Re: Time for new sequences, Monz! (Benedetti's ratios)

Hello, there, Monz, and maybe I can quickly clear up at least of the
Benedetti misunderstanding by explaining that both he and I -- and
also his commentator Claude Palisca -- take those ratios to refer to
_melodic intervals_ required for motion between vertical consonances
if one assumes that all intervals are classic 5-limit JI.

Evidently you are reading these same ratios as referring to names of
pitches, in a certain 20th-century fashion, e.g. "Which note of this
scale is the 16/15," as opposed to the actual meaning here: "The
semitone from C# up to D has a size of 16:15."

Note that I generally follow the convention of using ratios with colons
(16:15) rather than slant symbols or fraction bars (16/15) in an attempt
to minimize confusion on this point. Maybe signed numbers for these
ascending or descending melodic intervals may make this point clearer, and
further emphasize that these are sizes of steps, not names of notes.

Thus

D4 C#4 D4
-27:25 +16:15

means that at least in Benedetti's view, we have a descending 27:25 step
D4-C#4 followed by an ascending 16:15 C#4-D#4 step, resulting in a shift
downward of 81:80.

Again, these are all melodic intervals or distances, not names of notes --
the later convention one that I find curious. In a keyboard diagram,
for example, I pick some pitch as the measuring point, e.g. C or F as
"1:1" -- but that's just an arbitrary reference.

In the above example, for example, the mode might be D Dorian, among
other things -- but all the diagram says to me is that we have a
cadential semitone ascending from C# to D at a usual size of 16:15 for
this system.

As for something like

G4 A4
D4 D4
G3 D4

I would say that indeed if we assume that all melodic and vertical
intervals are pure 5-limit JI, then the interval G4-A4 must be a 9:8, and
if the upper part then descends to another G4 involving a 10:9 step, we
get an ascending 81:80 shift. In the example, we have a result of

+9:8 -10:9 = +81:80
G4 A4 G4

It seems to me that while some modern theorists might assign note-name
ratios to pitches, the intervals have their own logic.

Maybe I should proof these examples again, and certainly you and others
should feel free to point out alternative JI interpretations (a matter
which could be of great interest), but I hope that this clarification may
address at least a likely misunderstanding, opening the way for more
discussion on assumptions and solutions (classic or adaptive).

Most appreciatively,

Margo Schulter
mschulter@value.net