RE: Beginner hits comma, survives crash.

Steven, thanks for your comments.

>By taking the ii as a diminished triad, everything fell
>nicely into place.

This is something I never thought of. Which diminished triad, exactly,
sounded nice here?

(I know you already told me in e-mail; just thought the rest would like to
know).

What is interesting here is that in your solution with the diminished triad,
you circumvent the comma shift by moving one voice a chromatic semitone --
but both the comma and the chromatic semitone are represented by 54.5 cents!

>I presume it was more tolerable because
>the commas were only half as large as in the 22TET rendition. Can
>anyone offer an explanation for this?

Sounds like a good explanation to me. 21.5 cent shifts can happen a lot with
real instruments.


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Interesting experiment in general, Steve.

The most important effect of using V7 instead of V (other than the
well-known ones of dominant 7ths vs dominant triads) is that scale degree 4
in addition to 2 becomes a common tone between ii and V7.

Try out these subtly different 22TET four-part harmonizations, using a
similar progression:

1. Diminished ii (borrowed chord) (there's a pseudocomma error
on the b6, but no precedent for that pitch either):
13
9 9 9
7 7
4 4 4
0 0 0
0 0
20 20
16 15
13 13
9 9
0 0

I IV iio V V7 I


2. Scale-degree-2 common tone forced not to be common to avoid
pseudocomma-sharp fourth:
13
9 9 9
7 7
3 4 4
0 0 0
0 0
20 20
16 16
13 13
9 9
0 0

I IV ii V V7 I


3. Tonic allowed to wander to preserve all common tones and avoid
all pseudocomma errors:
12
9 9 8
7 6
3 3 3
0 0 21
0 21
19 19
16 16
12 12
9 9
0 21

I IV ii V V7 I


My personal impressions were these: The diminished and wandering tonic
(first and third) cases flow the most smoothly (smoothness being one, but
not the only, interesting consideration). The main difference between
those two was the obvious one: That the first obviously had the more
dissonant diminished ii instead the more typical minor ii, and the third
had that wandering-tonic sensation of "???!!! we did everything right but
we landed in the wrong place !!!???", right about on the V.

The other difference I heard between those two was that the V sounded a
little odd in BOTH based upon the small vs. large whole tone in the exposed
soprano line. The first one with its large upward whole-tone made that
chord seem over all oddly too high-pitched, even though it seemed to have
the right harmony, whereas the third one sounded a little flat even though
the harmonic effect seemed right.

In the second one, the upward movement from 3/22 to 4/22 struck me as
too freaky with the already "too-high" effect of the large whole-tone in
the soprano. To me the second one was too disconcerting (and discontiguous
from ii-V) for that reason.


By the way, to actually play these, I finally broke down and stuck
masking-tape labels of the numbers above on the keys and then sequenced
each part. I have no idea how keyboardists can possible play on stretched
keyboards! This woodwind & guitar player is just amazed that anybody can
do that!

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