I. 3.

ledge to bean invention of later times.
(a) Tieon. Smytn, Maih, cajk I.
CHAP. i.
Voice, its kinds.
SEGT. ir.
Mufick.
Fythagoreans define Mufitk an apt X compofition of contraries, and an uni¬ on of many, and confent of difterents. I^or it not only cq-ordinates rythms and modula- .tion, but all mannej of Syftems. Its end is to unite, and aptly conjoyn. God is the reconciler : > of things dilcordant, and tltis . ishis , chiefeft work
according to Mufick and Medicine,- to reconcile enmities. In Mufick, fay they, confifts. the agree¬ ment of all things, • and Ariftocracy of the Ilni- verle: For, what is harmony in the world, in a City is good Government, in a Family Tempe- iance. * r
If) Porptyr.'m (3) Of many Se$ls .(faith that were
Harm. Ptalo- converlant about hafrn[9ny, , the moft eminent
rt) Pint Ac Fythagorick and Arifoxenian \
Muf. Pirph. C to Vythagoras dijudicated it by real^/i, Arif ox- in ml. Harm, enuf by fenle.] ( d) The Fythagoreans not crediting the relation ot hearing in all thofe things, where- initis requifite, adapted reafons to the different ces of founds, contrary to thofe which are per-
loco dcato apud Pmphpr.
ceived by the fenfes *, fo that by this cnitery ( rea- Ibn^ they gave occalion of calumny 'to liich as were of a different opinion.
(e) PtdemAs Hence the Fythagoreans named that which we now call Harraonick, Canonick -,. not from the Canon or Inftrument, as fome imagine, but ftom reflitude, fince reafon finds out that which is right, by ufihg Harmonical Canons or Rules. Evea of all forts of Inflruments, framed by Harmonical Rules, (Pipes, Flutes, and the like) they call the Exercife, Canonick ; which although it be nor Canonick, yetis fo termed, becaufo it is made according to the Reafons and Theorems of Canonick. The Inftrument therefore foems to be rather denominated from its Canonick affoOiion. A. Canonick in general is a Harmonick, who is.coaverfant by ratiocination, about that which confifts of Harmony. Muficians and Har- monicks differ ; Muficians are thole Harmonicks who begin from fenfe, but Canonicks are Fytha- goreans^ whoarealfo called Harmonicks; both forts are termed by a general name, Mu¬ ficians.
Li:
(intermiffive) framing appellations from the aecir dentspertaining to each. The Diaftematick they conceived to be that which is fung, and refts up- on-every note, and raanifeft the mutation which' is in all its parts, which is inconfufed, anddiyi- and disjoined, by the magnitudes which are ininefeveral founds, as qoacerved, but not , com* mixt; the parts of the voice being apply»’d mu‘ tuaUy toone another, which may eafily be fepa- rated and diftinguilhed, and , are not deftroy’d togetter. Such is theMnfical kind of voice, which to the.'dtnowing,.. manifefts all foupds, of what magnitude every one participates : , For if a man ufe it not alter this manner, he is not laid to ftiig, bu^p, fpealt. .
Tne other kind ' they conceived tq be Qontinjf- (j/zr^Tiy which we difcourfe one to another, - aiid read, and. are not confirained to ufe .any mani^ left diftinfl ten'fiohs.. of founds, but . connefi the difoourfe, till we have finilhed that, .which we in- tended to Ipeak. Fc«: if any rrian in , difputipg, or apologizing, or reading, make diftinft magni¬ tudes, ' in the feveral founds, taking off and transferring the voice’ from one to another, he is nqt laid to read,, buf tofing.
'Human voice having in this > manner two parts, they conceived, that there aretwo places which each in paffingpoffelfeth. The plac^ of Continu.- pus voice, which is by nature infinite in magni¬ tude, receiveth its proper term from that, where¬ with the fpeakeibegan, until he ends, that is the place from' the feginning of his fpeech to his conclufive filence, ,fo that the variety thereof is in our power. , Bqt the place of Diaftematick voice is not in our power, but natural ; and this likewile is bound by different eftefls. The be¬ ginning is that which is firft heard, the end that which is laft pronounced ; for from thence we begin to perceive the magnitude a|.j4bundsj and their mutual commutations, from whence firft qur hearing feems toopera,te; whereas it is pof lible there may be fome more obfeure founds per- fefled in nature, which we cannot peiceive or hear. As for inftance, in things weigh’d there are fonhe bodies which foem to have no weight, as Straws, Bran, and the like ; but when as by ap- polition of fuch bodies, fome beginning of pon- derofity appears, then we lay, They firft come within the compafs of Statick. So, when a low found increafeth by degrees, that which firft all may be perceived by the ear, we make,ithp beginning of the place which mufical voice re.r quireth.
CHAP.
«
D d d
MtR ir IX.
3S6
p T T H A G 0 R J S.
CHAP. II.
\
Firji Miijick in the. ^laneti.
(4) Wca/w,
c^p. .3,
:i .im:.'
names of Sounds, in all prpbabiliry, X were derived from the feveifr Stars, wjllhh move circularlj^ih the Heavens, and tompfafs the
the inte')"ve‘n\ionqf the bi^^ti^jeitds'^^arthanmjl-^ 'Nature her aiding Fdaitjye- ■piolent editi- Jion of t'wo h>dies-Jh\uJ(^ e,n^^mjpuhd.
(c) NicomMA. f ^)Nowf :&^the-P^'/te^i?f. are carried Wisdd -With noii^'^^on*" yielding ^n^ gjfetly recedihg to' the other, "rntift hec^atH^ caufe iburids' dterent f?otn -each' other, in' the magniftide arid ‘fwiftnefs oF voice, and in place • which (‘according' 19 rhe realbn of. their proper founds, or'theirTwiftnelsy.'or the orbs of repref- fiohs, in which the, impetuous trarifportation of each is performed) 'are either rriore fluftuatihg, or on the contrary ■ rifiore relu£lant. .But thefe three (difterenceS of magnitude, celerity, and local di- ftance, are . manifeftly exigent in . the Planets, which are eoriftantly with ibi^nd 'ciTcumagiUted through the a:therial diffufion ; whence everyone is called aVjf, as void of rac-/?, ftation ; and^'etti
d Tone, by' how much the Mpon'is dill-ant' 'ftom the Earth * 'from. IRe Moonto ■ Meriury the IvalPof that Iphce, and from to alift&llas
.much ;^ frdnf tothe-S'i^^' •fef^uiple:H,'frohl'‘' ' ■ '
the Sun to Tone, that i's as far as the^Mbpn
is'from the Earth : ' from iMa/j to fiipter ■hal^-''arid Trom half^ and thence to the ZodiaCK thus there are made feyeh Topes, which 'they call a Diapafbn harm, oriy , ' that Is an pnly^rial concent: In which &7r«r;7 move's in file bi’fnr^'^'opd, Jupiter \ei fkQ. Fhryeim^ and In ther^ th^'li'ke' .0 . . •*, .
Thofe !^nds which the fweh' Planets, and the P- 21. Sphere of, fi.xfed; Scars, and f Hat which is'above us, terraedhyfherrf’id»rfcf;jr^^)?Vfriake3 Vyth^or^f 'affirmed tobe the'Nineltlufes : but the %)iiipo* ption, and fymfhony, and as it were conrij^io'n of them allf whereof as beibg erernai.and.iS^^ gotten, each fs a part and yiortion, B®S)eii Mnemofyne-y • -c
C‘H A P. - TH.
The OUoc\yord.
( ^)"\T 0 W Vythagoras fif'fl: of all, (Ir) the ,c., i'
middle Ibund by' conjunflion, being ir(*)
always in courfe, whence God and Aether I felf compared- to rfhe two Exfreams, f^oulcl rea- are called ©soV and 'A/Siip. • ' "■ I der only a Diatefiaron concent, both to the two Tecra-
Mamh. ibid. . Mof'eovtr the found uobivh' is ikade hy'ftnkingthe and to the Hypate : but ’ tftaf we might' HhVe chords, which nirpinduceth into the eorjo/mhingfvoeet and ni^f- greater variety, the two Extrerims maKfng ’tfte being con join-
Nieonit ibid.
oi/^ or harjh tinidijcordant : for ^ if a certain obfer- vation (f numbers moderate the biota it effe^s a har¬ mony cunjonant to it Jelf\ hut if it be temeraria^ts^ not governed by meafures^ there proceeds a troubled unpleafam noife which qffends:the ear. 'Now in hea¬ ven nothing is produced cafuaily \ nothing temerari¬ ous^ but all things there proceed according to divine Rules and felled Proportions : whence irrefragably- t s inf err that the founds which proceed from toe converfwnof the Cdtlefial Spheres^ .are mufical. For found neoejfdrily proceeds from motion., arid the proportion vehich is in all dipine things caufeth the harmony of this found. T/;ii: Pythagorasy^/)?^ dll the Greeks conceived in hf mind', and under - flood that the Spheres founded fomething concor- detnf- becaufe of the neceffity afpt^pportion^ z^feh neverforfakes Cxleflial Beings. ~ from the motion o{' Saturn, which is the high- eft and furtheft from us, the graveft found in the diapafon concord, is called ' Hypate ■, becaulc d-mm -fignifieth higheji : brif from the Lunary which *
fulleft Goncord'^each tp other, that is todSy,; the concord of pfapalbh, .Which confifts in a ‘ double end of one, proportion. 'Which iriafthuch as it Cquld nof'be andthebegm- doneby tw6Tetrachords,he added an eightli fbqnd, *jjng of the o« inlerting if betwixt the Mefe arid Paramefe, fwirig it from the ‘Mefe a whole Tone, and frofn 'ithe Paramefe a Semitone i fo, tint, which ,was ,Kf-- nierly the ,Piramefein the H^tkcfiord, is'fti’irthe ' third frond .-the Meate,' ‘bptH in^ame and;0rice^ but that which was now* inferted is- tb'e from the Neate, and hath a concent 11^; ft of DiatefTaroh, which before the .Mefe had ruijtothe •
Hypate ; but the Tone betwfeen them, that ds",fhe Mefe, and'the inferted,' Called the Parame^ in- , . ftead of the 'former, to whjehioever Teti^hcftd: it be added, whether to that which is at pate, being of the lower or.' to that of thh*Ke^, being of .the higher, will ' render Diapepi^rcdii- . cord^ which, is' eithet way a' lyftem j ' ci^ft- ing both of theTetrachord iffelf, and thdiddi- . tional Torie^ as the Diapente proportion'
is the loweft and neareft the earth, neate.-, fefquialtera)‘ds found to be a fytteni of iefqult^- Tui' fignifieth From thofe which' .rife and'f^uioflava, the (fone'therefoj^e^is 'feff
dor VtATVV
next thefe, viz. from the motion of fupiterwhQ is under Saturn, parypate and of Venus, who is above the Moon, paraneate. Again, from the mid¬ dle, which is the Suri% moyon, the fourth from each part, mefe, which is diftant by a diateffaron, 'In the Heptachord from both extreams, accor¬ ding tothe ancient way •, as the Sun is the fourth from each extream of the leven Planets, being in the midft. Again, from thofe which are neareft the Sun on each fide, from Mars who is placed betwixt Jupiter and the Sun, Hypermefe, which is likewife itirntTLichanus, and fromyW^r- cury who is placed betwixt Venus and the Sun, Paramefe.
by Mufical proportion calleththat
quio(ftava.^((r J T-h'us the interval, of four .*61101^, C O Meibmitt >
and of ftye, and of both corijoined t9g8t%r,
called Diapafbn, and the Tone mferted .b6tw,e^ meaning^of
the two Tetrachords, being after this manner ijf-
prehended by were determmed;tp
have this proportion in 'numbers. ' ' therefore puts
I ‘ . .. . . : . a Pome after
r - vrT'i.
C H A p. yi.
The Arithmetical Proportions of Ifarmo^^
(iz^pYtbagoras is faid'tahave frJlfoupdouttJ^ (a) Them.
1 proport i on and concord of Sounds one to ano- Smyvs. therfheDiateJfaron i n fefquitertiajhe Diapentein
fefqui-^
PTTHAGOK AS.
•P ART IX.
fefquialtera.^ the 1)1 apafon in duple. Theoccafi* (b) l>t die na- on and manner is related by (b) Cenforim/s.^ (c) caii. cap. lo. Boethius^ (d) Macrobiui^ and others j but more ^*2 exactly by {e) Nicomachus., thus:
(2) in'somn. Being an intenle thought, whether he might Scip. 2. invent any inftrumental help for the Ear, foiid ('e) Harm, en- intallible ; liich as the Sight hath by acom-
^eoefre^bv and a rule, and by a dioptre ^ or the Touch
yJOTi/icibw/de by a Ballance, or by the invention of meafures: vit. P)th. cap. As he paft by a Smith’s fliop, by a happy chance he heard the Iron Hammers ftriking upon the Anvil, and rend ring founds moft conlbnant one to another in all combinations except one. He . obfetved in them thele three concords, the Dia- pafon, the Diapente, and the Diateffaron •, but that which was between the Diateffaron and the Diapente, he found to be a difcord in it lelf, tho otherwile ufeful f or the making up of the greater of them, ( the Diapente ). Apprehending this come to him from God, as a moft happy thing, he halted into a Ihop, and by various trials, find¬ ing the difference of the founds to be according to the weight of the Hammers, and not accor¬ ding to the force of thofe who (truck, nor accor¬ ding to the falhion of the Hammers, nor accor¬ ding to the turning of the Iron which was in beat- ing out : Having taken exactly the weight^fthe Hammers, (f) he went ftraightway home, and to ^ihcrwife. One beam faftned to the walls, crofs from one corner of the room to the other, ( left any diffe¬ rence might arife from thence, or might be fuf- pe£ted to arife from the properties of feveral beams) tying four firings of the fame fubftance, (5) length,fwiftnefs, and (^) twift, upon each of them
which MeU)0‘ he hung a feveral weight, faftning it at the lower ond, and making the length of the firings alto- wodd change gether equal. Then ftriking the firings by two at unneceflariiy " a time interchangebly, he found out the afore- inco icTBfpo- laid concords, each in its own combination ^ for and ren- which was ttretched by the greateft weight, in refpefl of that which wasftretched by theleaft weight, he found to found a Diapafon. The greateft weight was of 1 2 Pound, the leaft of 6. Thence he determined, that the Diapafon did con- lift in double proportion, which the weights them- folves did Ihew. Next he found, that the greateft to the leaft but one, which was of eight pound, founded a Diapente, whence he inferred this to confift in the proportion, called, Sefquialtera, in which proportion the weights were one to ano¬ ther. But unto that which was left than it lelf in weight, yet greater than the reft, being of nine pound, he found it to found a Diateffaron ; and difcovered, that, proportionably to the weights, this concord wasSelquitertia,which firing of nine pound is naturally Sefquialtera to the leaft, for 9 to 6 is fo, {viz, Sefquialtera) as the leaft but one, which is 8, was to that which had the weight 6, in proportion Sefquitertia^ and 1 2 to 8 is Sefquialte¬ ra i and that which is in the middle between Dia¬ pente and Diateffaron, whereby Diapente exceeds Diateffaron, is confirmed to be in Selquioflava pro¬ portion, in which 9 is to 8. The fy Hem of both was called Dia^te, that is, both of the Diapente and Diateffaron joined together, as: duple pro¬ portion in compounded of Sefquialtera and Sef- quitertia fuch as are 2. 8. 6, Or on the contrary of Diateffaron and Diapente, as duple proportion is compounded of Sefquitertia and Sefquialtera, as 12. 9. 6. being taken in that order.
Applying both his hand and ear to the weights which he had hung on, and by them confirming the proportion of the relations, he did ingenioufiy transfer the common refult of the firings upon the crofs-beam, to the bridge of an ihftrument, which he called XofJbm®- and as for ftretching them proportionably to the weights, he did transfer that to an anfwerable fcrewing of the pegs. Making ufe of this foundation as an infallible rule, he ex¬ tended the experiment to many kinds of Inftru- ments. Cymbals, Pipes, Flutes, Monochords, Tri- gons, and the like ^ and he found, that this con- clufion made by numbers was confonant without variation in all. That found which proceeded from the number 5, he named Hypate-., that which from the number 8, Mefe., being Sefquiter¬ tia to the other , that from 9, Varan^e.^ being a tone fharper than the AJ^, viz. SelquioSlava ; that from 1 2, Neate. And fupplying the mid¬ dle Ipaces according to the Diatonick kind, with proportionable founds, he fo ordered theO£lo- chord with convenient numbers Duple, Sefquial¬ tera, Sefquitertia, and (the difference of tfiefo two laft) Sefquioflava.
Thus he found the progrefs by a natural necef fity, from the loweft to the highett, accofding to the Diatonical kind ^ from which again he did declare theChromatick and Enarmonick kinds.
CHAP. V.
The 'Divifion of the Diapafon,, according to the Diatonick kind,
THIS Diatonick kind feems naturally to havethefo degrees and progreffes, hemi- tone, tone and tone, (half note, whole note and whole note.) This is the. fyftem Diateffaron, con- fiftingof two tones, and that which is called a he- mitone ^ and then another tone being inferred, Dia¬ pente is made, being a fyftem of three tones and a hemitone. Then in order, after this there be¬ ing another hemitone, tone and tone, they make another Diateffaron, that is to fay, another Sefqui¬ tertia. So that in the antienter Heptachord, all fourths from the loweft, found a Diateffaron one to another, the hemitone taking the firft, fecondijf and third place, according to the progreflion irr the Tetrachord. But in the Fythagorical 0£lo- chord, which is by a conjun£lion a fyftem of the' Tetrachord, and the Pentachord, and that either D d d 2 joint-
388
ptthagoras.