Lecture 6A_HW_4_Solution

Lecture 6A_HW_4_Solution - EEL 4930 Audio Engineering...

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Unformatted text preview: EEL 4930 Audio Engineering Homework #4, Solution Lecture 6A 1. Fill in the missing column entries in the table below. JUST SCALE Note C C# D D# E F F# G G# A A# B C unison semitone major second minor third major third fourth tritone fifth minor sixth major sixth minor seventh major seventh octave Fraction 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 16/9 15/8 2/1 2.0000 Ratio 1.0000 1.0667 1.1250 1.2000 1.2500 1.3333 1.4063 1.5000 1.6000 1.6667 1.7778 Cents 0.0 111.7 203.9 315.6 386.4 498.0 590.2 702.0 813.7 884.4 996.1 1088.3 1200.0 Fraction 1/1 256/243 9/8 32/27 81/64 4/3 729/512 3/2 128/81 27/16 16/9 243/128 2/1 PYTHAGOREAN Ratio 1.0000 1.0535 1.1250 1.1852 1.266 1.3333 1.4238 1.5000 1.5802 1.6875 1.7778 1.8984 2.0000 Cents 0.0 90.2 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 Error* 0.0 -21.5 0.0 -21.5 21.5 0.0 21.5 0.0 -21.5 21.5 0.0 21.5 0.0 Ratio 1.0000 1.0595 1.1225 1.1892 1.2599 1.3348 1.4142 1.4983 1.5874 1.6818 1.7818 1.8877 2.0000 TEMPERED Cents 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Error* 0.0 -11.7 -3.9 -15.6 13.7 2.0 9.8 -2.0 -13.7 15.6 3.9 11.7 0.0 * Error in cents, compared to Just scale To convert to cents 1200 ln(ratio)/ln(2) Major third in Just is 1200 ln(5/4)/ln(2) = 386.3 cents Major third in Pyth is 1200 ln(81/64)/ln(2) = 1200 ln(1.266)/ln(2) = 407.8 cents Tritone in Just is 1200 ln(45/32)/ln(2) = 590.2 cents Tritone in Tempered is 1200 ln(26/12)/ln(2) = 1200 ln(1.4142)/ln(2) = 600.0 cents Minor sixth in Just is 1200 ln(8/5)/ln(2) = 813.7 cents Minor sixth in Pyth is 1200 ln(128/81)/ln(2) = 729.2 cents Difference = 813.7 - 729.2 = -21.5 cents Major sixth in Just is 1200 ln(5/3)/ln(2) = 884.4 cents Major sixth in Pyth is 1200 ln(27/16)/ln(2) = 905.9 cents EEL 4930 Audio Engineering Lecture 6A 2. Instrument builders in medieval times used the “Rule of 18” to determine where the frets on a guitar or other fretted instrument should go. The rule was to place the first fret at 1/18th of the distance from the nut to the bridge, the second fret at 1/18th of the distance from the first fret to the bridge, and so on. Ideally, the note at the twelfth fret should differ from the note of the unfretted string by one octave (frequency ratio = 2:1). By how many cents does the note at the 12th fret of such a medieval instrument differ from the ideal value? (18/17)12 = 1.0588, or 98.95 cents, whereas 21/12 = 1.0959 or 100.0 cents. The error per fret is -1.045 cents. The error at the 12th fret is -1.045*12 or -12.5 cents. 3. If the note A has a frequency of 220 Hz, what would the frequency be of a note E that is an interval of a fifth above it (a) in the Pythagorean scale? (b) … in the modern tempered scale? (c) … using the Rule of 18? a) 220*3/2 = 330.00 cents b) 220*25/12 = 329.63 cents c) 220*(18/17)5 = 328.24 cents ...
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