Dave Hill's Wolf sonority

Hello, there.

This post is a very enthusiastic response to Dave Hill's post in
Tuning Digest 1488. One of the greatest pleasures is to see how a
musical experience can be shared by people working with different
tunings and styles. Such experiences can add to understanding on all
sides while highlighting interesting common ground.

> The b - fsharp - d resolving chord with the wide fourth
> between b and fsharp is nearly the same as the wolf D minor chord
> which occurs with a twelve per octave keyboard tuned to a just scale
> on C. In first inversion, this chord from d down to a down to f has
> the d to a fourth wide from just by a 21.5 cent comma, while the
> interval from f to a is a just 5/4 major third.

One fascinating point that Dave's posts have brought home to me: what
tertian just intonation and the early 15th-century Pythagorean tuning
with the Wolf at F#-B (actually Gb-B) share is a keyboard with
prominent just fifths _and_ a number of pure or nearly-pure thirds.

When I posted about this Wolf in a 15th-century setting, I didn't
realize how similar a situation could arise with Dave's just piano
tuning, and the connection is indeed a delightful one.

> In the song My Country tis of Thee - God Save the Queen, which
> begins C C D B C D, the first two notes are harmonized with C major
> and the third note would normally be harmonized with D minor. On my
> piano, which I am here translating to the terms of the just scale on
> C, the D minor chord sounds bad if played with a D in the bass, but
> if the F is played in the bass, that D minor chord sounds all right
> to me - not perfectly consonant, but not egregiously out of tune,
> and possibly as effective musically as it would sound if a lowered D
> were used or raised F and A so that the chord would be a just minor
> triad.

Thanks to Dave for a very memorable example which neatly brings the
concept home, and suggests that the acoustical factors at play in this
Classic context might be similar to those in a late Gothic setting.

Just today, on a Yamaha TX802 tuned to Pythagorean and the keyboard
mapped to place the Wolf at F#-B (Gb-B), I tried the 15th-century
cadential sonority d-f#-b, holding it for a long time before resolving
it to c-g-c'. Curiously, it wasn't unpleasant at all, but just seemed
maybe to lend a bit of extra flavor or energy to the cadence. While
Mark Lindley's examples don't dwell on this sonority for too long, I
wouldn't hesitant to use it in a more prolonged cadential context.

This is radically different than the effect of a Wolf fourth (or
fifth) as a bare interval, as I definitely noticed -- maybe
nature's way of saying "12 notes per octave is not enough."

Anyway, the tunings and musical settings are in some ways very
different, but Dave's experience and mine "running with Wolves" seem
strikingly similar. It's interesting, for example, that in Dave's
setting f-a-d' or d-f#-b is the inversion of a minor triad; in mine,
it's an active and expansive combination with two _major_ intervals
both "stretching" as it were toward stable resolutions (M3-5, M6-8).

Thank you, Dave, for a great post -- I'm looking forward avidly to the
next chapter in your just intonation piano saga.

Most appreciatively,

Margo Schulter
mschulter@value.net