22-tone Equal and the Genera

As I have abundantly shown in previous posts, this temperament (and
analogous ones such as 29- 41- & 53-tone equal,) is impossible for the
diatonic, as well as for the chromatic modes (see the four chromatic
modal genera or scales I have contributed; these include several of the
forms of the 'minor' of the West) which obviously require thirds and
sixths vis-?-vis the tonic, it being granted that the fifth and/or
fourth must also be present.

The case of the enharmonic is less clear-cut, and requires more detailed
consideration. For after all, one might ask, surely the divers
enharmonic modes have fewer thirds or sixths vis-?-vis the tonic, and so
the inconsistency between the fifth cycle and those of the thirds would
appear to be less troublesome; in certain enharmonic modes it might
never arise at all.

For example, take one form of the enharmonic:
C C# Db F G G# Ab C (not the 22-tone equal notation)
or in 22-tone equal:
0 55 109 491 709 764 818 1200

There is a consonant major third between Db & F, and another between Ab
& C, and a consonant minor third between F & Ab.

In reality, of course, the notation of the 22-tone differs from the
above notation:
C Db B# F G Ab Fx C
which BTW is not a mere artefact of notation, but on the contrary
explains very succinctly the utterly non-diatonic and non-chromatic
nature of this system, but that is beside the point.

The effect of this particular Greek enharmonic mode - the most famous of
them as I recollect - is in 22-tone not wholly dissimilar to its effect
in 19-tone equal, except that the latter is much more singable because
of the wider 63.3 cent interval.

Other enharmonic modes in 22-tone equal such as:
C D D# F G G# Ab C (not 22-tone equal notation)
0 218 273 491 709 764 818 1200
have problems however. Here D-F is dissonant, and this would tend to
disagregate the already weak tonality of the enharmonic. If D is lowered
to make D-F consonant, D-G becomes dissonant. This is similar to the
break in the fifths that besets just intonation itself. This is not a
trivial matter; it is very noticeable to the ear.

Therefore ? and I am very glad to be able to give Paul Ehrlich some
reason for at least toying with the 22-tone equal ? some of its
enharmonic modes seem usable, so long as modulation, harmony, or the
drone are rigidly forbidden. As soon as these enter, the enharmonic
itself evolves toward the chromatic or the diatonic, with 1/3 tones used
not as basic modal members, but only as occasional ornaments. (And then,
in the diatonic and chromatic, the 22-tone equal becomes unusable).

And this indeed has always seemed to me ultimately the most valuable
r?le for the enharmonic; not as a fully independent genus, but as an
adjunct ? a very important adjunct ? to the chromatic and diatonic.

But I here hesitate, and have long hesitated, to pronounce with any kind
of certainty; if the Greeks used the enharmonic as a fully independent
genus, and they appear to have done so, there may well be a way to avoid
the difficulty, only I do not see it.

In fine, observe that _all_ enharmonic modes, whether infiltrated by the
chromatic or diatonic or kept pure, are available in the 19-tone equal,
and because of the 63.3 cent interval are much more easily sung in the
therein than in the 22-tone equal. The difference of 9 cents between
54.5 and 63.3 cents is very important, because it occurs near the
melodic threshold or limen of perception.


SMTPOriginator: tuning@eartha.mills.edu
From: mr88cet@texas.net (Gary Morrison)
Subject: A Higher-Level Comment on Gregg Gibson's Limen
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