Concerning Such Mechanism......

This is a transcription of pages 67 to 108 of "A Description
concerning such Mechanism as will afford a nice, or true Mensuration
of Time; together with Some Account of the Attempts for the Discovery 
of the Longitude by the Moon; and also An Account of the Discovery of 
the Scale of Musick". by John Harrison, London 1775. A special copy
containing an Appendix is in the Clockmakers' Library under the
Guildhall in the City of London.

This transcription, of the musical parts of this book, is not a
totally faithful copy. The original uses the same letter f, to
represent both s's and f's, and has many foot and margin notes. It uses a 
contemporary system of spelling and punctuation. Nevertheless this
rendition should be considerably easier for late twentieth century
readers to comprehend than a photocopy or facsimile of the original.
C.E.H.L. 3/90.

Now, in the former part of this book, I have treated about matters
pertaining to the strictness of measuring time; and have shewn the
deficiencies of such means as Mr. Graham had taken or made use of for 
that purpose; and I have also treated of the improper, troublesome,
erroneous - tedious method, which the professors at Cambridge and
Oxford would have to be for the longitude at sea:  And now I am about 
to treat of another concern, the which happened to fall in my way,
and the which [at least to the Royal Society of London, for in every
respect] must be well worth regarding when rightly considered [at
least I think I thought to be so] as being so secret a discovery; and 
that is the really true scale, or basis of musick; since for which
knowledge musicians might have played, or fiddled for ever, and
tuned, or have had the organ turned wrong in the church for ever, and 
the musical part of the mathematicians might have reasoned as they
have done, and wrote about it forever, and never have found upon what 
foundation the truth of the matter existed; and here, as in the first 
place, it may not be improper as in particular to remark, that Mr.
Huygens was in his conjecture, a great deal wrong; and my friend Dr.
Smith [Master of Trinity College, Cambridge] not knowing that I had
had any thing to do in the matter, though he and I had been pretty
intimately acquainted for two years, and had known each other much
longer, and as Mr.Graham afterwards told me, that he (the Doctor) had 
then had his book, Viz. upon this subject the scale of musick under
hand for longer than this time; but as finding reason to think, viz.
as from or upon an accidental conference which betwixt him and me,
that I was in the right, said, that he would drop his book, and that
I might make the best of mine, but instead of that, did some time
after, alter (viz. rather than perhaps to lose his labour) from what
he has grounded his work upon, and so as to come as near to me, as he 
himself told me demonstration would let him, and then published it;
whenas it is certain, that if he had not happened to converse with me 
upon the matter, he had printed his book upon his first ground or
principle, and then had been demonstratively sure of its being right, 
whenas it was far from being so, though not so far as Mr. Huygens's
conjecture was before him; and it is certain that neither theory,
demonstration, nor algebraic reasoning can have anything to do in the 
matter, his own proceeding being even a proof to the contrary, for
had such in the case been fact, why did he alter? or rather, how
could he have found room or occasion to have altered?  And as still
farther to remain a little wrong, not withstanding his alteration or
amendment; and as moreover to express what passed betwixt him and me, 
in his preface to his book, much wrong, instead of his being pleased
that there was, or is indeed, a firm and true foundation of musick;
but that, or all this, was not the worst jarr that happened betwixt
him and me, for as I could not adhere to him in the case, he
afterwards turned from being my friend in the longitude affair, to
his being therein no better than an enemy, and perhaps (as already
hnted in fearing that he should through me lose his labour, or that
he should become of low esteem, viz. from my foundation or discovery
of the scale of musick, as being indeed the only right one, and
should therefore in consequence be stronger than his;  [Footnote: So
he seemed, as it were, determined to keep me weak, if he could] for
indeed his neither is, nor can be, any better than an arbitrary
conclusion, for, as touching melody, the chief matter, it will not
afford a tune, when strictly put in executuion, to any right or true
content; neither, as touching harmony, will the fine chords, the
sharp 6ths and flat 3rds, rightly bear with his division or allotment 
(this is meaning after his alternation) whatever he might judge in
either respect from mean or false experiments to the contrary, and
his saying as near as demonstration would let him; the whole matter
[as I have verified, and can at any time verify] being as otherwise
established by providence, for I am very sure [and was then] from the 
most strict experience that can possibly be made or had, that my
foundation is true, and that it is impossible [from the nature and
niceness of the subject] for any thing else in the world to define
the matter; nay, besides myself, it has been allowed or attested by
several musical gentlemen, organists, &c. who heard the result [of,
or upon what it is grounded] to be in reality perfection itself,
whenas he [the doctor] was so obstinate in the matter as not to be
prevailed upon -- by all the inviting, or entreating speeches that I
could make, to come to hear it! viz. after I had sufficiently
provided the truth of the thing! [viz. more sufficiently than what I
had done from the first!] And indeed, the chief head or consequence
in the scale of musick, viz. the intervals of melody, are as I may
assert them truly sweet, or mathematically perfect, though never
before were thought to be so, or that there was such a field in
nature as wherein they could be so, but a foolish imagination sure!
Since a good voice never fails, but can always, and without any
difficulty, turn off a tune, or even a piece of tune, truly, viz. as
without any regard to the key, as hath been foolishly advanced, and
as even by Dr. Smith could not be; or indeed could it be, if the
perfection of the intervals of the melody were as the musical part of 
the mathematicians have thought they would be best, could they be so
had or admitted; as for instance, was the perfection of the 5th [as
an interval of melody] to be as 3 to 2 exact, the thing [objection or 
supposition] would be right, but as so a good voice never yet took
it, nor never will nor can, because, if it did, it would be very
unpleasant, or even ugly, viz. too wide considerably, [Footnote: But
then (as in supporting that the case) the out-of-tune ugliness or
unpleasantness (I am speaking as with respect to melody) would be
judged, as according to the common notion of the world [and that for
want of accurate experiments in the affair] to be as then, from its
not being as exactly as 3 to 2, but wider.] or as more especially to
be notified, the 4th to be taken by the voice, or by voices quite out 
of tune wide, viz. as with regard to harmony, or to the harmony of 4
to 3 [it is not bearing in that respect so much as the 5th] and
whereas true melody requires it so to be, and to which the voice
naturally adheres, yea if it wanted to take no more than as the
interval of one single 4th [Footnote: And here it may be notified,
that four 4ths and a sharp 3rd, each in the state nonsensically
stilted perfect, will not make two octaves, no not by a good deal;
whenas four natural 4ths, and a natural sharp 3rd, both must and will 
exactly do it.] and still again as with respect to harmony, [viz. as
in that extreme sharp state] to what amazing fineness it is when the
sharp 6th [viz. as when also in its respective properness of latitude 
sharp] is founded co-temporaneous with it, as I can now at any time,
and in each of these respects, certify from instrumental experience,
viz. to any who may be proper to hear the same, and as thence
consequently produce a proof, that there cannot be in the scale of
musick, or that the voice can have any thing to do with such
chimerical notes or intervals, as ones major or minor as imagined of
old; so the symmetry therefore as implied, of all the true intervals
of melody, and must in consequence thereof be also the most rational, 
or graceful chords of harmony, can have nothing to do with such
arbitary conjectures as have been advanced (viz. as according to
Holder's harmonical nonsense in the affair, surfeiting stuff sure!
though he speaks of it with great admiration) but are to the
contrary, and as I have verified from due experience, secretly
grounded upon the true relation, or as strictly touching this matter, 
may be said amazing proportion which the diameter and radius of a
circle bear respectively to the circumference: viz. as thus, as the
diameter and radius of a circle bear respectively to the
circumference; so do the sharp 3rd, and as here properly speaking,
larger note bear respectively to the octave (no tones major and minor 
being in nature, as of old imagined) and from hence all the others
are generated, have you as many keys, viz. by flats and sharps, as
you please; [Footnote: But here it may be noted, that there can be no 
occasion for so many flats and sharps in an organ for the church,
viz. any farther than for what key the whimsies of the organists may
want to play their voluntaries, &c. in, viz. Things that need to be
played there at all; Time in Divine Serivce, being to be otherwise
employed, and that, as not only more suitably, but even as more
takingly to the purpose to be done. But indeed, a more suitable
construction of the organ must be highly necessary, or else, a
consort of good psalm-singers must ever be disobliged by it, or not
come there, or to where it is at all, since there performances as
thence, could but seldom be as it were truly genuine, or naturally
good; but notwithstanding as in, or as such the said performances are 
not as now to be heard, neither to be remembered, that, viz. the
congregations, with the charity children, and in their paltry
piece-meal, hodge-podge manner, can bawl or squawl away along with
the organ, as if such the said children were the most proper
instruments, or assistants, for, or to the purpose, and are sometimes 
set at such a pitch, with, or by the organ (although but one part
sung) as to be even fit to split one's head (an absurdity sure) yet
still I say, to be as so thought the most proper; but it is not so
thought to be the case at the play-house, viz. as with children
there; but certainly, God Almighty never intended that the latter
should ever excel or over-set the former.] I say. as thence in the
whole, [and that as from the most strict experience, viz. as by or
from the most strict apparatusses to the purpose,] are generated to a 
mathematical degree of sweetness, if I may so term it, as well as to
be a surprising mathematical degree in proportion, as being seemingly 
from a thing quite foreign to the matter, yet still a wonderfully
strong, and stable foundation indeed! But certainly, as the works of
God are in all respects perfect, so his praise, as far as may ever be 
in relation to this [not meaning the play-house] must require it to
be so too; [Footnote: And to such purpose as it must be, that in or
from his completion of humane voices, they do not want as I have
shewn to take or make use of such nonsensically perfect intervals as
have been so weakly or foolishly imagined; for certainly, any one
note, whether taken in any tune or lesson of musick, and that whether 
by the voice or upon an instrument, ought always to be at exactly the 
same pitch as with respect to the rest, whereas, if such weakness is
intimated could take place, that would not be, nor consequently
musick to any scale at all; but still, for the sake as it were of
such that as that, it all along hitherto so happened, the violence,
as with respect to natural harmony, was in some measure put [as
thought for the better] to prey upon nature in tuning the organ, &c.
And whereas or as when, what was done for the best, as with quite a
contrary drift thereto, the whole being thereby for the worse
affected, and that as not in a very small degree, and yet the great
Mr. Handel among the rest [as not discovering the matter] had his
organ and harpsichord so tuned.] but still so long as the foundation
of musick lay his in secret, unknown of to the world, as also the
knowledge of such nice preparations or ways of proceeding as might or 
must be required, in or for its verification, i.e. so as whereby to
know whether, or if ever that was, or could be deemed as likely to be 
the case; but I say, so long as it lay hid, the consequence was, that 
it did not seem to have any absolute or real foundation at all, for
as in the musical part of the mathematicians, finding in computation, 
or in what they called theory, a defect of what they denominated a
comma, and to be as a thing unavoidable in the matter, they thought
that the beauty, or perfection of musick, must in some measure be as
thereby lost or prevented; whereas it is through the same that it is
indeed musick, and that to perfection, yea far surpassing our
imagination, as from the whole of this description is manifestly to
be perceived, and consequently the world to be but little obliged to
philosophy here, viz. in condemning the perfection of the thing, lor
the wisdom of God therein; but however they wanted to cloak that
deficiency [as they thought it to be] as much as they could, as
thinking that it was, or must be always in some measure, nay in great 
measure, so done by [or that it permitted so to be done by] or else
the thing [save only as thereafter through mistake upon the violin
and violincello] could not be so fine or taking as it was, viz. to be 
cloaked by various distributions of such and such parts of the said
comma, to such and such chords of harmony, and as at the same time,
without knowing what portion of which, each chord respectively, as
touching the matter, would bear; nay thought indeed, that such and
such a chord as with respect to harmony [not regarding, or notifying
what might belong to the melody altho' the chief] would bear the most 
[or the greatest share in that defect, as was thought to be the case
by Dr. Robert Smith, viz. before he conversed with me] whereas the
which in reality [or as on the contrary, under that supposition] will 
bear the least. Strange conjuring sure! As being in consequence
without any suitable experience to the purpose! and yet to prevail
through the ages! and as moreover, with the respective bearing of the 
sharp 3rd, or the result of that as with respect to melody [or as
even to the destruction of melody] quite the contrary way. viz. sharp 
instead of flat! But indeed the most part took it from the rest for
granted, as so and so be, viz.without thinking, or properly
experiencing the matters at all; nay, indeed to make experiments, as
thoroughly to the purpose, was quite out of the way or power of any
of them [or of all the learning or knowledge heretofore in the
world]; but to proceed, and though ever so far beyond the reason, I
do again certify, nay avouch or affirm, and that as without any
notice at all of the feigned term of a comma, that the intervals of
melody [the prinicpal matter] are from the circle &c. as here above
advanced, turned off exactly true, or strictly true, or strictly
perfect, i.e. as without any the least bearing, defect or
infringement at all, viz. as with respect to the most true, or
natural steps of any tune; whenas on the contrary, in the taking a
few of them [viz. by a true constructed monochord] according to what
we should think would, or ought to be perfect [I say a few of them,
for all cannot be so] each sure one in itself, as with respect to
melody, will then have such a bearing, or be so untrue as not to be
bore withal, yea so, besides the utter destruction of all the rest;
hence if the tuning of an instrument, but as most to be notified the
tuning of an organ, be false or varied from the result of the circle
as I have shewn [as indeed it has hitherto all along been, and that
in a pretty good degree, excepting through me, as of late, that some
tuners have altered] it is not at least fit for a psalm-tune or
anthem; for I am very sure, in its differing there from it cannot
afford a tune any more or better than a viol, &c. otherwise fretted
can do, and that is as nothing to the purpose, viz. in either anthem, 
psalm, or song tune; but still, as without the proof, assistance or
application of a perfect monochord, [Footnote: Nay, for this purpose
or all true purposes, there must as in the first place be two
monochords, in order as whence, by proper means or trials to prove
the truth of the string, or of each string.] viz. of such a one as I
have constructed, and divided upon the true foundation here shewn; or 
rather as the more easy, or as the most conveniently to be done, viz. 
as by the help of a proper set of forks tuned the most strictly to
such a monochord, for by which, the said forks or each fork can be
tuned to the thousandth part of a note or less; [Footnote: Nay, if a
set of forks so tuned, could be properly, or duly struck, how sweetly 
they play a psalm-tune slowly; nay if in two, three, or four parts,
nothing in the world so beat them; a monochord or monochords, as
under the same or such-like circumstances, to be expected.] and I
think that by a proper use of fire, viz. at a proper distance from
the organ, and as hence by means of a thermometer near, or not far
from the organ, that the same may be kept to the temperate heat [viz. 
to 55] for during such a time as may be required for tuning, by the
forks [meaning the same to be of a sufficiently large size for the
purpose] all the pipes included in the octave, viz. in what is called 
the principal; or at so long as by proper or due management of the
fire [as from strictly observing the thermometer] as that some of
them, as in chief may be so truly tuned, viz. so as whence or whereby 
to be checks upon such proceedings as may be thought necessary, or
more expedient to the purpose; [Footnote: And by the said forks [viz. 
of a lesser size] the harpsichord and spinet can also be so truly
tuned, that some players as well as others, have said, that they
never did hear the harpsichord, &c. before.] but I say, that without
something in this way it must be a very difficult matter to have it
right, or exactly tuned, yea though what is here above treated be the 
very voice of nature, it is not being to be expected, was there
nothing else in the matter, but that variations, or falsenes, must
arise or happen in the proceedings by 5ths, as according to Dr.
Smith; but the doctor says in his book that the voice part of the
anthem ought not to be played upon the organ, whereas I should think
it the most material, or else it might be very immaterial to have an
organ in a church, and there for a psalm-tune, which requires the
greatest truth of all: but however, be it as it will, our organists
generally there take care to blind imperfections with such stuff as
does not at all belong to the matter; but certainly a tune ought, as
in the first place, to be distinctly a tune, and consequently in a
special manner, for from such nonsense is as usually played before
they begin to sing, viz. as from whence but hardly to be known what
tune they are to sing, and therefore it would be much better if
imperfections did not want blinding, or to be blinded! But indeed the 
psalms in general, upon other accounts [viz. for want of better
discipline than that there is, and in which defect the persons are
much in fault] are no better than smothered, as will fairly appear
when I publish the treatise, which I have, as more particularly,
drawn up about the scale and use of musick, as therein unvieling that 
abuse or obsurity! [Footnote Viz. If, as according to Royal David's
Declarations, as touching his deliverences, as also of them of the
Israelites out of Egypt, as well as others the works of Almighty God, 
his Dispensations, &c. and as with praises, &c. thereupon pertaining, 
as in the psalms, be as still to be in rememberance or veneration,
and that as by the words or lines of the psalms to be [as now in the
new version] right duly handled, and as therein implying, for the
most part, by the going on in succession, with proper portions or
divisions of each, or any psalm in hand, viz. as when as so to be
done, or as so to be permitted by the parsons, viz. from a skilful
delivery of the clerk [meaning, the same as then, to be as fitly
chosen for the purpose, as it was for a playhouse], i.e.if their
dignity [viz. that of the priests] will so admit it; [Footnote: of
footnote: I say their dignity, not thinking the clerk to take any of
their business say their dignity, not thinking the clerk to take any
of their business from off their hands; notwithstanding, singing men
and boys in cathedrals have surplices: But as a tenor to this, Dr.
Smith [upon our discoursing] said to me that they sould send us
parsons, but where might we get good clerks? And indeed to have a
good clerk, must to great part be a gift of providence, whenas the
other is only as it were from learning] whenas if not, they ought,
and as with a suitable grace [or affinity to the tune, as well as at
the same time, by proper accents, &c. to enhance the nature of the
psalm] to do it themselves, but perhaps they might think it to be as
a thing almost repugnant to their preaching; but, no matter for that, 
they ought not to thing of it so, but otherwise, and that as truly
becoming thereto, viz. to be, as it were, with lower thought -- but
higher esteem, consequently without any the least pride in the
matter, so that as thence according to their drifts [viz. the whole
sacred drift, scope, or meaning of each psalm] as from their
contents, &c. as the which contents ought indeed to be, and that as
to a full intellegence, at the head of each psalm, that so the same
might, as at least with reason, vie with the musick, and that, as the 
most highly becoming such, that same part of the divine service, as
therein to do, or rather as in other words, as the most highly
fitting for such the highest part of the divine service, and as under 
the gospel's dispensation to be handled, or so as to be for the
better regarded: [Footnote of footnote: Not meaning the lines of the
148th and 149th psalms; nor them of the latter end of each verse of
the 136th, to be given out; neither do we sing the old 148th tune,
nor old 113th, such tunes being several others as of old being very
unsuitable to the purpose; neither as farther, do we use the 100th
psalm tune for any psalm but the 100th, having tune enough to suit
all other psalms, and their measures, as in the new version [and as
not over-looking therein the 96th and 87th, but for which psalms to
have fine and suitable tunes] and indeed it fitting that the 100th
should have a tune to itself, and none can suit it better than its
old tune, viz. as when sung eloquently or laudably, i.e. as when at a 
truly right or natural pitch, with good strength of voices -- in four 
parts rightly adapted.] no voices withal being to be played [or in
anywise to found] but what the voices sing, excepting the octave
below the bass: no repugnancy of thorough bass nonsense to be used in 
psalm singing. I speak from due experience, [Footnote of footnote:
That being no other, as with respect to psalmody, and as I have seen
fairly tried by a company of good singers, that as the devil's
invention, for they esteemed it as no better, as being, with regard
to them, a debar to any beauty in the matter.] and if at any time,
any of the three or four notes, the which the voices may sometimes
sing, cannot be reached or touched upon the organ, such an omission
would be no fault at all, because the voices may or can, do so well
without it; or if, instead of playing so many parts, they touch (at
least in the tenor) all the notes which the voices sometimes, or in
some places use, as in their passing from one note to another, i.e.
to act or to do in that point as doing in nature, and as letting the
upper parts be sung by the voices only, and as when in them, for a
verse or more, as best to suit the matter, or subject matter in hand
[and as to be instructed before-hand by the clerk] the treble to be
wholly omitted; I say in this matter the thing would be much better,
or they might do or act much better than to affect the making such a
strange confused noise, so foriegn to the matter, as they always do,
and therefore as in consequence of which [or of the whole I have
shewn] not the subject to remain, as under disguise, a mere nothing.
[Footnote of footnote : viz. as my the taking for singing [to the
praise and glory of God] here and there three or four verses, in a
nonsensical manner, as to be without any right drift or reason, and
as so, no matter in what version because, for such a going on; Dr.
Brady and Mr. Tate need not have made a new one, nor needs any parish 
deficient in the matter ever to choose it, but as still to their
shame keep on; I say as still to their shame, for it must be certain
that such proceeding can for the most pert signify nothing, save only 
for making a noise, or sham with organ, and thence putting as it were 
a slur upon David, just as if a psalm, though ever so well handled,
must or could be, as with respect to a sermon, nothing! But as
notwithstanding such impertinency, as with respect to the royal
psalmist, it may perhaps serve [as according to the paltry meaning of 
such adrift] to make the parson go up somewhere more brisk or
cheerful into the pulpit. &c. and as when it cannot be said, that
there is, or can be now, quite so much occasion here for preaching,
as when St. Paul &c. had to convert the world from such heathenism as 
was grown upon it, and whereas the praising of God (that everlasting
gospel) is to hold to eternity, and according to St. John, they sing
the song of Moses in heaven, as not being out of fashion there.] but
that as on the contrary by custom, the commendable matter here
imply'd to be rendered familiar, as the same, [viz. custom] has done
the badness of the play-house: for a psalm when at so low a degree as 
to be taken or handled as a nothing; and who can say the case is now
any better? Well may the play-house prevail, or even the buzzing
things in the street! Wherefore I say, if such as this, or the
contents of this, be to be regarded more than a play, then it is
certain that the smothering, as here above signified, will by my
writing be unvieled: but if the case here be not reasoned aright,
then David, who was a type of Christ, must be inferior to a priest;
for as farther, if Christ in the main contradicted David, [viz.as
touching the substance of his psalmody, as with respect to religion]
they could not both be as according to that same spirit of God, which 
was yesterday, to-day, and must be the same forever; but as in
consequence, if so, the best way would be to give religion over; but
still even from philosophy, God Almighty ought to be praised, or
highly praised for his works [yea, assuredly as from astronomy,
stupendous works indeed]; consequently if David's motives and ways be 
not sufficient, so as whereby to keep up his praise, there ought as
then to be others taken, but as in supposing it to be (as above)
reasoned aright, then, as in consequence of which, was this the
highest piece of worship, as here advanced, as with proper tunes and
compositions once to be right duly performed in churches [viz. as
with more proper taking, or suitable compositions, as well as to be
more properly used or handled, than as hitherto common in churches,
viz. as there to be performed by some proper choice of men in each
parish, and that as to their pleasure without any salaries, yea more
to their pleasure than running about in the fields, and as with their 
having a proper loft or gallery in the church - as suppossing by a
company of about fifty young men so situated, but that some of them
may be married men, [Footnote of footnote: and for which purpose
entire we had a loft erected.] and as to be right duly instructed by
the clerk, as I have known, and as whence in the whole, any one of
them would have almost thought himself half dead, if he could not
have got hmself to the church [footnote of footnote And I am very
sure that had there been an organ, and withal used in such a manner,
as hitherto used in churches, it would have been impossible in any of 
singers, ever for that to have been the case.]  and as so, together
with some boys for the upper parts of such compositions] how
wonderfully strange it will be! yea even to where unknown, or
accustomed thereto, as if they were Barbarians to it!
The psalms not being as only properly adapted to private meditation
or contemplation, were they, as now, in that way to be regarded, but
as, in chief David made use of their subject drifts and that to the
greatest advantage, in public singing; and who can, or dare say, that 
there is no occasion for any such method, or course, now to be
observed or taken, as there was in the royal psalmist's days? But
that as to the contrary, the drifts of the psalms, as with respect to 
singing, to lie under disguise above. So now, as in the whole, ought
not to be considered, whether it be not shame that these sacred
things should not be more punctually handled or better regarded, than 
what they are as now? or whether it was not a shame that David &c.
ever wrote tham at all, viz. as in behalf of a public worship? as the 
which latter, indeed seems to be -- by the parsons, tacitly thought
to be the case, or otherwise one would think that better care would
have been taken about them, viz. about such divine or sacred
precepts, yea even if less care was to be taken about a sermon.] 
But to return Dr. Smith says that the voice-part of an anthem ought
not to be played upon the organ: [Footnote: Not that I greatly mind
what we call an anthem; but a psalm, viz. with its tune or
composition of musick properly adapted (not such composition as
according to Mr. Handel's taste, of or for a psalm tune) and so to be 
pitch'd, as that exactly to suit the voices, and sung in three or
four parts by a company of singers as above - what a noble thing it
is! But it is to be notified, that a little bit too high or too low
in pitch, as the 1/8th part of a larger note, will greatly disoblige
the voices [viz. more than one would imagine]; I speak from the
experience of twenty years, and as with proper instrumental care of
pitching; and as in the same time [or long experience] I strictly
found or conformed [as in the time of divine service, or as therein
the best to suit] that one tune required to be pitched a little
flatter or sharper than another, and as when, without experience, one 
would have thought that the same pitch might have done right well;
nay and that any one, the same tune required to be pitch'd a little
flatter in the afternoon than in the forenoon; but still it must be
allowed that good voices for psalmody must have the preference before 
all other instruments; but then [and as here exhibited] they must
require to be exactly humour'd; [Footnote of footnote: not knowing
how it might be with the Hebrew musick, nor prehaps if we did, should 
we be therewith content.] but that is what the organ cannot do, save
only as in here or there a tune, and as at now and them? a season to
be expected, and as still with supposing it to be exactly in tune to
itself, or that it would keep so exact there to as to what it might
be set, and that they could touch or play thereon such notes, and
only such notes, as the voices sing, or rather as may, to the
greatest importance or enhancement by them be sung; and so as we had
not an organ, neither to help us, not to hinder us, [Footnote of
footnote: Viz. Not as in the main, - an organ instead of a psalm.] we 
had not our notes pitch'd to the fixed notes of an organ, nor of any
other instrument, but as only from an instrument whole pitch might be 
set exactly to where it was at any time required, and the which [as
from properly same divisions upon it] I noted, as from experience, to 
each note respectively, [Footnote of footnote: The instrument laid in 
its case untouch'd, save only for the time or times of its using] in
order that we might not ,in the least, ever be disobliged on that
account, viz. by being at all either too flat or too sharp: and here
it may be worthy remark, that an organist, who was out of place, came 
on purpose to hear our singing on three different Sundays, and
attended the church both fore-noon and after-noon, and said [or
owned] that it was impossible for a psalm [or the psalms] to be so
well handled by any instrumental musick whatever, and wondered how
the singers [the which consisted of plough-men, shoe-makers,
carpenters, smiths, taylors, weavers, &c. and as with some boys,
singing with their voices small, for the treble or highest part, and
with only two boys at full strength for the contra-part, viz. in such 
tunes as we used such a part] could ever be brought to such
perfection; for the first time he heard them; and upon the very first 
note he was astonished: [Footnote of footnote: And here it may be
notified, that nothing can be more handsome than for the parson to
sing bass along with the singers (and not to sit gazing about him, as 
knowing nothing of the matter); neither will it hurt or strain his
voice: as also here or there a man in the congregation who can not so 
well sing tenor.] Now I would instance of other gentlemen - strangers 
to us, besides this organist, who were also taken with our singing,
but I will only here mention one, who after the evening service, was
pleased to give the singers a treat and that because neither at St.
Paul's, the King's Chapel Royal, nor at the play-house had he heard
the like, though he had oft time frequented those places; and he also 
admired the decency of our singers, all standing when singing [facing 
the congregation] with their basses in the front, and in the next
pews the tenors, &c. and the trebles up behind; yea certainly a
finer, or a more graceful sight, than to see our gentry at the
play-house - a sight never designed by the dispensation of
providence; consequently, never [as a ceremony] for any pretended
psalmody - there to be sung in Lent.]
But why does he say so? Why the reason must be, because he never
found it to be rightly in tune [or to agree with what the voice and
ear wanted it to be; I am not speaking here about pitch] whereas I am 
very sure it can be so, or may be so, viz. if consisting only of such 
stops as may be said to be rightly proper for the purpose,
[consequently, not such stops, or musick thereon to be played, as to
be even repugnant to the design or nature of psalmody] [Footnote: No
such deficiency, and as hath been shewn in more respects than that,
surely wanting a great a regularity, as was instrumentally wanted in
the mensuration of time for the longitude.] but still needed to have
it exactly so, nothing more nice in the world! [Footnote: each
interval of melody requiring, if possible, to be even to a
mathematical point of exactness, and the same to be from or
accordingly (as I will once again affirm) to the result of the
circumference, diameter, and radius of a circle, for I am very sure
that no other points or stations will truly afford a tune; a most
surprizing, stupendious matter indeed! Consequently such stops as
call 10ths and 12ths [it tuned as they denominate perfect] can have
nothing to do with psalmody, nor rightly with any thing, save only so 
as whereby to help the organist to make a vast great, confused
noise.] And besides, as or without the foundation of the true or
perfect intervals of musick, as here spoken of [and as ought
certainly to be, nay must as in consequence be, the chief, or primary 
matter], it would have been a thing quite impossible, as with respect 
to consonancy, ever to have brought the respective bearings [as
denominated of the chords] to such and such their most proper or
respective distances or latitudes, viz. from each such ratio, as from 
which respectively they may be said to be generated [or, as
unqualified thence to issue]; and so as whence, not only to become as 
in the first place, as already avouched, true intervals of melody,
but also as at the same time, viz. from each, as it were their then
correspondent seasonings to afford the most lofty, or the most
elegant degrees of harmony; yea so I say, as touching this latter
point, as well as the other, and the which as otherwise would never
have been possible ever to have been brought to a true decision!
whenas, from the circumference, diameter, and radius of a circle,
that matter is withal undoubtedly, nay I am very sure undeniably,
decided, the chords having as thence, or from the allotement exactly
as thence, [viz. no one respectively to be in the least degree either 
flatter or sharper than as so allotted] they have, I say, as thence,
a much better relish, or a more lofty warbling, viz. in tunes or
lessons of musick, but if they could be had from what is thought
would be perfect, but still it is to be understood, that, to tune an
organ &c, only by the harmony of the chords, viz. as without any
other assistance (and although the common method hither to practised) 
must be quite insufficient for the matter of exactness, or as a
beginning at the wrong end of the work, and that for a want of a more 
proper means so as thereby to set out the steps, or to gauge the
matters more exactly, since as thence, by a good tuner, as without
such a proper gauging, all the chords may seemingly be had or
obtained to what they ought to be, and as at the same time not the
true intervals of tune; the intervals of melody being in themselves
much more nice or delicate than the consonances of harmony! As for
instance the 5th upon an instrument, may as a single consonance, be
thought to be very fine [nay, is indeed the most fine] when there set 
or taken exactly as 3 to 2, although voices never take it so [nor can 
ever take it so, that being only a foolish imagination, quite out of
the course of nature], and it may be thought to be good [viz. upon a
spinet, &c.] when anywhere taken betwixt that and the flat latitude,
at which it is as only, or as rightly to perfection to be admitted,
viz. as with regard to its mathematical point, or points of melody;
and the same may be said of all the rest, i.e. as strictly touching
their flat or sharp latitudes respectively, viz. from what has
thought would be perfect [could such have been their admittance];
consequently, it must be the true intervals or tune, or as in other
words, the true stepp'd passages among the different parts of melody
[though not to be fathomed by our reason] that gives to harmony its
true or fitness of relish, yea so, as well as to melody in itself, as 
in a single part tune or solo; Footnote: A meat pie (as here by the
by) will not be good, truly sweet, or relished, without some pepper
and salt: nay in a peal of five bells, i.e. where there is but one
5th, it, viz. that 5th, although seemingly under no restriction of
being otherwise than what we would think to be truly perfect, yet
will not be right truly sweet, unless it be no wider, but exactly
according to the result of the circle as above, as I the most
strictly know from experience, viz. by such means as by which, indeed 
it was right truly to be known; consequently, as even from thence
alone, was there nothing else, a full proof is had [as was also by my 
apparatus, testified by others as well as by me], viz. of what wrong
imaginations about the matter there has all along been, or prevailed, 
in the world! the true foundation of musick being unknown; but, as on 
the contrary, divers opinions and nonsense about it.] and as so, and
that that should be the case, what ought therefore, as once again, to 
be said of the foundation or existance of the natural notes, or
intervals of melody? and to what chief purpose must the same, as
thence, be said to be? But Dr. Smith speaks of perfection being in
the violin and violincello, as if upon them [at random] the
inconsistency, as hath been shewn, could be so truly humoured, as
whereby the chords and intervals to be rendered perfect [as according 
to what has foolishly been so styled], whenas, it is only their sort
of sound [or, as in part, surge] that is indeed excellent, or even
very excellent, for concealing the faults of musick; a famous
property indeed! And as when at the same time [as without fretts duly 
placed, viz. as according to the foundation from the circle as here
advanced, and the farther consequence of the truth of the strings, as 
to be acquired there from, and to be corrected, if or when occasion]
there can be no real perfection in them, no humouring to be in the
case (excepting as when, in a long note, they hear it wrong, and flip 
their finger a little to make it better for (as above) it is certain, 
that [as well as by the voice] any one note whatever, when taken in
any tune, ought always to be exactly at the same pitch as with
respect to the rest, or else (and still as above) no scale of musick
at all; and it is not possible, as pursuant to what has been said,
that the fingers can stop at all the sundry places at which they are, 
or ought at any time, to stop, and especially so, as with regard to
their playing in different keys, viz. so nearly hardly as to the 20th 
part of an inch, whereas to perfection much nearer, nay very much
nearer than so, is or must be required, as most especially on the
violin, where the strings are but short; or otherwise, and as chiefly 
touching any sort of psalmody, as an anthem, &c. there can be be no
such perfection in them as Dr. Smith seems, from these sort of
instruments, without fretts, to maintain; [Footnote: For supposing a
psalm-tune [viz. its tenor and bass] to be played slowly upon them,
and never in the least, at anytime, to flip the finger [or any
finger] from where at first stopp'd down, or pitch'd, what a bad
piece of work would be made! For even without fretts, they cannot (as 
above) right truly set their open notes at 5ths in tune, a 5th as a
single consonance, and chiefly upon them instruments, being good any
where, viz. betwixt and including where it is falsely said to be
perfect, and the flat latitude at which in tunes, or lessons of
musick, it only be as so, can be said to be; not but that they may
set them truer [viz. the open strings as 5ths] than they can always
stop other notes [the hand having withal sometimes a great way to
shift] but I am reasoning about perfection; and towards which (the
said perfection) in tuning by the use of fretts, mathematically
placed, and as a result of the circumference, diameter, &c. of a
circle, and thence on course, or as a very material matter in the
affair, the true or certain distance of the whole length of the
strings, viz. from the nut to the foreside of the top of the bridge,
to be, as by a lath or gauge, the most strictly kept or observed
[Footnote: of footnote: Now that is not to be done by hawling the
whole bridge at once, but as by discreetly jerking or pinching, at
the bridge, string by string.] and as together with such proper
dentings or small lengthenings respectively of the strings into the
nut [in the first string excepted] the whole length of a thick string 
not being rightly concerned in sounding close up to the nut [meaning
as from the thickness of the gut, viz, as without notifying when
stretched, the wire upon it, as in a covered string.] but that a
little bit of it, from its stiffness and lying flat in the notch,
will still as were remains at rest, or not (as again) be fairly
concerned in sounding; but, from the softness of the fingers, that is 
not the case at the fretts; I say as thus, and as together with Mr.
Bentinck's screws; for indeed without such screws, such experiments
on them instruments, as I am here about to speak of, could not well
be tried, whenas, as only then, in the making use of the larger note, 
or flat third fret from the nut, the strings in the first place being 
made correct [no easy matter to be done by musicians, at least at
present, it being as it were quite foreign to them; but I am still
talking about perfection] a touch or trial of the sharp 6th [the
which, as a single consonance, must be as sharp as the ear will
permit] as also of the 4th [the which must as still be sharper, or as 
rather with respect to consonance, out of tune wide or sharp] will
greatly rectify or decide the matter. viz. about the open 5ths &c.
nay, as not amiss, a touch of the fourth and first strings with the
bow under the strings, will, as a sharp 6th, [compound of the octave] 
as sharp or wide as ever the ear will permit -- give some
confirmation of the whole, nay sometimes by these, a small fault, or
faults in the string, if towards or near the nut end, may when
skilled in the matter, be discovered; and even hence it is withal (as 
farther) sufficiently proved, that what Dr. Smith asserts, as
touching the scale of music is not right; [Footnote of footnote: But
indeed, if a man be not able, or cannot be highly master in this
concern, viz. so as to make, and prove his strings to be right, right 
truly in order, he cannot make this [most highly good] experiment;
neither others, as belonging to the same purpose, and as also to be,
in the first place, as the most highly necessary.] for, in his making 
[or supposing] the 5th to be wider, must as in consequence, spoil the 
sharp 6th, because as thence, it must become wider or sharper than
what it will bear; now these are indeed very material matters, and
that besides the other proofs or truths which the fretts will afford; 
but still as overlooking all this, [as indeed, heretofore unknown or
unthought of, but that as on the contrary, being biassed and
prejudiced through false or foolish conjectures, viz. as touching
what was done, or might be done] these instruments, the violin and
violincello [not withstanding deficiency] were, and still are said,
and as without fretts, to be perfect; whenas it must be, that faults
by their voices are cloked or concealed. But here it may be proper to 
notify, that a viol [viz. with six strings], to anyone who may have a 
capacity to put it in order, or can be instructed to know what must
belong to that, and consequently to keep it so, or always to have, or 
make it so, will then afford [as in itself, and as so - the king of
instruments] the greatest proof of all, of what is the real scale of
musick! although an instrument now - of low esteem, nor was it worth
anything at all, for during all the while - the thing it was in
vogue; but I shall not here treat about its qualifications to the
purpose, for that would be as here too long: But I may here notify,
or certify, that an organist, who upon hearing me play some tunes
upon my viol owned that it spoke to perfection itself; and whenas,
without a monochord, a spinet or harpsichord can give no such proof
to the matter, viz. of what is the true, or real scale of musick - as 
the viol in itself can do.] but even without any farther to say, it
is certain that there must be greater faults embraced there, than
could be put up with on the organ, harpsichord or spinet; a famous
qualification indeed in them sort of instruments, as here above
advanced! And as very surprizing on the other hand, what ought there
to be said of the infamous, or monstrous division by the use of
fretts, as now in common upon the guitars? For certainly the
improvement of screw-work for the open-notes, cannot in the least do
any thing towards mediating or bettering the badness of the scale, or 
rudeness of the division thereupon used! viz. the same which was
foolishly, and for a long time, used upon the viols and lutes,
[Footnote: viz. the octave into twelve equal parts; two of which to
the whole tone, and one to the half.] but that there must be, as now
again, as well as were then for all the while, - infamous matters
indeed, viz. for the greatly abusing of musick; for now, from the
pretty voice of the guitar, viz. in its clokeing such stuff as can
have nothing to do in the matter, no, far from it, and though in that 
point (Viz. clokeing) much better than the viol, &c. could do, yet
still as with respect to musick (viz. in the condition intimated)
there can but be as it were a fine sort of janglement turned off,
for, was a psalm-tune or anthem to be played upon it [be such to be
notified] the beauty of holiness [as according to the royal psalmist] 
must, in the praising of God that way, be very much defaced, true
melody and harmony being - both as thence sacrificed, viz. to the
absurdity from, or by which the fretts are placed; and yet to this
ladies of quality must sing! But what must they sing? Why a shame on
themselves and their masters! because it can play nothing else! But
now to proceed, [the last piece as here above treated, viz. as about
the guitar, being as it were almost a digression, and but hardly
worth notice, but I say,] it ought certainly, as in a high degree, to 
be remarked, that Dr. Smith's endeavours, whereby to find the
bearings of each chord, viz. by the number of beats respectively in
any given time, and as thence to tune the organ exactly - could be
nothing, but were pretty much a-kin to the finding the longitude by
the moon; [Footnote: Now here it may be proper to notify, that no
beatings are to be heard from my viol when truly fretted, or rightly
in order, no, nor if you please from two viols, playing slowly a
psalm-tune and its bass, viz. any more or no more, than as from human 
voices; but indeed, not so the case with an organ, neither with
musical forks, but still not to be enumerated.] for, as nothing to
the purpose could be had that way, so in his tuning an organ,
harpsichord or spinet, and as not being on the other hand by an
accurate monochord, founded upon, what he calls his own principle,
neither as upon that of mine, how could he tell what was done, viz.
as touching any strictness or truth in either of them? [Footnote: But 
a monochord to perfection, to have been produced from Cambridge
education, would have been another thing [viz. something very
extraordinary indeed].] But as notwithstanding, whatever university
men write or do, it must be had in veneration, as was the case with
Mr. Huygens's division as touching the scale of musick; [Footnote: As 
likewise in his cycloid, viz. as with respect to any application of
such his demonstration to the pendulum of a clock, and where it (the
said pendulum) must move in the medium of air, and where,
consequently, the draught of the wheels of a clock must be concerned; 
and whenas, even without that [or else matters] it could not, for
other reasons which I have given, be as there -- for any good
applied.] viz. the octave into thirty-one equal parts, whereof five
of which was to go to what they call the whole note, and three to
what they call the half note major, where as if an organ,
harpsichord, or spinet, was to be tuned exactly thereto, viz. by a
monochord well executed, and truly divided or set off upon that
principal, i.e. each division to be thereupon true to its place, at
least to the 200th part of a 1/4th inch, as ought to be the case with 
a monochord, nay must to the purpose be so upon my principle [viz.to
set off with great accuracy from logarithmical calculations, and as
then together with such a string, as must still to the purpose be
required; [Footnote: for here I must notify, or rather certify, that
none of the common wire [viz. of the spinet wire &c.] will do for the 
string of a monochord: No. A string for a monochord is indeed
something very extraordinary, and of great moment, and that as to the 
world unknown before! But I shall not here treat of its properties or 
faculties; but however, it is very practicable to be produced, since
as now, after my discovery of such secrets or faults as would pertain 
thereto, and as not only so, but also how to prevent the same, and
render the matter perfect, it is no great difficulty to be had [but
still, not that every one will do] as is verified from divers sorts
of experiments by two monochords, truly perfect in other respects;
the old notions of a monochord being even as nothing at all towards
the matter.] or was a viol, &c. to be fretted accordingly as here
signified, viz. to what Mr. Huygens thought must be the best, they
would, viz. any or each of them respectively, be very confusingly out 
of tune, viz. more so by far than what Dr. Smith had imagined, and as 
farther upon his own conjectures had made, as he thought, very
accurate experiments about; and, as with respect to his book, no
doubt but that algebra was made a tool of, or rather (as in its
having nothing to do in the matter) a fool of. viz. before he took
occasion, through his conversing with me, to alter from what he
thought he had ascertained, not meaning that he altered from the
algebra, but only in the algebra, so as the better to suit with me;
but still as to his experience or application thereof to an
instrument (as already shewn) there could be no proof, either of his
principle or mine, or rather, as in other words, no proof at all of
what he had brought his principle to, as in comparison, or
consequence of mine! And yet to publish this on such a silly, weak
foundation, or insufficient, uncertain way of trying, as wherein (for 
ought he could prove or assure to the contrary) mine might be taken
or aimed at, instead of what he calls his own! O fie! Infamous
Cambridge craft indeed! Such experience as that, no being able to
verify the truth of what he thought, or might think, he had brought
the alteration of his book to!
[Footnote: But Dr. Smith says, that he directed Mr. Turner, an
organist so as put his way of tuning in execution, and that he [viz.
Mr.Turner] approved of it very much: But here, it must certainly be
worthy remark, that it had never been the doctor's way, had it not
been mine first.] for, from his conversing with me, be his book what
it will. or whether it had been wrote at all or not, or whether he
had so much as thought about it at all or not, he might, from that
way to work, had done the very same! University's ingenuity! Nor can
any the best player on the violin, &c. [viz. as without fretts, or
any adjusting, or correction of the strings as whence to be verified] 
ever as whence know what is the real scale of musick; for supposing
he could stop, or may stop exactly to, or in such places as to which
his ear may best like it, or even, as exactly to what he ought to
stop; yet I say, as thence, he can have no mathematical account of
the proportions or intervals of the scale, or of what is the scale of 
musick he makes use of: as for instance, no one, even any the best
player, could ever tell whether he played the sharp 3rd exactly to
what is said would be perfect, or whether he played it, as with
respect thereto, a little flat or sharp, in order that it really
should be so; no, no more than what a good singer as by nature can,
and that is at hence or thereby -- none at all; consequently, such a
performance can have nothing to do with the application of the real
scale of musick to the tuning the organ, the harpsichord, or spinet.
Now Mr. Graham never so much as offered to beset, bespatter, or
besiege my proceedings, after any such rate or manner; but, as
notwithstanding, Mr. Ludlam could: But now, upon my first telling Mr. 
Graham that the doctor and I could not chime in right about the scale 
of musick, and that I believed that I had lost a good friend as with
respect to the longitude affair, he [viz. Mr. Graham] was very much
displeased, and thought that, instead of the doctor using me ill
[viz. as my his taking or setting the accuracy of my labour as
nothing] he ought, as in an upright, ingenious man, so have been
pleased that musick had so good a foundation, and so as to put an end 
to all disputes and conjectures about the matter, and Lord
Macclesfield also expressed the same; however, I kept to my
integrity, not minding the loss of a friend, and who I had so great
an esteem for, and would very gladly have had him to have taken the
matter [as in its true light] quite off my hands [viz. before he
published his book, or as even from the first time we conversed about 
it] as thinking he had both more time and art than I, viz. so as
whereby the more handsomely to communicate both it and its use to the 
world: but that he would not do, as pretending [viz. after he had
altered his book] that demonstration would not let him, the which, as 
I have shewn, could be nothing; but as I was certainly in the right,
and standing in my integrity I lost his friendship, and instead it
was tears; [Footnote: Not that he had any skill in the matter [viz.
of machinery] but did me good, nay a great deal of good, from what
Mr. Graham said of it.] but this is the way of the university-men,
they want to suck the virtue out of everybody's works, and then to
call all their own; for through me, he [the doctor] brought his scale 
of musick very near to mine, or nearly to the truth, but as in the
main to be taken, left a little difference, that it might be called
his, and not mine -- besides this, and that a secret as this; and the 
which had never been discovered at all, had it not been through some
transactions I had with my third machine; consequently as so, and as
to be very weighty, or so highly useful a matter of discovery as it
was, and as never to have been known or discovered without it, it was 
therefore longitude enough of it, and worth all the money and time it 
cost (nay, it was even withal, as some requital towards the loss or
expence of the other two) viz. my curious third machine; and the
which; with the other two machines, was the most scandalously
sacrificed, viz. by a novice, as at, or to his pleasure employed --
by the board of longitude.

Now Mr. Graham allowed that his methods for a nice mensuration of
time, were insufficient as with respect to mine; but that was far
from being the case with Dr. Smith; he was a parson, and they are
strange things!

And now I think, that the drawing up of this book, and as together
with the drawing and other writings I shall leave [and especially
them of late] as illustrating why time-keeping can indeed be so truly 
had, must, if their virtue can be so kindly received--- be better to
the public than if I had finished or completed ten longitude
time-keepers; no (Lord Morton's) chance to take place in my
proceedings: For, towards a proof of which, Let it be remembered,
that I have said in this book, that if it pleased God to continue my
life and health a little longer, that then, from my last improvement, 
I would bring my watch or time-keeper so as to perform to a second in 
a fortnight; and now, since the drawing up of that part of the book,
I have indeed put the major part, but still not the most nice part
thereof, viz. of my last improvement, in execution, not venturing,
upon serious thought, to attempt the whole, lest I not live to see it 
perfected, and I now find the watch to perform as aboveexpressed, nay 
even to nearer than so! but still no astonishing matter, save only to 
them [or such philosophers] who cannot be able to weigh its
construction, or the main points of its contrivance, and as wherein
hardly to be influenced, whether any oil or not: But indeed, had I
continued under the hands of the rude commissioners, this completion, 
or great accomplishment, neither would, nor could, ever have been
obtained; but however, providence otherwise ordered the matter, and I 
can now boldly say, that if the provision for the heat and cold could 
properly be in the balance itself, as it is in the pendulum, the
watch [or my longitude time-keeper] would then perform to a few
seconds in a year, yea, to such perfection now are imaginary
impossibilities conquered; so the priests at Cambridge and Oxford,
&c. may cease their pursuit in the longitude affair, and as otherwise 
then to occupy their time.

I will now give some account how the real scale of Musick is indeed
generated from the proportion which the diameter and radius of a
circle bear respectively to the circumference; but as towards which,
this great, or secret discovery, it is as in the first place, to be
understood, that it was after I had made several strict experiments
of divers, or diverse divisions of the octave, and they as from or by 
such necessary, or proper preparations, or aparatusses to the
purpose, as from my other business I was enabled to make: yea, I may
boldly say as thence, from far more correct, or natural
qualifications to the purpose, than any before me were able to make
or have, (nay, and still are --- as yet the same), and that as so at
last, I found to my great surprize, or admiration (viz. as from the
same strictness of trial of the result of the properties of a circle, 
as here above specified, and as with such, the same apparatusses to
the purpose) the real foundation of the matter to exist, or be, as
thence, by the hand of providence established: and the which (as in
brief) I shall explain as followeth.

Let the ratio of the octave, or, as even here, as well as below to
the purpose, the octave itself, be represented by the logarithm of 2
(viz. .30103): and let that same number be also taken or supposed as
the circumference of a circle ----------
And then, (as in the margin) (.30103 x  2 = .60206 + .09582 = .69788) 
let the space of two octaves and a sharp 3rd be taken, or be as
chiefly, or rather as primarily to the purpose notified, viz. when
(as according to my discovery) the said sharp 3rd is in its most
strictly musical proportion, and that is as when, with respect to the 
octave the same is taken as the diameter of the circle (viz. here, as 
.09582): For the proportion which the circumference of the circle
bears to the diameter (and as true enough to this purpose, as well as 
to others) is about as 3.1416 to 1: So, as 3.1416 is to 1, so is
.30103 to .09582.
And then as five larger notes (but not with tones major and minor, as 
hath been imagined, and that from of old) and as together with two of 
the lesser notes (as all along foolishly styled half notes major)
are, or must be, exactly contained in the octave: so therefore, as in 
taking half the diameter for the larger note, viz. .04791, as I from
strict, or proper experience, found it to be -- as an interval of
melody, right truly pleasant (although, as barely in itself, as well
as the lesser note, nothing to do with harmony), and that four 5ths,
thence as below to be generated (viz. of each containing .17447),
(margin note .17447 x 4 = .69788) and as when, as I am proof sure, to 
be then in their most strictly musical proportion, will, as according 
to nature, be equal to the two octaves and sharp 3rd, and at the same 
time, as already intimated, each one of the four 5ths will also be as 
without any infringement in any case (viz. as with respect to the
product of nature) so generated by subtracting five times the radius
from the circumference, where will be left such a quantity or space,
as the two lesser notes must, with equal shares, take up; and that
will be .06148, so the half of which, viz. .03074 must be the lesser
note; and the lesser subtracted from greater will leave .01717,
properly to be called a flat or a sharp (or the difference of the
notes), and not non-sensically the half-note minor; the lesser note
having withal the same authority to be called a whole note as what
the other has; but they may respectively or properly be styled tone
major and tone minor, viz. without meaning the fictitious nonsense as 
of old: and (as well understood) a 5th must contain three of the
larger notes and one of the lesser (viz. as in the case or cases here 
.17447).
But as not withstanding, that from what is here above, are indeed the 
real steps or intervals of tune, or of natural melody, exactly
pointed out, or are to be thence truly generated (viz. accordingly as 
they out, or are to be as thence truly generated (viz. accordingly as 
they are taken by the voice or by voices); be also the real
consequence thereof, be also the real consonances, or chords of
natural harmony, truly limited or described; nay as so, in both
respects (viz. as touching both melody and harmony) I found to my
great surprize, to be confirmed upon strict instrumental musick, as I 
have shewn above. But still (and as has just been intimated) that
though from what is shewn above, the true steps of melody, as also
the true consonances of natural harmony, are as touching them all, or 
each of them, exactly to be defined, yet as from thence, no ratios at 
all can be said to be (that of the octave to be excepted), so the
said chords etc. must be denominated as they have all along been: and 
in the logarithm way, as here to the purpose the best way, as the
ratio of any chord is to be had by subtracting the logarithm of the
lesser number from that of the greater, so therefore, and as only
proper, viz. as in what is here, as first above -- may differ from
such ratios, so each chord, or interval, must to its properness or
sweetness of relish, in tunes or lessons of musick, be said to have
respectively such and such flatness or sharpness of latitude: as the
5ths to have, .00162 flat latitude, the 4th (its complement to the
octave) as much sharp; the flat 3rd to have .00109 flat latitude, the 
flat 6th as much sharp; and here I may notify, that the thirds will
bear their flat latitudes better than the 6th will bear their sharp;
nay the 5th will bear its flat latitude of .00162 as well or better
than the sharp 6th its sharp latitude of .00053: But to bear have I
said, as touching them all! whenas, as when in that, their exactly
right degrees, they are only as so rendered perfect! I speak from
strictly due experience (viz. from such as no man before me could
ever make, nay and are as still the same): and therefore, as each
interval respectively so results from the properties of a circle, as
I have shewn, they cannot each one, or any one, as by proof from
hence, be said to have a defect of any part or parts of a foolishly
feigned nonsensical comma; no for this, as here otherwise shewn, is
certainly the true essence of all that can be said of the matter,
whatever nonsense any book, as heretofore in the world may consist
of. Now whether my style of writing in this affair, be right proper
to the purpose or not, I thought it must be better than that the
contents of this book should be in danger of sleeping in oblivion;
yea, notwithstanding what I had -- as verbally communicated to the
world.
FINIS

Appendix to John Harrison's Later edition of Concerning Such
Mechanism....
Reproduced here with thanks to Damian Emanuel of Kirby Muxloe for
photocopying one of the rare copies in England and mailing it to me
here in Hawaii ...... April 1992.

The references are listed using the original page numbers and (in
bold) for the page and line numbers in section CSM transcription in
this edition of Pitch, Pi, .......)

APPENDIX
     
Let what is here following be joined to the secondary Note on Page
82.
(CSM page 6 of 16. line 27), viz. the words with good Strength of
Voices - in four Parts rightly adapted -----

And here it may be worthy Notice that (in any Psalm) as a Grace to
the Matter, the Trebles and Basses, at the End, or last Note of each
Line, do continue to sound a little after the Tenors and
Counter-Tenors have done, but the Basses, of the two, rather the
longest; and thus with the chief Matter, the Subject is, or must be
as thereby, very much set off or enhanc'd, and as whence withal the
Singers hacve Respites for their Voices; and certainly, as so in the
Whole, the Matter is or becomes very taking and good, or as according 
to St. Paul, very rich; but the Parsons take little Care about this
Sort of Richness, but for the most Part to render it Poorness;
consequently as so, or as whereby to become no small Contributors
towards the upholding of the Play-House; and at which Rate the
Devil's-Gloss that is upon the Play-House, must excel or outweigh the 
Divine Stamp, that is upon the Psalms.

And let this following be joined to the Note on Page 85 (CSM page 8
of 16. line 6), viz. after the Words, if less Care was to be taken
about a Sermon:----

for even in this Point it is withal, from their Carelessness, to be
observed, viz. the wrong Transmutation in some Places of the Psalms
[as in some Books or Impressions without Authority] since Tate and
Brady left 'em. But here it is as farther to be notify'd that an
Organist who plays in the Church, may or can also any Girl, who as
they think sings fine in the Church [though even there as nothing in
Comparison to what Church Singing ought to be] may also sing in the
Play-House; but then ought it not be asked. Why does not the Clerk.
&c. at the Church, or of the Psalmody there, go also to the
Play-House? - see Bedford's Abuse of Musick: and whenas his
Sentiments about the Matter are not according to what they ought to
be.


Since the publication of this book, a Monthly Review (for October
1775) was presented to me, wherein I found a great deal of Rancour,
yea, even to a high Degree, against me: but my Answer to the Matter
shall be but short.

As first, Let the Professors, Commissioners, for the Longitude, come
and shew me whatever Use there has as yet been made, or can ever be
made, for any good Purpose at Sea, in the numerous Columns, the which 
I spoke of in their Nautical Almanac.

And secondly, as touching the Scale of Musick, As they have (in the
same Review) said that it is confessed [meaning by all] that was
there but only one Key to be used, all its 8 Notes, should then be
tuned perfect Consonances, according to the Diatonick Scale, and as
only wherein, a Tune to be play'd perfect; whenas, in what they
therein think, --- that could not be: a foolish Imagination as of
old, even that there was such a Scale at all, for no such Thing is in 
Nature to be found, but what will have Clashes in it, and they very
great ones too! Whereas certainly, the real Scale of Musick, when so
far exhibited, as in proper Divisions necessary, must afford a Tune,
most truly sweet in itself, in any Key, in which as so they may be
pleased to play it, as is the Case in my Discovery: but they have
said (in the said Review) that my Scale only did, as it were
accidentally fall in with Dr. Smith's was vastly wide from mine,
before he had conversed with me, and altered his! Now these
Expressions are vastly different *; nor can it yet be proved, nor
ever will, that Dr. Smith, as without a proper apparatus, or rather
as without proper Apparatusses, could, as at his Pleasure, in tuning
the Spinet or Harpsichord [I say as so exactly tune either to
arbitary Scale, or as what is build upon assuredly true Foundation:
And now, if I have written wrong, as in there not being what they
would call a Diatonick Scale, let them come and prove that +, and I
will not only bear their Expences, but will also pay them for their
Labour. Therefore, to conclude, shall write no further about such
Nonsense, Spite and Poison [scandalously scurvy, dirty Work indeed]
as runs throughout the Whole of their maliciously groundless
Objections, as objected against Things which are really true and
done! Famous Fellows indeed! The like not being elsewhere to be
found; the Longitude not being to be right truly proved or completed, 
as long as such -- the said Fellows do reign.


* But still they agree, as well as what is in Dr. Smith's Preface
does, to the Words which passed betwixt the Doctor and me. Desperate
Priestcraft sure!

+ Neither were there ever such nonsensical Things, as Chromatick and
Enharmonick Scales, as being (all three) but only such imaginary
S(t)uff as was through Ignorance blaz'd, or buzz'd about in the
World, and that no good Purpose at all, but mere Confusion! There
being but one true Scale of Musick, and that a very stupendously
natural one indeed! Stupendious, I say, considering upon what the
same is grounded, or as from whence the same to exist!

Sat. 8th July 1995

Searching for something else through a battered metal trunk from 
Nigeria this afternoon, I discover the following notes which I had
transcribed from Dr. Robert Smith's "Harmonics" in the British Library 
I am posting it here for those interested in the history of tuning in
eighteenth century England. (Harrison's era)
(You may remember John Harrison wrote of how he felt that Smith had 
exploited his ideas. The mention of geometry and senses by Smith is interesting, although of course he is advocating JI
 logic.)

C.E.H.L.

excerpt from: Dr. Robert Smith "Harmonics" MDCCXLIX (1749)
Preface pages xi - xv

He told me he took a thin ruler equal in length to the smallest string 
of his base viol. and divided it as a monochord, by taking the interval 
of the major IIId, to that of the VIIIth, as the diameter of a circle, 
to its circumference.
The by the divisions on the ruler applied to that string, he adjusted 
the frets upon the neck of the viol. and found the harmony of the 
consonances so extremely fine that after a very small and gradual 
lengthening of the other strings at the nut, by reason of their 
greater stiffness he acquiesced in that manner the placing of 
the frets.

It follows from Mr Harrison's assumption that his IIId major is 
tempered flat by a full comma. My IIId determined by theory upon 
the principle of making all the concords within the extent of
every three octaves as equally harmonious as possible, is tempered 
flat by one ninth of a comma; or almost one eighth, when no more 
concords are taken into the calculation that what are contained
within one octave.

That theory is therefore supported on one hand by Harrison's 
experiment, and on the other by the common practice of musicians, 
who make the major IIId either perfect, or generally sharper than
 perfect, with a design I suppose, to improve the false concords, 
though to the manifest detriment of all the rest.
We may gather from the construction of the base viol, that Mr Harrison 
attended chiefly, if not solely to the harmony of the consonances 
contained within the octave; in which case the difference between 
his and my temperaments of the Major IId, VIth and Vth and their 
several dependents, are respectively no greater than 4, 3 and 1 
fiftieth parts of a comma. And considering that any assigned 
differences in temperaments of a system, will have the least 
affect in altering the harmony of the whole when at the best, 
I think a nearer agreement of that experiment with the theory 
could not be reasonably expected.
Upon asking him why he took the interval of the major IIId to
that of the VIIIth as the diameter to the circumference of a circle, 
he answered that a gentleman lately deceased had told him it would 
bring out the best division of a monochord whoever was the author
of that hypothesis for so it must be called, he took the hint, 
no doubt, from observing that as the octave, consisting of five
meantones and two limmas is a little bigger than six such tones,
or three perfect major IIIds, so the circumference of a circle 
is a little bigger than three of its diameters.

When the monochord was divided upon the principle of making the
major IIId perfect, or but very little sharper, as in Mr Huygen's 
system resulting from the octave divided into 31 equal intervals, 
Mr. Harrison told me that the major VIths were very bad and much
worse than the Vths and VIths major when equally tempered should 
differ so in their harmony, after various attempts I satisfied my 
curiosity; and this gave me the first insight into the theory of 
imperfect consonances.


With a view to some other inquiries I will conclude with the 
following observation. That, as almost all sorts of substances 
are perpetually subject to very minute vibrating motions, and 
all our senses and faculties, seem chiefly to depend upon such 
motions excited in the proper organs, either by outward objects
 or the power of the will, there is reason to expect that the 
theory of vibrations here given will not prove useless in promoting 
the philosophy of other things besides musical sounds.

Such readers as can only dip into this treatise must remember, 
that by the word vibration so often repeated I mean the time 
of a single vibration, which I notified once for all in 
sect I art. 8 31/12/1748.

Section I art 8.8 Harmonics

7. If two musical strings have the same thickness, density and 
tension and differ in length only, (which for the future I shall 
always suppose) mathematicians have demonstrated that the times
of their vibrations are proportional to their lengths (f).

8. Hence if a string of a musical instrument is stopt in the middle, 
and the sound of the half be compared to the sound of the whole, we
may acquire the idea of the interval of two sounds, whole single 
vibrations (always meaning the times) are in the ratio of 1 to 2; 
and by comparing the sounds of 2/3, 3/4, 3/5, 4/5, 5/6. 8/9, 9/10,
etc of the string with the sound of the whole, we may acquire the 
ideas of the intervals of the two sounds, whole single vibrations 
are in the ratio of 2 to 3, 3 to 4, 3 to 5, 4 to 5, 5 to 6, 8 to 9, 
9 to 10, etc.

Footnote:
(f) As a clear and exact demonstration of this curious theorem 
depends upon one or two more of no small use in harmonics, and 
requires a little of the finer sort of geometry, which cannot 
well be applied in few words, I have therefore reserved it to 
the last section of this treatise.

"Harmonics or the philosophy of musical sounds" 
by Robert Smith DD FRS
Cambridge MDCCXLIV (1744)


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