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

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

Author: Anonymous
Title: On the three manners of proportions
Source: New York, The Pierpont Morgan Library, B.12, ff. 65-68v. Used by permission.
Graphics: DEPRPC1A 01GF

[-f.65r-] Alia regula.

But ye schul vndirstande. that as proporcioun is a comparisoun betweyne diuerse quantiteis or [-f.65v-] noumbris. So is proporcionalitas a comparysoun or a lyknesse betwene .2. proporciouns in .3. diuerse quantiteis at the leste. the whoch quantiteis or noumbris be callid the termis of that proporcionalite. and that whanne be the firste terme passith the secunde it is callid the firste excesse. And whanne that be the secunde terme passith the thridde it is the secunde excesse. Thanne be ther .3. manir proporcionalites id est Geometrica. Arsmetrica. et Armonica.

Geometrica proporcionalitas is. whanne the same proporcioun is betwene the ferste terme and the seconde. that is betwene the seconde and the thridde id est whanne alle the proporciouns be lyk as betwene .8. and .4. and .2. that is proporcionalitas geometrice. for proporcio dupla is the ferste. And so is the secunde .9. and .6. and .4. sesquealtera. .16. and .12. and .9. sesquetercia. .24. and .20. and .16. sesquequarta. .36. and .30. and .24. sesquequinta. and foorth upward encresyng thi [noumbre] of differe[n]ce be on. the noumbre of difference and the excesse is al on. id est whanne be the ferste noumbre or terme passith the secunde. or the secunde the thridde. And aftir the lasse excesse. or difference. schal that proporcioun be clepid bothe the ferste and the secunde. as .9.6.4. the lasse difference [-f.66r-] is .2. and pars aliquota that is namid be .2. is callid the secunde or altera. put thanne to that excesse. or difference on vnite more and that is the more difference and tho too proporciouns be thanne bothe namid sesquealtera. Thanne take the moste noumbre of tho thre termis. and encrece a noumbre aboue with the more difference that was before. thanne hast thow .9. and .12. whos difference is .3. Encrece thanne the more noumbre be .3. and on vnite more id est be .4. thanne hast thu .16. so here be thre .9.12.16. in proporcione geometrica whethir of bothe proporciouns be [be bis] callid sesquetercia aftir the lasse difference. Werke thus foorth endelesly and thu schalt fynde the same. as sesquequinta. sesquesexta. sesqueseptima. sesqueottaua. sesquenona. sesquedecima. sesqueundecima et sic vlterius.

[New York, Pierpont Morgan, B.12, f. 66r; text: 5, 6, 7, 8, 9, 10, 11, 12, 20, 16, 36, 42, 49, 56, 64, 72, 81, 90, 100, 121, 132, 144, et sic sine numero] [DEPRPC1A 01GF]

Anothir general rule to fynde the proporcionalite that is callid geometrica is this. tak whoche .2 noumbris that thu wilt that be inmediat id est that on passith the tothir but be an vnite. multiplye the toon be the tothir and euerych be hymself and thu schal haue thre [-f.66v-] termis in proporcionalite geometrica that eythir proporcioun schal be namid in genere superparticularis be the lasse noumbre of tho tweyne that thu took ferst exemplum .3.4. multiplye .3. be .3. himself and that makith .9. multiplye .3. be .4. and it makith .12. multiplye .4. be himself and it makith.16. thanne thu hast thas thre .9.12.16. in proporcionalite geometrica. And thus thu scalt fynde the same of what two noumbris inmediat that euer thu take. And take this for a general rule. in this manir proporcionalite that the middil terme y multiplyid be himself. is no more ne lasse than two extremiteis y multiplyid eche be othir. as .12. multiplyid be himself id est .12. tymes .12. is .144. and so is .9. tymes .16. or .16. tymes .9. al that is on. and this rule faylith not of this manir proporcionalite in no kynde of proporcioun y warant.

De Arsmetrica proporcione

Proportionalitas Arsmetrica is. whanne the differens. or the excessis be lik id est whanne the more noumbre passith the secunde be as moch as the secunde passith the thridde and so foorth if ther be mo termis. thanne per exemplum .6.4. and .2. the ferst excesse or difference is. [-f.67r-] .2. betwen .6. and .4. And so is the secunde betwene .4. and .2. assay ho so wele.


Proporcionalitas Armonica is. whanne ther is the same proporcioun betwene the ferste excesse or difference. and the secunde that is betwene the ferste terme and the thridde exemplum .12.8.6. here is the ferste difference betwene .12. and .8. is .2. the secunde betwene .8. and .6. is .2. thanne the same proporcioun is betwene .4. and .2. that is betwene .12. and .6. for bothe be proporcio dupla.

De tribus predictis

For these proporcionaliteis. Boys. namith medietates id est middlis. and they haue these namis. Geometrica. Arsmetrica. Armonica. For the manir of tretynge of these .3. sciencis. Geometrye tretith of lengthe and brede of londe. Arsmetryk tretith of the morenesse and lassenesse of noumbre. Musyke. of acorde hynesse. and lowenesse. of voys

Thanne whanne thu biddist me yiue the a middil betwene .2. noumbris. y may aske the what manir of middil thu wilt haue. and aftir that schal be the diuersite of myn answere. for the noumbris mowe be referrid to lengthe and bridde of erthe or of othir [-f.67v-] mesuris that longith to geometrye. or [they add. sup. lin.] mowe be considerid as they be noumbris in hemself and so it longith to arsmetryk. Or they mowe be referrid to lengthe or schortnesse. or mesure of musical instrumentis. that causith hynesse and lowenesse of voys. and so it longith to Armonye. and to craft of musyk. [Exemplum in marg.] Exemplum of the ferste id est geometrye .9. and .4. if thu aske me whoche is the middil be geometrye. y sey .6. and for this skil. yf here ware a place that ware .9. feet long and .4. feet of brede. that be geometrye were .36. square feet. Thanne if thu bidde me yiue the a body or an othir place that were euene square that is callid quadratum equilaterum

[New York, Pierpont Morgan, B.12, fol. 67v; text: quadratum equilaterum] [DEPRPC1A 01GF]

wher in whauere nothir mor space ne lasse than is in that formir place that was ferst assynid thanne mose y avate of the lengthe of that formir place. and eche as moche his brede. so that it be no lengere than it is brode that moste be by proporcioun. so that the same proporcioun be bytwene the lengthe of the formere body. and a syde of the secunde that is betwene that same syde and the brede of the ferste body. and thanne hast thu the middil [-f.68r-] betwene the lengthe and the brede of the ferste body or place. and be that middil a place .4. squar that is euene ther to. as in this exsaumple that was ferste assynid id est .9. and .4. .6. is the middil and as many feet is in a body or a place that is euene .4. square .6. feet. as is in that that is .9. feet longe and .4. feet brode id est .36. in bothe.

Arsmetrica proporcio.

The secunde propocionalite is opin. Why it is callid the middil be arsmetryk the whoche tretith of morenese and lassenesse of noumbris in as moche that the more noum[bre] passith the secunde. be as moche as the secunde the thridde. as be no more ne lasse passith .12.9. thanne .9. doth .6. and therfore .9. is namid medium arsmetricum.

Armonica proporcio

The thridde proporcionalite is namid armonica or a middil be armonye for this skil. Diapason. that proporcio dupla. is the most perfyt acord aftir the Vnisoun. be the extremiteis of that diapasoun id est treble and tenor wil be yeue a middil. and that is callid a mene. the whoche is diapente id est sesquealtera to the tenor and dyatessaron id est sesquetercia to the treble. therfore that manir of [-f.68v-] middil is clepid Armonica medietas. Ensaumple of this. take a pype of .6. feet length with his competent brede. and it is a tenor in dyapason. to a pype of .3. feete with his competent brede. Thanne is a pype of .4. feet lengthe. the mene to hem tweyne. dyatessaron to the toon. and dyapente to the tothir. as thu schalt fynde more pleynly in the makynge of the monochorde. that is callid the instrument of pleynsonge. But this suffysith y now for knowlechynge of the manir of proporciouns.

Explicit. Amen.

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