Cardano, Geronimo, Opvs novvm de proportionibvs nvmerorvm, motvvm, pondervm, sonorvm, aliarvmqv'e rervm mensurandarum, non solùm geometrico more stabilitum, sed etiam uarijs experimentis & observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in V libros digestum. Praeterea Artis Magnae, sive de regvlis algebraicis, liber vnvs abstrvsissimvs & inexhaustus planetotius Ariothmeticae thesaurus ... Item De Aliza Regvla Liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum Geometricas quantitates inquirentis ...

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="id002985">
                <pb pagenum="170" xlink:href="015/01/189.jpg"/>
              ditonus: bis diapaſon
                <expan abbr="aũt">aut</expan>
              quindecim uo cibus. </s>
              <s id="id002986">Omnes igitur decem, &
                <lb/>
                <expan abbr="ſeptẽ">ſeptem</expan>
              uoces,
                <expan abbr="q̃">quae</expan>
              ſexdecim interuallis
                <expan abbr="diſtinguunt̃">diſtinguuntur</expan>
              , conſonantes ſunt: & ex
                <lb/>
              genere ditoni, & ſexquiquartæ, ſed paulo minus benè
                <expan abbr="ſonãt">ſonant</expan>
              quod ditonus
                <lb/>
              ipſe. </s>
              <s id="id002987">Igitur
                <expan abbr="quintuplã">quintuplam</expan>
              multiplicem ad ſex
                <expan abbr="quiquartã">quiquartam</expan>
              reduximus. </s>
              <s id="id002988">Verum
                <lb/>
              ut oſtenſum eſt & decimaſeptima,
                <expan abbr="q̃">quae</expan>
              bis diapaſon
                <expan abbr="cõſtat">conſtat</expan>
              , & ſemiditono
                <lb/>
              benè ſonat, hęc
                <expan abbr="aũt">aut</expan>
              inter nonaginta ſex & uiginti: quadrupla
                <expan abbr="igit̃">igitur</expan>
              eſt &
                <lb/>
              ſuperquadripartiens quintas. </s>
              <s id="id002989">Diapaſon quo que cum ſexta maiore & mi
                <lb/>
              nore eandem habent rationem quam 16 ad 5, & 10 ad 3, triplam utranque,
                <lb/>
              ſed altera ſexquiquinta, altera ſexquitertia: bis diapaſon uerò
                <expan abbr="cũ">cum</expan>
              eiſdem
                <lb/>
              ut uiginti ad tria, & 32 ad quin que ſexcupla utraque: ſed altera ſuperbipar­
                <lb/>
              tiens tertias, altera quintas. </s>
              <s id="id002990">
                <expan abbr="Manifeſtũ">Manifeſtum</expan>
              eſt igitur hanc diuiſionem
                <expan abbr="nõ">non</expan>
              ſo­
                <lb/>
              lum concinnam magis eſſe & ſuauem ſed omnem
                <expan abbr="tonorũ">tonorum</expan>
              & ſemitonio­
                <lb/>
              rum
                <expan abbr="neceſsitatẽ">neceſsitatem</expan>
              effugere. </s>
              <s id="id002991">Quòd uerò in cauſa fuit ut toni & ſemitonia
                <lb/>
              in uſu eſſent, id eſt, quoniam in
                <expan abbr="diſcẽdo">diſcendo</expan>
              neceſſe eſt eandem ſeruari ratio­
                <lb/>
              nem in
                <expan abbr="crementorũ">crementorum</expan>
              , ne que arithmeticam ſed
                <expan abbr="geometricã">geometricam</expan>
              . </s>
              <s id="id002992">Ideò
                <expan abbr="aſcẽſus">aſcenſus</expan>
              per
                <lb/>
              tonos & ſemitonia
                <expan abbr="cõmodus">commodus</expan>
              fuit, nam duplicem
                <expan abbr="ſolũ">ſolum</expan>
              differentiam pue
                <lb/>
              ri uſu aſſequi coguntur. </s>
              <s id="id002993">At uerò poterat & per ſexquiſextam diuidi dia
                <lb/>
              teſſaron, ut inter triginta ſex & quadraginta nouem interpoſitis 42, ue­
                <lb/>
              rùm triplex
                <expan abbr="ſequebat̃">ſequebatur</expan>
              in
                <expan abbr="cõueniens">conueniens</expan>
              : primum ut diateſſaron ad amuſsim
                <lb/>
              non ſeruaretur, ſed incidebat in cacophoniam, addita quadrageſima o­
                <lb/>
              ctaua parte: deficiente
                <expan abbr="aũt">aut</expan>
              in duabus ſexquiſeptimis numeris ſeu propor
                <lb/>
              tione ſexquitertia: ut inter 49 & 64 loco 48 & 64, uelut
                <expan abbr="etiã">etiam</expan>
              inter 48 ad
                <lb/>
              36, addita igitur monade in termino medio utrin que fit diſſonantia. </s>
              <s id="id002994">Se­
                <lb/>
              cundum inconueniens, eſt quae ſic diuidente non ſeruabatur ratio ſexqui­
                <lb/>
              quartæ & ſexquiquintæ ſeu ditoni & ſemiditoni, quæ uoces benè ſo­
                <lb/>
              nant. </s>
              <s id="id002995">Tertium inconueniens erat, quòd hæc ratio diuidendi diapentes
                <lb/>
              minimè ſatisfaciebat, uelut inter 324 & 216. Interponere enim neceſſe
                <lb/>
              erat 252 & 294, unde incongrua rurſus erat diuiſio. </s>
              <s id="id002996">His tot cauſis cum
                <lb/>
              proportiones maiores non fatisfacerent ut ſexqui quinta quæ diateſſa­
                <lb/>
              ron nullo modo æqualiter diuidere poteſt, & in diapente deficit ſexqui
                <lb/>
              uigeſima quarta, ut inter 25 & 36, coacti ſunt cum nec ſexquiſexta nec
                <lb/>
              ſexquiſeptima idoneæ eſſent ad ſexquioctauam confugere.</s>
            </p>
            <table>
              <table.target id="table21"/>
              <row>
                <cell>Diapaſon</cell>
                <cell>2</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>Bis diapaſon</cell>
                <cell>4</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>Diapaſon diapente</cell>
                <cell>3</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>Tris diapaſon</cell>
                <cell>8</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>Bis diapaſon
                  <expan abbr="diapẽte">diapente</expan>
                </cell>
                <cell>6</cell>
                <cell>1</cell>
              </row>
              <row>
                <cell>Hæmiolia</cell>
                <cell>3</cell>
                <cell>2</cell>
              </row>
              <row>
                <cell>Hæmitritæa</cell>
                <cell>4</cell>
                <cell>3</cell>
              </row>
              <row>
                <cell>Ditonus</cell>
                <cell>5</cell>
                <cell>4</cell>
              </row>
              <row>
                <cell>Semiditonus</cell>
                <cell>6</cell>
                <cell>5</cell>
              </row>
              <row>
                <cell>Sexta minor</cell>
                <cell>8</cell>
                <cell>5</cell>
              </row>
              <row>
                <cell>Sexta maior</cell>
                <cell>5</cell>
                <cell>3</cell>
              </row>
              <row>
                <cell>Bis diapaſon ditonus</cell>
                <cell>5</cell>
                <cell>1</cell>
              </row>
            </table>
            <p type="main">
              <s id="id002997">Eſt & alia diuiſio toni in ſemitonia,
                <expan abbr="q̃">quae</expan>
              eſt uaria
                <expan abbr="ponẽdo">ponendo</expan>
                <expan abbr="tonũ">tonum</expan>
              inter 18
                <lb/>
              & 16, media uox eſt 17 ſemitonium maius inter 17 & 16, ſed minus inter
                <lb/>
              18 & 17,
                <expan abbr="quorũ">quorum</expan>
              differentia eſt 1/288. Hic ſubit admiratio quomodo
                <expan abbr="ſemi­toniũ">ſemi­
                  <lb/>
                tonium</expan>
              minus
                <expan abbr="aptet̃">aptetur</expan>
              tam gratè in ſymphonijs, maius
                <expan abbr="aũt">aunt</expan>
                <expan abbr="nequaquã">nequaquam</expan>
              . </s>
              <s id="id002998">Ptole
                <lb/>
              męus hoc negaret, quia ſexquiquinta ſeu ſemiditonus
                <expan abbr="cõſtat">conſtat</expan>
              tono inte­
                <lb/>
              gro, qui eſt inter 90 & 80, & ſemitonio
                <expan abbr="pluſquã">pluſquam</expan>
              maiore quod eſt inter
                <lb/>
              96 & 90, & eſt ſexquiquinta decima:
                <expan abbr="q̃">quae</expan>
              maior eſt tono maiore 1/255. Pro­
                <lb/>
              pterea dicemus cauſam eſſe quae poſito ſemiditono inter 81 & 96, id eſt,
                <lb/>
              27 & 32 ſublato tono, id eſt, 234 & 216, remanebit 13 differentia 256 ad
                <lb/>
              243, ſeu qualis eſt 96 ad 91 & 1/8 quæ eſt ut 768 ad 729 et redit ad
                <expan abbr="idẽ">idem</expan>
              , ſcili</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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