Tone Spirals

8/20/2014 34

http://www.sacredscience.com/archive/ToneSpiral.htm 1/13

34.TONESPIRALSANDTONECURVES34.ToneSpiralsAndToneCurvesIn27,wediscussedthethreecharacteristiccurvesoftheseconddegree(parabola,hyperbola,andellipse)withintheconfigurationoftheP,i.e.usingasabasisonlytheplanePsystemoritsquantitativeandlogarithmicnumbers,towhichvaluesnaturallyalwayscorrespond.34.1.ToneSpiralsontheBasisofStringLengthsandFrequenciestheDecimalToneSpiralsWewillnowexamineafewtypicalcurvesthatareencounteredintheangle(vector)diagramsoftheP.Wealreadyknowthesedecimaltonespiralsfrom33.3and33a.(Ihavecalledthistypeoftonespiral decimaland theonebasedon tonelogarithmslogarithmic,but this caneasily lead toerrors interminologysee21.)WewillnowrecapitulatetheminFigures290and291,intwovariations:Fig.290withratiosaccordingtostringlengthsandFig.291withratiosaccordingtofrequencies.Ifwecreatesuchreciprocaldiagrams,wemustmaintainsomekindoforder.Here,thecommonelementisestablishedastheprogressionoftonestepsupwardswithin1octave(=thecircle),goingclockwisefrom360=0.Thus,thetwospiralsnecessarilygoinopposite(reciprocal,mirrorimage)directionstheirformsareexactlyreversed.Notsowiththegeometricintervalsofthetonesteps.Here,inthestringlengthdiagram,thedirectionofthediminution (the interval shortening) is clockwise, from left to right in the frequency diagram, it goescounterclockwise,fromrighttoleft.Weknowthatthisdiminutionisthecharacteristicelementofthelawofharmonicquantization,andwefindit,amongotherthings,individingthemonochord,wherethedivisionstepsgrowcontinuallyclosertogetherastheyascend.Thequestionnowis:whichtypeofdiminutionisinagreement with the diminution of the string length, if we think of the circles circumference as amonochord?Clearly,wemustnowuseasabasisthediagramofthestringlengthspiral,Fig.290(whichemergesfromthecomparisonwithFig.274),wherewefind,forexample,thenoteeinthesecondcircleofindex5atthecorrectplaceofstringdivision(5/5c0,6/5a72,7/8xfis144,8/5e216),i.e.at216.Aninstructiveoverviewofthereciprocalrelationshipoftheabovediagrams290and291isgiveninFig.292,wherewehavenotedthetoneses,e,f,g,as,anda.Thereciprocityisverynoticeablehere,aswellasintheintervals,thecorrespondingangles,andtheirdifferences.

8/20/2014 34


Figure290

Figure291

8/20/2014 34


Figure292Asforthemathematicalcharacterofthesetonespirals,itisbasedonanArchimedeanspiral.Whichvariantweuse(stringlengthorfrequency)depends,regardlessoftheirautonomousmeaning,onhowwecanusethemforektypicanalyses.Theanglesofbothvariantsarenotedinthetableofratiosattheendofthisbookothertablesshowonlythefrequencyangle,sincewearemostlyworkingwithfrequencies.34.2.ToneSpiralsontheBasisofLogarithms(theLogarithmicToneSpiral)Inanticipationofthenextchapter,wenowdiscussthelogarithmictonespiral.Comparingitwiththedecimaltonespiralallowsustoseeitsdifferencesandpeculiaritiesproperly.RefertoFig.293forthefollowing. Since the circles, distances, and angles here correspond not to the quantitative sizes of stringlengthsandfrequencies,butinsteadtothequalitativetonevalues,theoctavecircles1/1c2/1c4/1c8/1c ...mustbeequidistantbecause, indeed,wehear theoctavesas tonespacesofequaldistance.ThetoneangleiscalculatedaccordingtotheformulaatthebottomleftinFig.293.Thedistancesbetweentheremainingtonecirclesarealwaysbetween0and1000(with3logarithmicspaces)andcanmosteasilybeindicatedwithmillimeterpaperasanunderlay,using10cmforeachoctave01/12/1etc.Threeoctavesaresufficient, and just one for the position of the angle, as for all polar diagrams.However, several octavecircles have the advantage that they produce several rotations of the spiral, and thus show theirphysiognomymoreclearly.Asfor thedivisionof tonesonthecircularperipheryof the logarithmic tonespiralandthelogarithmicpolardiagram,thisisorientedaccordingtopsychicaldistances,i.e.accordingtointervalsaswehearthem,notaswecountthem.Theperspectiveelementofdiminutionfallsawayhere,and theeyesees the intervalsdistributed in thesamewayas theearhears them. It is interesting that thislogarithmic tonespiral is actuallynot a logarithmic spiral, but anArchimedeanone thereforewemustdifferentiateitfromlogarithmicspiralsinthepurelymathematicalsense.Wewilldiscussthisfurtherinthenextsection.34.3.TheToneCurvesofthePolarArrangementInsections1and2weconstructedtonespiralsstartingfromthefixedcenterofacircle.Ifwenowallow this center to wander regularly along the monochord string, a most remarkable curve appears,

8/20/2014 34


which I call the partialtone curve (Harmonia Plantarum, p. 127) and which is shown in Fig. 294.Consider theentire lengthof thecurveasamonochordstringof120cm.For theangle,weuseonly thestringlength angle.Halfway along the string, at the point 60 (cm),we place the angle 0 (= c), whosevectorwillcorrespondtotheupperhalfofthestring.Thenexttoneis16/15h(alwaysstringlengthratios).Wefirstdividehalfofthemonochordstringinto15parts,add1/15below,andfindthestringpositionfor16/15h(thevectorforthistonewaserroneouslyomittedinthedrawing).Itsangleisfoundaccordingtothescheme (x/y 360) 360, i.e. 16 360=5760 : 15=384360=24 (16/15h). This angle,with thecorrespondingvector,ismissingfromFig.294,asnoted.Thenextvalueis12/11h.Tofindtheplaceonthe string, we can now divide the half of the string again, in 11 parts, and add 1/11 to it. Or else wecalculate:60:11=5.45412=65.44cm,andsubtractthisamountfromthe120cmmonochordstring,gettingthelength54.5cmasaresult,whichwemeasurefromthebottomup,thusyieldingtheremainderof54.5cmforthetone12/11h.Nowwecalculatethecorrespondingangleanalogouslytotheabove(12/11360)360,yielding32.7.Weaddthisangle(totheleftorright),drawthevector(ray),andaddtothistheremainderofthestring,measuredfrombelow(thereforealwaysdiminishingwiththefollowingratios),from54.5.Weproceedinthesamewaywithalltheremainingtones,andsoconstructthepartialtonecurveofFig.294.Thetonevaluescannaturallybechosenarbitrarilyfromanypartialtonediagramonlytheymustbe reduced tooneoctave first,andmustbeselectedso that thevectorsaredistributedupon thecurveasevenly as possible, i.e. so that this curve, as the line connecting the endpoints of the vectors, can beconstructedasaccuratelyaspossible.Likethecircularperipheryoftheprecedingtonespirals, thepartialtonecurvecontainsallpossibletones,andthereforeaninfinitenumberoftonesreducedbyoctaves.

8/20/2014 34


Figure293

8/20/2014 34


Figure29434.3a.TheToneCycloidInsteadoftheshorterremaindersofthestring,wecanalsoremovethelongerones.Thatis,wetaketheupper,longersegmentofthemonochordstringinthecircle,insteadofthelower,shortersegment,anddrawitalongthevector.Ifwedothisinthesamewayforalltones,theresultisanevenmoreinteresting

8/20/2014 34


curve, which I call the tonecycloidan irregular ellipse almost circular in form (see Fig. 295). Thepartialtonecurve,towhichitisreciprocalintermsofthemonochordstring,isdrawninsidethecycloidforbettercomparison.Thistonecycloidisespeciallyinterestinginvariousways.Assumingthatitisindeedanellipse, I have first constructed the ellipse from the narrowest and widest diameters of the cycloidaprocessthatcanbefoundineverymathematicstextbook.Ifonelaysthisregularellipseuponthecycloid,onecanseethatwithafewwideningsandindentations,thecycloidalignswiththeellipse.InFig.307atheellipseisprintedontransparentpaper,allowingcomparisonwiththecycloid.Ifwehadsimplyderivedandestablished the cycloid as a curve drawn from observation data, as the expression of some naturalphenomenon (e.g. thepathsof theplanets),withoutknowledgeof its regularharmonic information, thenobviouslytheellipsewouldbehighlightedasamathematicalrelation.Forthedeviationsoftheellipse,onewoulddoubtlesslookfordisturbancefactorstostickwiththeexampleofplanetaryorbitsinthiscaseacceptinggravitationaleffectsfromotherplanets,etc.,asanexplanation.However,inthecycloidwehavealegitimateclarificationoftheseirregularitiesintheharmonicemergenceofthecurveitself. Between cycloid and corresponding ellipse, however, even closer relationships exist. Themonochordaxisisdividedbythecycloidinto3equaloctaves:the2octavesofthemonochorditselfandanadditionaloctaveofthesegmentlengthenedbelowfromthemonochord.ThisoctaveappearsagainintheellipseasthedistancesofthetwofocalpointsFandF1fromthetwoaxispointsBandAoftheellipsecurve.Theanglesatwhichthemajor(AB)andminor(CD)ellipseaxesintersectthemonochordlineatEandGareboth45, andwith the centerof the ellipse,S, an isosceles right triangleSEG is constructed,whoseheightEG(at thetonef)divides inhalf. Ihardlybelieve that theserelationshipscouldbemerelycoincidental.Remarkableaboveallistheverticalpositionofthemonochordlinethatproducesthecycloid,andthecenterSofthecorrespondingellipse,apparentlyexistingincompleteisolation.Assumingthatwecan see the prototype for the paths of the planets in this harmonic cycloida figure that the ancientharmonistsmusthaveknownof,considering theGreeksgreat talent forgeometricconstructions,even ifthey intentionally kept their other important harmonic theorems secretthen from the Pythagoreanviewpoint, the center S of the ellipsemust be given the name of the secret Pythagorean antihelion orcentral suna conceptwithwhich no one has been able to do anything up to this point, andwhichemergesinevitablyandobviouslyfromtheharmoniccycloidanditsellipse.Ifwepursuethisastronomicalsymbolicektypic further, thenwecome to further importantsphereharmonic realizations regarding thePythagoreanoctavethatplayedsuchagreatroleinancienttimes.Here,insidethecycloid,weseethisasagenerating element. Modern literature always mentions the scale and the 7 planets as the twofundamental concepts of the ancient harmony of the spheres. This is understandable on the basis of theexistingexotericancientsources,whosewritershadnoknowledgeofthetrueesotericbackgrounds.Butifwereturnto

8/20/2014 34


Figure295ancient Pythagorean thought and begin to study in Pythagorean terms, it is everywhere apparent thatPythagoreanism was a very different and exquisitely harmonically ramified domain of thought andobservation, which absolutely did not bow to such primitive idols as people today imagine. Thus it isevident tome that the generating space for the ancient harmony of the sphereswas not the scale thatoccupies the octave space, but instead the octave itself, and that the corresponding important diatonicstepsandtheir tonevaluesandvectors(circlespheres)werechosenwithinthisspace,soas toarriveatacomparisonandaninterpretationofthebodiesvisibleintheskies.Nowwesee,inFig.295,thattherearesignificantlymorethansevenimportanttonevalueswithintheoctave.Infutureharmonicstudy,however,thistonecycloidwithitsellipseisnotonlyvaluableforhistoricalanalyses,especiallythosearisingoutofthe harmony of the spheres, but farmore so for a prototypical interpretation of the planetary orbits. Todevelopeachplanetsharmonicvector,itsdistancefromthesun,anditscharacteristicellipsefromthetonecycloidwouldrequireaspecializedandintensiveharmonicundertakingonthepartofalearnedastronomer,forwhichtheabovecanonlyserveasanencouragement.Ifthisweretosucceed,acompleteunionbetweenmodern astronomy and Pythagoreanism would be achievedsomething that Kepler attempted in hisWeltengeheimnis,partiallyrealizedinhisThirdLaw,andinwhichhebelievedwitheveryfiberofhisbeing

8/20/2014 34


duringhislifetime.34.3b.ThePrimordialLeafIfwenowtaketheanglesthatwehavepreviouslysetononesideofthemonochord(heretheleft,butonecanalsousetherightside,obtainingamirrorimageofthesamefigures)forthepartialtonecurveandthecycloid,andplacethemsymmetricallyonthemiddleoftheaxis,thentheresultisthetonecurveoftheprimordialleafadescriptionthatwillberetainedhereforsimplicityssake,sinceithasalreadybeenshowninHarmoniaPlantarum(p.125)astheharmonicprototypeoftheleafingeneral.Hereagainwecanconstructtwodifferentfigures,dependingonwhetherweusethelongerorshorterplagalorauthentic(see 29.1)remainders on the vectors of the respective tonelocations on the monochord string. BothfiguresareshowntogetherinFig.297.Forreasonsofexactitude,theinnerfigureoftheprimordialleafisprintedseparately,anditsdevelopmentisdescribed(Fig.296).

PE7

Stringlengthsona120cmmonochordreducedbyoctaves,andtheirremaindersPE7

Tonevaluesandanglesbystringlengths

8/20/2014 34


Figure296

8/20/2014 34


Figure297Thefirstsmallsquareatthetopcontainsthetonevaluesandanglesofthepartialtoneplaneofindex7,basedonstringlengths,whichhaveareciprocalrelationshiptothevibrationnumbers(frequencies).Thesecond square at the top gives the corresponding string lengths, reduced by octavescalculated for amonochord120cmlongandtheremaindersofthesestringlengths.Forexample:1/3ofthestringlength

8/20/2014 34


willproducetheduodecimal(2ndupper fifth)g,and2/3of the string lengthproducesa toneanoctavelower,thusthe1stupperfifthg.1/2ofthecircumferenceofthecircle,i.e.thestringbentintoacircle(360),yields120,2/3,sincewereduceallthetoneswithinoneoctave,i.e.allgvaluesareonthevector120.Thesamegoesforthepositionoftonesonthemonochord,ifIwishtobringthemallintooneoctave.1/3gis40cmofthe120cmmonochordstring,andtheremainderofthestringis80cmlong.2/3gis240=80,theoctavebelowgbutsincewewanttobringalltonesintoanoctaveof11/2(060cmstringlengthor0360ofthecirclescircumference),allgvaluesremainatastringlengthof40cmandaremainderof80cm.Fig.296isnoweasytoconstruct.Draw40unitsfromthebottomuponamiddleaxisof60unitstofixthe toneg. Set the corresponding angle 120 symmetrically on this point, and set the lengths of the twoanglelegsequaltothecorrespondingstringlengthlikewise40units.Oneproceedsanalogouslywithalltones,thusgettingtheprimordialleafasthelineconnectingalltheendpointsoftheanglelegsobtained.Thegreaterthepartialtonecoordinateindexoneuses,i.e.themoreonefillsouttheoctavewithtones,themoreprecisely the primordial leaf can be constructed. Its form, however, always remains the same. But thismeans nothing other than that the primordial leaf is a formexpression of the very nature of tonesadiscovery that deepens and confirms the fundamental ideas of Goethes morphology of plants in acompletelynewway.Theconstructionoftheoutercurveoftheprimordialleaf(Fig.297)emergesofitselfaccordingtowhat issaidaboveandin34a.AsonecanseeinFig.297, theouterandinnercurvesof theprimordialleaf,incontrastwiththepartialtonecurveandthecycloid,haveamorphologicallyreciprocalrelationship,except that the apex of the smaller inner curve points upwards, while the outer curves apex pointsdownwards.34a.EktypicsInmanyofmyworks,Ihavediscussedthenatureofthespiralextensively,andhereIwillonlyrecapitulatefundamentalmattersandsummarizetheektypicdata.Mathematically,thespiralisimaginedasapointP,movingatagivenspeedalongastraightlinewhichiscontinuallyrotating,atanothergivenspeed,aroundacenterpointZ.Dependingonthemagnitudeandratioofthosetwospeeds,thevariousspiralsArchimedean,logarithmic,etc.thenemergewiththeirvariousformulae.Eveninthisdefinition,onecanrecognizeacertainparadoxofthespiral:itis,sotospeak,thegeometricsymboloftwodivergent,opposingmovements,akindof frozen timegeometry,acapturingof the temporal in thespatial.Asweconsider itfurther, the concept of speed separates into two components: something reaching outward, in onedirection,avector,andsomethingthatholdsitselfin,withacirculartendency.Oronecanalsosaythatinthepointmovingonthespiral,twoelementsconstitutingthespiralmeet:anelementofdirection(angle)andanelementofdistance(fromthecentralpoint),wherebytemporalturnsintospatialinbothcases.Thusthemore or less dynamic behavior of all spirals is understandable. It arises from those two divergenttendenciesofthelinearstrivingforwardandthecircumpolarcircling,theexpansiveandattractive. Characteristic for theharmonicdevelopments, then,are the spirals that result from the thoughtprocesses justdescribed,whichcanalsobe found inbothhalvesof thePdiagram.Think,also,of theverysignificantspiralsofthecochleainourinnerears!Onthebasisofharmonicdevelopments,aswehaveseenandwillseeagainin36,therearecharacteristictonespirals(takingthistermgenerally),andthesameappearsasinharmonicnumberanalysis:alltonevalueshaveapsychicalevaluation,andallowforanalysesof a certain type, especially through their octave reductions (a typical harmonic operation not known tomathematics or the mathematical sciences). (See Tonspektren and the atom model therein.) Theseanalysescanonlybearrivedatwithgreatdifficulty,ifatall,bymeansofthefamiliarmathematicalspirals. Theektypicsof thespiral innaturecanbeseen insomanyexamples, fromalmostallareasofknowledge,thatwewillgiveonlyafewexampleshere:thespiralcloudastheprototypeofgalaxies,spiralmovementsandlawsinphysics,thespiraloftheharmonicatommodel(tonespectra)asthemotoroftheopticalemissionofthespectra,spiralsinthemorphologicalconstructionofdiatoms,plants,andanimals(the

8/20/2014 34


curvesofblossoms,snails,theconstructionofthehelix,thespiralvascularstructureofplants),theideaofspiral developments of historicalmorphological isotopes, the idea of the spiral as a universal religioussymbol(P.Sarasin:HeliosundKeraunos,1924,p.67ff.),andmanyothers.Peoplehavetried,ifnotveryoften,tofigureouttheuniversalmorphologicalsignificanceofthespiral.Butapartfromthevariousmathematicalspirals,whichhavenoadvantageovereachotherintermsoftheirformulae,alltheseattemptshaveremainedmiredinthemathematicalconceptofquantity,similarlytothoseattemptedbymeansof thegolden section, etc.and from thisviewpoint,noonecan seewhy thespiralinparticularshouldhavesuchuniversalsignificance.However,ifwetracethetectonicsofthisformback to certain psychical values, as we can do in harmonics, and if we see this form not onlyphysiologically (the cochlea) anchored in the filter of this psychical value, but also in one of itsmostimportant modifications, namely the tonespiral and logarithmic spiral (see 18.3b) as a morphologicalexpressionofthispsychicalvalue,thenweseesomethingcompletelydifferentandmuchmoreauthoritativeineveryregardandwenowunderstandthatsuchapronouncedvalueformmustalsohaveitsvalueformalcounterpartsinallofnature,andthatittiesinwithourspiritualandreligiousimageconcepts.34b.Bibliography H. Kayser:Hrende Mensch, 8992 and Tables IV and VKlang, 8184 Abhandlungen:Tonspektren,pp.111189andtherelevanttablesGrundri,120,121,240253(groupspiral)HarmoniaPlantarum,124127,142ff.,152ff.,270ff.HarmonikaleStudienII(theviolinscroll).

Documents

Tone Spirals