9/7 and 45/32 clarifications

> > > > I think I've figured it out now; I suspect it has very much to do
> > > > with your choice of timbre in the orchestration! Those
> > > > pseudo-clarinets/chalumeaux in there have only half the overtone
> > > > series in them, skipping the odd partials as clarinets do.
> > >
> > > This can't very well be it; clarinets skip the even partial tones.
> >
> > Only on the distinction of starting the counting at 1 vs 0. Partials,
> > overtones, whatever; the half of them that would matter most here are
> > missing.
>
>That's probably what you meant, though. Skipping the even partials and
>superimposing a 7 and a 9 seems nice in clarinet terms.

No; I meant what I said, but evidently I didn't say it clearly enough. (Judging also from several off-list responses/questions from others....)

I'll try one more time.

9/7 is dangerously into "bad 4/3" territory. On harpsichord, when the 9/7 is not quite in tune, the rapid beats (really a nasty buzz) from the skanky 4/3 are very audible, and they are the main thing that is noticed. We don't hear a not-quite-resonant 9/7; we hear a thoroughly incompetent 4/3 protesting its error. This area is both lower in pitch, and louder, than the place in the overtone series where the 9th and 7th multiples are almost lining up.

But, in "clarinet" timbre where the odd overtones (OK, the "even partials") are missing, there's no 4th multiple in there from the lower note to bash against the 3rd multiple of the upper note. Therefore, the beats of the incompetent 4/3 are not audible and not in our way. Two "clarinets" making an almost-pure 9/7 sound much more harmonious than two harpsichord strings making the same almost-pure 9/7.

Of course, this is all moot most of the time, on harpsichord, because 9/7 doesn't come up at all except in a regular layout close to 2/7 syntonic comma....

=====

Clarification of my remarks about the "purity" of tritone 45/32 as a distinctive sound on harpsichord (when playing in 1/6 syntonic comma meantone): it has nothing to do, in this case, with the coincidence of the 45th multiple and 32nd multiple way up there. A perfectly tuned 45/32 sounds pure because the ear breaks it down to pure components 9/8 and 5/4.

In Baroque figured-bass terms, the ear fills in that missing note (for example, D between the pure C-F#) and the whole thing seems like a "4-2 chord", i.e. a dominant seventh in third inversion. That's why the tritone's resolution has to happen outwards: the F# going up to G, and the C going either to B or B-flat.

Brad Lehman

>Clarification of my remarks about the "purity" of tritone 45/32 as
>a distinctive sound on harpsichord (when playing in 1/6 syntonic
>comma meantone): it has nothing to do, in this case, with the
>coincidence of the 45th multiple and 32nd multiple way up there.
>A perfectly tuned 45/32 sounds pure because the ear breaks it down
>to pure components 9/8 and 5/4.

This 'breaking down' ability has been hotly contested here over a
period of years. I've been largely convinced that nothing like
this happens, unless one of the other tones is present, or was
recently present.

-Carl

> >Clarification of my remarks about the "purity" of tritone 45/32 as
> >a distinctive sound on harpsichord (when playing in 1/6 syntonic
> >comma meantone): it has nothing to do, in this case, with the
> >coincidence of the 45th multiple and 32nd multiple way up there.
> >A perfectly tuned 45/32 sounds pure because the ear breaks it down
> >to pure components 9/8 and 5/4.
>
> This 'breaking down' ability has been hotly contested here over a
> period of years. I've been largely convinced that nothing like
> this happens, unless one of the other tones is present, or was
> recently present.

I don't know anything that's been said about this particular
phenomenon of division, one way or another, here or elsewhere.

All I know is what I hear from the 45/32 interval itself, and its
usual functional position in the music of my specialty, when playing
in 1/6 syntonic comma meantone. It creates a very strong impression
(being comprised of those two very simple pure components) and
therefore draws extra attention to itself; and then it resolves in
the way it must do in 17th-18th century music.

It makes such cadences very strong harmonic events. It creates
tension because the in-tune-ness is so sudden and startling, in
context where no other available intervals (save octaves) are ever
pure. Then, the music relaxes from this dominant into the resolution
of a tonic that is gently less in tune (i.e. less intense, but with
medium-speed beating major thirds and fifth).

Or, if we're resolving from such a tritone into one of the augmented
fifths(!) C-G#, Eb-B, F-C#, or Bb-F# (being misspelled in the music as
a minor sixth), those too are very nearly pure, and therefore
startling. That is, the diminished fourths are nasty but their
inversions are extremely consonant in musical context.

We're safely out of range of 25/16 since we're 2/3 of a syntonic
comma sharp of it, and we're much too low from 8/5. 2/3 of a syntonic
comma from 25/16 puts us into the neighborhood a tiny bit sharp of
11/7, doesn't it? (Have I caught the correct ghost? Who in the JI
family lives nearest to 786.96495 cents?)

It's one of those freaky bits in 1/6 syntonic comma meantone that
lets one get away with some misspellings. For example, playing basso
continuo, if there's a notated D# in the middle of the bass staff
with figured "6" (and resolving next to E minor...), it works in a
B-F#-B spacing for the right hand; but it doesn't work if the F# is
placed in the same octave as the "D#". That is: the B major triad is
pretty lousy, but its first inversion (if spaced carefully!) is
usable.

Brad Lehman