Inconsistency and "incongruity" (non-ET's)

Hello, there, Paul Erlich and everyone, and this article is to share
some ideas about a concept of "incongruity," analogous to
inconsistency, which can apply to nonequal tunings of a given size.

The conventional category of "inconsistency" applies specifically to
equal temperaments, where the tuning system is mathematically closed,
and thus neatly defined as having an intrinsic maximum size.

In contrast, my analogous category of incongruity can apply to any
tuning system with a specified number of notes, whether the system is
intrinsically finite (i.e. a closed tuning), or arbitrarily given a
certain size (as with open tunings).

Here I would like to illustrate this wider concept of incongruity with
an example from the Peppermint 24 tuning system, which consists of two
12-note chains of fifths at about 704.096 cents (the Wilson/Pepper
temperament with a ratio between chromatic and diatonic semitones
equal to the Golden Section, Phi), with a distance of approximately
58.680 cents between the two chains (producing some pure 7:6 ratios).

When seeking the best approximation of the isoharmonic sonority
8:13:18:23, with differences of 5 between successive partials, I
arrived at this approximation, with C4 as middle C, and an asterisk
(*) showing a note raised by the 58.68-cent interval betwen the two
tuning chains:

|------------- ~8:23 (~1834.106, +5.832) -----------|
|--- ~13:23 (~991.809, +4.062) ---|
|--- ~4:9 (~1408.191, +~4.281) ---|
|----- ~8:13 -----|---- ~13:18 ---|---- ~18:23 -----|
B3 G*4 C#5 F*5
842.297 565.894 425.915
+1.770 +2.512 +1.550

Here, while 8:13, 13:18, and 18:23 -- and also the rather less
accurate 8:18 or 4:9 and 13:23 -- receive their best representations
in the Peppermint 24 system, the outer 8:23 does not.

The interval B3-F*5, at about 1834.106 cents, is ~5.832 cents wide of
a just 8:23 (~1828.274 cents). We might describe this interval as a
diminished twelfh plus the artificial "quasi-diesis" of ~58.680 cents.

In contrast, a regular augmented eleventh (e.g. F3-B4) at about
1824.574 cents is only ~3.701 cents narrow of 8:23, thus providing a
more accurate representation.

We therefore have a situation of _incongruity_: the best approximation
of 8:23 is not equal to the sums of the best approximations of 8:13,
13:18, and 18:23.

Suppose we seek to use the best representation of 8:23 in another
approximation of 8:13:18:23. The result is, again, incongruity:

|------------- ~8:23 (~1824.574, -3.701) -----------|
|--- ~13:23 (~982.276, -5.470) ---|
|--- ~4:9 (~1408.191, +~4.281) ---|
|----- ~8:13 -----|---- ~13:18 ---|---- ~18:23 -----|
D4 Bb*4 E5 G#5
842.297 565.894 416.382
+1.770 +2.512 -7.982

In this version, the representations of 8:13, 13:18, and 4:9 are
unchanged, and remain the best.

However, 18:23 is now represented by the regular major third E5-G#5 at
about 416.382 cents, or ~7.982 cents narrow -- the usual equivalent of
11:14 (~417.508 cents). In contrast, the diminished fourth plus
quasi-diesis of the previous example (C#5-F*5) is only about 2.512
cents wide of 18:23, providing a better representation.

Also, 13:23 is now represented by the augmented sixth less
quasi-diesis Bb*4-G#5 at ~982.276 cents, or ~5.470 cents narrow, in
contrast to the regular minor seventh G*4-F*5 of the previous example
at only ~4.062 cents wide.

Thus incongruity in a "real-world" open tuning with a given number of
notes presents dilemmas analogous to those of inconsistency in an
equal or other closed temperament with an intrinsically defined number
of notes.

Here is a possible definition of incongruity which seeks to include
inconsistency as a subset:

A tuning system of any arbitrarily defined size
exhibits incongruity when at no position is it possible
to approximate a given multi-voice chord or sonority
using the best representations available in the system
for each of the constituent intervals.

Inconsistency is incongruity occurring in the specific
setting of a closed tuning system, and resulting from
the intrinsically finite size of the closed system
rather than the arbitrary choice of a given size for
an open tuning system.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, Paul Erlich and everyone, and this article is to share
> some ideas about a concept of "incongruity," analogous to
> inconsistency, which can apply to nonequal tunings of a given size.
>
> The conventional category of "inconsistency" applies specifically to
> equal temperaments, where the tuning system is mathematically
closed,
> and thus neatly defined as having an intrinsic maximum size.

> In contrast, my analogous category of incongruity can apply to any
> tuning system with a specified number of notes, whether the system
is
> intrinsically finite (i.e. a closed tuning), or arbitrarily given a
> certain size (as with open tunings).

> Here is a possible definition of incongruity which seeks to include
> inconsistency as a subset:
>
> A tuning system of any arbitrarily defined size
> exhibits incongruity when at no position is it possible
> to approximate a given multi-voice chord or sonority
> using the best representations available in the system
> for each of the constituent intervals.

this ailment would seem to afflict even sonorities like major seventh
chords in traditional well-temperaments. are you sure this is
how "incongruity" should be defined? or should the approximations be
restricted to having to occur in the particular positions where they
would be available to combine in the right way with the other
intervals?

hoping to take another look at this tomorrow,
paul