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User:Guy vandegrift/Lectures/Just and equal tuning

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Lifted and modified from Special:Permalink/797723042#Comparison_to_just_intonation and stored here for educational purposes.

Name Equal
temperament
Decimal
approximation
Just
intonation
Difference
(cents)
Unison (C) 2012 = 1 1 11 = 1 0
Minor second (C/D) 2112 = 122 1.059463 1615 = 1.06666… −11.73
Major second (D) 2212 = 62 1.122462 98 = 1.125 −3.91
Minor third (D/E) 2312 = 42 1.189207 65 = 1.2 −15.64
Major third (E) 2412 = 32 1.259921 54 = 1.25 +13.69
Perfect fourth (F) 2512 = 1232 1.334840 43 = 1.33333… +1.96
Tritone (F/G) 2612 = 2 1.414214 75 = 1.4 ≈ 107 +17.49
Perfect fifth (G) 2712 = 12128 1.498307 32 = 1.5 −1.96
Minor sixth (G/A) 2812 = 34 1.587401 85 = 1.6 −13.69
Major sixth (A) 2912 = 48 1.681793 53 = 1.6666… ≈ (32)(32)(34) +15.64
Minor seventh (A/B) 21012 = 632 1.781797 169 = 1.77777… +3.91
Major seventh (B) 21112 = 122048 1.887749 158 = 1.875 +11.73
Octave (C) 21212 = 2 2 21 = 2 0

Audio examples

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This was lifted from Special:Permalink/795297653#Audio_examples

Equal temperament and just intonation compared A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound.

  • Other examples:
Just intonation An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.
Equal temperament An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. By listening to the above file, and then listening to this one, one might be able to hear the beating in this file.
Equal temperament and just intonation compared with square waveform A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just temperament between the two chords. In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent. The square waveform makes the difference between equal and just temperaments more obvious.

images

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Bach used the tritone in measure 9 of the Allemande of his Cello suite #2: https://www.youtube.com/watch?v=Q0hJWUpiE5w