Fn and Ft: DEPRPC3 TEXT
Title: On the three manners of proportions
Source: Sir John Hawkins, A General History of the Science and Practice of Music, 2 vols.  (London: Novello, 1853), 1:251-52.
[-251-] Thus over passid the reulis of proporcions, and of their denominacions, now shal ye understonde that as proporcion is a comparison betwene diverse quantiteis or their numbris, so is Proporcionalitas a comparison eyther a likeness be 2 proporcions and 3 diverse quantiteis atte last, the whech quantiteis or numbris been callid the termis of that proporcionalite; and whan the ferst terme passith the seconde than it is callid the ferst excesse; and whan the seconde terme passith the thyrd, than it is callid the seconde excesse: so ther be 3 maner of proporcionalities, scilicet Geometrica, Arithmetica, and Armonica. Proporcionalitas Geometrica is whan the same proporcion is betwene the ferst terme and the seconde, that is betwene the second and the thyrde; whan al the proporcions be like, as betwene 8, 4, 2, is Proporcionalitas Geometrica; for proporcion dupla is the ferst, and so is the seconde; 9 to 6, 6 to 4 Sesquialtera; 16 to 12, 12 to 9 Sesquitercia; 25 to 20, 20 to 16 Sesquiquarta; 36 to 30, 30 to 25 Sesquiquinta, and so forth upward, encresing the numbir of difference be one. The numbir of difference and the excesse is all one. Whan the ferst numbir eyther terme passith the seconde, eythir the seconde the thyrde, than after the lasse excesse or difference shall that proporcion be callid bothe the ferst and the seconde, as 9, 6, 4; the lasse difference is 2, and aliquota that is namyd be 2, is callid the seconde or altera: put than to the excesse or difference one unite more, and that is the more difference, and the tweyne proporcions be than bothe callid Sesquialtera. Than take the most numbir of the three termys, and increse a numbir above what the more difference that was before, than hast thou 9 and 12, whois difference is 3. Encrese than the more numbir be 3, and one unite, scilicet be 4, than hast thou 16. So here be 3, 9, 12, 16, in proporcionalite Geometrica, wherof bothe proporcions be called Sesquitercia, after the lesse difference. Werk thus forthe endlesli, and thou shal finde the same Sesquisexta, Sesquiseptima, Sesquioctava, Sesquinona, Sesquidecima, Sesquiundecima.
Another general reule to fynde this proporcionalite that is callid Geometrica is this, take whech 2 numbris that thou wilt that be immediate, and that one that passith the other be one unite, multiplie [-252-] the one be the other, and every eche be himselfe, and thou shalt have 3 termys in proporcionalite Geometrica, and eyther proporcion shal be namyd in general, Superparticularis, be the lasse numbir of the 2, that thou toke ferst. Exemplum, as 3, 4; multiplye 3 be himselfe, and it makyth 9; multiply 3 be 4, and it makith 12; multiplye 4 be himselfe and it makith 16; than thus thou hast 3, 9, 12, 16, in proporcionalite Geometrica, and thus thou shalt finde the same, what 2 numbris immediate that ever thou take.
And take this for a general reule in this maner proporcionalite, that the medil terme multiplied be himselfe is neyther mo ne lesse then the two extremyteis be, eche multiplied be other: exemplum, 12 multiplied be himselfe is 12 tymes 12, that is 144, and so is 9 tymes 16, or 16 tymes 9, that is al one. And this reule faylith never of this maner proporcionalite in no maner of keende of proporcion, asay whoso wil. Proporcionalitas Arithmetica is whan the difference or the excesse be like 1, whan the more numbir passith the seconde as moche as the seconde passith the thyrde, and so forthe, yf ther be mo termys than 3, exemplum 6, 4, 2. The ferst excesse or difference is 2 between 6 and 4, and thus the seconde betwene 4 and 2. Proporcionalitas Armonica is whan there is the same proporcion betwene the ferst excesse or difference and the seconde that is betwene the ferst terme and the thyrd, exemplum, 12, 8, 6. Here the firste difference betwene 12 and 8 is 4; the seconde betwene 8 and 6 is 2; than the same proporcion is betwene 4 and 2 that is betwene 12 and 6, for eyther is proporcion dupla. These 3 proporcionalites Boys callith Medietates, id est Midlis, and thei have these namis, Geometrica, Arithmetica, Armonica. As for the maner of tretting of these 3 sciences, Gemetrye tretith of lengthe, and brede of londe; Arithmeticke of morenesse and lassnesse of numbir; Musike of the highness and louness of voyse. Than whan thou biddest me yefe the a midle betwene 2 numbris, I may aske the what maner of midle thou wilt have, and after that shal be the diversite of myn answer; for the numbris may be referrid to lengthe and brede of erth, or of other mesore that longith to Geometrie; eyther thei may be considered as they be numbir in himselfe, and so they long to Arithmetike; eyther thei may be referrid to lengthe and shortnesse and mesure of musical instrumentis, the whech cause highnesse and lownesse of voyse, and so thei long to Armonye and to craft of musike: Exemplum of the ferst, id est, Gemetrye: of 9 and 4 yf thou aske me whech is the medle by Geometrye, I sey 6 for this skille; yf there were a place of 9 fote long and 4 fote brode be Gemetrye, that wer 36 fote square: than yf thou bade me yeve the a bodi, or another place that wer evyn square, that is callid Quadratum equilaterum, wherein wer neythir more space ne lesse than is in the former place that was ferst assigned, than must thou abate of the lengthe of the former place, and eke as moche his brede, so that it be no lengir than it is brode, that must be by proporcion, so that the same proporcion be betwene the lenthe of the former bodi and a syde of the seconde that is betwene the same syde and the brede of the ferst bodi; and then hast thou the medil betwene the lengthe and the bredth of the ferst bodi or place; and be that medle a place 4 square that is evyn thereto, as in this ensample that was ferst assignyd, 9 and 4 and 6 is the medil, and as many fote is in a bodi or a place that is evyn 4 square 6 fote, as in that that is 9 fote longe and 4 fote brode, videlicet, 36 in bothe. The seconde proporcionalite is opin whan it is callid the medil be Arithmetike, the whech trettyth of morenesse and lassenesse of numbir, in as moche as the more numbir passith the seconde be as moche as the seconde passith the thirde. Neyther more ne lesse passith 12, 9, than 9 passyth 6, and therefore 9 is Medium Arithmeticum. The thirde proporcionalite is callid Armonica, or a medil be armonye for this skille. Dyapason, that is proporcion dupla, is the most perfite acorde aftir the unison: betwene the extremyteis of the dyapason, id est the trebil and the tenor, wil be yeven a mydle that is callid the Mene, the whech is callid Dyapente, id est Sesquialtera to the tenor and dyatessaron, id est Sesquitercia to the trebil, therefore that maner of mydle is callid Medietas Armonica. Sequitur exemplum: a pipe of 6 fote long, with his competent bredth, is a tenor in dyapason to a pipe of 3 fote with his competent brede; than is a pipe of 4 fote the mene to hem tweyne, dyatessaron to the one and dyapente to the other. As thou shalt fynde more pleynli in the makyng of the monocorde, that is called the Instrument of Plain-song, the whech monocorde is the ferst trettyse in the begynnyng of this boke, but this sufficith for knowleeg of proporcions.
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