Yahoo Tuning Groups Ultimate Backup tuning Dan Stearns' pitch continuum

Warning: the following post is a waste of time.

Dan Stearns proposed 144-tET as a model of the virtual pitch continuum that
can be easily notated consistent with 12-tET notation habits (for which Joe
Maneri proposed 72-tET). As we have seen, for 11-limit JI, 72-tET is fine,*
so 144-tET can't hurt. But we should also be able to represent other
"kingdoms":
meantone and Erlich decatonic (~22tET) stuff, and maybe Blackwood decatonic
(~15-tET) and tetradecatonic (~26-tET) stuff. In 144-tET, one also has
708.33-cent fifths, great for the Erlich stuff. The next smaller fifths in
144-tET
after the 700-cent ones are the 691.67-cent fifths, at the extreme other end
of
the meantone spectrum. That's great for tetradecatonic stuff, but no real
meantone has such small fifths. It would be nice to have another fifth
halfway
in-between. Years ago on this list I proposed 288-tET for these very
reasons.
But 288-tET is not consistent with 11-limit JI, we've lost any reason to
hold onto
72-tET. So let's backtrack.

Starting with multiples of 12-tET again (due to cultural habit), 612-tET
(=12*17*3) presents itself. It was recommended by Bosanquet for its
0.01-cent-error 3-limit approximations and 0.04-cent-max-error 5-limit
approximations, and it happens to be consistent in the 11-limit with more
accuracy than any ET mentioned in this entire article. The gradations are
fine enough that a fifth suitable for any type of scale is guaranteed.
However, 17ths of a semitone would be hard to calculate with, reckon or
notate.

The exactly defined non-JI intervals we need the most are 1/5-octave (for
Blackwood decatonic stuff and since we already have 1/2-octave, 1/3-octave,
and 1/4-octave). That tells us we should work with a multiple of 60tET. The
first two multiples of 60tET that are consistent with the 11-limit are
480tET, 540tET, and 1200tET. The latter is cents, already used as notation
for Johnny Reinhard and the American Festival of Microtonal Music, and in
analysis by most of us. 480tET (=12*5*2*2*2) is a good choice too. 1/5-comma
meantone is 697.65 cents, 1/3-comma is 694.79 cents, the ~22tET decatonic
stuff suggests a fifth of around 709 cents, while the ~26tET tetradecatonic
stuff suggests a fifth of 692.31 cents; 480tET gives 697.5, 695, 710, and
692.5 respectively. Meanwhile, 540tET (=12*5*3*3) gives two good varieties
of meantone, one close to 1/5-comma and 43-tET, and the other close to
LucyTuning; a more tolerable 693.33-cent tetradecatonic fifth, and a superb
708.89-cent decatonic fifth. Furthermore, 540-tET contains the amazingly
good JI (through the 13-limit!) approximations of 270-tET, which Paul Hahn
has made note of before. The fascinating 27-tET is present too should anyone
find a good use for it.

Although using cents on the score can give musicians, particularly those not
trained to produce hundreds of intervals accurately, a false sense of
security, there is a major advantage to microtonal notation systems based on
numerals. Most people already know how to add and subtract numbers, and most
musicians know how to add and subtract 12-tET degrees. Thus the intervals in
a score will never be far from a musician's mind if the microtonal
alterations are numeric. With six steps to a semitone a glyph systems like
Sims' or Herf's allows easy enough calculation. But with 40, 45, or 100,
numbers seem the way to go.

Let's forget about 12-tET and see what else we can find. A great
representation of the pitch continuum is 342-tET (2*171). It's 19*3*3*2, so
contains 19tET, which is close to 1/3-comma meantone and could serve as a
basis for the notation (possibly even with a glyph system). It's even better
in the 11-limit than 270 & 540-tET (0.44 cents max error vs. 0.74 cents),
much better than 480-tET (1.19 cents max error), and it has a great
decatonic fifth of 708.77 cents and a good approximation of 1/6-comma
meantone (and a 691.23-cent fifth OK for tetradecatonics). Renewing an
interest in 1/5-octaves, 400-tET is also better in the 11-limit than 270 &
540-tET, and has decatonic, meantone, and tetradecatonic fifths of 708, 696,
and 693 cents, respectively. 400 is 50*8, so can be notated considering
50-tET (with its 696-cent fifths) as meantone, with a glyph system to
indicate halves, quarters, and eighths of these steps. The problem with that
is that notating 50-tET as meantone would require a quadruple sharp or a
quadruple flat somewhere and so might not be as concise as one would hope.

I must admit that I haven't found anything with steps larger than 4 cents
that really looks better than 144-tET.

*One could learn 72-tET as follows. Start with 12-tET. Learn 1/4-tones by
playing Arabic stuff, and listening to the essentially just 11:8, 11:6, and
11:9 intervals that 1/4-tones make with 12-tET. (6:8:9:11 chords are great
for this purpose). Then learn the 1/6-tones of 36-tET by playing the
essentially just 7:4, 7:6, and 7:9 intervals that 1/6-tones make with
12-tET. 1/12-tones can be learned by playing just 5:4, 5:3, and 5:9
intervals starting with 12-tET, and also as the spaces between 1/6-tones and
1/4-tones. Now one should have a total, integrated grasp of 72-tET and
11-limit JI.

Dan Stearns' pitch continuum

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗David J. Finnamore <dfin@xxxxxxxxx.xxxx>