Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Prime partis
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file
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0021
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21
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continuo ꝓportionabiles geometrice: vt ꝓportio
<
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dupla, q̈drupla, ſexdecupla, cētecupla vicecupla,
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octupla / et ſic ↄ̨ñter: quoue reperiūtur in his ṫmīs</
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<
p
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<
s
xml:id
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N11BC4
"
xml:space
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preserve
">¶ Hoc correlariū magis liquide patebit ex ſequē
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tibus. </
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<
s
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N11BC9
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xml:space
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preserve
">Proprietates huiꝰ medietas in ſequēti ca
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pite ponētur.
<
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xml:id
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N11DE6
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xml:space
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">Muſica
<
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medietaſ</
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</
s
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<
s
xml:id
="
N11BD3
"
xml:space
="
preserve
">¶ Harmonica autē muſicave medie
<
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tas ſiue ꝓportionalitas eſt quotienſcū diſpoſi-
<
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tis tribus termīs vel pluribus inter ipſos nec ſūt
<
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eedē ꝓportiones: nec differentie: ſed ſicut ſe habet
<
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maximꝰ terminꝰ ad minimū. </
s
>
<
s
xml:id
="
N11BDE
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xml:space
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preserve
">ita ſe hꝫ differentia
<
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/>
maiorū ad differentiã minoꝝ vt diſpoſitis his tri
<
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bus terminis .6.4.3. inter eos non reperiunt̄̄ eedē
<
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ꝓportiões: nec eedē differētie: ſed ſicut ſe hꝫ maxi-
<
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mus eoꝝ ad minimū: ita differētie maximi ad me
<
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diū et medii ad minimū ſeſe habēt: vt cõſtat. </
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>
<
s
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N11BEB
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">Aliq̄
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ꝓprietates ſignantur huic hermonice medietati:
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ſed ille in poſterū oſtendent̄̄
<
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xml:space
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machus.</
note
>
</
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<
s
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N11BF7
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xml:space
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">¶ Addit nichomachꝰ
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his tribus antiquis et famatis medietatibus ſiue
<
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ꝓportionalitatibus .7. recentiores proportiona-
<
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litates: vt cõpleretur numerus denariꝰ: qui apud
<
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antiquos pluris habebat̄̄:
<
note
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xlink:href
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xml:id
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xml:space
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">phūs .5.
<
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ꝑti: ꝓble
<
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matum.</
note
>
vt patꝫ ꝑ philoſophū
<
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decima quīta particula ꝓblematū: ſed has videre
<
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poteris apud Seuerinū boetiū in calce ſue arith
<
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metice: et apud alios recentes mathematicos: </
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>
<
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xml:space
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">Nõ
<
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em̄ huic operi ſunt interſerēde. </
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>
<
s
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N11C12
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xml:space
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">qm̄ philoſophan-
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tes nequā eis in ſuis phiſicis calculationibꝰ vtū
<
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tur.
<
note
position
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left
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xlink:href
="
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="
note-0021-04
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xml:id
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N11E00
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xml:space
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">Alia di-
<
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uſio me-
<
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dietatuꝫ.</
note
>
</
s
>
<
s
xml:id
="
N11C1E
"
xml:space
="
preserve
">¶ Hic tamē aduertendū eſt / duplex eſt ꝓpor-
<
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tionalitas quedã cõiuncta: quedã vero diſiiuncta.
<
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/>
<
note
position
="
left
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xlink:href
="
note-0021-05a
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="
note-0021-05
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xml:id
="
N11E0A
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xml:space
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">Cõiūcta
<
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medietaſ</
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>
</
s
>
<
s
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="
N11C2A
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xml:space
="
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">¶ Cõiuncta ꝓportionalitas eſt illa / q̄ in tribus vel
<
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pluribus termīs cõſiſtit cõtinue: vt ꝓportiõalitas
<
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reꝑta in his tribus termīs .3.6.12. </
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>
<
s
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="
N11C31
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xml:space
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">Et huic medie
<
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tati ꝓpriū eſt eſſe duarū ꝓportionū inter tres ter
<
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minos ad minꝰ. </
s
>
<
s
xml:id
="
N11C38
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xml:space
="
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">Inter tres terminos vti ſolum
<
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due ꝓportiones reperiuntur: nec poſſunt reperiri
<
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plures vtendo illis terminis et nõ aliis niſi cõpa-
<
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retur primus ad vltimum. </
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>
<
s
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="
N11C41
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xml:space
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">Sed tunc omnes termi
<
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ni bis capiuntur. </
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<
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">Quare notandum eſt / quando
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dicimus / inter tres terminos reperiuntur dum
<
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taxat due ꝓportiões vel ad ſummū tres: ſi vltimꝰ
<
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comparetur ad primū ītelligendū eſt dūmodo nõ
<
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vtamur niſi illis tribꝰ termīs: et nõ aliquibꝰ aliis
<
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virtualiter intermediis. </
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>
<
s
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="
N11C53
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">Inter .6. em̄ et .12. multe
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reperiuntur ꝓportiones dūmodo vtamur termīs
<
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intermediis puta octonario, nouenario, denario
<
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et vndenario.
<
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position
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xml:id
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xml:space
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">Propor
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tiõalitas
<
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diuiſa.</
note
>
</
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<
s
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="
N11C61
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xml:space
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">¶ Sed proportionalitas diuiſa ſiue
<
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diſiūcta eſt illa que cõſiſtit in .4. terminis aut plu
<
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ribus diſcõtinue: vt ꝓportionalitas que eſt in his
<
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quattuor termīs: 1.2.6.12. eſt ꝓportiõalitas diſiū
<
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ta </
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<
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xml:space
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">Et huic ꝓpriū eſt ī quattuor termīs ad mininꝰ
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cõſiſtere diſcõtinue ꝓportionabilibus: ita non
<
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eadem ſit proportio primi ad ſecundum et ſecundi
<
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ad tertium. </
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>
<
s
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xml:space
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">Hoc patet in exemplo dato.
<
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<
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medietaſ</
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</
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<
s
xml:id
="
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xml:space
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">¶ His
<
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tribus medietatibus addenda eſt quedam medie-
<
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tas ſiue ꝓportionalitas que a mathematicis ma
<
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xima et perfectiſſima dicitur. </
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>
<
s
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">Unde medietas per
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fectiſſima eſt illa que in quattuor terminis et tribꝰ
<
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interuallis cõſiſtit: in qua alie famate ꝓportiona
<
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litates reperiri poſſunt: vt in iſtis quatuor termīs
<
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6.8.9.12.
<
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xml:id
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xml:space
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">ꝓp̄etateſ
<
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medietaſ
<
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tis perfe
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ctiſſime.</
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>
</
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<
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">Ibi em̄ eſt maxima et perfectiſſima pro-
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portionalitas. </
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>
<
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">Per interuallū intellige propor-
<
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tionē que eſt inter duos terminos īmediatos. </
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>
<
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">Et
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ſic intelligēdo reperies dumtaxat inter quattuor
<
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terminos tria interualla: hoc eſt tres ꝓportiones
<
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ſereatim ſe habētes: vt in datis terminis reperies
<
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ꝓportiones .6. ad .8. et 8. ad .9. et .9. ad .12. </
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<
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medietas multas habet proprietates. </
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>
<
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<
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chead
="
Capitulum primū.
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proprietas eſt / ſi cõparetur tertius ad primū, et
<
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quartus ad tertium: reperitur proportionalitas
<
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arithmetica: quoniã reperiūtur eedem differentie
<
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et nõ eedem proportiones. </
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>
<
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</
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<
s
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N11CC0
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xml:space
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">Si comparetur quartus ad ſecūduꝫ, et tertius ad
<
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primū, reperietur proportionalitas geometrica /
<
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qm̄ vtrobi eſt ibi ſexq̇altera ꝓportio: differētie
<
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vero nõ vtrobi eedē: qm̄ vna differētia eſt nūerꝰ
<
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quaternariꝰ: alia vero ternariꝰ: igitur ibi eſt geo
<
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metrice medietas. </
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>
<
s
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="
N11CCD
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xml:space
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">Patet ↄ̨ña ex diffinitione geo-
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metrica medietatis. </
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<
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xml:space
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">¶ Tertia proprietas. </
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>
<
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xml:space
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<
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paretur numerus quartus ad ſcḋm, et ſecūdus ad
<
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primū, reperies harmonicam, ꝓportionalitatem
<
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</
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>
<
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">¶ Quarta ꝓprietas. </
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>
<
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xml:space
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">In iſta medietate perfectiſſi
<
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ma oēs cõſonantie ſimplices compariūtur.
<
note
position
="
right
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xlink:href
="
note-0021-09a
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="
note-0021-09
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xml:id
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xml:space
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">quatuor
<
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muſice cõ
<
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ſonãtie.</
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</
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<
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xml:space
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">Qua
<
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tuor em̄ ſunt muſice cõſonãtie ſimplices: videlicet
<
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tonus, diapente, diateſſeron, et diapaſon </
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>
<
s
xml:id
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N11CF1
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xml:space
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">¶ Unde
<
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tonus eſt duarū vocū quarum vna eleuatur ſuper
<
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alterã in ꝓportione ſexquioctaua vniꝰ ad alteran
<
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harmonica ↄ̨ſonãtia. </
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>
<
s
xml:id
="
N11CFA
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xml:space
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">vt inṫ duas voces quaꝝ vna
<
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ſe habet vt .8. et alia vt nouē: vel quaꝝ vna ſe ha-
<
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bet vt .16. et alia vt .18.
<
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position
="
right
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xlink:href
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="
note-0021-10
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xml:id
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N11E3A
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xml:space
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">Diateſſe
<
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ron.</
note
>
</
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>
<
s
xml:id
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N11D06
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xml:space
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">¶ Sed diateſſero eſt duarū
<
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vocum: quarum vna eleuatur ſuper alteram in ꝓ-
<
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portione ſexquitertia muſica conſonantia: vt in-
<
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ter duas voces ſe habentes vt .4. et .3.
<
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xlink:href
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xlink:label
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note-0021-11
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xml:id
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xml:space
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">Diapēte</
note
>
</
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>
<
s
xml:id
="
N11D14
"
xml:space
="
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">¶ Diapente
<
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vero eſt hermonica cõſonãtia duarū vocum: qua-
<
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rum vna eleuatur ſuper alterã in ꝓportõe ſexqui
<
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/>
altera. </
s
>
<
s
xml:id
="
N11D1D
"
xml:space
="
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">vt inter duas voces ſe habentes vt .12. et .8
<
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vt .3. et .2.
<
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xlink:href
="
note-0021-12a
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xlink:label
="
note-0021-12
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xml:id
="
N11E48
"
xml:space
="
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">diapaſõ</
note
>
</
s
>
<
s
xml:id
="
N11D27
"
xml:space
="
preserve
">¶ Diapaſon vero eſt conſonãtia harmo
<
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/>
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
<
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/>
illa harmonica que eſt inter duas voces ſe haben
<
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/>
tes ſicut .12. ad .6. eſt muſica conſonantia: que dia
<
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/>
paſon vocitatur.
<
note
position
="
right
"
xlink:href
="
note-0021-13a
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xlink:label
="
note-0021-13
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xml:id
="
N11E4E
"
xml:space
="
preserve
">Correla
<
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/>
riū ṗmū.</
note
>
</
s
>
<
s
xml:id
="
N11D3B
"
xml:space
="
preserve
">¶ Ex quo ſequitur / inter omēs
<
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/>
harmonicas ſimplices cõſonantias diapaſon eſt
<
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/>
maxima. </
s
>
<
s
xml:id
="
N11D42
"
xml:space
="
preserve
">Probatur / quia alie ſunt partes eius:
<
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/>
igit̄̄ ſūt ea minores: </
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>
<
s
xml:id
="
N11D47
"
xml:space
="
preserve
">Arguitur añs / q2 componitur
<
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/>
diapaſon ex tono, diateſſeron, et diapente, igitur
<
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/>
</
s
>
<
s
xml:id
="
N11D4D
"
xml:space
="
preserve
">Probatur antecedens / qm̄ .12. ad .6. eſt diapaſon
<
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/>
conſonantia: et talis conſonantia componitur ex
<
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/>
cõſonantia .8. ad .6. que eſt diateſſeron: et ex conſo
<
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/>
nantia .9. ad .8. que eſt tonus: et ex conſonantia .12
<
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/>
ad .8. que eſt diapēte: igitur diapaſon ex aliis tri
<
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/>
bus ſimplicibus concentibus conſtruitur ſiue con
<
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/>
ponitur. </
s
>
<
s
xml:id
="
N11D5C
"
xml:space
="
preserve
">Quare ſequitur diapaſon eſſe maximã
<
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/>
muſicã cõſonantiã inter ſimplices.
<
note
position
="
right
"
xlink:href
="
note-0021-14a
"
xlink:label
="
note-0021-14
"
xml:id
="
N11E56
"
xml:space
="
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">cõpoſite.
<
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ↄ̨ſonãtie</
note
>
</
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>
<
s
xml:id
="
N11D66
"
xml:space
="
preserve
">Dico inter ſim
<
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/>
plices / qm̄ multe ſunt cõpoſite conſonantie: vt di-
<
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/>
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
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>