User:Guy vandegrift/Lectures/Just and equal tuning

See also User:Guy vandegrift/Lectures

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)	2^0⁄12 = 1	1	1⁄1 = 1	0
Minor second (C♯/D♭)	2^1⁄12 = ¹²√2	1.059463	16⁄15 = 1.06666…	−11.73
Major second (D)	2^2⁄12 = ⁶√2	1.122462	9⁄8 = 1.125	−3.91
Minor third (D♯/E♭)	2^3⁄12 = ⁴√2	1.189207	6⁄5 = 1.2	−15.64
Major third (E)	2^4⁄12 = ³√2	1.259921	5⁄4 = 1.25	+13.69
Perfect fourth (F)	2^5⁄12 = ¹²√32	1.334840	4⁄3 = 1.33333…	+1.96
Tritone (F♯/G♭)	2^6⁄12 = √2	1.414214	7⁄5 = 1.4 ≈ 10⁄7	+17.49
Perfect fifth (G)	2^7⁄12 = ¹²√128	1.498307	3⁄2 = 1.5	−1.96
Minor sixth (G♯/A♭)	2^8⁄12 = ³√4	1.587401	8⁄5 = 1.6	−13.69
Major sixth (A)	2^9⁄12 = ⁴√8	1.681793	5⁄3 = 1.6666… ≈ (3⁄2)(3⁄2)(3⁄4)	+15.64
Minor seventh (A♯/B♭)	2^10⁄12 = ⁶√32	1.781797	16⁄9 = 1.77777…	+3.91
Major seventh (B)	2^11⁄12 = ¹²√2048	1.887749	15⁄8 = 1.875	+11.73
Octave (C)	2^12⁄12 = 2	2	2⁄1 = 2	0

Audio examples

Danse macabre

This computer generated recording (transcribed by Kevin MacLeod) illustrates the so-called "Devil's tritone", which might actually be "two devils", since 7⁄5 ≈ 10⁄7.

Problems playing this file? See media help.

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

External link

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