source file: m1514.txt
Date: Tue, 25 Aug 1998 14:41:07 -0400

Subject: RE: XH 17: n-tet's and harmony

From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

Margo,

Thanks for all the good press in your fine article -- I'm glad you found
my work interesting. There seems to be some inconsistency in your usage
of the word "limit". You appear to define the word thus:

>In describing just intonation systems, it is common to refer to the
>most complex prime number used in generating the intervals of a given
>system as its "n-limit."

But then you speak of a non-prime, 9, as a limit:

>Since the time of Debussy, more complex sonorities have been accepted
>as stable: for example, tetrads such as the "added sixth" (12:15:18:20
>in one possible 5-limit tuning) and "minor seventh" (12:14:18:21 in a
>9-limit tuning).

The prime limit of the first chord is indeed 5, but that of the latter,
7. However, as some of us have discussed before, there can be two
equally useful concepts, an odd limit (closer to Partch's usage and
useful for describing sonorities) and a prime limit (useful for
describing resources of JI systems). The chords 12:15:18:20 and
12:14:18:21 are what I call the two "saturated 9-limit tetrads," since
all the intervals in each chord are ratios of 9 or less, and no notes
can be added to either chord without increasing the odd limit, and there
are no other chords with these properties. (Incidentally, both chords
can be called minor seventh chords, as the first is octave-equivalent to
10:12:15:18.) This type of chord seems to have been missed by Partch,
who may not have realized that utonalities and otonalities are not the
only possible saturated chords for odd limits 9 and up. So perhaps you
were thinking of the odd limit of the second chord, and the prime limit
of the first chord?

But then you go on to say,

>Similarly, in Gothic style, a combination of three superimposed thirds
>(e.g. e-g-b-d') may resolve to a stable 3-limit sonority (e.g. f-c');
>in triadic harmony, it may resolve to a 5-limit sonority; in 7-limit
>harmony, it may serve as a stable tetrad.

Now any chord that could be notated as e-g-b-d' cannot be a stable
tetrad if the odd limit is 7, so you must be back to the prime limit
definition and thinking of 12:14:18:21, or you should have said 9
instead of 7, in which case you could have meant either 12:14:18:21 or
10:12:15:18.

On another point, you wrote,

>The striking "neo-late-Gothic" feature of 22-tet is its drastically
>contrasting versions of intervals I will here call in traditional
>fashion thirds and sixths, although Erlich's decatonic nomenclature is
>well worth studying and may give a better insight into the scale's
>structure

When discussing neo-late-Gothic applications of the tuning, there is no
need to consider the decatonic nomenclature; it is quite irrelevant. The
conventional (heptatonic) nomenclature predates any kind of equal
temperament and is not "derived from" or "most natural for" any ET; on
the contrary ETs (12, 19, 31) were developed to conveniently quantize
the ideas of composers using this nomenclature. Similarly, there has to
be a decatonic corpus (opus?) before that nomenclature can be useful; in
no way does 22-tET alone suggest or require it. It is true that I found
the decatonic scales in 22-tET, and 22-tET is very nearly the best
tuning for them. But as you demponstrate, the "Xeno-Gothic" paradigm can
suggest 22-tET just as strongly as the "hyper-diatonic" paradigm that
spawned the decatonic scales, and for the former case, some form of
Pythagorean notation is clearly more appropriate.

An easy way to get a feel for these and other possibilities on almost
any retunable synth is to use a 12-out-of-22 tuning where E-F and B-C
are 1/22 oct intervals and all the rest are 2/22 oct. Then:

1) The white keys are a "Pythagorean" diatonic scale
2) The black keys are a "Pythagorean" pentatonic scale.
3) All thirds and sixths formed by two keys of the same color are
"regular" and approximate ratios of 7 or 9.
4) All thirds and sixths formed by two different-colored keys are
"schismatic" and approximate ratios of 5.
5) A major, A-flat major, C minor, and C-sharp minor give approximate
5-limit just intonation, in a fashion analogous to the late Gothic
keyboard tuning.
6) Decatonic scales as in my paper can be formed by omitting C and F, B
and F, or B and E.
7) The interval B-f approximates 11:8 with an error of 6 cents and
sounds good in the chord B-a-b-eb'-f, which approximates 4:7:8:10:11. It
is interesting to resolve this interval by contrary motion through the
very incisive 1/22 oct intervals to c-e. The latter interval is
approximates a 9:7 with an error of 1 cent and is therefore more stable
than B-f. 9:7 sounds like a traffic noise to most people unless supplied
with a context, such as D-Gb-A-c-e, which approximates 4:5:6:7:9.

-Paul