Just and near-just systems

Hello, there, everyone, and given all the excitement recently about
the question of just and "near-just" tunings, I'd like to comment from
what could be a different viewpoint as someone who uses both types of
systems.

While discussions like this can often get rather general, I'd like to
approach the "just/near-just" question from my own concrete musical
perspective, and explain a bit about how I typically implement these
systems.

At least two important factors can influence the development of such
systems: what musical intervals and sonorities we happen to prefer;
and questions of "user-friendliness" in actual performance.

For example, as someone who tends to regard ratios around 14:11
(~417.51 cents) and 13:11 (~289.21 cents) as routine and ideal sizes
for regular major and minor thirds, I might reach very different
results than someone striving for ratios around 5:4 (~386.31 cents)
and 6:5 (~315.64 cents).

Likewise, as someone accustomed to a 24-note "regularized keyboard"
with two identical 12-note regular tunings mapped to standard manuals
at some convenient distance apart, I may have a very different
perspective than an expert user of a generalized keyboard, or a
composer who relishes a special Blackjack keyboard mapping.

For me, a 24-note just system tends to imply the following:

(a) There are two 12-note Pythagorean chains of
fifths (Eb-G#) placed at the distance of some
integer ratio; and

(b) Often the distance between chains is such as
to generate pure or near-pure versions of
sonorities such as 12:14:18:21 or 14:17:21.

The simplest system of this kind is a regular 24-note Pythagorean
tuning -- a simple chain of 23 fifths. From another view, this is the
special case of two 12-note chains which happen to be placed precisely
a Pythagorean comma apart (531441:524288, ~23.46 cents). A slight
modification of this scheme increases the distance between the manuals
to a 64:63 (~27.26 cents), or by about 3.80 cents, producing pure
ratios of 2-3-7-9.

A conceptually elegant feature of these two solutions is that we have
a regular or virtually regular system with free transposibility within
the range of the 24-note gamut. However, schemes with other distances
between the keyboards can have the advantage of making available other
types of intervals, or ratios of 2-3-7-9 in more positions.

Curiously, while such a Pythagorean-based system is, of course,
distinguished by its pure tuning of fourths and fifths, the primary
consonances in a neo-medieval setting, it can sometimes involve
delicate compromises regarding other types of stylistically favored
intervals.

For example, the Pythagorean tricomma tuning places the two keyboards
at a distance of three Pythagorean commas (each at 531441:524288, or
about 23.46 cents), or around 70.38 cents, emulating as it were
24-of-36-tET as a system combining approximate ratios of 2-3-7-9 and
14:17:21. A very notable difference is that the just system is much
less accurate for such ratios as 9:7, 7:6, and 7:4 -- as well as
making these ratios available in fewer positions!

While 36-tET might be considered a special kind of neo-medieval
"near-just" tuning -- and an excellent one -- such systems more
typically have the fifths tempered by about the same amount, but in
the opposite direction.

As with the Pythagorean-based just systems, a near-just system may
consist either of a simple 24-note regular tuning with a single chain
of fifths; or of two 12-note chains of fifths at some convenient
distance other than that of the enharmonic diesis (e.g. G#-Ab, the
amount which the chromatic semitone exceeds the diatonic semitone).

The usual parameters are about like this:

(1) The fifths are gently tempered in the wide direction
by about 1.5-2.65 cents;

(2) Regular major and minor thirds are around 14:11
and 13:11 or 33:28 (~284.45 cents);

(3) Augmented seconds and diminished fourths are not too
far from 17:14 (~336.13 cents) and 21:17 (~365.83 cents);

(4) There are often rather close approximations of ratios
of 2-3-7-9, with most of these intervals within 3 cents
or so of pure; and

(5) Other attractions can include sonorities involving narrow
fourths around 21:16 (~470.71 cents) or "superfourths"
around 11:8 (~551.32 cents); assorted neutral intervals
around ratios such as 11:9 (~347.41 cents) and 11:6
(~1049.36 cents); and so on.

While dialogues about the amount of temperament permitted or desirable
in a "near-just" system might focus on topics such as beat rates of
fifths, my range of around 1.5-2.65 cents could be taken as reflecting
certain pragmatic structural factors.

The lower amount of around 1.5 cents, as in 29-tET (fifths ~703.45
cents, ~1.49 cents wide), is about the minimum required to move
augmented seconds and diminished fourths within, say, 5 or 6 cents
of 14:17:21. Having some of these intervals available as part of a
regular 12-note chain helps in optimizing for item (3) above.

If we seek _all_ of the above optimization criteria in a regular
24-note tuning _with a single chain of fifths_, then the upper amount
of around 2.65 cents (fifths ~704.61 cents) provides ratios of 2-3-7-9
arguably approaching "near-just" accuracy, along with augmented
seconds and diminished fourths (around 341.46 cents and 363.14 cents)
still with a quality somewhat like that of 14:17:21, although leaning
somewhat toward the more central neutral region.

As it happens, 2.65 cents is close to the 2.7 cents or so which Dave
Keenan has mentioned as one possible maximum for a "near-just" tuning,
about that of 1/4-(syntonic)-comma meantone where fifths are impure by
around 5.38 cents in the opposite direction.

Here I should add that George Secor has proposed an approach to
"near-just" systems where fifths are considerably closer to pure, say
within about 1.625 cents -- or, from another viewpoint, where the
ratios 9:8 and 16:9 are impure by not more than twice this amount,
about 3.25 cents.

Given the attractions of such Gothic and neo-medieval sonorities as
6:8:9 and 4:6:9, Secor's point is a very cogent one as applied to the
systems we are now considering.

One response to these definitions might be to recognize various
degrees of "near-justness," with each proposed definition educating us
as to some of the compromises in an often fine balancing process.

To conclude, I might remark that the distinct attractions of just and
near-just systems of a similar kind can involve considerations other
than simple issues of "purity."

For example, one of the attractions of 36-tET for me is that it's a
bit of a "change of pace" -- having regular minor thirds and
diminished fourths the same size is a bit different. This 20th-century
kind of symmetry makes possible approximations of 2-3-7-9 and 14:17:21
in more positions -- and both Busoni's "tripartite tone" at 66.67
cents, and the 2/3-tone at 133.33 cents, are beautiful melodic
intervals.

With the Pythagorean tricomma tuning -- a kind of JI "emulation -- a
main attraction is having a 36-tET-like scheme with regular
Pythagorean intervals such as more expansive major thirds and compact
diatonic semitones. There's also the elegance of a system using all
"native" Pythagorean intervals -- the familiar medieval ones on each
keyboard, plus the novel ones to be found by mixing or moving between
notes on the two keyboards. Melodic steps at 70.38 cents (the tricomma
between the keyboards) and 133.53 cents, used in the same types of
progressions as their 36-tET counterparts, suggest the affinity of the
two systems.

Both systems, interestingly, start with a standard 12-note historical
tuning, and add another identical 12-note chain at some convenient
distance to generate new types of steps and sonorities.

Here I'd like to suggest that terms such as "just" and "near-just"
take shape in a range of stylistic settings, with the kind of
"optimization" sought in either approach -- or the very definitions of
these approaches -- varying along with musical preferences.

Most appreciatively,

Margo Schulter
mschulter@value.net