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 nature of proportions
Source: Sanford B. Meech, "Three Musical Treatises in English From a Fifteenth-Century Manuscript," Speculum 10 (1935): 235-69 at 265-66. Used by permission.

[-265-] Here beginneth tretises diuerse of musical proporcions and of theire naturis and denominacions fferst in Englissh and than in Latyne.

Now passid al maner sightis of descant and with him wel replesshid, yt natural appetide not saturate sufficientli but feruentli desirithe mo musical conclusions as nou in special or proporcions and of them to haue plein informacion, of the whech aftir myn vnderstonging ye shal haue opin declaracion. But for as moche as the namys of hem be more conuenientli and compendiusli sette in Latin than in Englissh, therfor the namis of hem shal stonde stille in Latin and as breueli as I can declare the naturis of them in Englissh.

Ferst ye shal vnderstonde that proporcion is a comparison of 2o thinges be encheson of numbir or of quantite like or vnlike eythir to othir, so that proporcion is seide in 2o maner of wyse scilicet equalitatis and jnequalitatis. Proporcion of equalite is whan 2o euyn thingis be likenyd eythir sette to-gedir in comparison, as 2o to 2o, or 4 to 4, and so of othir. Proporcion of jnequalite is whan the more thinge is sette in comparison to the lasse or the lasse to the more, as 2o to 4, or 4 to 2o, or 3e to 5e, or 5e to 3e. And this proporcion of jnequalite hathe 5 spicis or naturis or keendis, whois namys be these in general: 1. multiplex, 2. superparticularis, 3. superparciens, 4. multiplex superparticularis, 5. multiplex superparciens.

The ferst spice eythir keende of jnequalite is callid multiplex, that is to sey manifolde and that is whan the more numbir conteynythe the lasse manyfolde, as twies 1, and that is callid in special dupla id est tweyfold, as 2 to 1, or 4 to 2, or 6 to 3, and so forthe endlesli. Yyf the more numbir conteyne thries the lasse, than it is callid in sspecial tripla, as 3 to 1, 6 to 2, 9 to 3. Yyf it be 4 times the lasse conteinid in the more, than it is quadrupla, as 4 to 1, 8 to 2, 12 to 3, and so forthe qvindupla, sexdupla, sepdupla, ocdupla, and so vpward endlesli.

As for othir keendis, ye shal vnderstonde that ther be 2 maner of parties, one is callid [-266-] aliquota and a-nother is callid non aliquota. Pars aliquota is whan that partie be ony maner of multiplicacion yeldithe his hole, as whanne be-twene his hole and him is proporcio multiplex. As a vnite is pars aliquota of euery numbir, ffor be multiplicacion of that euery numbir wexith; and tweyne or dualite is pars aliquota of euery evyn numbir. and thus this partye shal be namyd in special after the numbir on whom he is multiplied and yeldithe his hole. ffor, yf he yeldethe his hole be multiplicacion of 2, it is callid altera, id est halfe; and, yyf he yeldithe his hole be multiplicacion of 3, it is callid tercia, id est the thirde parte. Sequitur exemplum: 2o is the thyrde parte of 6, and 3 of 9, and 4 of 12. And, yf he yeldith his multiplicacion be 4, than is callid Quarta, as 2o for 8te, for 4 tymys 2o is 8te. And, yf it yeldith his hole be multiplicacion of 5, than it is callid qvinta, an of 6 sexta, and so forthe endlesli.

Pars non aliquota is whan that party be no maner of multiplicacion may yelde his hole, as 2o is a parte of 5, but he is non aliquota, ffor hou-so-euer he be multiplied he makithe not euyn 5. For, yf ye take him twies, it makethe but 4, and, yf ye take him thries, he passithe and makithe 6.

Proporcio superparticularis is whan the more numbir conteynyth the lasse and more-ouer a party of him that is aliquota. and aftir the special name of that parties aliquota shal that proporcion be namyd in special, as be-twene 6 and 4 is proporcio sesquealtera. Ses in Grew, id est totum in Latin, al in Englisshe. So sesquealtera is for to sey al and halfe, ffor the more numbir conteynyth al the lasse and halfe therof more-ouer. Be-twene 8 and 6 is proporcio sesquetercia, ffor the more numbir conteynyth the lasse and his 3de parte ouer. Be-twene 10 and 8te is sesquequarta; be-twene 12 and 10 is sesquequinta; be-twene 14 and 12 is sesquesexta, et sic infinite.

Proporcio superparciens is whan the more numbir conteynythe the lasse and more-ouer, the whech excesse eyther superplu3 is not pars aliquota of the lasse numbir, as be-twene 5 and 3. But than thu must loke to that excesse whan the more numbir passithe the lasse and deuyde it in-to sweche parties that be aliquota. And loke hou many ther be therof, and what is her special namys, and whether they be 3e, 4te, or 5te, and so forthe. And, yf ther be 2o parties aliquote, than thu shalt sey in special superbiparciens; and, yf ther be 3, supertriparciens; and, yf ther be 4, superquartiparciens, and so forthe. And, ferthermore, tho parties that be 3cie, than thu shalt sey alwey atte last ende 3cias; and, yf thei be 4te, 4tas, and so forthe endlesli. Sequitur exemplum: be-twene 5 and 3 is proporcio superbiparciens 3as, for the more numbir conteynythe the lasse and 2o parties ouer that be 3cie. But they bothe to-gedir be not pars aliquota of the lasse numbir. Be-twene 7 and 5 is superbiparciens 5as; betwene 7 and 3 is superquartiparciens 3as; betwene 9 and 5 is superquartiparciens 5tas; betwene 10 and 6 is superbiparciens 3as. And loke ye take goode hede that ye deuyde the excesse in-to the grettest partyes aliquotas that ye may, as here in this last ensample 4 is deuydid in to 2o dualiteis that beene 3cie of six. And take this for a general rwle that the same proporcion that is betwene 2o smale numbris the same is be-twene here doubles, and treblis, and quatreblis, and quiniblis, and so forthe endlesly. Sequitur exemplum the same proporcion that is be-twene 5 and 3 is be-twene 10 and 6, be-twene 20 and 12, be-twene 40 and 24, be-twene 80 and 48, and so forthe endlesli.

Multiplex superparticularis is whan the more numbir conteynythe the lasse and a partye of him that is aliquota, as 5 and 2o is dupla sesque-altera. And so is 10 and 4, and so is 20 and 8te, but 7 and 3 is dupla sesque 3cia, and so is 14 and 6.

Multiplex superparciens is whan the more numbir conteynyth the lasse and the parties that be ouer aliquote. But thei alle to-gedir be not one parte aliquota, as 8te and 3 is dupla superbiparciens tercias. And so is 16 and 6, 32 and 12. Secuntur denominaciones.

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