Yahoo Tuning Groups Ultimate Backup makemicromusic MIDI example: Phi as interval (~833.09 cents)

Hello, there, everyone, and it's a pleasure to be posting again in
MakeMicroMusic after a "break" of some months.

My warmest thanks to Paul Erlich and Jon Szanto for helping to bring
about and expedite my return to this forum. As it happened, simply
subscribing at my new address via e-mail rather than the Web interface
seemed neatly to solve my technical problems, which might reflect my
use of a text-based Internet account and browser.

To start on the right foot, I'd like to share a musical example
addressing a question recently raised: the qualities of Phi as an
interval ratio, about 1.618 or ~833.09 cents. What you'll hear in this
example is an approximation in the Peppermint 24 tuning, ~832.76 cents:

<http://www.calweb.com/~mschulter/pmntphi1.mid>

This is one of my favorite intervals, and for me belongs to what I
would call the family of "supraminor and submajor" intervals. Other
examples would be supraminor thirds at around 17:14 (~336.13 cents),
with this passage also providing an example as I'll discuss shortly;
submajor thirds at around 21:17 (~365.83 cents); and submajor sixths
at around 28:17 (~863.87 cents).

To explain the notation for my example below (for people reading on
the Web, much better if you can view with the original formatting,
maybe "Expand Messages" or the like), I should say a bit about
Peppermint. Mostly it's a tuning very gratefully borrowed from Erv
Wilson and Keenan Pepper, the first of whom noted its position on his
Scale Tree, and the second of whom independently proposed it on the
Tuning list in September 2000.

As soon as I learned that the fifths were around 704.10 cents, I was
very excited, and indeed it's almost as if this were a custom
optimization designed especially for me. I could write a long essay on
this (and have written more than one), but suffice it to say that I
have a passion for regular major and minor thirds around 14:11 and
13:11 (about 417.51/289.21 cents), and supraminor and submajor thirds
around 17:14 and 21:17. In the Wilson/Pepper temperament, all four of
these intervals are within 1.5 cents of the just ratios.

This very nicely fits the "neo-medieval" music I make based largely on
13th-14th century Western European styles, where rather complex and
active thirds and sixths often resolve to stable fifths and fourths.
As has been discussed here, what you consider a "good" tuning can
often depend on what style you're following, and my passion for
Peppermint might be an apt example.

To get Peppermint 24, I took a basic 12-note tuning and added another
12-note chain of fifths at the distance of about 58.680 cents, so as
to produce some pure 7:6 minor thirds (~266.87 cents) as well as lots
of other new intervals quite close to just ratios of 2, 3, 7, 9, 11,
and 13.

In notating my example of the "Phi sixth," I use an asterisk (*) to
show a note on the upper 12-note keyboard, raised by 58.680 cents or
so. Here that becomes relevant for the next to last chord or sonority,
as we'll see.

(3/4) (2/4)
1 2 + 3 + | 1 + 2 + | 1 2 | 1 2 ||
F#4 G4 F#4 F4 F#4 G4 F#4 F4 F#4 F#*4
C#4 D4 C#4 C4 C#4 D4 C#4 C4 C#4 C#*4
Bb3 Bb3 G#*3 F*3

Mostly this passage is an example of what I call "Phi drift." The
upper voices start at the supraminor third and sixth (about 336 and
833 cents above the lowest voice), and move in parallel fourths: first
up to a sonority with the usual major third and sixth (around 416 and
912 cents), then back to the supraminor third and sixth, then down to
the second and fifth (approximating a JI chord of 8:9:12, another of
my favorites), then back up to the supraminor sonority, and so forth.

What I'm going for, to use a medieval European term, is "imperfect
consonance": a mildly unstable texture pleasant in itself, but with
some tension or inconclusiveness eventually leading to a resolution of
some kind. What I'd call a Phi sixth is not "dissonant" but a bit
"vague" or "cloudy" -- I once commented to Erv Wilson that it's the
kind of interval Debussy might have used around 1900 if it had been
part of the prevailing intonational palette. How's "impressionistic"?

Here the eventual resolution illustrates one of the features of the
24-note tuning: the lowest voice shifts from Bb on the lower keyboard
to G#*, forming a wide major third and sixth with the upper voices at
around 437.38 and 933.13 cents, with the sixth a pure 12:7, very close
to 7:9:12. This sonority resolves in a typical medieval fashion, with
the major third expanding to a fifth and the major sixth to an octave.

To put things less technically, this large major third is closer in
size to the fifth and so can resolve to it more efficiently, and
likewise the large major sixth to the octave; at the same time, these
intervals have ratios at or very close to a simple 9:7 or 12:7. For
me, "audibly just thirds with small integer ratios" often means 7:6
and 9:7, in which I find purity, efficiency, and excitement.

What I want to do here, in discussing Phi sixths, is to put my musical
impressions in a bit of context, as I hope the musical example itself
might help to do. Specifically, since I regard thirds and sixths as
active or "imperfect" consonances, a Phi sixth seems a natural part of
the landscape.

Here I might quickly comment that in addition to these supraminor or
Phi sixths, actually augmented fifths (e.g. Bb-F# in this example),
Peppermint 24 also has some close approximations of the 13:8 or
"harmonic" neutral sixth at about 842.30 cents, or about 1.77 cents
larger than the just ratio of ~840.53 cents. This is realized as a
regular minor sixth plus the 59-cent interval between the keyboards,
for example A-F*.

One of the consequences of my association with George Secor is that
I've acquired a taste for 13:8 to complement my predilection for the
nearby Phi sixth. In Peppermint 24, both sizes of sixths are
available. By the way, I would like warmly to thank George for lending
a lot of the creative impetus leading to and enriching my adventure
with this 24-note system, and to note that his HTT-29 not only
provided an inspiring precedent, but offers a near-just system with
greater overall accuracy for most of the ratios supported by
Peppermint, not to mention a wider range of intervals.

These remarks as to the Phi sixth and Peppermint might give some idea
of what is for me a "usual" musical style or tuning system, possibly
filling in a bit of perspective for my next intended post on some
delighted first impressions of what I would call a less conventional
and very beautiful tuning: 14-tET.

By the way, here's a link to a Scala file for Peppermint 24:

<http://www.calweb.com/~mschulter/peprmint.scl>

and here's one to Scala files for George Secor's HTT-29, HTT-17, and
the superset of HTT-41:

</tuning-math/messages/7574>

Again, thanks to the many people who have made this forum possible.

Most appreciatively,

Margo Schulter
mschulter@...

MIDI example: Phi as interval (~833.09 cents)

🔗Margo Schulter <mschulter@...>

🔗Aaron K. Johnson <akjmicro@...>