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L 8-9 Musical Scales, Chords , and Intervals, The Pythagorean and Just Scales

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. The ratio of the frequency of C4 to that of C2 is: a) 2 b) 3 c) 4 d) 8.

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L 8-9 Musical Scales, Chords , and Intervals, The Pythagorean and Just Scales

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  1. L 8-9Musical Scales, Chords , and Intervals,The Pythagorean and Just Scales

  2. History of Western ScalesA Physics 1240 Projectby Lee Christy 2010

  3. References to the History

  4. The ratio of the frequency of C4 to that of C2 is:a) 2b) 3c) 4d) 8

  5. One octave of the diatonic scale including the tonic and the octave note contains:a) 5 notesb) 6 notesc) 7 notesd) 8 notes

  6. One octave of the chromatic scale (including the octave note) contains:a) 8 notesb) 10 notesc) 11 notesd) 12 notese) 13notes

  7. 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.

  8. Intervals 12-tone scale (chromatic) 8-tone scale (diatonic) Note span Interval Frequency ratio C - C unison 1/1 C - C# semitone 16/15 C - D whole tone (major second) 9/8 C - D# minor third 6/5 C - E major third 5/4 C - F perfect fourth 4/3 C - F# augmented fourth 45/32 C - G perfect fifth 3/2 C - G# minor sixth 8/5 C - A major sixth 5/3 C - A# minor seventh 16/9 (or 7/4) C - B major seventh 15/8 C3 - C4 octave 2/1 C3 - E4 octave+major third 5/2

  9. Consonant intervals Overlapping harmonics tonic 120 240 360 480 600 720 840 960 1080 octave 240 480 720 960 fifth 180 360 540 720 900 1080 fourth 160 320 480 640 800 960 M third 150 300 450 600 750 900 1050 m third 144 288 432 576 720 864 1008 Dissonant intervals Perceived when harmonics are close enough for beating

  10. harmonic series Intervals between consecutive harmonics Fundamental f1 2nd 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

  11. 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

  12. IntervalsC- D, a secondC-E, a thirdC-F, a 4thC-G, a 5th,C-A, a 6thC-B, a (major) 7th,C-2C, an octaveC-2D, a 9thC-2E, a 10th,C-2F, an 11th, C-2G, a 12th,C-2A, a 13th, etc.

  13. C-Eb, a minor 3rdC-Bb, a dominant 7th, C-2Db, a flatted 9th, etc.

  14. Pythagorean ScaleBuilt on 5ths

  15. 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 5thDenote the frequency of C simply by the name C, etc.

  16. 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, andB=3/2 E = 243/128 C.

  17. 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, orF = 4/3 C

  18. Just Scale, Built on Major Triads

  19. 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.

  20. The major triad is the basis for the just scale, which we now develop in a way similar to that of the Pythagorean scale.

  21. F A C C E G G B D 4 5 6 4 5 6 4 5 6 Now take C to be 1

  22. 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

  23. 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

  24. 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

  25. major third + minor third perfect fifth compound intervals Adding intervals means multiplying frequency ratios octave perfect fourth + perfect fifth perfect fourth + major third major sixth perfect fourth + whole tone perfect fourth

  26. more compound intervals ratios larger than 2 can be split up into an octave + something perfect fifth + perfect fifth Octave + whole tone major seventh + minor sixth Octave + perfect fifth

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