{"id":35333,"date":"2026-05-29T11:56:35","date_gmt":"2026-05-29T11:56:35","guid":{"rendered":"https:\/\/www.lhi.is\/?post_type=threads&p=35333"},"modified":"2026-05-29T11:56:36","modified_gmt":"2026-05-29T11:56:36","slug":"oraedar-settings-beyond-the-attund","status":"publish","type":"threads","link":"https:\/\/www.lhi.is\/en\/thraedir\/thraedir-tolublad-11\/oraedar-stillingar-handan-attundar\/","title":{"rendered":"Irrational settings beyond the octave"},"content":{"rendered":"\n
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Phi-sp\u00edrall, upplausnir og \u00f3j\u00f6fn t\u00f3nbil \u00ed Fellingum t\u00edmans <\/h1>\n\t\t\t\t\t<\/div>\n\t
\n\t\t\t\n\t\t\t\t\t\t\t\t\t\t\"\"\n\t\t\t\t\n\t\t\t<\/figure>\n\t<\/section>\n\n\n\n
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\u00d3r\u00e6\u00f0ar stillingar handan \u00e1ttundar: Phi-sp\u00edrall, upplausnir og \u00f3j\u00f6fn t\u00f3nbil \u00ed Fellingum t\u00edmans<\/em><\/strong><\/p>\n

Einar Torfi Einarsson<\/p>\n

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 <\/p>\n

\u00cd \u00feessari grein er fjalla\u00f0 um \u00fe\u00e6r stillingar sem nota\u00f0ar eru \u00ed verkinu Fellingar t\u00edmans<\/em>. Verki\u00f0 er unni\u00f0 af undirritu\u00f0um \u00ed samvinnu vi\u00f0 t\u00f3nlistarh\u00f3pinn \u00deelgap og byggir m.a. \u00e1 t\u00edma\u00feenslu, lagskiptingu, bl\u00f6ndun raf- og ak\u00fast\u00edskra hlj\u00f3\u00f0a og vinnslu \u00e1 hlj\u00f3\u00f0- og kli\u00f0lit. Undir \u00feessum \u00f3l\u00edku l\u00f6gum liggur hins vegar \u00e1kve\u00f0in t\u00f3nfr\u00e6\u00f0ileg hugsun: leit a\u00f0 t\u00f3nr\u00fdmi sem er ekki loka\u00f0 \u00ed reglulegri \u00e1ttundarskiptingu heldur opnast sem breytileg, \u00f3samhverf dreifing. \u00cd sta\u00f0 \u00feess a\u00f0 ganga \u00fat fr\u00e1 jafnstillingu og reglubundnum endurtekningum er unni\u00f0 me\u00f0 \u00f3r\u00e6\u00f0 hlutf\u00f6ll, \u00f3j\u00f6fn t\u00f3nbil og \u00fea\u00f0 sem kalla m\u00e6tti phi-m\u00f3ta\u00f0an (\u03c6) sp\u00edral \u00ed t\u00ed\u00f0nir\u00fdmi. \u00deessi grein fjallar um \u00feessa hli\u00f0 verksins.<\/p>\n

\u00deetta setur verki\u00f0 \u00ed samtal vi\u00f0 lengri s\u00f6gu stillinga. Fr\u00e1 P\u00fd\u00feag\u00f3rasi til jafnstillingar hefur saga t\u00f3nfr\u00e6\u00f0innar a\u00f0 t\u00f6luver\u00f0u leyti sn\u00faist um hvernig skipta megi t\u00ed\u00f0nisvi\u00f0i \u00e1 noth\u00e6fan, sannf\u00e6randi og menningarlega sj\u00e1lfb\u00e6ran h\u00e1tt. \u00cd Fellingum t\u00edmans<\/em> eru stillingar hins vegar ekki hugsa\u00f0ar sem t\u00e6knilegt vandam\u00e1l, heldur formger\u00f0arlegt og heimspekilegt vi\u00f0fangsefni: hvernig m\u00e1 m\u00f3ta t\u00f3nr\u00fdmi sem endurspeglar heim sem er ekki loka\u00f0ur, ekki fullkomlega fanganlegur, heldur s\u00edfellt \u00ed fellingum og vindingum.<\/p>\n

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Af hverju stillingar?<\/strong><\/p>\n

Allar stillingar byggja \u00ed raun \u00e1 vali. \u00de\u00e6r segja ekki a\u00f0eins til um hvar t\u00f3nar lenda, heldur um hva\u00f0a tengsl eru talin gild milli \u00feeirra. \u00cd p\u00fd\u00feag\u00f3r\u00edskri hugsun var t\u00f3nbilum skipa\u00f0 eftir einf\u00f6ldum heilt\u00f6luhlutf\u00f6llum, einkum 2:1 fyrir \u00e1ttund og 3:2 fyrir fimmund o.s.frv. \u00deessi n\u00e1lgun byggist \u00e1 sterkri tengingu milli talna, hlj\u00f3ms og kosm\u00edskrar reglu v\u00edbrandi strengja. T\u00f3nr\u00fdmi\u00f0 er lesi\u00f0 sem r\u00f6\u00f0 einfaldra og st\u00f6\u00f0ugra st\u00e6r\u00f0fr\u00e6\u00f0ilegra sambanda.<\/p>\n

R\u00e9ttstilling (e. just intonation) heldur \u00feessari hugsun \u00e1fram, en leitast vi\u00f0 a\u00f0 skipuleggja fleiri t\u00f3nbil me\u00f0 hreinum hlutf\u00f6llum, svo sem 5:4 fyrir \u00fer\u00edhlj\u00f3m. \u00dear ver\u00f0a samhlj\u00f3mar sk\u00fdrir og \u201chreinir\u201d, en vandinn er s\u00e1 a\u00f0 sl\u00edkt kerfi er illf\u00e6ranlegt milli t\u00f3ntegunda; \u00fea\u00f0 sem hlj\u00f3mar vel \u00e1 einum sta\u00f0 brotnar au\u00f0veldlega annars sta\u00f0ar. Me\u00f0alt\u00f3nsstillingar (e. meantone) voru tilraun til a\u00f0 milda \u00feetta vandam\u00e1l me\u00f0 \u00fev\u00ed a\u00f0 dreifa fr\u00e1vikum, einkum \u00ed fimmundum, \u00feannig a\u00f0 \u00e1kve\u00f0in svi\u00f0 t\u00f3ntegunda yr\u00f0u noth\u00e6fari. Jafnstillingin (e. equal temperament) markar s\u00ed\u00f0an afgerandi breytingu: h\u00fan l\u00e6tur af \u00feeirri kr\u00f6fu a\u00f0 lykilt\u00f3nbil s\u00e9u fullkomlega hrein og velur \u00ed sta\u00f0inn jafna skiptingu \u00e1ttundarinnar \u00ed t\u00f3lf jafnst\u00f3r skref. \u00c1vinningurinn er gr\u00ed\u00f0arlegur sveigjanleiki, en kostna\u00f0urinn er s\u00e1 a\u00f0 \u00f6ll bil ver\u00f0a \u00ed \u00e1kve\u00f0num skilningi m\u00e1lami\u00f0lanir.<\/p>\n

\u00deessi \u00fer\u00f3un s\u00fdnir a\u00f0 stillingar eru aldrei hlutlausar. [1]<\/a> \u00de\u00e6r eru alltaf ni\u00f0ursta\u00f0a forgangsr\u00f6\u00f0unar: \u00e1 a\u00f0 h\u00e1marka hreinleika, flutningsh\u00e6fni, hlj\u00f3\u00f0fr\u00e6\u00f0ilegan einfaldleika e\u00f0a menningarlega notkun? \u00cd Fellingum t\u00edmans<\/em> vaknar \u00f6nnur spurning: hva\u00f0 gerist ef forgangsr\u00f6\u00f0in f\u00e6rist fr\u00e1 st\u00f6\u00f0ugleika yfir \u00ed mismun? Hva\u00f0 gerist ef t\u00f3nr\u00fdmi\u00f0 \u00e1 ekki a\u00f0 jafna \u00fat \u00f3reglu, heldur gera hana a\u00f0 virkri formger\u00f0?<\/p>\n

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Af hverju 12 t\u00f3nar?<\/strong><\/p>\n

T\u00f3lfskipting \u00e1ttundarinnar er ekki n\u00e1tt\u00farul\u00f6gm\u00e1l heldur s\u00f6guleg ni\u00f0ursta\u00f0a. H\u00fan var\u00f0 r\u00e1\u00f0andi vegna \u00feess a\u00f0 h\u00fan leysti \u00e1kve\u00f0in hagn\u00fdt og fagurfr\u00e6\u00f0ileg vandam\u00e1l \u00ed vestr\u00e6nni t\u00f3nlist. H\u00fan var\u00f0 a\u00f0 sameiginlegum m\u00e6likvar\u00f0a sem ger\u00f0i t\u00f3nflutning m\u00f6gulegan, opna\u00f0i skrif fyrir hlj\u00f3mbor\u00f0, hreyfinga milli t\u00f3ntegunda og ger\u00f0i t\u00f3nsk\u00e1ldum kleift a\u00f0 byggja st\u00f3r t\u00f3nverk \u00e1 st\u00f6\u00f0ugu undirlagi. En eins og m\u00f6rg kerfi sem n\u00e1 mikilli \u00fatbrei\u00f0slu, \u00fe\u00e1 ver\u00f0ur h\u00fan me\u00f0 t\u00edmanum ekki a\u00f0eins a\u00f0 verkf\u00e6ri heldur a\u00f0 sj\u00e1lfgefnum sj\u00f3ndeildarhring (stundum gagnr\u00fdnislausum).<\/p>\n

\u00dear liggur \u00e1st\u00e6\u00f0an fyrir \u00fev\u00ed a\u00f0 tilraunir me\u00f0 a\u00f0rar stillingar skipta m\u00e1li. \u00de\u00e6r gera ekki a\u00f0eins r\u00e1\u00f0 fyrir fleiri e\u00f0a f\u00e6rri t\u00f3num, heldur afhj\u00fapa sj\u00e1lfa forsenduna a\u00f0 t\u00f3nr\u00fdmi \u00feurfi a\u00f0 vera jafnt og loka\u00f0. \u00deegar t\u00f3nsk\u00e1ld og fr\u00e6\u00f0imenn kanna kerfi handan \u00e1ttundar (e. non-octave), \u00f3r\u00e6\u00f0ar stillingar (e. irrational tunings) e\u00f0a svok\u00f6llu\u00f0 EDN-kerfi[2]<\/a>, \u00fear sem annar \u201eendurtekningarkvar\u00f0i\u201c er valinn \u00ed sta\u00f0 2:1, er veri\u00f0 a\u00f0 opna t\u00f3nfr\u00e6\u00f0ina aftur sem ranns\u00f3knarsvi\u00f0 fremur en a\u00f0 sam\u00feykkja hana sem loka\u00f0an sta\u00f0al.[3]<\/a><\/p>\n

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Af hverju \u00e1ttundir?<\/strong><\/p>\n

\u00c1ttundin hefur au\u00f0vita\u00f0 gr\u00ed\u00f0arlega s\u00e9rst\u00f6\u00f0u vegna \u00feess a\u00f0 h\u00fan er b\u00e6\u00f0i skynr\u00e6n og formger\u00f0arleg. T\u00f3nar sem eru a\u00f0skildir me\u00f0 hlutfallinu 2:1 eru ekki eins, en \u00feeir skynjast samt sem skyldir me\u00f0 svo afgerandi h\u00e6tti a\u00f0 \u00feeir mynda einingu. \u00deetta hefur gert \u00e1ttundina a\u00f0 meginramma t\u00f3nr\u00fdmisins. H\u00fan lokar kerfinu. H\u00fan segir: h\u00e9r endurtekur sig sami flokkur, \u00e1 n\u00fdju h\u00e6\u00f0arstigi.<\/p>\n

En hva\u00f0 gerist ef ma\u00f0ur hverfur fr\u00e1 \u00feessari hringr\u00e1s? Hva\u00f0 gerist ef t\u00f3nr\u00fdmi\u00f0 lokast ekki \u00e1 1200 centum[4]<\/a>, e\u00f0a ef \u00fea\u00f0 er yfirleitt ekki skipulagt sem kerfi me\u00f0 endurkomu? Sl\u00edkar spurningar hafa legi\u00f0 a\u00f0 baki m\u00f6rgum \u201enon-octave\u201c tilraunum \u00e1 20. og 21. \u00f6ld, \u00fear \u00e1 me\u00f0al Bohlen\u2013Pierce kerfinu sem hefur 3:1 sem lotu (e. period) og \u00fdmsum phi-tengdum sk\u00f6lum \u00fear sem gullinsni\u00f0i\u00f0 (\u03c6) e\u00f0a \u00f6nnur \u00f3r\u00e6\u00f0 hlutf\u00f6ll koma \u00ed sta\u00f0 e\u00f0a trufla hina hef\u00f0bundnu \u00e1ttund.[5]<\/a><\/p>\n

H\u00e9r er gagnlegt a\u00f0 staldra vi\u00f0 hugtaki\u00f0 \u00f3samm\u00e6lanleiki (e. incommensurability). \u00deegar t\u00f6lur e\u00f0a hlutf\u00f6ll eru \u00f3samm\u00e6lanleg, er ekki h\u00e6gt a\u00f0 leggja \u00feau undir einn einfaldan sameiginlegan m\u00e6likvar\u00f0a. \u00deetta leiddi a\u00f0 \u00f3r\u00e6\u00f0um t\u00f6lum (e. irrational numbers), \u00fe.e. engin almenn brot n\u00e1 a\u00f0 fanga \u00fe\u00e6r (t.d. \u03c0 og e<\/em>). \u00cd t\u00f3nfr\u00e6\u00f0ilegu samhengi merkir \u00feetta a\u00f0 bilin ganga ekki upp \u00ed snyrtilegu, reglulegu neti. Gullinsni\u00f0i\u00f0 er s\u00e9rlega \u00e1hugavert \u00ed \u00feessu tilliti, b\u00e6\u00f0i vegna \u00feess a\u00f0 \u00fea\u00f0 er \u00f3r\u00e6tt og vegna \u00feess a\u00f0 \u00fea\u00f0 er erfi\u00f0ara a\u00f0 n\u00e1lgast \u00fea\u00f0 me\u00f0 einf\u00f6ldum heilt\u00f6lubrotum heldur en flest \u00f6nnur hlutf\u00f6ll (og \u00fev\u00ed er \u03c6 oft s\u00f6g\u00f0 \u00f3r\u00e6\u00f0asta talan[6]<\/a>). \u00dev\u00ed getur \u00fea\u00f0 virka\u00f0 sem t\u00f3nfr\u00e6\u00f0ilegur \u201ctruflari\u201d: \u00fea\u00f0 dregur t\u00f3nr\u00fdmi\u00f0 fr\u00e1 einfaldri endurkomu og yfir \u00ed opnara, sn\u00fanara og s\u00edbreytilegt ferli.[7]<\/a><\/p>\n

Raft\u00f3nverki\u00f0 Stria<\/em> eftir John Chowning er gott d\u00e6mi. Verki\u00f0 s\u00fdnir hvernig h\u00e6gt er a\u00f0 yfirgefa heim \u00e1ttundarinnar \u00e1n \u00feess a\u00f0 falla \u00ed \u00f3skipulagt t\u00f3mar\u00fam. \u00cd Stria<\/em> er gullinsni\u00f0i\u00f0 nota\u00f0 sem fast skipulagsafl b\u00e6\u00f0i \u00ed t\u00ed\u00f0nir\u00fdmi og hlj\u00f3\u00f0h\u00f6nnun. Chowning bygg\u00f0i verki\u00f0 \u00e1 \u00e1tta \u201egervi-\u00e1ttundum\u201c, hverri skipt ni\u00f0ur \u00ed n\u00edu t\u00f3na, og nota\u00f0i \u03c6 l\u00edka sem hlutfall \u00ed FM-hlj\u00f3\u00f0myndun, \u00feannig a\u00f0 t\u00f3nr\u00fdmi og hlj\u00f3\u00f0r\u00f3f ur\u00f0u hlutar af einni og s\u00f6mu skipan. \u00datkoman er ekki hef\u00f0bundi\u00f0 \u201et\u00f3nal\u201c r\u00fdmi, heldur heill hlj\u00f3\u00f0heimur sem \u201csn\u00fdst\u201d um \u03c6.[8]<\/a><\/p>\n

\u00dearna opnast ein m\u00f6guleg lei\u00f0 fr\u00e1 heimi \u00e1ttundarinnar yfir \u00ed a\u00f0rar stillingar og a\u00f0rar n\u00e1lganir. Stria<\/em> er ekki fyrirmynd \u00ed \u00feeirri merkingu a\u00f0 Fellingar t\u00edmans<\/em> endurtaki a\u00f0fer\u00f0 Chownings heldur s\u00fdnir okkur a\u00f0 \u00feegar \u00e1ttundin er ekki lengur sj\u00e1lfgefinn endapunktur, ver\u00f0ur t\u00f3nr\u00fdmi\u00f0 a\u00f0 ranns\u00f3knarsvi\u00f0i. \u00cd \u00fev\u00ed samhengi ver\u00f0ur spurningin ekki lengur \u201ehvernig \u00e1 a\u00f0 skipta \u00e1ttundinni?\u201c heldur \u201ehvernig \u00e1 a\u00f0 m\u00f3ta dreifingu?\u201c<\/p>\n

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Leit a\u00f0 \u00f3j\u00f6fnum t\u00f3nbilum: <\/strong>\u03c6-sp\u00edrall sem a\u00f0fer\u00f0<\/strong><\/p>\n

Einfaldasta lei\u00f0in til a\u00f0 nota gullna sni\u00f0i\u00f0 fyrir stillingar v\u00e6ri a\u00f0 l\u00e1ta t\u00ed\u00f0nir fylgja hreinni veldisr\u00f6\u00f0, til d\u00e6mis:<\/p>\n\"\"\n

Sl\u00edk a\u00f0fer\u00f0 er \u00e1hugaver\u00f0, en h\u00fan lei\u00f0ir ekki a\u00f0 \u00feeirri ger\u00f0 dreifingar sem leita\u00f0 var a\u00f0 \u00ed Fellingum t\u00edmans<\/em>. \u00c1st\u00e6\u00f0an er s\u00fa a\u00f0 \u00feegar sl\u00edkri veldisr\u00f6\u00f0 er varpa\u00f0 yfir \u00ed cent-r\u00fdmi ver\u00f0a bilin \u00e1 milli samliggjandi t\u00f3na \u00ed reynd j\u00f6fn e\u00f0a n\u00e1nast j\u00f6fn. Dreifingin vir\u00f0ist \u00f3j\u00f6fn \u00ed t\u00ed\u00f0nir\u00fdmi, \u00fev\u00ed bil \u00ed Hz st\u00e6kka me\u00f0 hverju skrefi en \u00feegar s\u00f6mu r\u00f6\u00f0 er varpa\u00f0 yfir \u00ed cent-r\u00fdmi, sem er l\u00f3garitm\u00edskt r\u00fdmi, ver\u00f0ur h\u00fan l\u00ednuleg: \u00f6ll skref eru jafnst\u00f3r. \u00deess vegna dug\u00f0i einf\u00f6ld phi-veldisr\u00f6\u00f0 ekki ein og s\u00e9r til a\u00f0 framlei\u00f0a \u00feau \u00f3j\u00f6fnu t\u00f3nbil sem leita\u00f0 var a\u00f0 \u00ed verkinu. Einf\u00f6ld mynd af \u00feessu m\u00e6tti setja fram svona:<\/p>\n

Hz:<\/p>\n

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cent:<\/p>\n

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Cent-r\u00fdmi\u00f0 felst \u00ed \u00fev\u00ed a\u00f0 sko\u00f0a veldisv\u00f6xt \u00ed l\u00f3garitm\u00edsku formi. R\u00f6\u00f0 sem er veldisleg \u00ed t\u00ed\u00f0num (hz) ver\u00f0ur l\u00ednuleg \u00ed centum, \u00fev\u00ed cent m\u00e6la hlutfallslega fjarl\u00e6g\u00f0 milli t\u00ed\u00f0na fremur en einfaldan mismun \u00ed Hz. \u00deetta samr\u00e6mist skynjun t\u00f3nbila, \u00fear sem eyra\u00f0 bregst fyrst og fremst vi\u00f0 hlutf\u00f6llum (sbr. 220\u2013440 Hz og 440\u2013880 Hz sem sams konar bil, \u00fe\u00f3tt mismunurinn \u00ed Hz s\u00e9 mj\u00f6g \u00f3l\u00edkur).<\/p>\n

Af \u00feeirri \u00e1st\u00e6\u00f0u dug\u00f0i einf\u00f6ld phi-veldisr\u00f6\u00f0 ekki til a\u00f0 framlei\u00f0a \u00fe\u00e1 \u00f3j\u00f6fnu t\u00f3nbilaskipan sem leita\u00f0 var a\u00f0 \u00ed verkinu. \u00dev\u00ed \u00feurfti a\u00f0 \u00fer\u00f3a a\u00f0fer\u00f0 sem m\u00f3ta\u00f0i dreifinguna sj\u00e1lfa \u00e1 annan h\u00e1tt.<\/p>\n

\u00dea\u00f0 sem s\u00f3st var eftir var annars konar heg\u00f0un: ekki einfaldlega n\u00fdr \u201egenerator\u201c e\u00f0a n\u00fd lotust\u00e6r\u00f0, heldur \u00f3j\u00f6fn t\u00f3nbilaskipan (e. nonuniform intervals) \u00fear sem sumir t\u00f3nar dragast saman \u00ed \u00fe\u00e9tta klasa en a\u00f0rir dreifast meira. Me\u00f0 \u00f6\u00f0rum or\u00f0um: markmi\u00f0i\u00f0 var ekki a\u00f0eins a\u00f0 fara handan \u00e1ttundarinnar, heldur a\u00f0 m\u00f3ta t\u00f3nr\u00fdmi \u00fear sem mismunur \u00ed \u00fe\u00e9ttleika yr\u00f0i a\u00f0 formger\u00f0.<\/p>\n

Af \u00feeirri \u00e1st\u00e6\u00f0u voru ger\u00f0ar prufur me\u00f0 \u00fdmsum a\u00f0fer\u00f0um. \u00cd sta\u00f0 \u00feess a\u00f0 skilgreina t\u00f3nana beint var byrja\u00f0 \u00e1 skrefar\u00f6\u00f0 sem s\u00ed\u00f0an var m\u00f3tu\u00f0 og afl\u00f6gu\u00f0 (e\u00f0a truflu\u00f0). Eftir nokkrar tilraunir sem misheppnu\u00f0ust kom \u00ed leitirnar form\u00fala sem reyndar leggur upp fr\u00e1 j\u00f6fnum skrefum, en \u00feau eru bj\u00f6gu\u00f0 me\u00f0 \u03c6 og s\u00ednus-falli:<\/p>\n\"\"\n

H\u00e9r er 1200\/N einhver upphafleg j\u00f6fn skipting, en h\u00fan er ekki l\u00e1tin standa \u00f3hreyf\u00f0. \u00cd sta\u00f0 \u00feess kemur log2<\/sub>(\u200b\u03c6) inn sem m\u00e6likvar\u00f0i \u00e1 \u00e1hrif gullna sni\u00f0sins \u00ed l\u00f3garitm\u00edsku t\u00ed\u00f0nir\u00fdmi, \u00e1 me\u00f0an \u03b1 er stu\u00f0ull sem stj\u00f3rnar styrk skekkjunnar. S\u00ednus-li\u00f0urinn s\u00e9r s\u00ed\u00f0an um a\u00f0 beygingin ver\u00f0i ekki brotakennd heldur sl\u00e9tt og samfelld yfir r\u00f6\u00f0ina (samt \u00f3samhverft vegna \u03c6-\u00e1hrifa).<\/p>\n

Skrefin (t\u00f3nbilin) Sn<\/sub> eru \u00fev\u00ed ekki j\u00f6fn, heldur breytileg. \u00deau eru s\u00ed\u00f0an l\u00f6g\u00f0 saman \u00ed uppsafna\u00f0a r\u00f6\u00f0 C (fyrir cent):<\/p>\n\"\"\n

\u00dear me\u00f0 ver\u00f0ur til r\u00f6\u00f0 cent-sta\u00f0a e\u00f0a punkta sem l\u00fdsir upphaflegri fer\u00f0 um t\u00ed\u00f0nir\u00fdmi\u00f0. N\u00e6sta a\u00f0ger\u00f0 f\u00f3lst \u00ed \u00fev\u00ed a\u00f0 fella \u00feessa r\u00f6\u00f0 ni\u00f0ur \u00ed eitt svi\u00f0 me\u00f0 modulo-a\u00f0ger\u00f0:<\/p>\n\"\"\n

A\u00f0 lokum er punktunum ra\u00f0a\u00f0 \u00ed h\u00e6kkandi r\u00f6\u00f0 e\u00f0a vi\u00f0 getum sagt a\u00f0 \u00feeir s\u00e9u felldir saman (e. folded) sem l\u00edkist \u201espectral\u201c a\u00f0fer\u00f0um \u00feegar yfirt\u00f3num er varpa\u00f0 ni\u00f0ur \u00ed eina \u00e1ttund nema h\u00e9r erum vi\u00f0 a\u00f0 tala um r\u00f6\u00f0 t\u00f3nbila \u00ed centum. \u00deetta skref er afgerandi, \u00fev\u00ed \u00fear breytist framlei\u00f0slur\u00f6\u00f0in \u00ed n\u00fdja dreifingu. T\u00f3narnir birtast \u00fe\u00e1 sem punktasafn innan gervi-\u00e1ttundar (sem er misst\u00f3r), og bilin \u00e1 milli n\u00e6stu t\u00f3na ver\u00f0a aflei\u00f0ing af samspili \u00feriggja \u00fe\u00e1tta:<\/p>\n