Margo's mysterious sequence

A couple of weeks ago, Margo sent me a copy of a short but interesting
sequence she had prepared, sesci001.mid. The progression of scale
degrees with her tunings (cents relative to 12-tET) is as follows:

forward in time ---->

0 (C): 0.00* -27.27
1 : -37.04 +13.70*
2 (D): +3.91* -23.36
3 : -5.86* -33.13 -5.86
4 (E): +7.81* -19.43
5 (F): -1.95* -29.22
6 : -38.99 +11.72*
7 (G): +1.95* -25.32
8 : -35.08 +15.65*
9 (A): +5.86* -21.41
10 : -3.91* -31.18 -3.91
11(B): +9.77* -17.48

*These start earlier, then are joined by the other notes.

There is a circle along the pythagorean chain: Eb-Bb-F-C-G-D-A-E-B-F#-C#
-G#, with all the tunings consistent, but the final chord snaps back to
Eb-Bb rather than continuing the trend to D#-A#; thus there is a melodic
fifth degree (G# to Eb) which is flat by a full pythagorean comma as the
final chord sounds.

The fifths are joined (except at the end) by two other notes forming
in each case a 12:14:18:21 chord. As the above chart shows, this
results in notes on the same nominal scale degree being differently
tuned by 63:64 (about 27.3 cents) over a short period of time.

The challenges, on paper, of this small sequence are many. The ear must
accept melodic shifts of about 1/4 semitone, very different from
"normal" music heard by Western ears. Then there is the pythagorean
comma (23.5 cents) at the end, though this time it's at least a
deviation from some interval other than unison!

To my surprise, none of these challenges is audible to my ear. The
sequence sounds "fine", howbeit the chords are deliciously dark in a
way not possible without microtuning. I don't hear any objectionable
motion whatever.

Just for fun, I thought I'd see how closely my adaptive tuning program
could be made to approximate Margo's tuning of this sequence. This
represents a _serious_ challenges to my methods, because I am unable to
recognize some important features:

. Many scale degrees (in the 12-tET world) need to be recognized as
multiple scale degrees in the target sequence for grounding
purposes.

. The maintaining of consistency of the tuning through the circle of
fifths would depend upon some kind of melodic springs across the
fifth degree.

These challenges, especially the first, represent severe limitations to
my methods, ones which over time I want to overcome, just to be able
to handle sequences such as this (and meantone scales with more than
12 notes, and...).

I had to alter many of my input parameters to extreme values just to
make a sound at all similar, and _still_ didn't do a very good job:

. To allow a given scale degree to have markedly different tunings
in a very short time span, I had to make horizontal springs very
weak (I used 1/100 of usual values).

. For the same reason, grounding springs had to be made very weak
(again I used 1/100 of usual values).

. I made up a tuning file that targets septimal intervals (7:6 and
9:7) in order to match Margo's chord tunings; this at least I've
got fairly well covered in the methods I already support.

Here's how close I came (file sesci001mar1.mid):

0 (C): +15.82* -18.85
1 : -16.55 +13.40*
2 (D): +16.16* -18.53
3 : +12.23* -16.28 -1.42
4 (E): +16.19* -18.33
5 (F): +12.50* -15.94
6 : -19.48 +16.48*
7 (G): +12.84* -15.92
8 : -19.21 +16.70*
9 (A): +12.87* -15.60
10 : +15.55* -18.87 +0.56
11(B): +13.18* -15.41

*These start earlier, then are joined by the other notes.

Even with severely weakened horizontal and grounding springs, vertical
intervals are compromised by a few cents. All sense of a pythagorean
chain in the circle of fifths is lost. Also, not shown, there is
significant horizontal motion (about 4 cents) as each dyad becomes a
tetrad, in notes continuously sounding, a byproduct of the weakened
horizontal springs. At least, to my ear, the progression does sound
superficially similar to Margo's.

I am indebted to Margo for this illustrative sequence: the extent to
which it sounds natural is a great surprise to me. It also provides an
excellent test sequence to use as I attempt to make my methods more
generally applicable to sequences with more than 12 scale degrees per
octave.

Oh, and here are Margo's and my tunings:

http://value.net/~mschulter/sesci001.mid (original JI sequence)
http://value.net/~mschulter/sesciat1.mid (adaptive tuning version)

Thanks, Margo!!

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_25717.html#25717
>
> Even with severely weakened horizontal and grounding springs,
vertical intervals are compromised by a few cents. All sense of a
pythagorean chain in the circle of fifths is lost. Also, not shown,
there is significant horizontal motion (about 4 cents) as each dyad
becomes a tetrad, in notes continuously sounding, a byproduct of the
weakened horizontal springs. At least, to my ear, the progression
does sound superficially similar to Margo's.
>
> I am indebted to Margo for this illustrative sequence: the extent to
> which it sounds natural is a great surprise to me. It also
provides an excellent test sequence to use as I attempt to make my
methods more generally applicable to sequences with more than 12
scale degrees per octave.
>
> Oh, and here are Margo's and my tunings:
>
> http://value.net/~mschulter/sesci001.mid (original JI sequence)
> http://value.net/~mschulter/sesciat1.mid (adaptive tuning version)
>
> Thanks, Margo!!
>
> JdL

Well, all the "fudging" seemed to pay off, since I do believe that
John deLaubenfels' version is perceptibly smoother. However, at
first listen, both versions *do* still sound rather similar (!!)

Anybody else have any comments?? Sometimes I wish I had
a "blindfold" test of such exercises... I wonder if I'm
not "influenced" to a degrees.

Thanks to John and Margo for this interesting experiment!

__________ ________ ________
Joseph Pehrson