Yahoo Tuning Groups Ultimate Backup tuning Re: A JI sequence -- Adaptive tuning questions

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A JI sequence -- Adaptive tuning questions
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About a month ago, two currents of my musical life converged to
produce a kind of progression in just intonation (JI) which I somehow
suspected might prove interesting for John deLaubenfels, adaptive
tuner extraordinaire.

In the weeks since, I have been pleased not only to discover through
John's keen analysis some facets of my progression of which I had not
been aware, but to learn that this example may raise some significant
questions regarding melodic perception and adaptive tuning
algorithms.

In his related article, John discusses these questions, sharing with
us his expertise in the theory and practice of adaptive tuning. He
offers an adaptively tuned version of my JI sequence based on current
methods, and suggests some possible further developments.

For my part, I'll explain a bit about the musical background of my JI
example, and also some of the melodic features which John called to my
attention.

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1. Sequence for a JI circumambulation
-------------------------------------

Around the end of May, I was excitedly immersed in two musical
currents. The first was a JI system with two 12-note Pythagorean
manuals tuned a pure 7:6 apart, which I named Sesquisexta after the
medieval Latin term for this ratio ("and again a sixth"). For people
who may be interested in this tuning, here is a link to a Scala file:

http://value.net/~mschulter/sesquisx.scl

The second current was my discovery of the EXAMPLE command in Scala
version 1.41 for MS-DOS, Manuel Op de Coul's outstanding free program
for scale design and analysis, which can take a text file defining an
example or musical composition along with a specified Scala scale file
and produce a MIDI file with pitchbend messages.

These currents reached their confluence when I came across an
interesting form of JI "circumambulation" in the Sesquisexta tuning
and decided to use Scala to create a MIDI sequence which could be
shared with others.

One of the special features of Sesquisexta is that in addition to the
usual Pythagorean intervals available on either manual, there are no
less than 11 pure sonorities of 12:14:18:21 or 14:18:21:24 available
by mixing notes from the two manuals.[1] Counting these two related
forms separately, we have 22 such "7-flavor" sonorities to choose from
(featuring just ratios of 9:7, 7:6, 12:7, 7:4, or 8:7).

One idea that occurred to me was a "circumambulation" through a
sequence of six _almost_ identical cadences, for example with unstable
12:14:18:21 sonorities resolving to stable fifths. Here I'll give such
a sequence in a notation with C4 as middle C, and a "v" symbol to show
a note a septimal comma (64:63, ~27.26 cents) lower than the usual
Pythagorean degree:

Bb3 Dbv4 C4 Ebv4 D4 Fv4 E4 Gv4 F#4 Av4 G#4 Bv4 Bb4
Bb3 C4 D4 E4 F#4 G#4 Bb4
Eb3 Gbv3 F3 Abv3 G3 Bbv3 A3 Cv4 B3 Dv4 C#4 Ev4 Eb4
Eb3 F3 G3 A3 B3 C#4 Eb4

MIDI example: <http://value.net/~mschulter/sesci001.mid>

In this type of circumambulation we start at the fifth Eb3-Bb3, and
conclude an octave higher at Eb4-Bb4, with the lowest voice moving
through a "circle" -- or better "near-circle" -- of six whole-tone
steps: Eb3-F3-G3-A3-B3-C#4-Eb4.

As the spelling may suggest, the first five steps are in fact
identical whole-tones at the usual 9:8 (~203.91 cents), but the last
step C#4-Eb4 is actually a diminished third at the rather complex
ratio of 65536:59049 (~180.45 cents), a Pythagorean comma smaller than
the others (531441:524288, ~23.46 cents).

For me, the main musical issue was: what effect might this melodic
unevenness have on the overall progression? Listening at the keyboard,
I found that it gave me no problem at all -- but then, I'm accustomed
to comma shifts of various kinds, especially those involving a
septimal comma or some interval serving a similar role in a tempered
system.

An historical bonus for me was that this sequence made the
Pythagorean comma a kind of poetry in motion, as I'll explain in a
footnote for those curious, demonstrating two "unusual" intervals
described around 1325 by the great medieval European theorist Jacobus
of Liege.[2]

Apart from this "creative asymmetry" introduced by the Pythagorean
comma at the conclusion of the progression[3], I didn't notice any
other special issues.

From another historical viewpoint, by the way, I looked at this
sequence as a kind of homage to Claude Debussy and his use of the
whole-tone scale around 1900, a century ago. The idea of an equivalent
scale in Pythagorean-based JI (e.g. Eb-F-G-A-B-C#-Eb), complete with a
touch of JI "unevenness," really appealed to me. I was tempted to
describe the effect as "Impressionistic," maybe almost "panmodal."

At this point, I found it both natural and easy with Scala's help to
produce a MIDI file for sharing with John deLaubenfels, master of
adaptive tuning and also a known connoisseur of septimal sonorities
such as 12:14:18:21.

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2. John's response: looking at things diagonally
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When I eagerly sent my MIDI file to John, my guess was that the
Pythagorean comma adjustments might be the main adaptive tuning
question. The news I got back was at once very encouraging, and most
intriguing from an analytical standpoint.

John found that none of the adjustments bothered his ear, an opinion
with special significance coming from someone with much experience in
hearing and weighing such fine points of intonation.

This, then -- at least as far as John and I were concerned -- was one
case where JI using pure vertical sonorities and rational intervals
only could succeed in very practical musical terms.

Having shared his verdict on the musical effect, however, John didn't
stop there: he noted some features of the sequence which I hadn't even
noticed, raising questions both for microtonal theory and for future
adaptive tuning algorithms.

While I had been focusing on the Pythagorean comma, John pointed out a
more pervasive issue which caused no problem in listening but raised
some issues for theory: the use in this example of lots of notes a
septimal comma apart.

Since the septimal comma of 64:63 didn't occur as a _direct_ melodic
step, I hadn't considered the more subtle matter of notes in different
voices and in relatively close musical proximity. As soon as I
diagrammed the piece, however, John's point became perfectly evident.

Here, for example, are the highest and lowest voices, with "+" signs
used to connect notes a septimal comma apart -- or, more precisely, an
octave less a septimal comma:

Bb3 Dbv4 C4 Ebv4 D4 Fv4 E4 Gv4 F#4 Av4 G#4 Bv4 Bb4
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + + + +
Eb3 F3 G3 A3 B3 C#4 Eb4

Since John couldn't hear anything "wrong," this example raised for him
an interesting question: how much might the ear accept "different
scale degrees which are a small fraction of a semitone apart."[4]

Here, for example, degrees such as Eb3 and Ebv4 are a septimal comma
apart, around 27 cents, or a bit more than 1/8-tone. More generally,
this presence of two "versions of the same basic note" is a feature of
various JI systems, and also of many regular temperaments when carried
to enough notes to generate small "commalike" intervals.

As the last diagram suggests, this kind of relation between notes in
two separate voices might be described as "diagonal," involving
different voices (the vertical dimension) at different points in time
(the horizontal dimension).

John also took note of another kind of curious relation which didn't
have any audible impact on him as a listener: the presence within a
single melodic line of scale degrees at the interval of a fifth a
septimal comma smaller than pure. As I then found, there were also
fourths of this kind a comma smaller than pure.

Here, for example, are the narrow fifths formed by notes in the
highest voice, with ratios of 189:128 (~674.69 cents):

Bb3 Dbv4 C4 Ebv4 D4 Fv4 E4 Gv4 F#4 Av4 G#4 Bv4 Bb4
|-------------------------|
|------------------------|
|------------------------|
|------------------------|

Here are the narrow fourths formed by notes in the same voice, with
ratios of 21:16 (~470.78 cents):

Bb3 Dbv4 C4 Ebv4 D4 Fv4 E4 Gv4 F#4 Av4 G#4 Bv4 Bb4
|---------------|
|--------------|
|--------------|
|-------------|
|---------------|

These diagonal and indirect melodic intervals involving the septimal
comma were something which had escaped my notice -- until I read
John's analysis, to which he brought his musician's ear and theorist's
insight.

From my perspective, John's feedback at once lent me much
encouragement in exploring this kind of JI progression, and enriched
my understanding of some of the subtle relationships involved.

At the same time, I was delighted to have shared an example which
might have some significant ramifications for adaptive tuning
theory. As John wrote:

"[I]t is in my mind a very open question how
much the ear will accept different scale degrees
which are a small fraction of a semitone apart.
Your sequence would seem to suggest, `quite
a lot!'"[5]

In his related article, John focuses on this kind of question,
considering not only what an adaptive tuning program might ideally
seek to model, but some of the complications in implementing
"real-world" algorithms more closely approximating such an ideal.[6]

For people interested in comparing my original JI sequence with John's
adaptive tuning version, here are links to both MIDI files:

original JI version: <http://value.net/~mschulter/sesci001.mid>
John's adaptive tuning: <http://value.net/~mschulter/sesciat1.mid>

Please let me conclude by thanking John, and also Manuel Op de Coul,
for their most patient and friendly assistance in helping me get
started with Scala's EXAMPLE feature, showing once again the
generosity and mutual aid so often exemplified by members of our
tuning community.

-----
Notes
-----

1. The 12:14:18:21 sonority is a rounded 0-267-702-969 cents; and
the 14:18:21:24 a rounded 0-435-702-933 cents.

2. This may be easiest to explain using rounded numbers. The first
five ascending whole-tones in the lowest part, Eb-F-G-A-B-C#, at
around 204 cents each, add up to 1020 cents, an interval (Eb-C#) which
Jacobus calls the _pentatonus_ ("five whole-tones"), and might also
be termed a Pythagorean augmented sixth. Taking the 1200-cent octave
Eb3-Eb4 and subtracting this interval Eb3-C#4 of about 1020 cents, we
have left about 180 cents for the remaining step C#4-Eb4. Jacobus
terms this the _tonus minor_ or "small tone," more precisely at a
ratio of 65536:59049 (~180.45 cents). As Jacobus points out, six pure
9:8 whole-tones would produce not a 2:1 octave but a _hexatonus_ at
the ratio of 531441:262144 (~1023.46 cents). In rounded terms, six
whole-tones at 204 cents each are equal to about 1224 cents --
exceeding a pure octave by around 24 cents, the Pythagorean comma, or
more precisely ~23.46 cents.

3. For the faster-moving voices, the highest and third-highest, the
Pythagorean comma adjustment at the end of the progression introduces
another "unusual" interval. These voices otherwise share a pattern of
ascending by a pure 7:6 minor third and then descending by a 28:27
semitone (~62.96 cents), two motions adding up to ascend by a 9:8
whole-tone, e.g. Bb3-Dbv4-C4-Ebv4-D4-Fv4-E4-Gv4-F#4-Av4-G#4-Bv4-Bb4 in
the highest voice. However, the last descending interval Bv4-Bb4 is
actually a Pythagorean comma larger than 28:27, or 137781:131072
(~86.42 cents). This interval is just slightly smaller than the usual
Pythagorean diatonic semitone or limma at 256:243 (~90.22 cents), the
difference being equal to the "3-7 schisma" or amount by which the
septimal comma exceeds the Pythagorean comma, ~3.80 cents (an integer
ratio of 33554432:33480783).

4. John deLaubenfels, personal communication.

5. Ibid.

6. Of course, there is the additional complication that different
listeners may vary widely in their tolerance of -- or positive liking
for -- various kinds of "small interval" shifts and relationships in
JI or similar systems. One approach might be an adaptive tuning
application with "User Preferences" to vary such parameters to taste.

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: A JI sequence -- Adaptive tuning questions

🔗mschulter <MSCHULTER@VALUE.NET>

🔗Paul Erlich <paul@stretch-music.com>

🔗George Zelenz <ploo@mindspring.com>

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗monz <joemonz@yahoo.com>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗Paul Erlich <paul@stretch-music.com>