The Origin of the Chromatic ScaleOutline• Revisit Lissajous Shapes• Pythagoras’ Hypothesis– The Magic of Integer Ratios• The Circle of Fifths and the Chromatic Scale• Imperfect Tuning and Compromise:– Equal TemperamentLissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-11Lissajous Shapesxy1-1-111stharmonicand 1stharmonic1stharmonicand 2ndharmonic344.0 Hzand 486.6 Hz1stharmonicand 3rdharmonicLissajousShapesHarmonicVisualizationPythagoras’ Hypothesis• The Magic of Whole Number Ratiosl1l2monochordPythagoras’ Hypothesis• The Magic of Whole Number Ratiosl1l21, 2, 3...Pythagoras’ Hypothesis• The Magic of Whole Number Ratiosl1l2=l2l132diapentePythagoras’ Hypothesis• The Magic of Whole Number Ratiosl1l2=l2l121diapsondiapente=l2l132diatesaron=l2l143Pythagoras’ Hypothesis• The Magic of Whole Number Ratiosl1l2=l2l121diapsondiapente=l2l132diatesaron=l2l143octaveperfect 5thperfect 4thRatios, Harmonics and Intervals• Work out octave, fifth, third and fourth on boardMake a scale with octaves101001000100000246810121 octavefrequency (Hz)Make a scale with fifths1010010001000002468101212 fifths = 7 octavesfrequency (Hz)Slide the some of the fifths down one octave10100100010000024681012frequency (Hz)Slides fifths down again10100100010000024681012frequency (Hz)Etc....etc.10100100010000024681012Chromatic Scale10100100002468101214frequency (Hz)octaveChromatic Scale10100100002468101214frequency (Hz)octavef2/f1= 2.0272912Oops!Chromatic Scale10100100002468101214frequency (Hz)1.067871.053501.067871.05350Imperfect Tuning10100100002468101214frequency (Hz)octave f2/f1= 2.0272912No combination of integer ratios (such as 3/2)can produce an EXACT octave interval1.067871.05350PythagoreanEqualTemperament110.000104.41499.11292.81388.09982.50078.31173.33369.60966.07561.87558.73355.000110.000103.82697.99992.49987.30782.40777.78273.41669.29665.40661.73558.27055.000A1• Lissajous Shapes help us see harmony• Pythagoras’ Hypothesis– Integer ratios are harmonious• A sequence of fifths producesthe twelve pitches of theChromatic Scale• Imperfect Tuning (7 octaves ~ 12 fifths) and Compromise (Equal
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