School of Music
University of Nebraska--Lincoln
Lincoln, NE 68588-0100
(phone: [402] 472-2507; Internet: plefferts1@unl.edu)

Data entry: Jonathan Haupt
Checked by: Peter Slemon
Approved by: Peter M. Lefferts

Author: Anonymous
Title: On the three manners of proportions
Source: Sanford B. Meech, "Three Musical Treatises in English From a Fifteenth- Century Manuscript," Speculum 10 (1935): 235-69 at 268-69. Used by permission.

[-268-] Thus ouerpassid the rwlis of proporcions and of their denominacions, nou shal ye vnderstonde that as proporcion as a comparison betwene diuerse quantites oythir numbris, so is proporcionalitas a comparison eythir a likenes be 2o proporcions in 3re diuerse quantiteis atte last, the whech quan[ti]teis or numbris been callid the termis of that proporcionalite. And whan the ferst terme passithe the 2de, than it is callid the ferst excesse. And whan the 2de terme passith the thirde, than it is callid the 2de excesse.

So ther be 3 maner of proporcionalitees, scilicet geometrica, arithmetrica, and armonica. Proporcionalitas geometrica is whan the same proporcion is betwen the ferst terme and the 2de that is betwene the 2e and the 3de, whan al tho proporcions be like. As betwene 8, 4, 2 is proporcionalitas geometrica, ffor proporcio dupla is the ferst, and so is the 2de; 9 to 6, 6 to 4, sesquealtera; 16 to 12, 12 to 9, sesquetercia; 25 to 20, and 20 to 16, sesque 4ta; 36 to 30, 30 to 25, sesque 5ta; and so forthe vpward, encresing the noumbir of difference be one. The noumbir of difference and the excesse is all one. Whan the ferst numbir eythir terme passith the 2de eythir the 2de the 3de, than aftir the lasse excesse or difference shal that proporcion be callid bothe the ferst and the 2de. As 9, 6, 4 the lasse difference is 2o and aliquota, that is namyd be 2o is callid the 2de or altera. But than to the excesse or difference one vnite more and that that is the more difference, and tho tweyn proporcions be than bothe jcallid sesquealtera. Than take the most numbir of tho 3 termys and encrese a noumbir aboue with the more difference that was before, than hast thu 9 and 12 whois difference is 3. Encrese than the more numbir be 3 and one vnite, scilicet be 4, than hast thu 16 so here be 3, 9, 12, 16 in proporcionalite geometrica, wherof bothe proporcions be callid sesquetercia aftir the lesse difference. Werk thus forthe endlesli and thu shalt fynde the same: sesque 6ta, sesque 7ma, sesque 8ua, sesque 9na, sesque 10ma, sesque 11ma.

Anothir general rwle to fynde this proporcionalite that is called geometrica is this. Take whech 2o numbris that thu wilt that be immediate and that one that passith the othir be one vnite. Multiplie the one be the othir and euerych be himselfe, and thu shalt haue 3e termys in proporcionalite geometrica, and eyther proporcion shal be namyd in genere superparticularis be the lasse numbir of tho 2o that thu toke ferst, exemplum: as 3, 4, multiplye 3 be himselfe and it makyth 9, multiple 3 be 4 and it makith 12, multiplie 4 be himselfe and it makith 16. Than thus thu hast 3, 9, 12, 16 in proporcionalite geometrica, and thus thu shalt fynde the same, what 2o numbris immediate that euer thu take.

And take this for a general rwle in this maner proporcionalite that the medil terme multiplied be himselfe is neyther more ne lesse than the 2o extremyteis the eche multiplied be othir. Exemplum: as 12 multiplied be himselfe is 12 tymes 12, that is 144; and so is 9 tymes 16, or 15 tymys 9, that is al one. And this rwle faylith neuer of this maner proporcionalite in no maner of kende of proporcion, asay whoso wil.

Proporcionalitas arithmetica is whan the difference or the excesse be like, id est whan the more numbir passithe the 2de as moche as the 2de passith the 3de, and so forthe yf ther be mo termis [-269-] than 3e. Exemplum: 6, 4, 2, the ferst excesse or difference is 2o betwene 6 and 4, and thus the 2de betwene 4 and 2.

Proporcionalitas armonica is whan ther is the same proporcion betwene the ferst excesse or difference and the 2de that is betwene the ferst terme and the thirde. Exemplum: 12, 8, 6. Here the firste difference betwene 12 and 8te is 4, the 2de betwene 8te and 6 is 2o. Than the same proporcion is betwene 4 and 2o that is betwene 12 and 6, for eythir is proporcio dupla.

These 3 proporcionalitees Boys callith medietates, id est midlis, and thei haue thes namis geometrica, arithmetrica, armonica. As for the maner of treting of thees 3 sciencis, gemetrye tretith of lengthe and brede of londe; arithmetrike of morenes, and acorde, and the lasnes of numbir; musike of the highnes and lounesse of voyse. Than whan thu biddist me yefe the a middle betwene 2o numbris. I may aske the what maner of midle thu wilt haue, and aftir that shal be the diuersite of myn answere. Fffor the numbris may be referrid to lengthe and brede of erthe or of othir mesour that longith to gemetrie; eythir thei may be considerid as they be numbir in hemselfe, and so they long to arithmetrike; eythir bei may be referrid to lengthe and shortnesse and mesure of musical jnstrumentis the whech cause highnesse and lownesse of voyse, and so thei longe to armonye and to crafte of musike.

Exemplum of the ferst, id est gemetrye: of 9 and 4 yf thu aske me whech is the medle be gemetrye, I sey 6 for this skille. Yf here were a place of 9 fote long and 4 fote brode be gemetrie, that were 36 fote square. Than yf thu bad me yeve the a bodi or another place that were euyn square that is callid quadratum equilaterum, where in were neythir more space ne lesse than is in the former place that was ferst assignyd, than must thu abate of the lengthe of the formere place and eke as moche his brede, so that it be no lenger than it is brode. That must be by proporcion, so that the same proporcion be betwene the lengthe of the formir bodi and a syde of the 2de that is betwene the same side and the brede of the ferst bodi. and than hast thu the medil betwene the lengthe and the brede of the ferst bodi or place, and be that medle a place 4 square that is euyn therto, as in this ensaumple 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 euyn 4square 6 fote as in that that is 9 fote longe and 4 fote brode, id est 36 in bothe.

The 2de proporcionalite is opin. Why? It is callid the medil be arithmetrike, the whech tretyth of mornesse and lasnes of numbir, inasmoche as the more numbir passith the 2de be as moche as the 2de passith the 3de. Neythir more ne lesse passith 12 9 than 9 passith 6, and therfore 9 is medium arithmeticum.

The 3de proporcionalite is callid armonica or a medil be armonye for this skille. Dyapason that is proporcio 2la is the most perfite acorde aftir the vnisoun betwene the extremyteis of the dyapason, id est the trebil. And the tenor wil be yeue a mydle that is callid the mene, the which is callid a dyapente, id est sesquealtera, to the tenor, and dyatessaron, id est sesquetercia, to the treble. Therfor that maner of mydle is callid medietas armonica.

Sequitur exemplum: a pipe of 6 foote longe with his competent brede 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 diapente to the othir, as thu shalt fynde more pleynli in the makyng of the monacorde that is callid the jnstrument of plainsong, the whech monacorde is the ferst tretyse in the begynnyng of this boke. But this sufficith for knowlech' of proporcions. Secundum Chilston.

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