Alvarus, Thomas, Liber de triplici motu, 1509

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              <p xml:id="N11B85">
                <s xml:id="N11B9D" xml:space="preserve">
                  <pb chead="Prime partis" file="0021" n="21"/>
                continuo ꝓportionabiles geometrice: vt ꝓportio
                  <lb/>
                dupla, q̈drupla, ſexdecupla, cētecupla vicecupla,
                  <lb/>
                octupla / et ſic ↄ̨ñter: quoue reperiūtur in his ṫmīs</s>
              </p>
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              <p xml:id="N11BC3">
                <s xml:id="N11BC4" xml:space="preserve">¶ Hoc correlariū magis liquide patebit ex ſequē­
                  <lb/>
                tibus. </s>
                <s xml:id="N11BC9" xml:space="preserve">Proprietates huiꝰ medietas in ſequēti ca­
                  <lb/>
                pite ponētur.
                  <note position="left" xlink:href="note-0021-01a" xlink:label="note-0021-01" xml:id="N11DE6" xml:space="preserve">Muſica
                    <lb/>
                  medietaſ</note>
                </s>
                <s xml:id="N11BD3" xml:space="preserve">¶ Harmonica autē muſicave medie­
                  <lb/>
                tas ſiue ꝓportionalitas eſt quotienſcū diſpoſi-
                  <lb/>
                tis tribus termīs vel pluribus inter ipſos nec ſūt
                  <lb/>
                eedē ꝓportiones: nec differentie: ſed ſicut ſe habet
                  <lb/>
                maximꝰ terminꝰ ad minimū. </s>
                <s xml:id="N11BDE" xml:space="preserve">ita ſe hꝫ differentia
                  <lb/>
                maiorū ad differentiã minoꝝ vt diſpoſitis his tri­
                  <lb/>
                bus terminis .6.4.3. inter eos non reperiunt̄̄ eedē
                  <lb/>
                ꝓportiões: nec eedē differētie: ſed ſicut ſe hꝫ maxi-
                  <lb/>
                mus eoꝝ ad minimū: ita differētie maximi ad me­
                  <lb/>
                diū et medii ad minimū ſeſe habēt: vt cõſtat. </s>
                <s xml:id="N11BEB" xml:space="preserve">Aliq̄
                  <lb/>
                ꝓprietates ſignantur huic hermonice medietati:
                  <lb/>
                ſed ille in poſterū oſtendent̄̄
                  <note position="left" xlink:href="note-0021-02a" xlink:label="note-0021-02" xml:id="N11DEE" xml:space="preserve">Nicho-
                    <lb/>
                  machus.</note>
                </s>
                <s xml:id="N11BF7" xml:space="preserve">¶ Addit nichomachꝰ
                  <lb/>
                his tribus antiquis et famatis medietatibus ſiue
                  <lb/>
                ꝓportionalitatibus .7. recentiores proportiona-
                  <lb/>
                litates: vt cõpleretur numerus denariꝰ: qui apud
                  <lb/>
                antiquos pluris habebat̄̄:
                  <note position="left" xlink:href="note-0021-03a" xlink:label="note-0021-03" xml:id="N11DF6" xml:space="preserve">phūs .5.
                    <lb/>
                  ꝑti: ꝓble­
                    <lb/>
                  matum.</note>
                vt patꝫ ꝑ philoſophū
                  <lb/>
                decima quīta particula ꝓblematū: ſed has videre
                  <lb/>
                poteris apud Seuerinū boetiū in calce ſue arith­
                  <lb/>
                metice: et apud alios recentes mathematicos: </s>
                <s xml:id="N11C0D" xml:space="preserve">Nõ
                  <lb/>
                em̄ huic operi ſunt interſerēde. </s>
                <s xml:id="N11C12" xml:space="preserve">qm̄ philoſophan-
                  <lb/>
                tes nequa̄ eis in ſuis phiſicis calculationibꝰ vtū­
                  <lb/>
                tur.
                  <note position="left" xlink:href="note-0021-04a" xlink:label="note-0021-04" xml:id="N11E00" xml:space="preserve">Alia di-
                    <lb/>
                  uſio me-
                    <lb/>
                  dietatuꝫ.</note>
                </s>
                <s xml:id="N11C1E" xml:space="preserve">¶ Hic tamē aduertendū eſt /  duplex eſt ꝓpor-
                  <lb/>
                tionalitas quedã cõiuncta: quedã vero diſiiuncta.
                  <lb/>
                  <note position="left" xlink:href="note-0021-05a" xlink:label="note-0021-05" xml:id="N11E0A" xml:space="preserve">Cõiūcta
                    <lb/>
                  medietaſ</note>
                </s>
                <s xml:id="N11C2A" xml:space="preserve">¶ Cõiuncta ꝓportionalitas eſt illa / q̄ in tribus vel
                  <lb/>
                pluribus termīs cõſiſtit cõtinue: vt ꝓportiõalitas
                  <lb/>
                reꝑta in his tribus termīs .3.6.12. </s>
                <s xml:id="N11C31" xml:space="preserve">Et huic medie­
                  <lb/>
                tati ꝓpriū eſt eſſe duarū ꝓportionū inter tres ter­
                  <lb/>
                minos ad minꝰ. </s>
                <s xml:id="N11C38" xml:space="preserve">Inter tres terminos vti ſolum
                  <lb/>
                due ꝓportiones reperiuntur: nec poſſunt reperiri
                  <lb/>
                plures vtendo illis terminis et nõ aliis niſi cõpa-
                  <lb/>
                retur primus ad vltimum. </s>
                <s xml:id="N11C41" xml:space="preserve">Sed tunc omnes termi­
                  <lb/>
                ni bis capiuntur. </s>
                <s xml:id="N11C46" xml:space="preserve">Quare notandum eſt /  quando
                  <lb/>
                dicimus /  inter tres terminos reperiuntur dum­
                  <lb/>
                taxat due ꝓportiões vel ad ſummū tres: ſi vltimꝰ
                  <lb/>
                comparetur ad primū ītelligendū eſt dūmodo nõ
                  <lb/>
                vtamur niſi illis tribꝰ termīs: et nõ aliquibꝰ aliis
                  <lb/>
                virtualiter intermediis. </s>
                <s xml:id="N11C53" xml:space="preserve">Inter .6. em̄ et .12. multe
                  <lb/>
                reperiuntur ꝓportiones dūmodo vtamur termīs
                  <lb/>
                intermediis puta octonario, nouenario, denario
                  <lb/>
                et vndenario.
                  <note position="left" xlink:href="note-0021-06a" xlink:label="note-0021-06" xml:id="N11E12" xml:space="preserve">Propor­
                    <lb/>
                  tiõalitas
                    <lb/>
                  diuiſa.</note>
                </s>
                <s xml:id="N11C61" xml:space="preserve">¶ Sed proportionalitas diuiſa ſiue
                  <lb/>
                diſiūcta eſt illa que cõſiſtit in .4. terminis aut plu­
                  <lb/>
                ribus diſcõtinue: vt ꝓportionalitas que eſt in his
                  <lb/>
                quattuor termīs: 1.2.6.12. eſt ꝓportiõalitas diſiū­
                  <lb/>
                ta </s>
                <s xml:id="N11C6C" xml:space="preserve">Et huic ꝓpriū eſt ī quattuor termīs ad mininꝰ
                  <lb/>
                cõſiſtere diſcõtinue ꝓportionabilibus: ita  non
                  <lb/>
                eadem ſit proportio primi ad ſecundum et ſecundi
                  <lb/>
                ad tertium. </s>
                <s xml:id="N11C75" xml:space="preserve">Hoc patet in exemplo dato.
                  <note position="left" xlink:href="note-0021-07a" xlink:label="note-0021-07" xml:id="N11E1C" xml:space="preserve">maxima
                    <lb/>
                  medietaſ</note>
                </s>
                <s xml:id="N11C7D" xml:space="preserve">¶ His
                  <lb/>
                tribus medietatibus addenda eſt quedam medie-
                  <lb/>
                tas ſiue ꝓportionalitas que a mathematicis ma­
                  <lb/>
                xima et perfectiſſima dicitur. </s>
                <s xml:id="N11C86" xml:space="preserve">Unde medietas per­
                  <lb/>
                fectiſſima eſt illa que in quattuor terminis et tribꝰ
                  <lb/>
                interuallis cõſiſtit: in qua alie famate ꝓportiona­
                  <lb/>
                litates reperiri poſſunt: vt in iſtis quatuor termīs
                  <lb/>
                6.8.9.12.
                  <note position="left" xlink:href="note-0021-08a" xlink:label="note-0021-08" xml:id="N11E24" xml:space="preserve">ꝓp̄etateſ
                    <lb/>
                  medietaſ­
                    <lb/>
                  tis perfe­
                    <lb/>
                  ctiſſime.</note>
                </s>
                <s xml:id="N11C96" xml:space="preserve">Ibi em̄ eſt maxima et perfectiſſima pro-
                  <lb/>
                portionalitas. </s>
                <s xml:id="N11C9B" xml:space="preserve">Per interuallū intellige propor-
                  <lb/>
                tionē que eſt inter duos terminos īmediatos. </s>
                <s xml:id="N11CA0" xml:space="preserve">Et
                  <lb/>
                ſic intelligēdo reperies dumtaxat inter quattuor
                  <lb/>
                terminos tria interualla: hoc eſt tres ꝓportiones
                  <lb/>
                ſereatim ſe habētes: vt in datis terminis reperies
                  <lb/>
                ꝓportiones .6. ad .8. et 8. ad .9. et .9. ad .12. </s>
                <s xml:id="N11CAB" xml:space="preserve">¶ Iſta
                  <lb/>
                medietas multas habet proprietates. </s>
                <s xml:id="N11CB0" xml:space="preserve">¶ Prima
                  <cb chead="Capitulum primū."/>
                proprietas eſt /  ſi cõparetur tertius ad primū, et
                  <lb/>
                quartus ad tertium: reperitur proportionalitas
                  <lb/>
                arithmetica: quoniã reperiūtur eedem differentie
                  <lb/>
                et nõ eedem proportiones. </s>
                <s xml:id="N11CBC" xml:space="preserve">¶ Secūda proprietas
                  <lb/>
                </s>
                <s xml:id="N11CC0" xml:space="preserve">Si comparetur quartus ad ſecūduꝫ, et tertius ad
                  <lb/>
                primū, reperietur proportionalitas geometrica /
                  <lb/>
                qm̄ vtrobi eſt ibi ſexq̇altera ꝓportio: differētie
                  <lb/>
                vero nõ vtrobi eedē: qm̄ vna differētia eſt nūerꝰ
                  <lb/>
                quaternariꝰ: alia vero ternariꝰ: igitur ibi eſt geo­
                  <lb/>
                metrice medietas. </s>
                <s xml:id="N11CCD" xml:space="preserve">Patet ↄ̨ña ex diffinitione geo-
                  <lb/>
                metrica medietatis. </s>
                <s xml:id="N11CD2" xml:space="preserve">¶ Tertia proprietas. </s>
                <s xml:id="N11CD5" xml:space="preserve">Si cū-
                  <lb/>
                paretur numerus quartus ad ſcḋm, et ſecūdus ad
                  <lb/>
                primū, reperies harmonicam, ꝓportionalitatem
                  <lb/>
                </s>
                <s xml:id="N11CDD" xml:space="preserve">¶ Quarta ꝓprietas. </s>
                <s xml:id="N11CE0" xml:space="preserve">In iſta medietate perfectiſſi­
                  <lb/>
                ma oēs cõſonantie ſimplices compariūtur.
                  <note position="right" xlink:href="note-0021-09a" xlink:label="note-0021-09" xml:id="N11E30" xml:space="preserve">quatuor
                    <lb/>
                  muſice cõ­
                    <lb/>
                  ſonãtie.</note>
                </s>
                <s xml:id="N11CEA" xml:space="preserve">Qua­
                  <lb/>
                tuor em̄ ſunt muſice cõſonãtie ſimplices: videlicet
                  <lb/>
                tonus, diapente, diateſſeron, et diapaſon </s>
                <s xml:id="N11CF1" xml:space="preserve">¶ Unde
                  <lb/>
                tonus eſt duarū vocū quarum vna eleuatur ſuper
                  <lb/>
                alterã in ꝓportione ſexquioctaua vniꝰ ad alteran
                  <lb/>
                harmonica ↄ̨ſonãtia. </s>
                <s xml:id="N11CFA" xml:space="preserve">vt inṫ duas voces quaꝝ vna
                  <lb/>
                ſe habet vt .8. et alia vt nouē: vel quaꝝ vna ſe ha-
                  <lb/>
                bet vt .16. et alia vt .18.
                  <note position="right" xlink:href="note-0021-10a" xlink:label="note-0021-10" xml:id="N11E3A" xml:space="preserve">Diateſſe­
                    <lb/>
                  ron.</note>
                </s>
                <s xml:id="N11D06" xml:space="preserve">¶ Sed diateſſero eſt duarū
                  <lb/>
                vocum: quarum vna eleuatur ſuper alteram in ꝓ-
                  <lb/>
                portione ſexquitertia muſica conſonantia: vt in-
                  <lb/>
                ter duas voces ſe habentes vt .4. et .3.
                  <note position="right" xlink:href="note-0021-11a" xlink:label="note-0021-11" xml:id="N11E42" xml:space="preserve">Diapēte</note>
                </s>
                <s xml:id="N11D14" xml:space="preserve">¶ Diapente
                  <lb/>
                vero eſt hermonica cõſonãtia duarū vocum: qua-
                  <lb/>
                rum vna eleuatur ſuper alterã in ꝓportõe ſexqui­
                  <lb/>
                altera. </s>
                <s xml:id="N11D1D" xml:space="preserve">vt inter duas voces ſe habentes vt .12. et .8
                  <lb/>
                vt .3. et .2.
                  <note position="right" xlink:href="note-0021-12a" xlink:label="note-0021-12" xml:id="N11E48" xml:space="preserve">diapaſõ</note>
                </s>
                <s xml:id="N11D27" xml:space="preserve">¶ Diapaſon vero eſt conſonãtia harmo­
                  <lb/>
                nica duarum vocum vel ſonorum (quod in preſen­
                  <lb/>
                tiarum pro eodem capio) quarū vna eleuatur ſu-
                  <lb/>
                pra alteram in ꝓportione dupla. / vt conſonatia
                  <lb/>
                illa harmonica que eſt inter duas voces ſe haben­
                  <lb/>
                tes ſicut .12. ad .6. eſt muſica conſonantia: que dia­
                  <lb/>
                paſon vocitatur.
                  <note position="right" xlink:href="note-0021-13a" xlink:label="note-0021-13" xml:id="N11E4E" xml:space="preserve">Correla­
                    <lb/>
                  riū ṗmū.</note>
                </s>
                <s xml:id="N11D3B" xml:space="preserve">¶ Ex quo ſequitur /  inter omēs
                  <lb/>
                harmonicas ſimplices cõſonantias diapaſon eſt
                  <lb/>
                maxima. </s>
                <s xml:id="N11D42" xml:space="preserve">Probatur / quia alie ſunt partes eius:
                  <lb/>
                igit̄̄ ſūt ea minores: </s>
                <s xml:id="N11D47" xml:space="preserve">Arguitur añs / q2 componitur
                  <lb/>
                diapaſon ex tono, diateſſeron, et diapente, igitur
                  <lb/>
                </s>
                <s xml:id="N11D4D" xml:space="preserve">Probatur antecedens / qm̄ .12. ad .6. eſt diapaſon
                  <lb/>
                conſonantia: et talis conſonantia componitur ex
                  <lb/>
                cõſonantia .8. ad .6. que eſt diateſſeron: et ex conſo­
                  <lb/>
                nantia .9. ad .8. que eſt tonus: et ex conſonantia .12
                  <lb/>
                ad .8. que eſt diapēte: igitur diapaſon ex aliis tri­
                  <lb/>
                bus ſimplicibus concentibus conſtruitur ſiue con­
                  <lb/>
                ponitur. </s>
                <s xml:id="N11D5C" xml:space="preserve">Quare ſequitur diapaſon eſſe maximã
                  <lb/>
                muſicã cõſonantiã inter ſimplices.
                  <note position="right" xlink:href="note-0021-14a" xlink:label="note-0021-14" xml:id="N11E56" xml:space="preserve">cõpoſite.
                    <lb/>
                  ↄ̨ſonãtie</note>
                </s>
                <s xml:id="N11D66" xml:space="preserve">Dico inter ſim­
                  <lb/>
                plices / qm̄ multe ſunt cõpoſite conſonantie: vt di-
                  <lb/>
                tonus, ſemitonus, tritonus, bis diateſſeron, bis
                  <lb/>
                diapēte, bis diapaſon, et ter, et quater diapaſon /
                  <lb/>
                et ſic conſequenter.
                  <note position="right" xlink:href="note-0021-15a" xlink:label="note-0021-15" xml:id="N11E5E" xml:space="preserve">Stentoꝝ</note>
                </s>
                <s xml:id="N11D76" xml:space="preserve">Sed cum difficultate maior cõ­
                  <lb/>
                ſonantia bis diapaſon reperitur in voce humana
                  <lb/>
                niſi ſtētor ab inferis rediret cuiꝰ mire vocis et ho-
                  <lb/>
                merus et philoſophus ſeptimo politicorū capite
                  <lb/>
                quarto meminit. </s>
                <s xml:id="N11D81" xml:space="preserve">Si tamen vox humana in aſcen­
                  <lb/>
                dendo in infinitū augmētaretur ſiue intenderetur
                  <lb/>
                vel aliquod inſtrumentū harmonicū: in infinitum
                  <lb/>
                duplicarentur harmonice conſonantie: et ſemper
                  <lb/>
                harmonicam ꝓportionalitatem ſeruarent </s>
                <s xml:id="N11D8C" xml:space="preserve">¶ Sed
                  <lb/>
                de his hactenus. </s>
                <s xml:id="N11D91" xml:space="preserve">Parum em̄ philoſophie deſer-
                  <lb/>
                uiūt: ſed introducuntur omnia iſta vt clare inſpi-
                  <lb/>
                ciat phiſicus rerum naturalium indagator velo-
                  <lb/>
                citatem motuū non penes harmonicas conſonan­
                  <lb/>
                tias: aut muſicas equalitates ſiue proportionali-
                  <lb/>
                tates attendi debere: que vti concluſio niſi ter-
                  <lb/>
                minos predictos intelligeret ei perſpicua nõ eſſet
                  <lb/>
                  <note position="right" xlink:href="note-0021-16a" xlink:label="note-0021-16" xml:id="N11E64" xml:space="preserve">Correla-
                    <lb/>
                  riū ſcḋm</note>
                </s>
                <s xml:id="N11DA7" xml:space="preserve">¶ Patet ſecundo ex dictis hanc medietatem / quã </s>
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