Posted: November 28th, 2015
Intervals, Chords, and Scales Worksheet
The perfect musical intervals, as discovered by Pythagoras, are considered to have the following frequency ratios:
2:1 octave (C/C)
3:2 perfect fifth (G/C)
4:3 perfect fourth (F/C)
5:3 major sixth (A/C)
5:4 major third (E/C)
8:5 minor sixth (Ab/C)
6:5 minor third (Eb/C)
Triads are three-note chords built on intervals of a major or minor third. There are four possible triads: major (major third plus minor third), minor (minor third plus major third), diminished (two minor thirds), and augmented (two major thirds).
The scale of equal temperament divides the octave up into 12 equal frequency ratios, so that each successive semitone may be found by multiplying the previous frequency by 21/12 or 1.05946.
Problem 1 – perfect intervals
Consider a C4 note with frequency 261.63 Hz. Calculate the frequencies of the following notes assuming they make perfect intervals with C.
a) C5 (one octave higher than C4)
b) G4 (a perfect fifth higher than C4)
c) F4 (a perfect fourth higher than C4)
d) A4 (a major sixth higher than C4)
Problem 2 – triads
a) Calculate the frequencies of a perfect C major triad (C, E, G) as follows. Start with a C4 (261.63 Hz) and then find the E4 a major third above it. Then take that E4 frequency and multiply by 6/5 to get the G4 a minor third above it. These two intervals (major third plus minor third) make a C major triad chord.
b) Calculate the frequencies of a perfect C minor triad (C, Eb, G). This time start with a minor third to calculate Eb then do a major third to get G.
c) Calculate the frequencies of a C diminished triad (C, Eb, Gb). This triad has two minor thirds.
d) Calculate the frequencies of a C augmented triad (C, E, G#). This triad has two major thirds.
Problem 3 – equal tempered scale
a) Calculate all twelve frequencies of the notes in the equal tempered scale, starting on C4 and going in semitones to C5. Start by multiplying 261.63 Hz by 21/12 to get C# and repeat until you reach the octave.
b) How do the frequencies of Eb, E, Gb, and G compare to the frequencies of these notes that you calculated in problem 2? Why are they different?
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