Blog 19 Piano Maths - mathematicalwhetstones.com
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1 Piano Maths When the frequency of vibration of two piano strings are in simple ratio with each other they produce noticeably harmonious notes when struck together. The reasons for the perception of consonance (and for that matter dissonance) are, for the most part, thought to be a cultural phenomenon. However the combination of each sound’s multiple wave patterns through the air must also contribute to that perception. For example if the piano key below middle on a regular piano is struck then the string associated with the corresponding note will begin to vibrate with a pattern that repeats at the rate of 220 cycles per second (220 Hz) causing a sound to be heard by a listener. The A key above middle C when struck will cause another somewhat shorter string to vibrate at 440 Hz. The two notes, A below and above middle C, span what is known as an octave, a musical interval between one musical pitch and another with double its frequency. When struck together they sound harmonious. Musical intervals are measured as ratios of pitch frequencies. In the 6 th century BCE, in one of his famous experiments, the Greek philosopher Pythagoras showed that when a stretched string was divided in certain simple ratios (such as 2:1, 3:2 and 4:3) and each of the two string intervals were plucked, the two distinct sounds were perceived by the ear as harmonious. Diagram 1 (Pythagorean intervals) Diagram 1 depicts three important ratios for western music, commonly referred to as the fourth, the fifth and the octave. The octave relationship between a tone of frequency and another of frequency 2 is regarded as a natural phenomenon, sometimes referred to as ‘the basic miracle of music’. So harmonious are these tones that the notes corresponding to these tones are given identical names. The Octave 2: 1 2 1 The Fifth 3: 2 3 2 The Fourth 4: 3 4 3
Transcript of Blog 19 Piano Maths - mathematicalwhetstones.com
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PianoMaths
Whenthefrequencyofvibrationoftwopianostringsareinsimpleratiowitheachothertheyproducenoticeablyharmoniousnoteswhenstrucktogether.Thereasonsfortheperceptionofconsonance(andforthatmatterdissonance)are,forthemostpart,thoughttobeaculturalphenomenon.Howeverthecombinationofeachsound’smultiplewavepatternsthroughtheairmustalsocontributetothatperception.
Forexampleifthepianokey𝐴belowmiddle𝐶onaregularpianoisstruckthenthestringassociatedwiththecorrespondingnotewillbegintovibratewithapatternthatrepeatsattherateof220cyclespersecond(220Hz)causingasoundtobeheardbyalistener.TheAkeyabovemiddleCwhenstruckwillcauseanothersomewhatshorterstringtovibrateat440Hz.Thetwonotes,AbelowandabovemiddleC,spanwhatisknownasanoctave,amusicalintervalbetweenonemusicalpitchandanotherwithdoubleitsfrequency.Whenstrucktogethertheysoundharmonious.
Musicalintervalsaremeasuredasratiosofpitchfrequencies.Inthe6thcenturyBCE,inoneofhisfamousexperiments,theGreekphilosopherPythagorasshowedthatwhenastretchedstringwasdividedincertainsimpleratios(suchas2:1,3:2and4:3)andeachofthetwostringintervalswereplucked,thetwodistinctsoundswereperceivedbytheearasharmonious.
Diagram1(Pythagoreanintervals)
Diagram1depictsthreeimportantratiosforwesternmusic,commonlyreferredtoasthefourth,thefifthandtheoctave.
Theoctaverelationshipbetweenatoneoffrequency𝑓andanotheroffrequency2𝑓isregardedasanaturalphenomenon,sometimesreferredtoas‘thebasicmiracleofmusic’.Soharmoniousarethesetonesthatthenotescorrespondingtothesetonesaregivenidenticalnames.
TheOctave2: 12 1
TheFifth3: 23 2
TheFourth4: 34 3
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Forexample,asmentionedabove,thestringsofthe𝐴sbelowandabovemiddle𝐶onamodernpianovibratewithfrequenciesof220and440cyclespersecond.Theintervalbetweenthispairof𝐴s(orindeedanyotherconsecutivepairoftwoidenticallynamednotes)iscalledanoctave.Ofthe52whitekeysonthepiano,thereareseven𝐴keyslinkingsixoctaves.Thelowest𝐴note(theveryfirstwhitekeyonthekeyboard)hasafrequencyof27.5Hzwithapitchjusthigherthanthenormalthresholdofhearingbelowwhichacousticvibrationstendtobefeltratherthanheard.Onemoreoctaveupandwefindthenext𝐴withafrequencyof55Hz.Thepatternofdoublingthefrequencycontinuesrightuptotheseventh𝐴withafrequencyof3520Hz.
Sofar,wehaveconstructedjustsevenmusicalpitches,withfrequenciesofvibrationrelatedbybeingmultiplesofpowersof2ofabasefrequency.Whythegapbetweentwosuccessivemembersofthissetoffrequenciesshouldbefilledwithsevenwhiteandfiveblackkeysonakeyboard,withsevenletternamestolabelthem,isamysterywemustexplain.
Wemustshowwhyitseemsnaturaltoconstructscalesofpitchesinthewaywedoandwhyitisreasonabletospeakofmusicalintervalsintermsofnumbersofscalestepsalongthestandardkeyboardratherthanasfrequencyratiosbetweenpairsofpitches.
Asafirststepinfillingthegapbetweenpitcheswithfrequenciesfand2fweobserve,asdidPythagoras,thatthepitchthathasafrequencyoneandahalftimesthefrequencyofaparticularnotewillsoundharmoniouswithit.Itwillbeanexcellentcandidateforinclusioninourscaleofpitches.
Similarly,thepitchthathasafrequencyofvibrationoneandathirdtimesthatofachosennotewillsoundalmostasharmoniouswithitasthepitchinthepreviousexample.Weshouldchoosetoincludeitalsotohelpfillthegapbetweenfand2f.Thuswenowhavepitchescorrespondingtofrequencies𝑓, !
!𝑓, !
!𝑓, 2𝑓.
Diagram2(Findingtwointervals)
𝑓
Base
2𝑓
32𝑓
43𝑓
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Thepitchwithfrequency!!𝑓willsoundharmoniouswiththepitchoneandonehalf
timeshigherthanit,but!!× !
!𝑓 = !
!𝑓isafrequencygreaterthan2𝑓correspondingtoa
pitchoutsidetheoctave.Thesolutionthoughisstraightforwardbecauseweknowthatanypitchisharmoniouswithanyanotherpitchiftheirfrequenciesaretheratio2: 1.Thereforehalvingthefrequency!
!𝑓to!
!𝑓providesathirdstepinthescale.
Diagram3(halvingtheoctave)
Again,knowingthatapitchwithafrequencyoneandhalftimesanotherisharmonious,thetwopitcheswithfrequencies!
!𝑓and!
!𝑓areharmonious(checkthat!
!× !!𝑓 = !
!𝑓)
then,usingthesamestrategyabove,anothersuitablestepis2× !!𝑓 = !"
!𝑓.
Diagram4(Dividinganddoublingforintervals)
Thistakesourlistto𝑓, !!𝑓, !
!𝑓, !
!𝑓, !"
!𝑓, 2𝑓correspondingtothenotes𝐴,𝐵,𝐷,𝐸 and 𝐺as
whitekeysonthepiano,withthepitchoffrequency2𝑓the𝐴ofthenextoctave.
Diagram5(Thesetoffiveintervals)
32𝑓
𝑓
Base
2𝑓
94𝑓
43𝑓
98𝑓
𝑓
Base
2𝑓
169𝑓
43𝑓
98𝑓8
9𝑓
32𝑓
𝑓
Base
2𝑓
169𝑓
43𝑓
98𝑓
32𝑓
𝐴 𝐵 𝐷 𝐸 𝐺
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Thissimplesetoffiveharmoniousnotesisanexampleofwhatisknownasapentatonicscale.ThesescalesareoftenthebasisofpleasantcatchymelodiessimplybecausetheyareconstructedfromthenaturallyoccurringnaturalPythagoreanintervals.Infactitisofnosurprisethatpentatonicscalesweredevelopedindependentlybymanyworldwidecivilisations.
Therearealsoheptatonicorsevennotescales,andthemajorandminorscalesofWesternmusicareperhapsthemostcommonlyknown.Heptatonicscalescomeindifferentformscalledmodes,eachhavingtheircharacteristicpatternofwholeandhalfsteps,andwereinusebythemedievalchurchandstillexistinsomeEuropeanfolkmusic.Forourpurposeswecouldconstructaheptatonicscalebyaddingtwoharmoniousnotesinthefollowingway.
Pythagoras’sshowedthatpitcheswithfrequenciesthatwereinsimpleratiowitheachothertendedtobeharmonious.Goingupanother!
!stepfrom𝐵bringsthepitchto!"
!"
abovethebasenote.Thenumber!"!"isapproximately!"
!"= !
!andso,toleratingthis
compromise,thefrequencyofaharmoniouspitchworthusingwouldbeoneandaseventhtimeshigherthanthatofnote𝐴.Thatis,!
!𝑓.Thisnewnotewouldsitas𝐶in
between𝐵and𝐷.Wecouldalsoapplythesamefractiontothefrequencyof𝐷(to!!𝑓),
creating𝐹atafrequencyof!!
× !!𝑓 = !"
!"𝑓.Thesecalculationsthen,takentogether,
explaintheoriginalconstructionofthewhitekeysshownhere(thepentatonicnotesarehighlightedinred).
Diagram6showsthesevennotesandfrequenciesofourheptatonicscale.
Diagram6(Sevennotes)
Theratioofthetwopitchfrequenciescorrespondingtothenotes𝐴and𝐵is!!orabout
1.11.Likewisewecouldconsidertheratiosofconsecutivenotesinourentirescaleincludingintothenextoctave.
Table1onthenextpagesummarisestheresults.
𝑓
Base
2𝑓
169𝑓
43𝑓
98𝑓
32𝑓
𝐴 𝐵 𝐷 𝐸 𝐺𝐹
3221 𝑓
87 𝑓
𝐶
5
Pair 𝐵,𝐴 𝐶,𝐵 𝐷,𝐶 𝐸,𝐷 𝐹,𝐸 𝐺,𝐹 𝐴,𝐺
Ratio 98
6463
76
98
6463
76
98
Approximate 1.125 1.016 1.167 1.125 1.016 1.167 1.125
Table1(Intervalfrequencies)
Table1measuresstepsizesfromintervaltointervalandalthoughtheintervalsthemselvesareharmonious,itmightbebeneficialtolookforotherharmoniousintervalstoevenouttheclimbuptheoctave.Indeedthisintervalinconsistencymostlikelypromptedtheconstructionofhalfstepstofillinthelargerscalegaps,buthowcouldthesehalfstepsbeengineered?
Inmodernpianostherearetwelveintervals,calledsemitones,peroctavewithblackkeys(knownassharpsandflats,butwe’llcallthemsharpsinthisdiscussion)clumpedineitherpairsortriplesasshownhere.
Diagram7(Apianooctave)
Anoctaverunsfromanykey,say𝑨,uptoanequivalentlynamedkey,𝑨,inthesamepositionrelativetothepositionoftheblackkeys.Gettingfromthelower𝐴tothehigher𝐴takestwelvesteps.Thustherearealwaysfiveblackkeysperoctavenomatterwhatthestartingkeyis.
Notethattherearenoblackkeysbetweenthepairofnotes𝐵and𝐶 andthepairofnotes𝐸and𝐹andthingsbegintomakesenseonceyoulookbackatthetable.Thetwostepsizes,both1.016,andhighlightedinred,arethesmallestintheoctaveandthusitseemsreasonabletospliteachoftheotherfivestepsintotwobyapplyingabitofPythagoreanlogic.
𝑨 𝑩 𝑫 𝑮𝑬
𝐹#
𝑪 𝑭
𝐺#𝐴# 𝐷#𝐶#
𝑨
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Forexample,thestepbetween𝐶 and𝐷is1.167,soanewnotewithapitchfrequencymorethanthatof𝐶butlessthanthatof𝐷needstobefound.Supposeweincreasethefrequencyofnote𝐵byafactorof!
!to!
!× !!𝑓 = !"
!"𝑓.Thisfrequencyisverycloseto!
!𝑓
andthebenefitofusing!!𝑓isthatitinvolvesasimplerratioandyetstillliesbetween
thefrequencyof𝐶and𝐷.Thusweadopt!!𝑓asthefrequencyofthenewnote𝐶#.
Exactlythesamereasoningappliesinordertosplitthegapbetween𝐹and𝐺.Increasingthefrequencyof𝐸by!
!to!
!× !!𝑓 = !"
!"𝑓 ≈ !
!𝑓sothefrequencyofanewnote𝐹#becomes
!!𝑓.
‘Worthycandidate’frequenciesareonesthatgeneratefairlyevenstepsyetatthesametimeinvolvefractionsofthebasefrequency𝑓whosenumeratorsanddenominatorsareminimallysmall.Thetrickistofindabalancebetweenthosetwocompetingqualities,andperhapstherearenobestanswers.
Ifweapplytheratio!!tothenote𝐸withfrequency!
!𝑓wecanconstruct𝐺#as!"
!𝑓.
Increasingthefrequencyof𝐺#by!!andhalvingconstructsthefrequencyfor𝐴#as
!!× !"
!𝑓 ÷ 2 = !"#
!"#𝑓 ≈ !"
!"𝑓.Finallyfor𝐷#wecanincreasethefrequencyof𝐴by!!
!sothat
thefrequencyof𝐷#becomes!!!×𝑓 = !!
!𝑓.
Wecanrepresentallofthesefrequenciesaboveandbelowthepianokeys
Diagram8(Apianooctavewithintervalsshown)
3221𝑓
𝑨 𝑩 𝑫 𝑮𝑬
𝐹#
𝑪 𝑭
𝐺#𝐴# 𝐷#𝐶#
118𝑓
1716𝑓
158𝑓
74𝑓
54𝑓
98𝑓
87𝑓
43𝑓
32𝑓
169𝑓
𝑨2𝑓𝑓
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Table2showsthecompletesetofstepsizesacrossallsemitonesofanoctavewhenthegapsarefilledbytheproceduredescribed.Thedecimalapproximationsaregiventomakecomparisonseasier.
Step 𝐴 𝑡𝑜 𝐴# 𝐴# 𝑡𝑜 𝐵 𝐵 𝑡𝑜 𝐶 𝐶 𝑡𝑜 𝐶# 𝐶# 𝑡𝑜 𝐷 𝐷 𝑡𝑜 𝐷#
StepRatio 1716
1817
6463
3532
1615
3332
Approximate 1.0625 1.0588 1.0159 1.0938 1.0667 1.0313
StepRatio 𝐷# 𝑡𝑜 𝐸 𝐸 𝑡𝑜 𝐹 𝐹 𝑡𝑜 𝐹# 𝐹# 𝑡𝑜 𝐺 𝐺 𝑡𝑜 𝐺# 𝐺# 𝑡𝑜 𝐴
Ratio 1211
6463
147128
6463
145128
1615
Approximate 1.0909 1.0159 1.1484 1.0159 1.1328 1.0667
Table2(Stepsizesbetweenintervals)
Theoctaveisthemainbuildingblockofthemodernpiano.Thefirstkeyofthe88keysofthekeyboardisan𝐴withpitchfrequency27.5Hz.Thefrequencyofthenext𝐴is55Hz,andthenextafterthat110HzandsoonuptotheseventhAat3520Hz.
Choosinganyparticularnote,saythefifthA(the49thkey)alongthekeyboardwithpitchfrequencyof440Hz,wecouldbeginconstructkeyfrequenciesforeachofthesemitonesintheoctave.Thusfor𝐴#,thecorrespondingfrequencyis467.5Hz,forBthefrequencybecomes495Hz,for𝐶,502.9Hz,etc.rightthroughto825Hz.Thetwelvestepsizes,correcttotwodecimalplaces,rangefromabout1.02toabout1.15,andsowhilenotexactlythesame,roughlyfollowageometricprogression.Infact,sinceeveryoctaveonthepianostepsupinthesamemanner,thefrequenciesofthepitchesacrosstheentirekeyboardare,atleastapproximately,ingeometricprogression.
Withoutgoingintotoomuchdetail,theslightvariationinthestepsizesmeantthatanymusicalpiecethatwasplayedindifferent‘keys’(thatistosayusingoctavescalesthatbeginwithdifferentbasekeys)soundeddifferently.Forexample,theratiostepsinvolvedbyplayingaprogressionof12semitonesfrom𝐴tothenext𝐴areapproximately1.00,1.06,1.06,1.02,1.09,1.07,1.03,1.09,1.02,1.15,1.02,1.05and1.07.However,theratiostepsofaprogressionofsemitonesfrom𝐸tothenext𝐸become1.09,1.02,1.15,1.02,1.05,1.07,1,1.06,1.06,1.02,1.09,1.00,1.06,1.06and1.02.Thesounddifferencescausedbyanytranspositionofkeysbecomesimmediatelyobvious,andsoitbecamethepracticethatmusicalpieceswerewrittenforcertainspecifiedbasekeys.
Thisallchangedwiththeadventofequaltemperedtuning.
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Moderntuning
Anytruetuningregimeisoneinwhichintervalsareexpressedastheratiooftwointegers.Moreconsonantsoundsoccurwhentheseratiosinvolvesmallintegers.AccordingtoJMurrayBarbour2atemperamentisamodificationofatuningandirrationalnumbersarerequiredtoexpressratiosofsomeorallofitsintervals.
Asmusicalstylesdeveloped,specificfactorsandharmonictendenciesledtothegradualadoptionofequaltemperamentpianotuning.Inanequaltemperedregimeall12semitonesinanoctavehavefrequenciesinstrictgeometricalprogressionbasedonthecommonratio𝑟 = 2!" ≈ 1.059463.(Agoodrationalapproximationtothisnumberisthefraction!"#
!"#.)Perfectharmonygavewaytopracticality.Musicalpiecescouldbe
playedinanykeywithidenticaltonalrelativities.ThecomposerJSBachwho,intheearly18thcentury,wrotetwosetsof24preludesandfugues,oneineachmajorandminorkey,exhaustivelydemonstratedthis.
Thegeometricratioisappliedbothforwardsandbackwardsfromwhatisreferredtoasthestandardpitch,whichforthepianoisthepitchoffrequency440Hzcorrespondingtothenote𝐴directlyabovemiddle𝐶.Theprogressionclimbstotherightfrom𝐴440usingtheratio𝑟 = 2!" anddescendstotheleftfrom𝐴440using𝑟 = !
!!" .Inthissenseit
isopenendedalbeitwiththeobviousphysicallimitationsoftheinstrumentandtheear.
WecancomparethefrequenciesofPythagoreantuningwithtemperedtuningfortheoctavewithbasenoteA440
Note A A# B C C# D
Tuning 440 467.5 495.0 502.9 550.0 586.7
Tempered 440 466.2 493.9 523.3 554.4 587.3
Note D# E F F# G G# A
Tuning 605.0 660.0 670.5 770.0 782.2 825.0 880
Tempered 622.3 659.3 698.5 740.0 784.0 830.6 880
Table3(Comparisonofintervaltuningandequaltemperedfrequencies)
Thenoticeablecomparativefrequencydifferencesarethoseassociatedwiththenotes𝐶,𝐷#,𝐹and𝐹#butotherdifferencesareminimal.Theremightwellbeotherwaystoconstructtuningintervalsthatreducethefourroguedifferencesbutthepriceoftemperedtuninghastobepaidsomewhere,andsoit’sakintothemousechasingitstail.Afterall,iftherewasawaytodoit,itwouldhavebeendonebynow.
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Furtherreading
1.TheBattleBetweenImpeccableIntonationandMaximizedModulation
(TimTrue,CedarvilleUniversity,Ohio,USA)
2.J.MurrayBarbour,TuningandTemperament:AHistoricalSurvey
(EastLansing:MichiganStateCollegePress,1953)
3.https://en.wikipedia.org/wiki/Piano_key_frequencies
