ROCHESTER PHY 103 - Lecture Notes - Scales

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ScalesSlide 2Diatonic scaleHow universal is this scale?Neanderthal fluteOvertones of the stringSlide 7Pythagorus and the circle of fifthsPythagorean scale (continued)Slide 10Pythagorean scale (continued)The circle doesn’t exactly closeGetting all the notesCircle of fifthsEquivalency over the octaveSlide 16Perfect fifths and thirdsWhat do they sound like?What do the waveforms look like?Perfect intervals and periodic waveformsMajor and minor chordsTriadsJust temperamentThe Just ScaleListening example (from Butler)Meantone temperamentJust intonationEqual temperament12 tonesEqual TemperamentEqually spaced by multiplicative factorsListening examples, log and linear spacingsTempered scale octavesFrequenciesCentsTurning notes and cents into frequenciesEqual temperamentAttempts at new scales Harry Partch’s tuningSlide 39Partch’s tonality diamondPartch’s and Partch-era compositionsMicrotonal musicSlide 4319 tone tempered scaleAdaptive tuningsGamelan tonesPierce ScaleRagasIndian music and pitchAlhaiya bilaval time: late morning –major scaleAbhogi - Early nightRecommended ReadingScalesPhysics of Music PHY103 Image from www.guitargrimoire.com/Diatonic scaleW W H W W W HMajor scaleHow universal is this scale?9,000 Year Old Chinese Flutes JUZHONG ZHANG et al. Nature 1999Excavations at the nearly Neolithic site of Jiahu in Henan Province China have found the earliest completely playable tightly dated multinote musical instrumentsOne of these flutes can be played: check the file K-9KChineseFlutes.ramNeanderthal flute•This bear bone flute, found in Slovenia in 1995, is believed to be about 50,000 years old. Differing hole spacing suggest a scale with whole and half note spacingsOvertones of the stringDiatonic scale•Notes that are octaves apart are considered the same note. x2 in frequency•An octave+ 1fifth is the 3rd harmonic of a string, made by plucking at (1/3) of the string•If f is the frequency of the fundamental then the third harmonic is 3fW W H W W W H•Drop this down an octave (divide by 2) and we find that a fifth should have a frequency of 3/2fFifthMajor scalePythagorus and the circle of fifthsImage from Berg + StorkEvery time you go up a fifth you multiply by 3/2. Every time you go down an octave you divide by 2.Pythagorean scale (continued)third fourthThe major third is up a fifth 4 times and down an octave twice 4 2463 1 31.26562 2 2� � � �� � = =� � � �� � � �The fourth is up an octave and down a fifth2 42 1.33333 3� ��� = =� �� �Pythagorean scale (continued)sixthThe sixth is up a fifth 3 times and down an octave once 3 1343 1 31.68752 2 2� � � �� � = =� � � �� � � �The minor third is down a fifth three times and up and octave twice. 35232 22 1.18523 3� �� � = =� �� �minor thirdPythagorean scale (continued)1 2433234324632octavefifthfourththirdsixthThe musical scale related to pure fractions! Mathematical beauty is found in music.frequencyThe circle doesn’t exactly closeWhat happens if you go up by a fifth 12 times? The circle of fifths leads us to expect that we will get back to the same note. (3/2)12=129.746337890625 An octave is 2 times the base frequency7 octaves above is 27 times the base frequency. 27=128.0000However 27≠ (3/2)12Getting all the notesOne fifth has to be bad. The Wolf fifth.So the subdominant chords can be played the bad fifth is placed between C# and ACircle of fifthsGuides harmonic structure and key signatures for baroque, romantic and much folk music.Keeping chords in tune in one key is sufficient for much of baroque and folk music.Equivalency over the octaveWe associate notes an octave apart as the same noteIt is difficult to determine the octave of a note. Some people sing an octave off by mistake.Other animals (birds) with not recognize songs played in a different octaveplucked is a major chord2 x 2/3fifth of long string2 x 4/5third of longest stringPythagorean third: In frequency 34/26 In length2x 26 /34Perfect fifths and thirdsWhat’s so perfect about the perfect fifth?•We started with a harmonic 3f and then took it down an octave to 3f/2 Frequency ratio 1:3/2Is the Pythagorean third perfect?•Fifth harmonic 5f is 2octaves +a major third•Take this down two octaves and we find a major third should be 5f/4 = 1.25f. Frequency ratio 1:5/4•However the Pythagorean major third is 1:(34/26)=1.2656What do they sound like?I used the generate tones function in the old AuditionSequence: Sum of 2 sines prefect, interval followed by Pythagorean intervalThen sum of 2 sets of 5 harmonics, perfect inteval followed by Pythagorean interval Perfect fifth with frequency ratio 1:3/2Wolf fifth (between C# and A ) 1:218/311=1.4798Perfect third 1:5/4Pythagorean third 1:34/26=1.2656What do the waveforms look like?One of these is a perfect third, the other is the Pythagorean third. Which one is which?Perfect intervals and periodic waveforms•A sum of perfect harmonics adds to a periodic wave --- for sum of frequencies that are integer ratios - Fourier series•However 5/4 (prefect major third) is not an integer times 1 --- so why does the sum of two frequencies in the ratio 1:5/4 produce a periodic waveform?Major and minor chordsPerfect major third and fifthFrequency ratios 1:5/4:3/2For the triadMinor thirdMajor thirdMinor chord: set the major third.Divide the G frequency by 5/4 so that the G and E give a perfect third.3/2*4/5=6/5Frequency ratios 1:6/5:3/2 for the triadMajor MinorTriads•Major triad with perfect major third and fifth 1:5/4:3/2 The minor third in the triad has ratio 3/2 divided by 5/4 = 6/5 and so is perfect or true.•However other thirds and fifths in the scale will not be true.Just temperamentMore than one tuning system which use as many perfect ratios as possible for one diatonic scale.Image From Berg and StorkThe Just ScaleTwo different whole tonesT1 = 9/8 = 1.125T2 = 10/9 = 1.111One semitoneS = 16/15 = 1.067Listening example (from Butler)a) An ascending major scale beginning on C4, equal temperament followed by the chord progression I-IV-V7 in Cmajor and in F# major.b) The same scale but with Pythagorean tuning, followed by the same chord progressionc) The same scale and chords, quarter comma meantone temperament (errors distributed in a few tones)d) Just intonationMeantone temperamentIn general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect


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