GustavGustav HolstHolst (1874-1934)(1874-1934)

TheThe PlanetsPlanets -- Saturn,Saturn, thethe BringerBringer ofof OldOld AgeAge MusicalMusical IntervalsIntervals llReferRefer toto TableTable 10.310.3 inin thethe text.text. llIntervalsIntervals areare ratiosratios ofof frequenciesfrequencies oror lengths.lengths. MusicalMusical IntervalsIntervals llOctaveOctave ((naturalnatural interval)interval) uuStringString LengthLength RatioRatio == 2/12/1 == 2.002.00 oror ½½ == 0.500.50 uuFrequencyFrequency RatioRatio alsoalso == 2.002.00 oror ½½ == 0.500.50 MusicalMusical IntervalsIntervals MusicalMusical IntervalsIntervals llPerfectPerfect FifthFifth uuSevenSeven semitonessemitones uuStringString LengthLength RatioRatio == 3/23/2 =1.50=1.50 oror 2/32/3 == 0.66670.6667 uuFrequencyFrequency RatioRatio alsoalso == 1.501.50 oror 2/32/3 == 0.66670.6667 MusicalMusical IntervalsIntervals MusicalMusical IntervalsIntervals llPerfectPerfect FourthFourth uFiveFive semitonessemitones uStringString LengthLength RatioRatio == 4/34/3 == 1.331.33 oror ¾¾ == 0.7500.750 uStringString LengthLength RatioRatio == 1.3331.333 oror ¾¾ == 0.7500.750 MusicalMusical IntervalsIntervals PythagoreanPythagorean ScaleScale llPythagorasPythagoras (582-507(582-507 BC)BC) llRatiosRatios forfor intervals:intervals: 1.000,1.000, 1.333,1.333, 1.500,1.500, 2.0002.000 (unison,(unison, fourth,fourth, fifth,fifth, octave)octave) llUsedUsed thesethese ratiosratios toto constructconstruct aa mathematicalmathematical scale.scale. PythagoreanPythagorean ScaleScale llUsedUsed stringstring lengthslengths sincesince frequenciesfrequencies werewere notnot known.known. llHowHow dodo youyou dividedivide anan octaveoctave (1.00(1.00 toto 2.00)2.00) intointo 88 equalequal parts?parts? llOrOr inin termsterms ofof frequenciesfrequencies anan intervalinterval suchsuch asas 220220 –440–440 Hz?Hz? Pythagoras’sPythagoras’s RuleRule llMultiplyMultiply oror dividedivide anan existingexisting lengthlength (ratio)(ratio) byby 3/23/2 (=1.500),(=1.500), factorfactor ofof fifths.fifths. llIfIf thethe resultresult lieslies betweenbetween 11 andand 2,2, leaveleave itit asas itit is.is. Pythagoras’sPythagoras’s RuleRule llIfIf thethe answeranswer isis lessless thanthan 1,1, doubledouble itit (up(up anan octave)octave) llIfIf thethe answeranswer isis greatergreater thanthan 1,1, halvehalve itit (down(down anan octave)octave) Pythagoras’sPythagoras’s RuleRule StepStep #1#1 ll StartStart withwith DD44 == 1.0001.000 (293.7(293.7 Hz)Hz) ll MultiplyMultiply DD44== 1.001.00 byby 1.501.50 toto getget 1.51.5 (the(the fifth)fifth) whichwhich

isis AA44 (440(440 Hz).Hz). Pythagoras’sPythagoras’s RuleRule StepStep #2#2 ll StartStart withwith DD44 == 1.0001.000 (293.7(293.7 Hz)Hz) ll DivideDivide DD44 == 1.001.00 byby 1.501.50 toto getget 0.6660.666 andand doubledouble toto getget 1.3331.333 (the(the fourth)fourth) whichwhich isis GG44 (392(392 Hz).Hz). Pythagoras’sPythagoras’s RuleRule StepStep #3#3 ll StartStart withwith AA44 == 1.5001.500 (440(440 Hz)Hz) ll MultiplyMultiply AA44 == 1.501.50 byby 1.501.50 toto getget 2.2502.250 andand halvehalve toto getget 1.1251.125 (the(the majormajor second)second) whichwhich isis EE44 (229.6(229.6 Hz).Hz). Pythagoras’sPythagoras’s RuleRule StepStep #4#4 ll StartStart withwith GG44 == 1.3331.333 (292(292 Hz)Hz) ll DivideDivide GG44 == 1.3331.333 byby 1.501.50 toto getget 0.888880.88888 andand doubledouble toto getget 1.7771.777 (the(the minorminor seventh)seventh)

whichwhich isis CC55 (523.3(523.3 Hz).Hz). PentatonicPentatonic ScaleScale llTheseThese firstfirst 55 notesnotes D,D, E,E, G,G, A,A, C,C, andand DD againagain constituteconstitute thethe 5-note5-note ChineseChinese scalescale calledcalled pentatonicpentatonic (5(5 tones)tones) llGreekGreek scalesscales hadhad 77 notesnotes calledcalled septatonicseptatonic Pythagoras’sPythagoras’s RuleRule StepStep #5#5 ll StartStart withwith EE44 == 1.1251.125 (229.6(229.6 Hz)Hz) ll MultiplyMultiply EE44 == 1.1251.125 byby 1.501.50 toto getget 1.68751.6875 (the(the majormajor

sixth)sixth) whichwhich isis BB44 (493.9(493.9 Hz).Hz). Pythagoras’sPythagoras’s RuleRule StepStep #6#6 ll StartStart withwith CC55 == 1.7771.777 (523.3(523.3 Hz)Hz) ll DivideDivide CC55 == 1.7771.777 byby 1.501.50 toto getget 1.18511.1851 (the(the minorminor third)third)

whichwhich isis FF44 (349.2(349.2 Hz).Hz). Pythagoras’sPythagoras’s RuleRule StepStep #7#7 ll StartStart withwith GG44 == 1.3331.333 (292(292 Hz)Hz) ll MultiplyMultiply GG44 == 1.3331.333 byby 1.501.50 toto getget 2.002.00 (the(the octave)octave)

whichwhich isis DD55 (587.3(587.3 Hz).Hz). MusicalMusical IntervalsIntervals

TwoTwo differentdifferent ratiosratios betweenbetween adjacentadjacent notes;notes; semitonesemitone andand tone.tone. DorianDorian ModeMode ToneTone == TT andand SemitoneSemitone == ss

TT ss TT TT TT ss TT DD EE FF GG AA BB CC DD ModesModes llScalesScales basedbased onon thethe whitewhite keyskeys ofof thethe pianopiano llSinceSince therethere areare sevenseven differentdifferent namednamed keyskeys A,A, B,B, C,C, D,D, E,E, F,F, G,G, therethere areare sevenseven modes.modes. IonianIonian ModeMode oror MajorMajor ScaleScale

TT TT ss TT TT TT ss CC DD EE FF GG AA BB CC AoelianAoelian ModeMode oror MinorMinor ScaleScale

TT ss TT TT ss TT TT AA BB CC DD EE FF GG AA MajorMajor andand MinorMinor ScalesScales llMajor:Major: J.S.J.S. BachBach “Well“Well TemperedTempered ClavierClavier BookBook II”II” PreludePrelude II inin CC major.major. (Track(Track #1)#1) llMinor:Minor: PreludePrelude IVIV inin C#C# minor.minor. (Track(Track #7)#7) SevenSeven ModesModes llCC == IonianIonian (major(major scale)scale) llDD == DorianDorian llEE == PhrygianPhrygian (Spanish(Spanish oror Oriental)Oriental) llFF == LydianLydian (funny,(funny, comical)comical) llGG == MixolydianMixolydian llAA == AeolianAeolian (minor(minor scale)scale) llBB == LocrianLocrian (not(not used)used) PhrygianPhrygian ModeMode (starts(starts onon E)E) VaughnVaughn WilliamsWilliams FantasiaFantasia onon aa ThemeTheme byby ThomasThomas TallisTallis MathematicalMathematical BasisBasis forfor ScalesScales MultiplesMultiples ofof 1.5001.500 generategenerate thethe samesame 88 notenote scalescale thatthat waswas foundfound byby musiciansmusicians toto bebe thethe “right”“right” onesones forfor aa musicalmusical scale.scale. MathematicalMathematical BasisBasis forfor ScalesScales llTheThe fifthfifth isis aa multiplemultiple ofof 1.500.1.500. llTheThe fifthfifth isis thethe 33rdrd

harmonic.harmonic. (3h(3h11/2h/2h11 == 1.50)1.50) MathematicalMathematical BasisBasis forfor ScalesScales l ThirdThird harmonicharmonic ofof AA isis E.E. l ThirdThird harmonicharmonic ofof EE isis B.B. l ThirdThird harmonicharmonic ofof BB isis F#.F#. l ofof F#F# isis C#,C#, ofof C#C# isis G#,G#, ofof G#G# isis D#,D#, ofof D#D# isis A#,A#, ofof A#A# isis F,F, ofof FF isis C,C, ofof CC isis G,G, ofof GG isis D,D, andand ofof DD isis backback toto A.A. l ThisThis isis thethe entireentire chromaticchromatic scale!scale! AestheticAesthetic BasisBasis forfor ScalesScales l TheThe 33rd harmonicharmonic isis thethe lowestlowest andand strongeststrongest harmonicharmonic thatthat isis notnot anan octave.octave. l StringedStringed instrumentsinstruments havehave thethe 33rd harmonic.harmonic. l AA scalescale basedbased onon 33rd harmonicsharmonics shouldshould bebe thethe mostmost “natural”“natural” oror pleasing.pleasing. Equal-TemperedEqual-Tempered ScaleScale l 77 GreekGreek modesmodes oror ChurchChurch modesmodes useuse allall ofof thethe whitewhite keyskeys l CompriseComprise 77 combinationscombinations ofof T=toneT=tone andand s=semitones=semitone sequencessequences l UsingUsing aa particularparticular modemode requiresrequires thethe scalescale toto startstart onon oneone andand onlyonly oneone note.note. l NeedNeed toto placeplace semitonessemitones anywhere.anywhere. Equal-TemperedEqual-Tempered ScaleScale

llE-FE-F isis aa semitonesemitone andand soso isis B-C.B-C. llAddAdd 55 moremore (the(the blackblack keys).keys). Equal-TemperedEqual-Tempered ChromaticChromatic ScaleScale

ll1212 equallyequally spacedspaced semitonessemitones 12 __ llÖÖ22 »» 1.059463...1.059463... MajorMajor ScalesScales CC MajorMajor

TT TT ss TT TT TT ss CC DD EE FF GG AA BB CC MajorMajor ScalesScales

FF MajorMajor TT TT ss TT TT TT ss FF GG AA BBbb CC DD EE FF

Studio - Scales.sng MajorMajor ScalesScales QualityQuality llHappyHappy llStrongStrong llSereneSerene MinorMinor ScalesScales AA MinorMinor

TT ss TT TT ss TT TT AA BB CC DD EE FF GG AA MinorMinor ScalesScales

AA MinorMinor TT ss TT TT ss TT TT AA BB CC DD EE FF GG AA Studio - Scales.sng MinorMinor ScalesScales QualityQuality llSadSad llErieErie llTroublingTroubling DiatonicDiatonic vsvs.. ChromaticChromatic llDiatonicDiatonic ScalesScales -- majormajor andand minorminor scalesscales llDiatonicDiatonic notesnotes -- notesnotes ofof aa particularparticular scalescale llChromaticChromatic notesnotes -- thethe otherother notesnotes EqualEqual TemperedTempered vsvs.. PythagoreanPythagorean llPythagoreanPythagorean -- thethe whitewhite keyskeys determineddetermined byby thethe rulerule ofof 3/2.3/2. llEqualEqual TemperedTempered -- 1212 evenlyevenly spacedspaced intervalsintervals byby thethe factorfactor ofof 12 __ ÖÖ22 »» 1.059463...1.059463... llTheyThey areare similarsimilar butbut notnot thethe same!same! EqualEqual TemperedTempered vsvs.. PythagoreanPythagorean Note Equal Tempered Pythagorean C 261.6 260.7 D 293.7 293.3 E 329.6 330.0 F 349.2 347.7 G 392.0 391.1 A 440.0 440.0 B 493.9 495.0