1

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

2

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

3

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

4

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

6

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