Vitruvius, De architectura libri decem, 1567

Table of Notes

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            <p style="it">
              <s xml:id="echoid-s13990" xml:space="preserve">
                <pb o="182" file="514.01.214" n="214" rhead="LIBER"/>
              tus ſuauitatem: </s>
              <s xml:id="echoid-s13991" xml:space="preserve">uel ratione maioris, et abſolutæ conſonantiæ, quæ cæteras continet. </s>
              <s xml:id="echoid-s13992" xml:space="preserve">Veræ igitur cõſonantiæ, aut
                <lb/>
              ſimplices ſunt, aut cõpoſitæ. </s>
              <s xml:id="echoid-s13993" xml:space="preserve">Simplices ſunt tres, diateſſaron proportione ſeſquitertia conſtans, diapente quæ ſeſ-
                <lb/>
              quialtera cõparatione eſſicitur, diapaſon dupla ratione confecta. </s>
              <s xml:id="echoid-s13994" xml:space="preserve">Cõpoſitæ ſunt diapaſon diapente, diapaſon dia
                <lb/>
              teſſaron, diſdiapaſon. </s>
              <s xml:id="echoid-s13995" xml:space="preserve">Nunc ſingulæ declarabuntur. </s>
              <s xml:id="echoid-s13996" xml:space="preserve">Diateſſaron concentus a noſtris quarta dicitur. </s>
              <s xml:id="echoid-s13997" xml:space="preserve">Constat to
                <lb/>
              nis duob. </s>
              <s xml:id="echoid-s13998" xml:space="preserve">et hemitonio minori, ſeſquitertia ſcilicet proportione conſtans. </s>
              <s xml:id="echoid-s13999" xml:space="preserve">Diapente dicitur quinta, quoniã quem-
                <lb/>
              admodũ diateſſaron, ideſt quarta, ſcandit a linea ad ſecundum ſpatium, uel a ſpatio ad ſecundã lineã, quatuor uo
                <lb/>
              cis gradus amplectens. </s>
              <s xml:id="echoid-s14000" xml:space="preserve">Ita diapente ſcandit a qualibet linea ad tertiã, & </s>
              <s xml:id="echoid-s14001" xml:space="preserve">a quolibet ſpatio ad tertium per quin-
                <lb/>
              que uocis gradus, poniturq́; </s>
              <s xml:id="echoid-s14002" xml:space="preserve">in proportione ſeſquialtera. </s>
              <s xml:id="echoid-s14003" xml:space="preserve">Ideo quemadmodum in monachordo diateſſaron po-
                <lb/>
              nitur in quatuor partibus diuiſo neruo, ita diapente ponitur tripartito. </s>
              <s xml:id="echoid-s14004" xml:space="preserve">atque ut in ſumma dicam, quicquid ſo-
                <lb/>
                <note position="left" xlink:label="note-514.01.214-01" xlink:href="note-514.01.214-01a" xml:space="preserve">10</note>
              num emittere potest, ſiue nernus, ſiue calamus, ſiue aliud ex qua uis materia conſtans corpus, cum uelimus
                <lb/>
              concentum ab eo reddi, neceſſe est uel magnitudines, uel ſpatia proportione illa diſtinguere, quam concentus
                <lb/>
              requirit. </s>
              <s xml:id="echoid-s14005" xml:space="preserve">Ita ijs regulis organorum artifices utentes, non temere, nec caſu, ut plerique faciunt, ſed linearum
                <lb/>
              & </s>
              <s xml:id="echoid-s14006" xml:space="preserve">corporum proportiones inuenientes quàm primum rem ipſam conſequuntur, nec experiundo tentant, quæ
                <lb/>
              certitudine præcognoſcunt. </s>
              <s xml:id="echoid-s14007" xml:space="preserve">Sed nos ad rem. </s>
              <s xml:id="echoid-s14008" xml:space="preserve">Quemadmodum diateſſaron non progreditur uſque ad tres tonos,
                <lb/>
              & </s>
              <s xml:id="echoid-s14009" xml:space="preserve">ditonum hemitonio minori excellit, & </s>
              <s xml:id="echoid-s14010" xml:space="preserve">eſt amplius ſeſquitono per interuallum toni, ſexq́; </s>
              <s xml:id="echoid-s14011" xml:space="preserve">dieſes, & </s>
              <s xml:id="echoid-s14012" xml:space="preserve">duo com
                <lb/>
              mata complectitur, ita diapente trium tonorum ſpatio, & </s>
              <s xml:id="echoid-s14013" xml:space="preserve">dieſi una conſtat, a qua ſi tonus auferatur, diateſſa-
                <lb/>
              ron relinquitur, & </s>
              <s xml:id="echoid-s14014" xml:space="preserve">quarta dempta relinquitur tonus. </s>
              <s xml:id="echoid-s14015" xml:space="preserve">his ſtantibus, eſt, quod etiam cognoſcamus diapente mi-
                <lb/>
              norem eße octo dieſibus, & </s>
              <s xml:id="echoid-s14016" xml:space="preserve">conſtare ditono, & </s>
              <s xml:id="echoid-s14017" xml:space="preserve">ſeſquitono, & </s>
              <s xml:id="echoid-s14018" xml:space="preserve">diſcrimen inter diateſſaron, & </s>
              <s xml:id="echoid-s14019" xml:space="preserve">diapente tonus
                <lb/>
              eſt, unde ſi diateſſaron tonus addatur, fit diapente, prædictæ hæ conſonantiæ ſupraparticularibus maiori-
                <lb/>
                <note position="left" xlink:label="note-514.01.214-02" xlink:href="note-514.01.214-02a" xml:space="preserve">20</note>
              bus proportionibus constant, quoniam nulla proportio ſupraparticularis maior est ſeſquialtera, uel ſeſ-
                <lb/>
              quitertia, quod ex earum denominationibus haberi poteſt, quemadmodum in tertio libro dictum est.
                <lb/>
              </s>
              <s xml:id="echoid-s14020" xml:space="preserve">Præterea neque duæ diateſſaron, neque duæ diapente conſonantiam efficere poſſunt, quoniam neque in
                <lb/>
              multiplici, neque in ſupraparticulari proportione reperiuntur, in quibus diximus concentus poni, ſed pro-
                <lb/>
              portione ſunt ſuprapartienti, ex qua nullus concentus fieri potest, cuius ratio eſt. </s>
              <s xml:id="echoid-s14021" xml:space="preserve">Quod ſymphoniæ, concen. </s>
              <s xml:id="echoid-s14022" xml:space="preserve">
                <lb/>
              tusq́; </s>
              <s xml:id="echoid-s14023" xml:space="preserve">in ijs uocum, & </s>
              <s xml:id="echoid-s14024" xml:space="preserve">ſonituum comparationibus reperiuntur, in quibus manifesta, & </s>
              <s xml:id="echoid-s14025" xml:space="preserve">clara est earum com-
                <lb/>
              munis menſura, quemadmodum in multiplici proportione est dupla, cuius illa pars eſt menſura, quæ inter
                <lb/>
              duos terminos pro differentia collocatur, quemadmodum inter duo, & </s>
              <s xml:id="echoid-s14026" xml:space="preserve">quatuor, binarius eſt utriusq́; </s>
              <s xml:id="echoid-s14027" xml:space="preserve">menſura. </s>
              <s xml:id="echoid-s14028" xml:space="preserve">
                <lb/>
              Inter nouem & </s>
              <s xml:id="echoid-s14029" xml:space="preserve">octo unitas, & </s>
              <s xml:id="echoid-s14030" xml:space="preserve">in ſupraparticularibus etiam ut in ſeſquialtera inter quatuor, & </s>
              <s xml:id="echoid-s14031" xml:space="preserve">ſex binarius
                <lb/>
              cadit, tanquam menſura cognita utriusq; </s>
              <s xml:id="echoid-s14032" xml:space="preserve">termini, ita etiam inter ſex, & </s>
              <s xml:id="echoid-s14033" xml:space="preserve">octo, quæ ſeſquitertia proportione
                <lb/>
                <note position="left" xlink:label="note-514.01.214-03" xlink:href="note-514.01.214-03a" xml:space="preserve">30</note>
              comparantur, binarius quoque communis menſura eſt utriuſque numeri, quod in ſuprapartientibus non eſt repe
                <lb/>
              rire, quemadmodum inter tria, & </s>
              <s xml:id="echoid-s14034" xml:space="preserve">quinque binarius non eſt menſura, neque ternarius, quomam binarius non
                <lb/>
              æquat, ternarius excellit quinarium, ſimilis ratio est in reliquis ſuprapartientibus. </s>
              <s xml:id="echoid-s14035" xml:space="preserve">Diapaſon a noſtris octana
                <lb/>
              dicitur, ea ponitur in dupla comparatione. </s>
              <s xml:id="echoid-s14036" xml:space="preserve">ita integer neruus ad dimidium comparatus diapaſon reddit. </s>
              <s xml:id="echoid-s14037" xml:space="preserve">Ea
                <lb/>
              autem in ſcala ordinatur ab una linea ad quartum ſpatium, uel ab uno ſpatio ad quartam lineam. </s>
              <s xml:id="echoid-s14038" xml:space="preserve">Dicitur
                <lb/>
              diapaſon, quod per omnes concentus eat, amplectitur enim diateſſaron, & </s>
              <s xml:id="echoid-s14039" xml:space="preserve">diapente, & </s>
              <s xml:id="echoid-s14040" xml:space="preserve">eſt terminus omnium
                <lb/>
              ſimplicium concentuum, continetur aut em interuallo maiori, quàm quinque, & </s>
              <s xml:id="echoid-s14041" xml:space="preserve">minori quàm tonis. </s>
              <s xml:id="echoid-s14042" xml:space="preserve">Oritur a
                <lb/>
              ſeſquialtera, & </s>
              <s xml:id="echoid-s14043" xml:space="preserve">ſeſquitertia proportione, quemadmodum tertio libro diximus. </s>
              <s xml:id="echoid-s14044" xml:space="preserve">conſtat igitur quinque tonis, & </s>
              <s xml:id="echoid-s14045" xml:space="preserve">
                <lb/>
              duobus hemitonijs minoribus, caditq́; </s>
              <s xml:id="echoid-s14046" xml:space="preserve">a ſex integris tonis commate uno. </s>
              <s xml:id="echoid-s14047" xml:space="preserve">Eſt autem comma illud amplius quo
                <lb/>
              apotome dieſim excellit. </s>
              <s xml:id="echoid-s14048" xml:space="preserve">Quod ſi a diapaſon diateſſaron auferatur, relinquitur diapente, & </s>
              <s xml:id="echoid-s14049" xml:space="preserve">ſi diapente tol-
                <lb/>
                <note position="left" xlink:label="note-514.01.214-04" xlink:href="note-514.01.214-04a" xml:space="preserve">40</note>
              las diateſſaron reliqua erit, & </s>
              <s xml:id="echoid-s14050" xml:space="preserve">ablato tono, & </s>
              <s xml:id="echoid-s14051" xml:space="preserve">diapente restat ſeſquitonus. </s>
              <s xml:id="echoid-s14052" xml:space="preserve">Animaduertendum uero est nul-
                <lb/>
              lam è ſimplicibus conſonantijs poſſe æque partiri certo, & </s>
              <s xml:id="echoid-s14053" xml:space="preserve">integro numero. </s>
              <s xml:id="echoid-s14054" xml:space="preserve">hoc primum in concentu diateſſa-
                <lb/>
              ron, & </s>
              <s xml:id="echoid-s14055" xml:space="preserve">diapente manifeſtum eſt, quoniam utraque in comparatione ſuperparticulari ponitur. </s>
              <s xml:id="echoid-s14056" xml:space="preserve">Simile iudicium
                <lb/>
              diapaſon habetur, nam cum eius minimi termini ſit unitas, & </s>
              <s xml:id="echoid-s14057" xml:space="preserve">binarium, cumq́; </s>
              <s xml:id="echoid-s14058" xml:space="preserve">binarius non ſit è numeris qua-
                <lb/>
              dratis, continuo ſequitur diapaſon, quod comparatione constat unius ad duo non pofſe æque partiri, nec etiam
                <lb/>
              in plures partes, quoniam in arithmeticis probatum est, quod inter duos quadratos numeros proportione re-
                <lb/>
              ſpondens medium cadit, & </s>
              <s xml:id="echoid-s14059" xml:space="preserve">aliàs dictum eſt, ignotas, & </s>
              <s xml:id="echoid-s14060" xml:space="preserve">irregulares eas rationes haberi, quæ certo, & </s>
              <s xml:id="echoid-s14061" xml:space="preserve">deter-
                <lb/>
              minato numero deſignari non poſſunt. </s>
              <s xml:id="echoid-s14062" xml:space="preserve">Cum igitur ex arithmeticis habeatur ex multiplicatione non quadra-
                <lb/>
              ti numeri in quadratum, non effici quadratum, & </s>
              <s xml:id="echoid-s14063" xml:space="preserve">ubihoc non datur, ibi non reperiri medium inter eos duos
                <lb/>
              numeros proportione reſpondens: </s>
              <s xml:id="echoid-s14064" xml:space="preserve">Sequitur nullam comparationem mediam inter multiplices reperiri. </s>
              <s xml:id="echoid-s14065" xml:space="preserve">cum me-
                <lb/>
                <note position="left" xlink:label="note-514.01.214-05" xlink:href="note-514.01.214-05a" xml:space="preserve">50</note>
              dietas arithmetica nil aliud ſit, quàm uinculum extremorum ex ea comparatione, quam utrunque habet ad
                <lb/>
              medium. </s>
              <s xml:id="echoid-s14066" xml:space="preserve">Diateſſaron diapente eſt compoſita conſonantia, una inquam non duæ, uocaturq́; </s>
              <s xml:id="echoid-s14067" xml:space="preserve">a nostris undecima.
                <lb/>
              </s>
              <s xml:id="echoid-s14068" xml:space="preserve">Alij eam non admittunt, licet ſuauiſſime ad aures perueniat, quoniam ſuprapartienti comparatione constat. </s>
              <s xml:id="echoid-s14069" xml:space="preserve">
                <lb/>
              Eſto a 1, & </s>
              <s xml:id="echoid-s14070" xml:space="preserve">b 2. </s>
              <s xml:id="echoid-s14071" xml:space="preserve">minimi diapaſon numeri, ſit c 4. </s>
              <s xml:id="echoid-s14072" xml:space="preserve">d 3. </s>
              <s xml:id="echoid-s14073" xml:space="preserve">minimi diateſſaron. </s>
              <s xml:id="echoid-s14074" xml:space="preserve">T uco c. </s>
              <s xml:id="echoid-s14075" xml:space="preserve">in b 4, in
                <lb/>
              2, redduntur 8, & </s>
              <s xml:id="echoid-s14076" xml:space="preserve">ſic e 8, duco a. </s>
              <s xml:id="echoid-s14077" xml:space="preserve">in d. </s>
              <s xml:id="echoid-s14078" xml:space="preserve">i, unum in tria, redduntur, 3, quæ ſint f, certum eſt e,
                <lb/>
              ad f, ideſt octo ad tria continere duplam, & </s>
              <s xml:id="echoid-s14079" xml:space="preserve">ſeſquitertiam, quoniam ſi proportio una addit ad aliam tantũ,
                <lb/>
              quantum tertia eſt ſupra quartam, ſit ut quæ erit compoſita ex prima, & </s>
              <s xml:id="echoid-s14080" xml:space="preserve">quarta æqualis ſit ijs, quæ ex alijs
                <lb/>
              componentur. </s>
              <s xml:id="echoid-s14081" xml:space="preserve">Eſto igitur ut quantum proportio inter 1. </s>
              <s xml:id="echoid-s14082" xml:space="preserve">& </s>
              <s xml:id="echoid-s14083" xml:space="preserve">2. </s>
              <s xml:id="echoid-s14084" xml:space="preserve">addit ſupra 3, & </s>
              <s xml:id="echoid-s14085" xml:space="preserve">4. </s>
              <s xml:id="echoid-s14086" xml:space="preserve">tantum addat
                <lb/>
              proportio inter 2. </s>
              <s xml:id="echoid-s14087" xml:space="preserve">& </s>
              <s xml:id="echoid-s14088" xml:space="preserve">4. </s>
              <s xml:id="echoid-s14089" xml:space="preserve">proportioni, quæ eſt inter 8, & </s>
              <s xml:id="echoid-s14090" xml:space="preserve">ſex. </s>
              <s xml:id="echoid-s14091" xml:space="preserve">Dico proportionem compoſita ex 1. </s>
              <s xml:id="echoid-s14092" xml:space="preserve">ad
                <lb/>
              2, & </s>
              <s xml:id="echoid-s14093" xml:space="preserve">ex ſex, ad octo fore æqualem ex alijs compoſitæ, ſcilicet 3. </s>
              <s xml:id="echoid-s14094" xml:space="preserve">& </s>
              <s xml:id="echoid-s14095" xml:space="preserve">4. </s>
              <s xml:id="echoid-s14096" xml:space="preserve">2. </s>
              <s xml:id="echoid-s14097" xml:space="preserve">& </s>
              <s xml:id="echoid-s14098" xml:space="preserve">4. </s>
              <s xml:id="echoid-s14099" xml:space="preserve">ut in Arithmeticis
                <lb/>
              probatur. </s>
              <s xml:id="echoid-s14100" xml:space="preserve">Dico igitur e 8, non eſſe multiplicem f 3, neque ſupraparticularem eſſe eorum comparationẽ. </s>
              <s xml:id="echoid-s14101" xml:space="preserve">
                <lb/>
                <note position="left" xlink:label="note-514.01.214-06" xlink:href="note-514.01.214-06a" xml:space="preserve">60</note>
              </s>
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