Stretching the 19-tone Equal

I have found that the 19-tone equal temperament gives noticeably
smoother consonant harmony if its octave is stretched by 2 to 3 cents,
and the tuning degree widened slightly from 63.16 to 63.3 cents.

The reason is, I take it, as follows. In 19-tone without an octave
stretch (octave of exactly 1200 cents) the fifth and fourth are both
tempered by 7.2 cents. The fifth is flat from just, the fourth sharp.
But it is almost universally admitted that the fourth is decidedly less
sensitive to mistuning than the fifth, I mean primarily in harmony, but
perhaps also in melody. Indeed, before this century the fourth was more
often than not considered as not so consonant even as the major third.

Therefore we should be able to improve the 19-tone equal fifth by
stretching the octave, without making the fourth objectionable. Indeed,
merely to make the fifth equally deteriorated with the fourth, we _must_
stretch the octave. As for the octave itself, I do not imagine that many
will object to a stretch so small as 2-3 cents, which is all that is
possible before the fourth begins to become objectionable.

Fokker, reported in Mandelbaum, used a different method to decide on how
much the 31-tone equal octave should be tempered. He divided the cent
total corresponding to each of the six consonances by the number of
corresponding degrees in the 31-tone equal, in order to find what
31-tone equal tuning degree would give a pure value for each of the six
consonances. Then he averaged these values according to divers weighting
schemes in order to arrive at an ideal value for the octave.

I shall perform a similar operation here for the 19-tone equal.

The ideal value for the octave itself is of course 1200/19 = 63.16 ;
here are the values for the other six consonances.
3:2 701.96/11 = 63.82
4:3 498.04/8 = 62.26
5:4 386.31/6 = 64.39
5:3 884.36/14 = 63.17
6:5 315.64/5 = 63.13
8:5 813.69/13 = 62.59

If one considers the consonances to be all of equal importance, one can
simply take the average of the above seven figures, which is 63.22, to
arrive at a figure for the octave of 1201.2. However, it seems to me
that some weighting in called for, because the smoother a consonance,
the more susceptible it is to mistuning. If one weights the octave at 7,
the fifth at 6, the fourth at 5, the major third at 4, the major sixth
at 3, the minor third at 2, and the minor sixth at 1, one arrives at a
tuning degree of 63.29 cents, giving an octave of 1202.5 cents.

I use a tuning degree of 63.3 cents and an octave of 1202.7 cents, and
find the resulting consonant harmony to be very agreeable indeed -
noticeably better than that of the 19-tone equal without an octave
stretch. The fifth is 696.3, 5.7 cents flat of just, and the fourth is
506.4, 8.4 cents sharp of just.

The consonances within the octave are not the only ones to be
considered, however. Beyond the octave, both the major tenth 5:2 and the
perfect twelfth 3:1 are well known to be scarcely less definitely fixed
by beats than the octave itself. Now it so happens that as we stretch
the octave we improve these two intervals, critical to harmony, even
more than the major third and perfect fifth. With a tuning degree of
63.3 cents, we have a major tenth only 3.8 cents flat of just, and a
perfect twelfth only 3.0 cents flat of just. The perfect eleventh 8:3
meanwhile becomes 11.4 cents sharp of just, but the perfect eleventh is
a very weak consonance at most, and is of little concern.

I know it can be very annoying to retune one's synth, or even to store
alternate tunings, but here is one variation on the 19-tone equal that
is definitely worth hearing and using. The resulting consonant harmony
is very rich and brilliant. The dissonant harmonies remain extremely
grinding and aggressive.


SMTPOriginator: tuning@eartha.mills.edu
From: Gregg Gibson
Subject: 61 Intervals of Just Intonation
PostedDate: 14-12-97 17:31:22
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