Tuning and temperament (3): Just Intonation and Equal Temperament

We have seen in previous posts on this topic that:

  1. The perfect fifth corresponds to a frequency ration of 3/2 or 1.5
  2. The octave corresponds to a frequency ration of 2.
  3. It is impossible to have fully accurate fifths and octaves together on a keyboard, as 12 fifths overshoot 7 octaves by an amount known as the Pythagorean Comma.
  4. Pythagorean tuning tries to have as many fifths perfect as possible: it has 11 perfect fifths and one which is flat by the comma, and this flat fifth will sound noticeably out of tune.

Just intonation

Just Intonation is the ascribing of small and simple fractions to all intervals within a single octave. We can do this by making a whole lot of fractions using powers of 3 and 5, and adjust them by multiplying or dividing by a power of two so that the result lies between 1 and 2. This gives the following ratios:

Interval Semitones Name Fraction Cents
C – C 0 Unison 0 0.0
C – C♯ 1 Semitone 16/15 111.73
C – D 2 Whole tone 9/8 203.91
C – E♭ 3 Minor third 6/5 315.64
C – E 4 Major third 5/4 386.31
C – F 5 Perfect fourth 4/3 498.04
C – F♯ 6 Tritone 45/32, 64/45 590.22
C – G 7 Perfect fifth 3/2 701.96
C – G♯ 8 Minor sixth 8/5 813.69
C – A 9 Perfect sixth, Major sixth 5/3 884.36
C – B♭ 10 Minor Seventh 16/9, 9/5 996.09
C – B 11 Major Seventh 15/8 1088.27
C – C’ 12 Octave 2 1200

You can see one way of doing this on the wikipedia page above. This particular group of fractions is known also as 5-limit tuning, as there is no prime number greater than 5 used. If we allow the use of 7, we the following list of fractions and cents is one of many possible (the other intervals stay as for 5-linit tuning):

Interval Semitones Name Fraction Cents
C – C sharp 1 Semitone 15/14 119.44
C – D 2 Whole tone 8/7 231.17
C – F sharp 6 Tritone 7/5 582.51
C – B flat 10 Minor Seventh 7/4 968.83
C – B 11 Major Seventh 28/15 1080.56

There’s a whole page of possible 7-limit tunings here.

Just intonation is no better than Pythagorean temperament, but it’s a handy reference chart with which to compare other temperaments.

Equal temperament

This is the temperament in which almost all modern pianos are tuned, and the one in which piano tuners are trained. It is very simple to describe: each semitone is exactly 100 cents. So we can easily determine the cent values of intervals by listing all the necessary multiples of 100:

Interval Semitones Name 5-limit Just Equal
C – C 0 Unison 0.0 0
C – C♯ 1 Semitone 111.73 100
C – D 2 Whole tone 203.91 200
C – E♭ 3 Minor third 315.64 300
C – E 4 Major third 386.31 400
C – F 5 Perfect fourth 498.04 500
C – F♯ 6 Tritone 590.22 600
C – G 7 Perfect fifth 701.96 700
C – G♯ 8 Minor sixth 813.69 800
C – A 9 Perfect sixth, Major sixth 884.36 900
C – B♭ 10 Minor Seventh 996.09 1000
C – B 11 Major Seventh 1088.27 1100
C – C’ 12 Octave 1200 1200

We notice first that aside from the unison and the octaves, no other intervals are pure, although some are closer than others. The fifth is 700 cents instead of 701.955: that’s close enough for the human ear (in fact there are beats, but they are too slow to be noticed). Other intervals are quite poor: the major and minor thirds are both about 15 cents out, as are the sixths.

As before, we can see this by plotting major thirds for both temperaments:

[In ]: s = 44100
[In ]: dur = 1
[In ]: dpi = 2.0*np.pi
[In ]: hz = 283
[In ]: t = np.linspace(0,dur,dur*s)
[In ]: x = np.sin(dpi*hz*t)
[In ]: y = np.sin(1.25*dpi*hz*t)
[In ]: z = np.sin(2.0**(300.0/1200)*dpi*hz*t)
[In ]: pyplot.plot(t,x+y)
[In ]: pyplot.plot(t,x+z)

So here is a justly tuned minor third:

just_min3rd

This is a rich, multi-layered output, and the effect of the sound would be consonant -it would still sound very good.

And here is the equal tempered minor third:

eq_min3rd

As you see, it’s a jagged thing with internal beats.

Equal temperament of course has the advantage of allowing play equally well in any key: a difference of 7 semitones will be 700 cents no matter where you are on the keyboard. It has the disadvantages of having no pure intervals, and no differences in “colour” between keys: in equal temperament, the only difference between keys is their pitch. The frequency ratios between harmonies never changes.

We can consider equal temperament as another way of dealing with the Pythagorean comma. In Pythagorean tuning we kept 11 fifths pure and flattened the last fifth by the comma. In equal temperament, all fifths are equally flattened by one twelfth of the comma. Since an equal tempered fifth is 700 cents, and a pure fifth 701.955 cents, we find that

    \[ 12\times(701.955-700)=23.46 \]

which is the amount of the comma.

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