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L 8-9 Musical Scales, Chords , and Intervals, The Pythagorean and Just Scales History of Western Scales A Physics 1240 Project by Lee Christy 2010 References to the History Musical Intervals (roughly in order of decreasing consonance) Name of Interval Notes (in key of C major) Pythagorean Frequency Ratios Just Frequency Ratios # Semitones (on equaltempered scale) Octave C↔C 2 2 12 Fifth C↔G 3/2 6/4 = 3/2 7 Fourth C↔F 4/3 Major Third C↔E 81/64 Minor Third E↔G Major Sixth C↔A Minor Sixth E↔C Tonic C 5 5/4 4 3 27/16 9 8 1 4/4 = 1 none * a semitone interval corresponds to a frequency difference of about 6% C D E F G A B C * The white notes of the piano give the seven notes of the C-major diatonic scale. The ratio of the frequency of C4 to that of C2 is: a) 2 b) 3 c) 4 d) 8 One octave of the diatonic scale including the tonic and the octave note contains: a) 5 notes b) 6 notes c) 7 notes d) 8 notes One octave of the chromatic scale (including the octave note) contains: a) 8 notes b) 10 notes c) 11 notes d) 12 notes e) 13notes A musical scale is a systematic arrangement of pitches Each musical note has a perceived pitch with a particular frequency (the frequency of the fundamental) Going up or down in frequency, the perceived pitch follows a pattern One cycle of pitch repetition is called an octave. The interval between successive pitches determines the type of scale. Intervals 12-tone scale (chromatic) 8-tone scale (diatonic) Note span C-C C - C# C-D C - D# C-E C-F C - F# C-G C - G# C-A C - A# C-B C3 - C4 C3 - E4 Interval Frequency ratio unison 1/1 semitone 16/15 whole tone (major second) 9/8 minor third 6/5 major third 5/4 perfect fourth 4/3 augmented fourth 45/32 perfect fifth 3/2 minor sixth 8/5 major sixth 5/3 minor seventh 16/9 (or 7/4) major seventh 15/8 octave 2/1 octave+major third 5/2 Consonant intervals Overlapping harmonics tonic 120 240 octave fifth fourth 360 480 240 480 180 160 600 360 M third 150 300 m third 144 288 840 720 900 640 800 450600 750 900 576 720 1080 960 480 432 960 720 540 320 720 864 1080 960 1050 1008 Dissonant intervals Perceived when harmonics are close enough for beating harmonic series Intervals between consecutive harmonics Fundamental 2nd f1 harmonic f2 = 2f1 octave 3rd harmonic f3 = 3f1 perfect fifth 4th harmonic f4 = 4f1 perfect fourth 5th harmonic f5 = 5f1 major third 6th harmonic f6 = 6f1 minor third f2 2 f1 1 f3 3 f2 2 f4 4 f3 3 f5 5 f4 4 f6 6 f5 5 CT 2.4.5 What is the name of the note that is a major 3rd above E4=330 Hz? A: G B: G# C: A D: A# E: B Intervals C- D, a second C-E, a third C-F, a 4th C-G, a 5th, C-A, a 6th C-B, a (major) 7th, C-2C, an octave C-2D, a 9th C-2E, a 10th, C-2F, an 11th, C-2G, a 12th, C-2A, a 13th, etc. C-Eb, a minor 3rd C-Bb, a dominant 7th, C-2Db, a flatted 9th, etc. Pythagorean Scale Built on 5ths A pleasant consonance was observed playing strings whose lengths were related by the ratio of 3/2 to 1 (demo). Let’s call the longer string C, and the shorter G, and the interval between G and C a 5th Denote the frequency of C simply by the name C, etc. Since f1= V/2L, and LC= 3/2 LG, G =3/2C. Similarly a 5th above G is 2D, and D= 1/2 (3/2G)= 9/8 C. Then A is 3/2 D= 27/16 C. Then 2E= 3/2 A or E= 81/64 C, and B=3/2 E = 243/128 C. We now have the frequencies for CDE… GAB(2C) To fill out the Pythagorean scale, we need F. If we take 2C to be the 5th above F, then 2C= 3/2F, or F = 4/3 C Just Scale, Built on Major Triads We take 3 sonometers to play 3 notes to make a major triad, e.g. CEG. This sounds consonant (and has been the foundation of western music for several hundred years), and we measure the string lengths required for this triad. We find (demo) that the string lengths have ratios 6:5:4 for the sequence CEG. The major triad is the basis for the just scale, which we now develop in a way similar to that of the Pythagorean scale. F A C C E G G B D 4 5 6 4 5 6 4 5 6 Now take C to be 1 CT 2.4.5 Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th above this? A 1650 Hz B: 220 Hz C: 495 Hz D: 660 Hz E: None of these CT 2.4.5 What is the frequency of the note that is a (just) major 3rd above E4=330 Hz? A: 660 Hz B: 633 Hz C: 512 Hz D: 440 Hz E: 412 Hz CT 2.4.5 Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th below this? A 165 Hz B: 220 Hz C: 110 Hz D: 66 Hz E: None of these compound intervals Adding intervals means multiplying frequency ratios major third + minor third perfect fourth + perfect fifth perfect fourth + major third perfect fourth + whole tone 5 6 30 3 4 5 20 2 4 3 12 2 3 2 6 1 4 5 20 5 3 4 12 3 5 16 80 4 4 15 60 3 perfect fifth octave major sixth perfect fourth more compound intervals ratios larger than 2 can be split up into an octave + something perfect fifth + perfect fifth major seventh + minor sixth 3 3 9 2 9 2 2 4 1 8 15 8 15 3 2 3 8 5 5 1 1 2 Octave + whole tone Octave + perfect fifth