Re: Quick bulletin -- e-based tuning and 7-flavored valleys

Hello, there, everyone.

What with all the excitement this week on the List about harmonic
entropy, Vicentino, 24-note archicembalos, adaptive JI, and (curiously
just last night, as we'll see) approximations of 7:4 in 19-tone equal
temperament (19-tET) and elsewhere, the following quick bulletin may
fit in nicely.

In an article I posted last night about a 24-note archicembalo version
of Keenan Pepper's Phi-based neo-Gothic tuning, there's a comparison
with intervals in other 24-note tunings sharing the same general
region around 704 cents (fifth size, of course).

One of the tunings included is a temperament I first described this
summer in an article on neo-Gothic tunings, now referred to as an
"e-based" tuning. In this tuning, the ratio between the whole-tone and
diatonic semitone or limma is equal to Euler's _e_, about 2.71828.

This definition results in a fifth of around 704.607 cents. As it
happens, this tuning was my introduction to the region between 29-tET
and 17-tET, so it would be a fond one in any event.

However, as noted in the 24-note archicembalo article on Pepper's
tuning, the e-based temperament has a feature only brought out when we
carry it beyond 12 notes: a major third (13 fourths up) and minor
third (14 fifths up) at ~440.110 and ~264.497 cents, about 5.026 cents
and 2.374 cents from 9:7 and 7:6. The latter approximation,
especially, caught my attention.

Then, just yesterday, I checked out the 15th fifth -- and found that
it generates a minor seventh at ~969.104 cents, only ~0.278 cents from
a pure 7:4. It was interesting that same evening to read on the List
about approximations of 7:4, and also about the relationship of that
interval to 7:6 or 9:7 in various multi-voice sonorities.

Anyway, the e-based tuning now has a musical claim to fame apart from
the general beauty of its region and its august mathematical
derivation honoring the great Leonhard Euler: what I might call very
accurate "7-flavor" ratios inviting some great neo-Gothic cadential
resolutions.

Yes, as John deLaubenfels might say, this is an instance where "the
musical context makes it unstable anyway." Explaining the musical
logic of such intervals in a neo-Gothic context requires a bit of
background material on cadential progressions, and this is a fine
opportunity for a "gentle introduction" to the musical patterns
involved. I hope to supply one in the next few days.

To make clear the neo-Gothic context, I prefer to speak of these
intervals as "7-flavor" rather than "7-limit"; similarly, the
neo-Gothic context for the use of ratios such as 14:11 or 17:14 might
be surprising to someone who associates different expectations with
"11-limit" or "17-limit" music.

As I said in the latest 24-note archicembalo tuning article, either a
Renaissance meantone like Vicentino's temperament (likely 1/4-comma)
or a neo-Gothic tuning like Pepper's has secrets revealed only when we
go beyond 12 notes.

With the tuning based on Euler's _e_, however, it is only when we go
beyond 12 notes that we encounter the distinguishing mark of this
tuning lending it a special "degree of suavity," to borrow a
felicitous phrase.

A comment based on some "harmonic entropy" remarks of Paul's: with
fifths only 2.65 cents from pure, a virtually pure 7:4 (is that called
"a desirable slight detuning," Paul?), and a 7:6 within 2.5 cents of
pure, maybe the variance of 5 cents in the 9:7 isn't such a problem,
especially since we are discussing an unstable sonority here. How
timely, Paul, that this topic should come up in your posts just
yesterday on "Synths and Centsibility," as if in response to the
question I was asking myself: "Is 440 cents close enough?"

In sum, for now, the special quality of the e-based tuning, in
comparison to other tunings in the same general neighborhood, is that
it offers variations on the usual and rather complex "plateau"
intervals of the region, _plus_ a set of 7-flavor "valley" intervals
on a 24-note archicembalo inviting very efficient cadential
resolutions. It's the combination of features which gives the e-based
tuning its special place on the continuum.

Most respectfully,

Margo Schulter
mschulter@value.net