First Steps in Music Theory - 145%
September 22, 2022 | Author: Anonymous | Category: N/A
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Ijrst [tfps iu`
Etrsjk
Tmfary
Ijrst[tfps Eursjk
Tmfary Gr`dfs
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Kahtfhts
Ijrst pubcjsmfd jh 0::: by Tmf @ssakj`tfd Ba`rd ai tmf Ray`c [kmaacs ai Eusjk (^ubcjsmjhg) Cjejtfd, ` wmaccy awhfd subsjdj`ry ai @BR[E Rfprjhtfd jh 2>>>, 2>>4,2AA6,2>>:
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efkm`hjk`c, pmatakapyjhg, rfkardjhg, ar atmfrujsf, wjtmaut tmf prjar pfrejssjah ai tmf kapyrjgmt awhfr
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tmjs baal js `v`jc`bcf irae Tmf Brjtjsm Cjbr`ry
Kavfr dfsjgh by @d`e M`Z, Cahdah Tfxt dfsjgh `hd sfttjhg by Gfaiirfy W`dscfy Eusjk arjgjh`tjah by O`kl Tmaepsah ^rohtfd jh Fhgc`hd by E`rtjhs tmf ^rjhtfrs Ctd, Bfrwjkl upah Twffd
[fktjah
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Tjef Y`cufs? Kratkmfts, Ejhjes `hd [fejbrfvfs
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Tjef Y`cufs?
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`hd [fejqu`vfrs Pu`vfrs Cfctfr H`efs ai Hatfs
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Tjef [jgh`turfs? ic, Trjpcfts
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Tmf Graupjhg ai Hatfs
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Tmf Graupjhg ai Rfsts
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Cfdgfr Cjhfs
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Tmf [k`cf ai @ E`oar
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Tmf [k`cfs ai B Ic`t `hd
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Dupcfts
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Daubcf [m`rps `hd Ic`ts< Fhm`reahjk Fqujv`cfhts
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Tmf [k`cfs ai B E`oar `hd G [m`rp Ejhar
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Tmf [k`cfs ai D Ic`t E`oar `hd B IJ`t Ejhar
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Tmf Tfkmhjk`c H`efs ai Hatfs jh Dj`tahjk [k`cfs
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Jhtfrv`cs
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Trj`ds ah tmf Tahjk, [ubdaejh`ht `hd Daejh`ht
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Kmards ah tmf Tahjk, [ubdaejh`ht `hd Daejh`ht
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Tmf Kmrae`tjk [k`cf
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Arh`efht [jghs
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Tmf @cta Kcfi
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Gfhfr`c Pufstjahs
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0 Jrrfguc`r Tjef [jgh`turfs 2 crrfguc`r Tjef Djvjsjahs ; Tmf Tfhar Kcfi< Akt`vf Tr`hspasjtjahs 4 Tmf [k`cfs ai I [m`rp E`oar `hd D [m`rp Ejhar 6 Tmf [k`cfs ai G Ic`t E`oar `hd F Ic`t Ejhar 9 Dj`tahjk `hd Kmrae`tjk Jhtfrv`cs 7 Kaepauhd Jhtfrv`cs = Jhvfrsjahs ai Trj`ds : Jdfhtjiyjhg Kmards t> Tmf 22 ^ragrfssjah 00 Kmards `t K`dfhkf ^ajhts 02 Tr`hspasjtjah by Jhstruefhts jh B Ic`t, @ `hd I t; [@iB jh [mart `hd Apfh [karf 04 Arh`efhts 06 Gfhfr`c Pufstjahs
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^hfi`kf E`hy tf`kmfrs m`vf s`jd tm`t jt waucd bf usfiuc {ar studfhts ta cr`vf ` gr`df-by-gr`df prfsfht`tjah ai tmf b`sjk i`kts ai eusjk ccrfary, p`rtjkuc`rcy ahf jh ` paklft-sjzf iare`t. Ijrsi [ifps jh Eusjk Tmfary m`s bffh wrjttfh jh rfspahsf ta tmjs hffd. trjrsi [tfps jh Eusjk Tmfary js ef`ht `s ` suppart iar tmf atmfr @ssakj`tfd Ba`rd tmfary pubcjk`tjahs Tmf @B Gujdf ta Eusjk Tmfary `hd Gr`dfs 0 ta 6 ai Eusjk Tmfary jh ^r`ktjkf hat `s ` substjtutf iar tmfe. Jt df`cs sjepcy wjtm tmjhgs wmjkm rhust bf lhawh. E`ttfrs sukm `s kcfis, sk`cfs, lfy sjgh`turfs,
jrrtfrv`cs, tjef v`cufs, tjef sjgh`turfs, `hd tmf graupjhg ai hatfs `hd rfsts hffd ta bf lhawh `hd uhdfrstaad bfk`usf tmfy .rrf fssfhtj`c fcfefhts ai eusjk. Jt js, ai kaursf, iar tmjs rf`sah ccr`t tmf e`oarjty ai qufstjahs jh tmf @ssakj`tfd Ba`rd Gr`dfs 0 ca 6 tmfary fx`ejh`tjahs df`c wjtm tmfsf fcfefhts. Pufstjahs wmjkm rfqujrf grf`tfr eusjk`c krf`tjvjty `hd jhsjgmt, sukm `s r?aepasjhg `hswfrjhg rmytmes, rfkaghjsjhg pmr`sf p`ttfrhs
,rhd sfttjhg wards ta eusjk, `rf df`ct wjtm fxtfhsjvfcy `t tmf ,rppraprj`tf pc`kfs jh Eusjk Tmfary jh ^r`ktjkf `hd `rf hat kavfrfd mfrf. Jt js jepart`ht ta hatf maw tmjs baal kaepcfefhts batm Tmf @B Gujdf ta Eusjk Tmfary `hd Gr`dfs 0 ta 6 ai Eusjk Tmfary jh ^r`ktjkf. @ctmaugm Tmf @B Gujdf jhkcudfs `cc tmf jhiare`tjah
rfqujrfd iar tmf @ssakj`tfd Ba`rd's tmfary fx`ejh`tjahs, jt w`s
hat jhtfhdfd iar fx`ejh`tjah k`hdjd`tfs `cahf, `hd jt js hat bralfh up jhta gr`dfs. Eusjk Tmfary jh ^r`ktjkf, ah tmf atmfr m`hd, w`s spfkjijk`ccy dfsjghfd `s ` sfrjfs ai warlbaals cf`djhg ca tmf v`rjaus gr`dfs ai tmf tmfary fx`ejh`tjahs. Jt `ssuefs tm`t studfhts wjcc m`vf studjfd tmf rfcfv`ht sfktjahs aJ Tmf @B Gujdf, `hd js e`jhcy kahkfrhfd wjtm `dvjkf ah maw ta sft `baut
^rfiakf
Gr`df 0
`hswfrjhg tmf typfs ai qufstjahs sft jh tmf fx`ejh`tjahs, `hd wjtm pravjdjhg fxfrkjsfs iar pr`ktjkf. Ijrsi [tfps jh Eusjk Tmfary wjcc, J mapf, pravf ta bf jhv`cu`bcf ta eusjk studfhts ai `cc `gfs `hd `bjcjtjfs wmjcf iucijtcjhg jts prje`ry iuhktjah a{ pravjdjhg suppart iar tmf prfp`r`tjah ai
tmf ijrst ijvf gr`dfs ai eusjk tmfary fx`ejh`tjahs.
Frjk T`ycar
c'kapcf aitfh t`p tmfjr ifft'jh tjef' wmfh tmfy `rf cjstfhjhg ta ` crjkkf ai eusjk, fspfkj`ccy ji jt js ` e`rkm ar ` d`hkf. Tmfy `rf c,)ppjhg tmf stf`dy bf`ts ai tmf eusjk. Jh Gr`df 0, bf`ts `rf scrawh cjlf tmjs?O J ) ) ) ) O U Tmfsf sjghs `rf k`ccfd kratkmfts ar'qu`rtfr hatfs'. @ scjgmtcy djiifrfht sjgh, O, js usfd car ` sauhd wmjkm c`sts `s cahg `s iaa kratkmfts. Jt js k`ccfd ` rhjhje ar'm`ci hatf'. @ sauhd wmjkm c`sts `s cahg `s /aur r'ratkmfts js wrjttfh . ? tmjs js ` sfejbrfvf ar 'wmacf hatf'. Mfrf kratkmfts, ejhjes `hd sfejbrfvfs `rf cjhfd up tagftmfr?
rratkmfts(qu`rtfrhatfs)
O ) ) ) ) ) ) ) ) ) ) )
rhjhjes (m`ci hatfs) rkejbrfvfs (wmacf hatfs)
a
a
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@ sfejbrfvf c`sts `s cahg `s twa ejhjes ar iaur kratkmfts, ar wf ( ,ch s`y tm`t ` sfejbrfvf m`s tmf s`ef 'tjef v`cuf' `s twa rrrjhjes ar iaur kratkmfts. Tmfrfiarf ` ejhje m`s tmf s`ef cjrrrf v`cuf `s twa kratkmfts, but ahcy m`ci tmf tjef v`cuf ai ` ,
[fktjah 7>
7
c0
Ji yau pc`y `cc tmf wmjtf hatfs irae K ta tmf hfxt K ta tmf
rjgmt, ahf `itfr tmf atmfr, yau wjcc m`vf pc`yfd ` sk`cf?
t2;4697= Tmfh wf kmfkl tmf jhtfrv`cs. @cc tmf jhtfrv`cs `rf karrfkt,
Tmjs p`rtjkuc`r sk`cf js tmf sk`cf ai K e`oar? 'K' bfk`usf tm`t js tmf hatf jt bfgjhs `hd fhds ah, `hd 'e`oar' bfk`usf ai tmf w`y jts sfejtahfs `hd tahfs `rf `rr`hgfd' Mfrf js tmf s`ef sk`cf (K -c? e`oar) wjtm tmf sfejtahfs e`rlfd r
jhkcudjhg tmf sfejtahf bftv
Gr`df 2
[fktjah 2
Ji tmfrf js ` tjef sjgh`turf ai it, tmf eusjk wjcc bf ifct ta m`vf iaur bf`ts jh ` b`r, `s e`rlfd t mfrf?
e`y bf `ddfd. Tmjs js hat rf`ccy hfkfss`ry ji tmf hatfs bf`efd tagftmfr, but jt js usu`c jh `cc atmfr kjrkuest`hkfs?
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But ji tmfrf js ` tjef sjgh`turf ai ij jt wjcc bf ifct ta m`vf twa bf`ts jh ` b`r?
tj Eusjk wjtm twa bf`ts jh ` b`r js s`jd ta bf jh dupcf tjef. Eusjk wjtm tmrff bf`ts jh ` b`r js jh trjpcf tjef, `hd eusjk wjtm iaur bf`ts js jh qu`drupcf tjef. Tmfrfiarf jh Gr`df 2?
. . .
tmf dupcf tjef sjgh`turfs
`rfi`hdi
@ trjpcft js ` graup ai tmrff hatfs pfriarefd jh tmf tjef ai twa. Iar fx`epcf? ;
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pfriarefd jh tmf tjef
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Tmfsf `rf fx`epcfs ai ejhjes `hd sfejqu`vfrs iarejhg trjpcft graups?
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tmf trjpcf tjef sjgh`turfs `rf $, i `hd i, < tmf qu`drupcf tjef sjgh`turfs `rf it`hd $.
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jh [fktjah ;.
Tmf sjgh iar ` trjpcft js tmf ijgurf B wrjttfh fjtmfr `bavf ar bfcaw tmf ejddcf hatf. @ kurvfd cjhf (cjlf ` scur) ar ` squ`rf br`klft
Jt k`h `csa jhkcudf hatfs wjtm djiifrfht v`cufs pravjdfd tmfy `dd up ta tmf fqujv`cfht ai tmrff fqu`c hatfs?
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Gradf 2
kx`epcfs tmf hatfs `rf bf`efd jh bf`ts? twa ejhje bf`ts jh i , `hd iaur kratkmft bf`ts jh i. Rfsts e`y bf jhkcudfd `eahgst graups ai hatfs wmjkm `rf bf`efd tagftmfr?
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Tmfrf `rf ahcy ` ifw pajhts ta `dd ta wm`t w`s s`jd jh Gr`df
ch c, 0
rfst?0
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`hd $, ` kaepcftfcy sjcfht b`r js smawh by ` sfejbrfvf
`baut `vajdjhg tjfs `hd `baut bf`ejhg hatfs?
0 @vajd tjfs wmfrf passjbcf Jh
i `hd it, wrctf
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haru
Wmfh b`rs kaht`jh batm hatfs `hd rfsts, jt js ` gfhfr`c rucf tm`t ` hfw rfst smaucd bf usfd `t tmf st`rt ai fvfrv sjcfht bf`t?
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wjtm kratkmft bf`ts?
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Bf`e tagftmfr
jh $, sjx sfejqu`vfrs ijctjhg ` kaepcftf b`r ai $,
(b) jh ir ,ij`hd it, fjgmt sfejqu`vfrs wmjkm kaucd bf rfpc`kfd by
` erhce?
*r-tiijt Kaep`rf tmf c`st fx`epcf wjtm?
it) c J O 0," 2r J O cO r J 1,)r c cc 1) c Tmf s`ef
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js truf ai
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hatit) O c'' hat iX O cO -
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acc tjef sjgh`turfs fxkfpt i , wmjkm rfqujrfs ` brfvf
rfst. Mawfvfr, yau wjcc hat bf `slfd `baut jmjs jh Gr`df 2 (Pufstjahs jhvacvjhg tmf usf ai brfvf hatfs `hd rfsts `rf hat sft uhtjc Gr`df 4.)
Tmf fxpc`h`tjah iar tmf djiifrfht bf`ejhg js tm`t jh batm
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Gradf 2
wjtm qu`ufr bf`ts?
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rrjtm ejhje bf`ts?
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wjtm ejhje bf`ts?
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But atmfrwjsf tmfrf smaucd bf `s ifw rfsts `s passjbcf? wjtm kratkmft bf`ts?
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Tmfrf js `h jepart`ht fxkfptjah ta tmf rucf tm`t '` hfw rfst smaucd bf usfd `t tmf st`rt ai fvfry sjcfht bf`t'. Wmfh tmfrf `rf iaur bf`ts jh ` b`r, ` twa-bf`t rfst smaucd bf usfd iar tmf ijrst twa bf`ts, ar iar tmf c`st twa, but hat jh tmf ejddcf?
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Hatjkf tm`t twa-bf`t rfsis jh tmf ejddcf ai ` iaur-bf`t b`r `rf trf`tfd djiifrfhtcy irae twa-bf`t hatfs?
t) J t O J bij it) O J ir) ) ) J bh t) V O J
wjtm ejhje bf`ts?
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wjtm kratkmft bfats?
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Wmfh ` sjcfhkf c`sts iar /fss tm`h ` iucc bf`t, f`km subdjujsjah ai tmf bf`t eust st`rt wjtm ` hfw rfst. Mfrf, 'subdjvjsjah' ai tmf bf`t ef`hs f`km m`ci bf`t ar fvfh f`km qu`rjfr bf`t. Bfcaw `rf fx`epcfs ai m`ci-bf`t rfsts. Ta e`lf tmfe kcf`rfr, tmf iucc bf`ts `rf e`rlfd
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(Hatjkf tm`t jt js hai st`hd`rd pr`ktjkf ta wrjtf tmf b`r `s
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smawh by `s ifw rfsts `s passjbcf, but tmfy stjcc iaccaw tmf rucfs fxpc`jhfd `bavf. Tmf bfst w`y ai uhdfrst`hdjhg tmjs js ta sff maw tmf bf`ts `rf `rr`hgfd jh ` b`r wmjkm kaht`jhs ahcy ` sjhgcf
hatf, fjtmfr `t tmf fhd ar `t tmf bfgjhhjhg ai tmf b`r. Iar
J
kvfh tmaugm yau e`y saeftjefs sff tmjs, `hd
jt js kfrt`jhcy
sjepcfr.) Ji ` b`r ai
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tc `t
@ sjcfhkf wmjkm c`sts iar earf tm`h ` huebfr ai iucc bf`ts js
fx`epcf, jh ` b`r ai
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[fktjah 4
Gradf 2
fhdjhg wjtm ` sfejqu`vfr?
it st`rts wjtm ` sfejqu`vfr, tmf s`ef rfsts `rf usfd ta kaepcftf tmf b`r, but tmfy m`vf ta bf `rr`hgfd jh rfvfrsf
ardfr?
.
tmf sfejqu`vfr rfst kaepcftfs tmf ijrst (qu`vfr) subdjvjsjah ai tmf ijrst bf`t< . tmf qu`vfr rfst pravjdfs tmf sfkahd (qu`vfr) subdjvjsjah ai tmf ijrst bf`t< . tmf kratkmft rfst kaepcftfs tmf ijrst m`ci ai tmf b`r< . tmf ejhje rfst pravjdfs tmf sfkahd m`ci ai tmf b`r. Tmf rfsts `rf `rr`hgfd tmf s`ef w`y jh i , fxkfpt tm`t tmf ejhje rfsts bfkaef kratkmft rfsts?
`hd it , ai kaursf, tmf bf`t js ` ejhje. Tmf subdjvjsjahs ai tmf ejhje bf`t `t tmf fhd,/bfgjhhjhg ai ` b`r `rf smawh by tmf br`klfts `bavf tmfsf fx`epcfs, Jh
tmf rfsts `rf `rr`hgfd
ijrst,
` ejhje rfst (iar tmf ijrst twa kratkmft bf`ts)<
tmjrd,
` qu`vfr rfst (iar tmf ijrst m`cj ai tmf iaurtm kratkmft bf`t)<
c`st,
` sfejqu`vfr rfst (ta e`lf up tmf sfkahd m`ci ai tmf
J
sfkahd, ` kratkmft rfst (iar tmf tmjrd kratkmft bf`t)<
bf`t).
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tmf hatfs `hd rfsts waucd bf fx`ktcy tmf s`ef `s jh tmf
i, fx`epcfs.
;=
Gradf 2
[fktjah 7
;:
Zau e`y bf `slfd ta h`ef ar wrjtf hatfs hffdjhg up ta tr,r"ra cfdgfr cjhfs `bavf ar bfcaw tmf st`vf. Bfcaw js tmf kaepcftf r`hgf ai tmf Gr`df 2 hatfs. Tmf hfw hatfs `rf wrjttfh `s bc`kl hatf-mf`ds?
Tmfsf tmrff sm`rps iare tmf lfy sjgh`turf a{ @ e`oar?
BKDF IG@BKDF IG@BKDF IG@BKDF IG@BKD
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Wmfh tmf sk`cf js wrjttfh wjtm jts lfy sjgh`turf, tmf sm`rps `rf, ai kaursf. hat hffdfd bfiarf tmf jhdjvjdu`c hatfs? @s yau k`h sff, saef ai tmfsf hatfs k`h bf wrjttfh fjtmfr jh tmf trfbcf kcfi ar jh tmf b`ss kcfi. Tmfsf twa p`ss`gfs sauhd fx`ktcy tmf s`ef?
@cc tmf sk`cfs wf m`vf m`d sa i`r st`rtfd ah ` wmjtf hatf ai tmf batm st`rt Bb `hd Fb e`oar lfyba`rd. But tmfsf twa sk`cfs ah ` bc`kl hatf. Tmjs js tmf sk`cf ai Bb e`oar? Rfefebfr tm`t jh fvfry e`oar sk`cf tmfrf js ` sfejtahf bftwffh tmf ;rd `hd 4tm dfgrffs, `hd `hatmfr bftwffh tmf 7tm `hd [tm (?0st) dfgrffs. @cc tmf atmfr jhtfrv`cs `rf tahfs. [a ji wf st`rt ` e`oar sk`cf ah @, tmf hatfs K, I `hd G `cc m`vf ta bfkaef K{, I# `hd Gi?
4>
Gr`df 2
4t
[fktjah =
Hatjkf tm`t jt hffds `h Ft `s wfcc `s tmf Bb ta lffp tmf sfejtahfs jh tmf rjgmt pc`kfs. Tmfsf twa ic`ts iare tmf lfy sjgh`turf ai Bb, e`Uar?
J
cfrf js tmf dfskfhdjhg sk`cf ai Fb e`oar wrjttfh wjtm jts lfy
sjgh`turf?
Mfrf js tmf sk`cf ai Bb e`oar wrjttfh wjtm jts lfy sjgh`turf?
Tmf lfy sjgh`turfs ai tmf e`oar sk`cfs jh Gr`df 2 kaht`jh up ta tmrff sm`rps ar up ta tmrff ic`ts. Lfy sjgh`turfs hfvfr ejx sm`rps `hd ic`ts' tmfy kaht`jh fjimfr sm`rps ar ic`ts. Fvfry hfw lfy
# # # # #
sjgh`turf `dds ahf earf sm`rp (ar ahf earf ic`t)? [marps? ? G e`oar 0 sm`rp (I+)
Tmf sk`cf ai FI e`oar hffds `h @b `s wfcc `s B `hd F\?
2 sm`rps (I# Kij)
5 D e`oar
; sm`rps (Iij K# G#) 5 @ e`oar
#+
Ic`ts? 0 ic`t
Tmf tmrff ic`ts iare tmf lfy sjgh`turf ai Fb e`oar?
;
(Br)
5 I e`oar
ic`ts
(Bt Ft)
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Bb e`oar
ic`ts
(Bb
Ft
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Fb e`oar
w
F
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Gradf 2
[fktjah :
Tmf sm`rps ar ic`ts jh ` lfy sjgh`turf `rf `cw`ys wrjttfh jh tmf ardfr smawh `bavf. Jh tmf lfy ai @ e`oar, {ar jhst`hkf, tmf Ii js wrjttfh ijrst, tmfh tmf Ki, `hd c`stcy tmf Gjc. Hatjkf, taa, wmfrf tmf sm`rps ar ic`ts `rf pc`kfd ah tmf st`vf. @hy atmfr `rr`hgf-
Hatjkf tm`t? . tmfrf js a/u-rays ` sfejtahf bftwffh tmf 2hd `hd ;rd dfgrffs
42
.
efht waucd bf jhkarrfkt. (Mfrf `rf saef WRAHG w`ys ai
-
.
but tmfrf `rf pcfhty earf )
.
wrjtjhg tmfsf lfy sjgh`turfs
iij,,, +'t,, #r,%,,0
-
@s wfcc `s tmf e`oar sk`cf tmfrf js `hatmfr typf ai sk`cf wjtm sfvfh hatfs? tmf ejhar sk`cf. But uhcjlf tmf e`oar sk`cf, tmf ejhar sk`cf fxjsts jh twa djiifrfht iares. Tmfsf `rf tmf m`reahjk ejhar sk`cf `hd tmf efcadjk ejhar sk`cf. Ta jccustr`tf tmf sjejc`rjtjfs `hd djiifrfhkfs bftwffh m`reahjk
`hd efcadjk ejhar sk`cfs, mfrf js tmf sk`cf ai @ ejhar jh batm iares. Tmf sfejtahfs `rf e`rlfd r --r , `hd tmf dfgrffs a{ tmf sk`cf `rf smawh uhdfrhf`tm.
`skfhdjhq " @ m`reahck echar
- rr;4691t
+F
- rr;46910 --
dfskfhdjhq
- 07964;r' @ efcadjk ejhar
- 07964;rt
ai tmf sk`cf< tmf m`reahjk ejhar sk`cf (cjlf tmf e`oar sk`cf) js tmf s`ef batm `skfhdjhg `hd dfskfhdjhg< tmf s`ef kaejhg dawh `s sk`cf efcadjk tmf ejhar js hat gajhg up. Jh tmf `skfhdjhg sk`cf tmf 9tm `hd 7tm dfgrffs `rf r`jsfd by ` sm`rp (Ii `hd G#), but jh tmf dfskfhdjhg sk`cf tmf 7tm `hd 9tm dfgrffs `rf uh`ctfrfd (tmf hatfs `rf G h`tur`c `hd I h`tur`c)<
tmf m`reahjk ejhar sk`cf jhkcudfs tmrff jhtfrv`cs ai ` sfejtahf, `hd jt `csa m`s `h fxtr`-wjdf jhtfrv`c (fqu`c ta tmrff sfejtahfs) bftwffh tmf 9tm `hd 7tm dfgrffs'
Tmf gaad hfws js tm`t jh Gr`df 2 yau wjcc hat bf fxpfktfd ta lhaw `baut baim iares ai tmf ejhar sk`cfs. Zau e`y kmaasf fjtmfr m`reahjk wmjkm iare yau w`ht ta `hswfr qufstjahs ah ar efcadjk. But yau eust lhaw wmjkm iare yau `rf usjhg< `hd yau eust stjkl ja yaur kmasfh iare, hat kmap `hd km`hgf bftwffh tmf twa. Tmf ejhar sk`cfs iar tmjs Gr`df `rf sft aut jh iutc jh tmf hfxt twa [fktjahs. Tmf m`reahjk sk`cfs `rf jh [fktjah
0> `hd tmf efcadjk sk`cfs jh [fktjah 00, sa yau wjcc hffd ta
study ahcy ahf ai tmfsf sfktjahs.
Jh batm sfktjahs tmf sk`cfs `rf prjhtfd wjtmaut `hd wjtm tmfjr lfy sjgh`turfs fxkfpt {ar tmf sk`cf ai @ ejhar, wmjkm m`s ha lfy sjgh`turf (cjlf tmf sk`cf ai K e`oar). Tmf @ ejhar sk`cfs `rf gjvfh ahcy jh tmf b`ss kcfi sjhkf wf m`vf `crf`dy m`d tmfe jh tmf trfbcf kcfi.
44
iij%
wjtmaut lfy sjgh`turfs?
`skfhdjha " @ m`reahjk ejhar
iijiij
wjtmaut lfy sjgh`turfs?
F m`reahjk ejhar
46
[fktjah 77
Gradf 2
dfskfhdjha
D m`reahjk ejhar
wjtm lfy sjgh`turfs?
iij%
iijiij wjtm lfy sjgh`turfs?
`skfhdjhg @ efcadjk ejhar
##
dfskfhdjhg
49
iijiij
wjtmaut lfy sjgh`turfs?
[fktjah 72
Gr`df 2 F efcadjk ejhar
Tmf wards 'm`reahjk' `hd 'efcadjk' dfskrjbf ejhar skacfs, hat ejhar lfgs. Tmjs js bfk`usf ` pjfkf ai eusjk jh ` ejhar lfy e`y usf hatfs t`lfh irae batm iares ai tmf ejhar sk`cf. Tmfrfiarf ` pjfkf js sjepcy'jh F ejhar'ar'jh D ejhar'ftk., hat'jh F m`reahjk ejhar' ar 'D efcadjk ejhar' ftk. Zau k`h sff jh tmf ejhar sk`cfs prjhtfd jh [fktjahs 0> `hd 00
iij%
`bavf tm`t?
.
iijiij wjtm lfy sjgh`turfs?
.
tmf lfy ai F ejhar, cjlf tmf lfy ai G e`oar, m`s ` lfy sjgh`turf ai ahf sm`rp? I{<
tmf lfy ai D ejhar, cjlf tmf lfy ai I e`oar, m`s ` lfy sjgh`turf ai ahf ic`t? Bb.
D efcadjk ejhar
Wmfh ` ejhar lfy m`s tmf s`ef lfy sjgh`turf `s ` e`oar lfy jt js tmf 'rfc`tjvf ejhar' ai tmf e`oar lfy. [jejc`rcy, ` e`oar lfy js tmf 'rfc`tjvf e`oar' ai tmf ejhar lfy wjtm tmf s`ef lfy sjgh`turf. Tmf s`ef js truf ai tmf lfys ai @ ejhar `hd K e`oar, wmjkm m`vf ha lfy sjgh`turfs. Tmfrfiarf?
.
wjtm lfy sjgh`turfs?
tmf lfy sjgh`turfs ai ejhar lfys da hat kaht`jh `cc tmf sm`rps ar ic`ts wmjkm e`y bf hffdfd< tmf lfy ai @ ejhar, cjlf tmf lfy ai K e`oar, m`s ha lfy sjgh`turf<
?-
wjtmaut lfy sjgh`turfs?
47
tmf lfy ai @ ejhar js tmf rfc`tjvf ejhar ai tmf lfy ai K e`oar, `hd tmf lfy ai K e`oar js tmf rfc`tjvf e`oar ai tmf lfy ai @ ejhar< tmf lfy ai F ejhar js tmf rfc`tjvf ejhar ai tmf lfy ai G e`oar, `hd tmf lfy ai G e`oar js tmf rfc`tjvf e`oar ai tmf lfy ai F ejhar< tmf lfy ai D ejhar js tmf rfc`tjvf ejhar ai tmf lfy ai I e`oar, `hd tmf lfy ai I e`oar js tmf rfc`tjvf e`oar ai tmf lfy ai D ejhar.
4:
Gradf 2
[fktjah 74
@ctmaugm ` e`oar lfy `hd ` ejhar lfy e`y m`vf tmf s`ef lfy sjgh`turf, tmfy `rf djt'cfrfht lfys. Tmfy m`vf djiifrfht lfyhatfs (tahjk hatfs) `hd djiifrfht tahjk trj`ds. Kaep`rf tmf tahjk hatfs (jh br`klfts) `hd tahjk trj`ds ai tmfsf lfys?
ct e`lfs ha djiifrfhkf ta cmf huebfr ai tmf jhtfrv`c tm`t K js K# jh tmf ijrst fx`epcf but Ki jh tmf sfkahd fx`epcf. Rfefebfr tm`t ji tmf twa hatfs iarejhg `h jhtfrv`c `rf pc`yfd tagftmfr (`s `bavf) tmf jhtfrv`c js'm`reahjk'. Ji tmfy `rf pc`yfd ahf `itfr tmf atmfr, tmfy e`lf ` 'efcadjk' jhtfrv`c, `hd jt dafs hat e`ttfr wmjkm ai tmf tvua js pc`yfd ijrst? batm ai tmf jhtfrv`cs e`rlfd r mfrf `rf ` 6tm?
4`
I -9--
uhd
@ ejhar
Ge`oar
iij QO-
uhd
F ejhar
Ie`oar
iij
""d
ke`oar
@4
5
D ejhar
Zau wjcc sff jh c`tfr Gr`dfs tm`t fvfry e`oar lfy m`s ` rfc`-
tjvf ejhar lfy `hd, sjejc`rcy, fvfry ejhar lfy m`s ` rfc`tjvf e`oar lfy.
Bfcaw js ` cjst ai Jt`cj`h wards `hd tmfjr ef`hjhgs wmjkm, jh `ddjtjah ta tmasf jh Gr`df 0, yau smaucd lhaw jh Gr`df 2. (ct js tmf s`ef `s tmf cjst prjhtfd jh Eusjk Tmfary jh ^r`ktjkf, Gr`df 2.) `
Tmf qufstjahs ah jhtfrv`cs jh Gr`df 2 wjcc jhkcudf qufstjahs usjhg `hd ejhar. @s jh tmf hfw sk`cfs jh tmjs Gr`df - batm e`oar bf tmf lfyhatf hatf ai tmf jhtfrv`c `cw`ys 0, tmf wjcc Gr`df cawfr ai tmf sk`cf, `hd yau wjcc m`vf ta lhaw ahcy tmf huebfr ai tmf jhtfrv`c. Tmf huebfr ai `h jhtfrv`c js tmf huebfr ai cfttfr h`efs wmjkm jt kaht`jhs, `hd tmjs st`ys tmf s`ef wmftmfr tmf sk`cf js e`oar ar ejhar. Iar fx`epcf, batm ai tmfsf `rf jhtfrv`cs ai ` ;rd, sjhkf tmfy batm jhkcudf tmrff cfttfr h`efs?
iijI @
(B) K#
`c, `cc` `cc`rg`hda `hd`htjha assac
kah, kac dackf f, fd
`t, ta, by, iar, jh, jh tmf stycf ai ta tmf, jh tmf e`hhfr ai (`cc` e`rkj`? jh tmf stycf ai ` e`rkm) bra`dfhjhg (gfttjhg ` cjttcf scawfr `hd prab`bcy ` cjttcf caudfr) scjgmtcy i`stfr tm`h `hd`htf (but e`y `csa ef`h scjgmtcy scawfr) vfry Nccfgra assaj? vfry qujkl) wjtm swfft, sait `hd
fsprfssrua
@
(B) Km
(ar fsprfss. ar fspr.)
ip (? iartfpj`ha) gjakasa
fxprfsscvf
caud, tmfh jeefdj`tfcy sait pc`yiuc, efrry
e`fstasa
vfry scaw, sacfeh gr`kfiuc r`tmfr scaw (but hat `s scaw `s carga) scaw, st`tfcy but e`ofstjk
efha
cfss
eassa, eata
eavfefht (efha eassa? scawfr< kah eaia? wjtm eavfefht)
hah
hat
gr`uf gr`ztasa
c`rgmftta c`rga e`
vfry, eukm
eacta
pjj
earf i`st (i`stfr tm`h accfgra)
prfsta
wjtmaut
sfhz` st',
siz (?
scarz`hda
ar siarz`ta) sjejcf (ar sje.) sastfhuta tfhuta trappa uju`kf, ujua
iarkfd, `kkfhtfd
jh tmf s`ef w`Z sust`jhfd
mfcd taa eukm (hah tra^^a? hat taa eukm) cjvfcy, qujkl
Zau smaucd `csa lhaw tmfsf sjghs?
JOJ `t`
)
Tmf sjgh 8 avfr ar uhdfr ` hatf ef`hs tm`t jt js ta bf `kkfhtfd. ,r (avfr) `hd h (uhdfr) ef`h tmf s`ef, ar fvfh strahgfr `kkfhts.
ttt ))) -/
[fktjah 74
Gradf 2
6>
Dats jhsjdf ` scur ef`h tm`t tmf hatfs smaucd bf scjgmtcy sfp`r`tfd (sfejst`kk`ta), but cfss sa tm`h hatfs wjtm
ardjh`ry st`kk`ta dats.
JO `t
60
@ wfdgf sjgh ' jhdjk`tfs ` supfrst`kk`ta (st`kk`tjssjea)? tmf hatf js ta bf pc`yfd `s brjficy `s passjbcf `hd pfrm`ps `kkfhtfd `s wfcc.
j,
Tmf sjgh
JO
gjvfh ` scjgmt prfssurf (`hd gfhfr`ccy scjgmtty sfp`r`tfd).
-
ef`hs tm`t tmf hatf js ta bf
Gr`df ;
Eusjk js jh sjepcf tjef wmfh tmf bf`ts djvjdf jhta twa fqu`c hatfs. Tmf tjef sjgh`turfs usfd jh Gr`dfs 7 `hd 2 wfrf `cc fx`epcfs ai sjepcf tjef sjgh`turfs?
jhi ,i`hd ir tmf )bf`t
djvjdfs
jhta
jhij ,ij`hd $ tmf ) bf`t djvjdfs jhta jh $ tmf
)
) ) ,Oc, O )
,.,1,
bf`t djvjdfs jhta .H .H
Gr`df ; jhtradukfs tjef sjgh`turfs ai ` djiifrfht ljhd? kaepauhd tjef. Jh kaepauhd tjef tmf bf`ts djvjdf jhta tmrff fqu`c hatfs, `hd tmf bf`ts `rf wrjttfh `s dattfd hatfs. Tmf kaepauhd tjef sjgh`turfs jh Gr`df ; `cc usf ` dattfd kratkmft (dattfd qu`rtfr hatf) `s tmf sjgh iar ` sjhgcf bf`t. Mfrf, iar jhst`hkf, tmfrf `rf twa dattfd kratkmft bf`ts jh ` b`r?
Tmf tjef sjgh`turf iar twa dattfd kratkmft bf`ts jh ` b`r js $ , ef`hjhg tm`t tmfrf `rf sjx qu`vfrs jh ` b`r. Tmjs k`h bf kahiusjhg, bfk`usf tmfrf `rf `csa sjx qu`vfrs jh ` b`r ai H < but jh i tmf sjx qu`vfrs waucd bf tmf fqujv`cfhc aJ tmrff kratkmft
bf`ts. Kaep`rf
=
O-i?
O70c
). )
7rDO0 O0c)
| wjtm
))
J
Gr`df ;
[fktjah 2
Wf s`w jh Gr`df 2 tm`t jt js passjbcf ta djvjdf ` sjepcf tjef bf`t jhta tmrff fqu`c hatfs by wrjtjhg tmf tmrff hatfs `s ` trjpcft. [a tmfsf twa rmytmes, iar fx`epcf, sauhd fx`ktcy tmf s`ef, fvfh tmaugm tmfy `rf wrjttfh djiifrfhtcy?
[jepcf trjpcf tjef
Trih
Kaepauhd trjpcf tjef
ihc) OJ )c
Hfvfrtmfcfss, ` trjpcft js `h fxkfptjah ta tmf gfhfr`c rucf tm`t sjepcf tjef bf`ts djvjdf jhta twa, hat tmrff. Tmf kaepauhd tjef sjgh`turfs jh Gr`df ; `rf $ , $ `hd i. $ m`s twa (dattfd kratkmft) bf`ts jh ` b`r, sa jt js jh kaepauhd dupcf tjef. $ m`s tmrff (dattfd kratkmft) bf`ts jh ` b`r, sa jt js jh kaepauhd trjpcf tjef. i m`s iaur (dattfd kratkmft) bf`ts jh ` b`r, sa jt js jh kaepauhd qu`drupcf tjef. Mfrf yau k`h kaep`rf tmfsf kaepauhd tjef sjgh`turfs wjtm tmf sjepcf dupcf, trjpcf `hd qu`drupcf tjef sjgh`turfs wmjkm yau studjfd jh Gr`df 0?
;O )
i,
jO
g = ).
CCi
co
).
)
`O |
).
JJ
Vj `cj
66
{th'"" kratkmft
bf`ts jh ` b`r)
(tmrff dattfd kratkmft bf`ts jh ` b`r)
[jepcf qu`drupcf tjef
Cc cj `-i `o J )
4) h
)
O
i
(iaurkratkmftbf`tsjh uuu.r
Kaepauhd qu`drupcf tjef
^ ). =
).
).
)
| J CCi Vo `-cO `?o
(iaur dattfd kratkmft bf`ts jh ` b`r)
[jepcf dupcf tjef
4o
'-
|
{,*a kratkmft bf`ts jh ` b`r)
Kaepauhd dupcf tjef
g
c-i-0,
c-
|
t*a
dattfd kratkmft bf`ts jh ` b`r)
Twa iucc bf`ts jh kaepauhd tjef `rf wrjttfh O. hat tc TmfrfJarfO. smautd hat bf rfpc`kfd by U. .U. jh uhy ai tmfsf b`rs?
=O J FO OJ FOO
J '&) ) J 'gO ) )c
Gr`df ;
[fktjah ;
67
Jh $ `hd
i
` hatf c`stjhg tmrff iucc bf`ts js wrjttfh tmus?
FO. ) |
hat
c,cttr i
) )
c.,a' FO
? [jepcf dupcf
tjef
ic J
Kaepauhd dupcf tjef
gr
0 t tr.Hc.HZv, t '/ 0ty.Hc)ytv v c' Mfrf tmf apfhjhg iaur-b`r rmytme (e`rlfd
eusjk e`y bfgjh wjtm ` hatf ar ` graup ai hatfs bfiarf tmf ijrst b`r cjhf?
@ pjfkf ai
[kmue`hh, 'Tmf Efrry ^f`s`ht', iuae @cbue Oar tmf Zauhg
"ck Tr`djtjah`c? 'Tmf Ijrst Hawfcc'
0
@0
)
js jeefdj`tfcy rfpf`tfd (@2). Tmfh tmfrf js ` djiifrfht iaur-b`r rmytme (B), `ctmaugm jt taa st`rts wjtm `h upbf`t. Ijh`ccy 40 js rfpf`tfd `g`jh (@;).0 Hatjkf tm`t @,0 dafs hat ijhjsm `t tmf fhd aJ b`r 4? jt ijhjsmfs wjtm tmf qu`vfr rfst. [jejc`rcy, @2 ijhjsmfs wjtm tmf qu`vfr rfst jh b`r =, `hd @; wjtm tmf qu`vfr rfst jh tmf c`st b`r. @s ` rfsuct, tmf c`st b`r js hat ` iucc b`r (` iucc b`r waucd m`vf ` kratkmft rfst). Tmf upbf`t qu`vfr `t tmf bfgjhhjhg e`lfs up iar tmf 'ejssjhg' qu`vfr `t tmf fhd. Tmjs fxpc`jhs saeftmjhg wmjkm yau wj\U aitfh ijhd jh pjfkfs wmjkm st`rt wjtm `h upbf`t? tmf v`cuf ai tmf hatf ar hatfs bfiarf tmf b`r cjhf `t tmf bfgjhhjhg js dfduktfd irae tmf c`st b`r. Jh @; m`s bfkaef bfttfr lhawh jh ` v`rjfd iare
Gradf ;
[fktjah 9
atmfr wards, tmf tjef v`cufs ai tmf upbf`t apfhjhg pcus tmf tjef v`cufs ai tmf jhkaepcftf b`r `t tmf fhd '`dd up' ta ` iucc b`r. Tmfrfiarf?
Ji ` sfktjah ai eusjk bfgjhhjhg wjtm `h upbf`t js rfpf`tfd, tmf rfpf`t k`h bf smawh by tmf usu`c rfpf`t sjgh (sff Gr`df 0, p. 2=).'Tmf Brjtjsm Grfh`djfrs', quatfd `bavf, js `h fx`epcf. @2 js fx`ktcy tmf s`ef `s @0, sa jt dafs hat m`uf ta bf wrjttfh aut `g`jh? jt kaucd bf smawh by ` rfpf`t e`rl jhstf`d. Mawfvfr, tmf rfpf`t e`rl eust bf jh fx`ktcy tmf rjgmt pc`kf, wmjkm js `itfr tmf qu`vfr rfst jh b`r 4. Hatf tm`t jt js iaccawfd by ` daubcf b`r cjhf, `hd tm`t tmf p`ss`gf e`rlfd r"-r '`dds up' ta ` iucc b`r?
94
ic
`pjfkfbfgjhhjhg
`pjfkfbfgjhhjhg
J kaucd".d J O
cc
O ckaucdf"dc) c
`pjfkfbfgjhhjhgi `pjfkfbfgjhhjhg
O
$
it
)
|
)
ckaucd".d
kaucd
".> | OQ,QQO
cO.
cc ftk.
cc
Ai kaursf, `h jhkaepcftf b`r `t tmf fhd k`h jhkcudf earf tm`h ahf hatf, ar ` ejxturf ai hatfs `hd rfsts, pravjdfd tmfjr tat`c v`cuf js karrfkt. But tmfy eust bf graupfd jh tmf usu`c w`y up ta tmf pajht wmfrf tmf b`r staps smart. Wm`t tmjs ef`hs jh pr`ktjkf js tm`t tmfy eust bf graupfd `s tmaugm tmfy wfrf iaccawfd by tmf upbf`t hatfs `t tmf bfgjhhjhg. Iar jhst`hkf?
t t cc uuthat cO (.a-pu,"cO J t O cc hat cO -
b_.
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