Re: Welcome to Patrick Mullen -- Adaptive JI and Gabrieli

Hello, there, and welcome to Patrick Mullen.

As Paul Erlich has suggested, I would consider the likeliest tuning
for a minor seventh in the music of Giovanni Gabrieli, for example the
_Sonata pian' e forte_ of 1597 (in Phrygian as I seem to recall -- or
possibly Aeolian?), to be somewhere around 9:5 (3:2 fifth plus 6:5
minor third) or 16:9 (two pure 4:3 fourths).

Following Nicola Vicentino (1555) and Paul, I might suggest that a
5-limit "adaptive JI" (just intonation) scheme, possibly based like
Vicentino's on 1/4-comma meantone with adjustments to achieve pure
5-limit concords, could be very attractive for vocalists and
performers on non-fixed-pitch instruments. Of course, for such
performers, _any_ precise mathematical model is just that, a guide
rather than a literal and mechanical description of what is actually
sung or played.

On a keyboard, my solution would be simple: meantone, for example
1/4-comma, with as many notes per octave as necessary to include all
the accidentals called for by a given piece.

Incidentally, Zarlino's 2/7-comma meantone might be an attractive
choice for some pieces in modes (Dorian, Phrygian, Aeolian) featuring
an arithmetic division of the fifth above the bass -- that is, with
the minor third below and major third above. Lindley recommends it for
some organ pieces by Andrea Gabrieli, and the same considerations
might apply to comparable pieces by Giovanni Gabrieli also.

In analyzing a piece like _Sonata pian' e forte_ -- and thanks,
Patrick, for drawing my interest to this piece -- I might focus both
on two-voice progressions and on multi-voice sonorities. While I don't
seem specifically to associate bold seventh sonorities with this
piece, I should take another look.

Apart from an interval such as the meantone augmented sixth (very
close to 7:4), I would tend to take a minor seventh in this era as
somewhere around 16:9 or 9:5.

Here I might just add that in determining "consonance" in the sense of
aesthetic stability/instability or restfulness/tension, I find that
categorical perception can often be more important than the abstract
mathematical simplicity or complexity of an interval.

Thus in a 14th-century context (e.g. Machaut), either the usual 81:64
or 5:4 is unstable because "thirds are unstable intervals in this
music" -- and likewise either the usual meantone minor seventh midway
between 16:9 and 9:5, or 7:4, in a 16th-century setting such as
Gabrieli.

Interestingly, Vicentino finds the "proximate minor third" at ~11:9
quite concordant, but either ~7:6 or ~7:4 on the dissonant side, the
first leaning toward a major second and the latter toward a minor
seventh, both regarded as clear dissonances in a 16th-century
context.

Of course, I have my biases: I like _large_ integer ratios, for
example, a commonplace in 3-limit or Pythagorean JI. This might
influence my views on various issues, so maybe I'm wise to put people
on notice now and then of this proclivity.

Most appreciatively,

Margo Schulter
mschulter@value.net