd to the contrary, as will be sufficiently confirm'd before I have done; and as the same is to be understood of all the rest, viz. as chiefly touching any Succession of Notes or Tune, they cannot therefore, as with respect to Melody be said to have any Latitude at all, for when thus render'd, they can be no otherwise look'd upon than as being, or as becoming to be, in or from a perfectly Melodious Scale; Therefore, when ever (in this Treatise) I make use of the term Latitude, 'tis always to be understood as with respect to Harmony, or to the highest degrees thereof, tho' the words may sometimes seem to imply the contrary, whereas there is no such thing in the Scale of Melody, as being entirely perfect in itself; and so as thence, I may farther assert, that by the term Latitude is only represented the distances from such Mathematical points of exactness in single Consonancy, as still can have nothing to do in the more high Source of Harmony, save only, so far as from whence to be notify'd (as always imply'd) the naturalness or Virtue from the distance of each corresponding Chord therefrom.
neither indeed [or as without which quantity of flatness] as touching its very best, and as I have already affirm'd in its Secondary case, i.e. as a Consonance of Harmony; ++++++ (this connects to +++ "and surely it cannot be amiss")
And now I shall observe, as here in the first place, that the 4th by its having as much sharp Latitude as the 5th has flat, viz., ,00162 is thereby also [as I found] render'd a perfect Interval of Melody or true Step of Tune, as said above of all in general; but what I am here to notify about the 4th is, that it will not so well bear Harmony, unless its Relation the Sharp 6th be founded co-temporaneous with it, and then it is as ought to be, that it becomes as styl'd above to the most elevated height of Perfection, and would not then be so with any less Latitude; neither at the same time, must the Sharp 6th be impair'd by its having either more or less than ,00053 Sharp, and it is also to be observ'd that the Sharp 3rd, which is the upper Consonance in this Harmony, will, and as then terminated by the others have also its due, Seasoning, viz. of ,00109 flat Latitude, but [as already affirm'd] no suffering or bearing is now amongst these to be perceived, as might be imagin'd, or as hath been observ'd from what might be thought a Temperament, but on the contrary, by a fine or lofty Warbling are all advanced to the highest degree of Perfection, as being therefore refin'd from the dullness that what has been called perfection would make: And the same is to be observ'd in the Sharp 3rd and 5th, where the Flat 3rd is the upper Consonance in the Harmony, for the which then, from their due or correspondent Latitudes (as I found) did also afford the most lofty Air that is possible to be, and most especially so, in both these Cases, viz. if in this Harmony, as well as in that of the 4th and Sharp 6th (as just above spoken of) the Octave or Octaves are also sounded; hence, what can be said of the tuning of the Sharp 3rd. really wide or sharp, and is thereby Sacrificing the Flat 3rd, and as in consequence the Flat 3rd, and as in consequence thereof its Complement to the Octave the Sharp 6th, and that to a great degree? Why, such a proceeding may well be called a Temperament, yea even when to a less degree than that, bad enough! So the truth of the Matter is (as all along to be seen) That the Scale of Melody as the Principal, or as in the first place chiefly to be observ'd is entirely perfect in itself, that is without any the least bearing or suffering, as being so render'd by, as it were its free and eternal Inheritance of such proper Latitudes of the Chords, as secretly permitted, or rather mysteriously ordain'd for the Purpose; or as is very suprizingly to be express'd of it in other words, viz. as that each Note in the Scale of Melody should so have its respectively exact and domestick d egree in the Borders of the Chords of Harmony, as whereby to be situated [as touching matter of real, tho' secret truth] even to the greatest p reheminence ?? i n the Field of Nature, otherwise, each Interval therein must have been under the untunableness o f bearing or suffering, and that to a much greater degree than ever was imagin'd, and in consequence of which, no true Tune or fine Melody ever to have been had [viz. neither by the Voice, nor upon any Instruments whatever, no far from it] whenas the contrary is really fact, (Continued at **** below)
As for Instance, no Master of Musick in playing or hearing a Single Part, as upon the Violin &c. ever judging whether each Note be taken according to any certain
degrees of Temperament, but that whether or not each Note be taken truly in Tune; and it is certain that that truth never did, nor can consist of Such Intervals as have been imagin'd perfect; that of the Octave only to be excepted.
( from **** above)
and not only so, but moreover as above in such an astonishing manner as not to occasion any Diminuation of the Harmony of the said Chords, but on the other hand, as when duly, truly, or fairly try'd, to the Advancement of the same, or so the rendering the Harmony more Graceful. But farther as the Scale Of Melody is indeed truly Melodious or perfect in itself, there is something more, and that of great importance therein , and the which I shall here treat of, and that is the Latitude of the larger Note, could that, as touching that Interval be properly so Denominated, the which would be as with respect to the imaginary Tone Major ,00324 flat, and with respect to the imaginary Tone Minor ,00215 Sharp, but as in fact, in its having no Affinity to either of them [that Imagination being, as I found with respect to it no better than a meer Chimera] it cannot therefore be said to have any Latitude, since it is not at all to be observ'd as with regard to its Ratio, that being betwixt 48 to 43 and 19 to 17, that is in plain terms nothing, was that the light in which it is to be viewed; well, but what then can, or is to be said as touching its Perfection? Why according to the most accurate Experiments that I am sure is possible to be made, it is to the Octave exactly as the Radius of a Circle to the Circumference, and only as to [viz. without any the least Deviation] a fine Interval of Melody or true Step of Tune, and is naturally so taken by good Voices, that is, with as great facility as any other Interval; and besides my Experiments as touching this upon the Spinet and Viol, in the tuning of Bells also [viz. by a Monochord] nothing but this [or exactly this] as the larger Note, will render a Peal to be truly Melodious, as which I have had many Experiments undeniable of the fact: Now as Nature or Providence has order'd [and that as a chief Matter of Proportion in the Scale of Musick] that this should be so, and truly good, [when exactly put in execution] to the highest perfection that can be yea [when exactly put in execution] to the highest perfection that can be desir'd, as also the same with respect to the better Note as above, and from the exactness of these, as in a surprizing manner, the truth of the whole Scale to consist, Who ought therefore, for the Future to be so bold as to presume to diversify the order, as hath been hitherto ignorantly surmis'd or pretended, since neither our Reason or Theory can have to do in the Matter?
T he Log. of 8 to 9, or that of |
the imaginary Tone Major ----- | ,05115,25
The truly melodious Note, viz. |
from the Radius of Circle as above | ,04791
The Log. of 9 to 10, or that of |
the imaginary Tone Minor ---- | ,04575,75
The Log. of the fictitious |
Comma, or of the difference betwixt|
the imaginary Tone Major and the | ,00539,5
Minor - The Log. of 80 to 81 ---- |
c onnects from ++++ above
and surely it cannot be amiss to denominate Melody and Harmony after this manner, for as a Proof thereto, an Instrument may be so tun'd, or maybe found to have stood so in tune as to afford all the Chords tolerable well as with respect to Harmony, when at the same time, as with regard to a Single Part [and then as best discover'd in playing slowly] not to speak a Tune truly as suppose a Psalm-Tune or its Bass, neither very short, but proper Succession of Notes as some one, or any one select Change &c. five or more tunable Bells might strike,
(goes to ++++ below)
* Or even, not truly the four Notes here following, as the last of our Notes in the Bass of the Psalm-Tune,
viz.
the which Notes, according to the secret law of melody, are by, the Voice turn'd off perfect, and not according to what has been thought the perfect laws of harmony, as may be verify'd by the strict Application of a Perfect Monochord to a Spinet, for playing them as a single part slowly, they will be found to the Ear as fine or distinct as when a good Voice sings 'em and therefore, only true or truly harmonious when exact to the secret Principle of melody, as in this Book advanc'd; the last four Notes in the tenor of the same Tune, as may be proper to be notify'd, being
L ucy Note: Notation Clef called Tenor, but this is ambiguous not shown, # on central line (assuming central line is C) Notes therefore are A-A-G#-A.
(connects from ++++ above)
I say not then, or in these Cases to speak the truth as with respect to Melody and that in any proper key, or in any of their proper Keys, and from any part of the Instrument, if (as again) from any one part, as is, or even ever will be too commonly the Case in the fine Instruments of Musick, the organ, harpsichord and Spinet, for even as agreeable to which, a two stopt Harpsichord may be so in Tune, or as above, may sometimes be found to have stood so in Tune as to afford all the Chords tolerably well, and yet at the same time but few of its two and two Strings to be strictly found at Unison, whence the Melody must as must chiefly be impair'd, the Intervals thereof being in themselves far more nice and delicate than the Consonances of Harmony, and yet these latter [the World being as in the first place full of Harmony,
++ Footnote starts here
Yea in reality full, but as nevertheless we are oft times, viz. in other matters hard set to rightly understand it, but still as thence the more at last to admire it, or the Kingdom of God therein.
Footnote ends here ++
and as notwithstanding in this Matter with wrong Notions concerning the highest degree and degrees thereof] are what the others [as without a Monochord] are try's or Tun'd by, and when as in consequence at the same time, with a great desire to put such Violence, or as I may call it beloved Violence upon Nature, and especially in tuning the Organ as to make the Chords seem perfect as single Consonances, when as on the contrary they must not there seem so, Nature [in any Tune] as I have shewn, and that to much greater, tho' secret Perfection chusing other measures in that of the Octave only to be excepted; consequently the beatings as heard in single Consonances upon the Organ, viz. when at the best that can, or may according to Nature be set, are not then to be esteem'd as being from false degrees of the Intervals [i.e. as touching the Perfection of Musick] as hath been been imagin'd, but from the Nature of the Instrument, since no such beatings are perceptible from Voices sounding at the same degree or degrees of Latitude, as may be attested from the same not being then perceptible when upon, or from the Nature of the Viol, neither as from the Spinet, or Harpsichord,