Mathematics, Music, and Means Math Teacher s Circle Tuesday, October 4, 2016
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1 Wrm-up Mtemtics, Music, nd Mens Mt Tecer s ircle Tuesdy, Octoer 4, 2016 If 0 < xx re te coordintes of points on te rel numer line, solve ec of te followin equtions for xx in terms of nd. 0 xx 1. xx xx = 2. xx xx = xx. xx xx = Just Intontion Just Intontion is ny musicl tunin in wic te frequencies of notes re relted y rtios of smll wole numers. ny intervl tuned in tis wy is clled pure or just intervl. Pure intervls re importnt in music ecuse tey nturlly tend to e perceived y umns s "consonnt": plesin or stisfyin. Intervls not stisfyin tis criterion, conversely, tend to e perceived s unplesnt or s cretin dissonnce : disstisfction or tension. Te two notes in ny just intervl re memers of te sme rmonic series. requency rtios involvin lre inteers suc s 1024:729 re not enerlly sid to e justly tuned. "Just intontion is te tunin system of te lter ncient reek modes s codified y Ptolemy; it ws te estetic idel of te Renissnce teorists; nd it is te tunin prctice of ret mny musicl cultures worldwide, ot ncient nd modern." qul Temperment Just intontion cn e contrsted nd compred wit equl temperment, wic domintes Western instruments of fixed pitc (e.., pino or orn) nd defult MII tunin on electronic keyords. In equl temperment, ll intervls re defined s multiples of te sme sic intervl, or more precisely, te intervls re rtios wic re inteer powers of te smllest step rtio, so two notes seprted y te sme numer of steps lwys ve exctly te sme frequency rtio. However, except for doulin of frequencies (one or more octves), no oter intervls re exct rtios of smll inteers. c just intervl differs different mount from its nloous, eqully tempered intervl. 1
2 Semitones Solfèe Note Just Intontion Sclin ctor requency (Hz) qul Temperment Sclin ctor requency (Hz) 0 o 1/ # 16/1 2 Re 9/8 6/ Mi # /4 4/ 6 # 4/ So / / 9 L # / 10 9/ 11 Ti # 1/8 12 o 2/ Te expression for x in wrm-up exercise #1 is known s te ritmetic men (sometimes clled te vere) of nd. lculte te ritmetic men of te notes nd. 2. Te expression for x in wrm-up exercise #2 is known s te eometric men of nd. lculte te eometric men of te notes nd.. Te expression for x in wrm-up exercise # is known s te rmonic men of nd. lculte te rmonic men of te notes nd. Solfèe Note Solfèe Note Solfèe Note o o o Re Re Re Mi Mi Mi So So So L L L Ti Ti Ti o o o 2
3 4. or 2 numers <, Vlerie sys tt you cn compute te ritmetic men wit te formul +, ut 2 sley sys se is wron nd tt you ve to use te formul. Wo is correct? xplin. 2. Write n expression for te ritmetic men of tree numers,,, nd c. 6. ould you clculte te ritmetic men of numers y first verin two numers, nd ten verin tis result wit te tird numer? Sow wy or wy not. 7. To clculte te eometric men of n numers, multiply te numers toeter nd tke te nt root of te result. Write n expression for te eometric men of,, nd c. 8. ould you clculte te eometric men of numers y first findin te eometric men of two numers, nd ten findin te eometric men of tis result wit te tird numer? Sow wy or wy not. 9. One definition of te rmonic men is te reciprocl of te vere of te reciprocls. Use tis definition to write n expression for te rmonic men of nd. Once you ve trnslted te expression, rewrite it s simple frction. 10. Use te definition for rmonic men iven in 9 to write n expression for te rmonic men of,, nd c rewrite it s simple frction. 11. o te sme (clculte te rmonic men) for,, c, nd d.
4 12. n you mke conjecture out te rmonic men of numers,,, c, d, nd e? 1. ould you clculte te rmonic men of numers y first findin te rmonic men of two numers, nd ten findin te rmonic men of tis result wit te tird numer? Sow wy or wy not. 14. ind te ritmetic men, eometric men, nd rmonic men of nd. 1. Joe cn pint ouse in ours. Sm cn pint te sme ouse in ours. How lon will it tke tem to pint te ouse toeter? 16. Joe cn pint ouse in ours. Sm cn pint te sme ouse in ours. How lon will it tke tem to pint te ouse toeter? 17. How re questions 1 nd 16 relted to te rmonic men? 4
5 1. In te semicircle, sement of te dimeter s lent, nd sement of te dimeter s lent. ind te lent of te rdius m of te semicircle. 2. In trpezoid, se = nd se =. is te midpoint of, nd is te midpoint of. ind te lent m of sement. m m. In te semicircle, sement of te dimeter s lent, nd sement of te dimeter s lent. ind te lent of te rdius m of te semicircle in terms of nd. 4. In trpezoid, se = nd se =. is te midpoint of, nd is te midpoint of. ind te lent m of sement in terms of nd. m m
6 6. ind te lent of ltitude in rit trinle. 6. ind te lent of sement. 7. ind te lent of ltitude in rit trinle, in terms of nd. 8. ind te lent of sement, in terms of nd.
7 7 9. Trpezoid is similr to trpezoid. ind te lent of sement. 10.,, nd re tnent to te semicircle. ind te rdius of te semicircle. 11. Trpezoid is similr to trpezoid. ind te lent of sement, in terms of nd. 12.,, nd re tnent to te semicircle. ind te rdius of te semicircle, in terms of nd.
8 1. Isosceles trinle is similr to isosceles trinle. = nd =. ind te lent of. 14. ircle nd circle re externlly tnent to ec oter t point. Te dimeter of circle is, nd te dimeter of circle is. ind te lent of te common tnent. 1. Isosceles trinle is similr to isosceles trinle. = nd =. ind te lent of, in terms of nd. 16. ircle nd circle re externlly tnent to ec oter t point. Te dimeter of circle is, nd te dimeter of circle is. ind te lent of te common tnent, in terms of nd. 8
9 17. = nd =. isects nle. is perpendiculr to. ind te distnce etween points nd. 18. Te dionls of trpezoid intersect t, nd sement is prllel to te ses trou. = nd =. ind te lent of sement. 19. = nd =. isects nle. is perpendiculr to. ind te distnce etween points nd, in terms of nd. 20. Te dionls of trpezoid intersect t, nd sement is prllel to te ses trou. = nd =. ind te lent of sement, in terms of nd. 9
10 21. Pole is feet tll, nd pole is feet tll. Wires from te top of ec pole to te ottom of te oter pole cross t point. ind te eit of te point of intersection of te wires. 22. Pole is feet tll, nd pole is feet tll. Wires from te top of ec pole to te ottom of te oter pole cross t point. pole trou touces te wire connectin te top of te poles t. ind te eit of te pole. 2. Pole is feet tll, nd pole is feet tll. Wires from te top of ec pole to te ottom of te oter pole cross t point. ind te eit of te point of intersection of te wires, in terms of nd. 24. Pole is feet tll, nd pole is feet tll. Wires from te top of ec pole to te ottom of te oter pole cross t point. pole trou touces te wire connectin te top of te poles t. ind te eit of te pole, in terms of nd. 10
11 2. Squre is inscried in rit trinle. Side = nd =. ind te lent of te side of te squre. 26. Trinle s cute nles t nd. Squre H is inscried in trinle wit side H correspondin to side. Side = nd te ltitude to side s lent. ind te lent of te side of te squre. H 27. Squre is inscried in rit trinle. Side = nd =. ind te lent of te side of te squre, in terms of nd. 28. Trinle s cute nles t nd. Squre H is inscried in trinle wit side H correspondin to side. Side = nd te ltitude to side s lent. ind te lent of te side of te squre, in terms of nd. H 11
12 29. Te semicircle wit endpoints of te dimeter t nd is inscried in trpezoid s sown. se = nd se =. ind te lent of te perpendiculr sement drwn from te point of tnency. 0. In semicircle, te sement of te dimeter s lent, nd te sement of te dimeter s lent. is perpendiculr to te dimeter t, nd is rdius. is perpendiculr to te rdius. ind te lent of sement. 1. Te semicircle wit endpoints of te dimeter t nd is inscried in trpezoid s sown. se = nd se =. ind te lent of te perpendiculr sement drwn from te point of tnency, in terms of nd. 2. In semicircle, te sement of te dimeter s lent, nd te sement of te dimeter s lent. is perpendiculr to te dimeter t, nd is rdius. is perpendiculr to te rdius. ind te lent of sement, in terms of nd. 12
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