Re: Exploring tunings -- from Pythagorean to 23-tET

"How do YOU compose in X tuning"

Hello, there, Robert Valentine and everyone.

Your question of how to approach a new scale is a really captivating
one, with many directions to go in.

First of all, to offer a direct answer: my procedure for exploring a
tuning may vary dramatically depending on the tuning.

Note that for people with a synthesizer linked to a computer, or a
computerized music system, the steps might be quite different -- for
example, simply specifying a tuning in a program like Manuel op de
Coul's Scala and letting the computer set the tuning.

With Pythgorean for music in usual Gothic styles, or 1/4-comma
meantone for music in usual Renaissance styles, on a Yamaha TX-802
it's mainly a matter of selecting the right preset tuning and playing
or improvising more or less as I had been doing for the previous 30
years or so -- but with a more idiomatic tuning.

At the other extreme, my latest experiment in using 23-tone equal
temperament (23-tET) in a Sethares-inspired metallophone-like timbre for
neo-Gothic music has required careful preparation and "mapping out" of
intervals and cadences before figuring out a tuning table and then
entering the tuning at the front panel -- or actually, two 12-note
tunings, one for each MIDI controller keyboard, giving a 17-note system in
all.

Of course, "tuning it up" is only the beginning of actually exploring
the scale, seeing how my anticipations based on calculations in cents
and so forth match up with musical experience. For example, while a
crude experiment in Pythagorean persuaded me that an interval of
around 678 cents (the diminished sixth or "Wolf fifth," e.g. G#-Eb)
could indeed sound like a recognizable and acceptable fifth in a
vibraphone-like timbre, I was really excited when I confirmed that in
23-tet, 13/23 octave could actually likewise serve as a stable fifth
in practice as well as theory.

Improvising or composing in a new tuning is for me often a process of
moving from the known to the unknown. The familiar Gothic sonorities
and cadences (for a neo-Gothic tuning), or Renaissance progressions
(for a tuning like a 24-note archicembalo in 1/4-comma meantone),
provide a point of ready orientation. From here, I can branch out into
altered intervals, direct chromaticism or shifts of a comma or diesis,
and so forth.

In what follows, I'll try to address some of these questions in more
detail, maybe focusing especially on my "24-out-of-N" paradigm I tend
to apply to theoretically open as well as closed tunings: Pythagorean,
1/4-comma meantone, 29-tET, and Keenan Pepper's beautiful new
neo-Gothic temperament based on Phi, for example.

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1. Context: A prologue in three short illustrations
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As a prologue to what follows, I would start with a word that may not
be so surprising: _context_. How one approaches a scale may vary
radically with how one approaches intervals, modes, and progressions.
People who come from backgrounds other than medieval and Renaissance
European music shouldn't be surprised if some of my own views and
"natural" ways of approaching certain tunings seem quite strange.

For example, your recent mention of 53-tET reminded me of how people
can view a tuning through different musical lenses and make different
assumptions about its basic intervals, let alone the fine points of
stylistic applications.

One of my favorite "litmus tests" might be to ask people: "How many
steps does a diatonic semitone have in 53-tET?"

From my medievalist perspective, the answer is 4 steps, and we have a
regular tuning almost identical to 3-limit or Pythagorean JI, e.g.

9 9 9 4 9 9 4
F G A B C D E F

For someone oriented mainly to a 5-limit or higher standard of
stability, however, 53-tET just as "naturally" suggests an
approximation of Ptolemy's or Zarlino's syntonic diatonic with its
unequal whole-tones of either 8 or 9 steps, and large diatonic
semitones of 5 steps.

Two other amusing illustrations of "context" involve the way that a
major third in a given tuning can mean quite different things to
different people.

When I read James Barbour's comments that when all things are said and
done, the thirds of 29-tET are still not very good, I want to reply:
"What do you mean: the major third of 29-tET is around 413.7 cents,
quite close to a usual Pythagorean 81:64, but just enough larger to
make a bit of a musical distinction?" The humor, of course, is that
Barbour is very likely referring to what I would term diminished
fourths, or "alternative thirds," vis-a-vis 5:4. Then, again, he was
hardly evaluating these tunings in terms of Gothic or neo-Gothic
aesthetics <grin>.

Similarly, when Paul Erlich recently talked about "stretching" the
major thirds of 22-tET, my first reaction was, "At 436.36 cents, do
they really need 'stretching' -- they're already a tad larger than
your beloved 'mininum entropy' valley at 9:7." Of course, Paul was
doubtless referring to the alternative or "schisma" third of 7 steps,
a bit smaller than 5:4. I was reflecting on the "regular" (to me)
major third of 8 steps, a rather stretched 81:64 which can have a very
pleasant effect when it expands to a fifth.

Enough of contextual musings, the main point being that how one
approaches and explores a tuning may depend on what musical
"questions" one is asking at the outset -- not that experience can't
broaden these questions or raise new ones.

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2. A personal tuning pattern: from-12-to-24-out-of-N
----------------------------------------------------

In explaining my approach to how-many-out-of-N questions, I might
begin by noting that for historical medieval and Renaissance music I
play, and for more or less "conventional" improvisations in similar
styles, a 12-note range of Eb-G# or the like is typically ample. Thus
12-out-of-N with two conventional MIDI keyboards tuned in unison can
seem like a natural rather than constrained solution.

However, I also tune 13-to-24-out-of-N either to emulate late medieval
or Renaissance/Manneristic schemes such as the proposed 17-note
Pythagorean keyboards of the early 15th century or a 24-note subset of
Vicentino's 31-note meantone cycle for the archicembalo; or to devise
schemes of this kind for compositions or improvisations in derivative
styles.

Although I'm trying to come up with some scheme of "circulating"
14th-century-style cadences for 17-tET (easier done on paper than
mastered fluently at the keyboard), this is almost like a solution in
search of a problem. Circulating around a complete tuning circle isn't
generally a part of the medieval/Renaissance/Manneristic tradition,
and so is something of a novelty to me. There are a few pieces from
the 16th and early 17th centuries which fall into this kind of
category, including Colonna's "example of circulation" moving through
all 31 steps on his instrument, but these are rather like the
exception which underscores the rule.

Thus my main reason for using an n-tET is not necessarily that it
circulates, but mainly that it represents an interesting shade of
temperament (e.g. 29-tET for neo-Gothic music) or an attractively
"different" setting (e.g. 23-tET "Sethareanized" with a metallophone
timbre for neo-Gothic purposes).

Note, by the way, that various theoretically open tunings lend
themselves to "virtual" or musical closure in a certain number of
notes: for example a 53-note cycle in Pythagorean, or a 31-note cycle
in 1/4-comma meantone. Here, as with an n-tET, someone regarding
circulation as a usual musical feature rather than a "special effect"
would have a strong incentive to tune a full cycle.

However, circularity aside, there are two powerful motivations for
tuning more than 12 notes per octave, motivations applying both to
experimental late medieval and Renaissance composers, and to
derivative styles:

(1) Having standard sonorities and cadences available on
more steps of the gamut; and

(2) Having available new kinds of vertical and melodic
intervals and progressions.

To illustrate the first point, let's consider what happens when we
take a 12-note Pythagorean or neo-Gothic tuning with a range of Eb-G#
and add a 13th note Ab. Now we have available, for example, two new
cadences of the following typical variety. Here C4 is middle C:

Eb4 F4 G4 G4 Ab3
C4 D4 D4 Eb3
Ab3 G3 Bb3 Ab3

In the first example, Ab serves to permit a new kind of regular
cadence with a descending diatonic semitone to G (Ab-G). In the second
example, maybe more radically, Ab itself serves as a new stable goal
for a cadence. In our expanded 13-note system, the unstable sonority
Bb3-D4-G4 now has _two_ possible regular resolutions where a major
third expands to a fifth and a major sixth to an octave, the familiar
one to A by descending semitonal motion or the new one to Ab by
ascending semitonal motion:

G4 A4 G4 Ab3
D4 E4 D4 Eb3
Bb3 A3 or Bb3 Ab3

The second enticement to larger tunings, the availability of new kinds
of intervals both for vertical sonorities and for melodic
progressions, may be even more compelling from a xenharmonic
viewpoint.

For example, I love 29-tET, and enjoy 24-out-of-29 because it makes
available those striking major thirds at 11/29 octave or ~455.17
cents, a diesis wider than the usual 10/29 octave (~413.79 cents),
which can expand to a fifth with a characteristic melodic motion of
only a diesis or 1/5-tone (~41.38 cents) in one of the voices.

Why not _always_ tune 24-out-of-N, rather than any smaller number, for
tunings where precise or virtual closure involves 24 or more notes?

My main answer might be that when using the two keyboards as manuals
with contrasting timbres, having all or most of the keys in unison can
make things simpler. With tuning sets of 13-16 notes, the moderate
complications can be charming, but for 17-24 notes, treating both
manuals as part of a monotimbral "archicembalo" is generally more
straightforward.

Incidentally, my choice of 24 notes as a "large" tuning size mainly
reflects the logistics of performing polyphonic music live on two
12-note keyboards with conveniently repeating octaves on each
manual. The TX-802 itself supports any arbitrary mapping of notes to
keys. Since many compositions or improvisations in medieval or related
styles can easily fit in a two-octave range, for example, a 48-note
tuning mapping 24 notes per octave to each 49-note keyboard would be
quite technically possible. My main question might be: "How do I play
fifths with a single hand, let alone octaves?"

Of course, it remains true that each note brings new possibilities,
and that 24 is only one possible number in a cycle such as 29-tET,
Pythagorean 53, or a 121-note circulating version of Keenan Pepper's
tuning.

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3. A "parallel keyboard" paradigm: 12-times-2-out-of-N
------------------------------------------------------

When tuning 24-out-of-N, I have a leaning more specifically to a
"generalized" (or maybe better "parallel") scheme in which the two
12-note keyboards have identical interval arrangements at the distance
of a comma, diesis, or n-tET scale step apart.

In a 24-note Pythagorean or "Xeno-Gothic" scheme, this distance is
equal to the Pythagorean comma of 531441:524288 (~23.46 cents). For a
24-note subset of Vicentino's or Colonna's 31-note cycle in 1/4-comma
meantone, it is likewise equal to a diesis (e.g. G#-Ab) of 128:125, or
~41.06 cents.

For 29-tET, a neo-Gothic tuning with qualities quite different than
Vicentino's 16th-century meantone, the distance between manuals is
interestingly an almost identical diesis (here 1/29-octave or
1/5-tone) of ~41.38 cents.

For Keenan Pepper's regular tuning where the ratio of the chromatic to
the diatonic semitone is equal to Phi, with fifths of ~704.096 cents,
the distance is again equal to the comma or diesis between Ab and G#,
here ~49.15 cents.

This kind of "12-times-2-out-of-N" arrangement might be described as a
"two parallel universes" approach to composition or improvisation.

Each keyboard, in itself, offers a complete and familiar 12-note
tuning. Additionally, the two "universes" combined offer two enhanced
dimensions of musical creativity:

(1) Progressions which make the "jump" or "quantum leap"
between the two parallel keyboards; and

(2) Sonorities combining notes from both keyboards, either
making intervals already present within a 12-note tuning
available at more locations, or generating new kinds
of intervals.

Both dimensions manifest themselves, for example, in the small sample
of "enharmonic" or fifthtone music we have from Nicola Vicentino
(1555). Stunning diesis shifts in cadential or other progressions,
and novel intervals such as the "proximate minor third" which
Vicentino describes as having a ratio of approximately 11:9, beckon us
to a new realm of musical creativity.

Similarly, a "two parallel universes" arrangement for a neo-Gothic
tuning such as 29-tET makes available both progressions from one
12-note keyboard to another (involving melodic intervals altered by
a diesis or 1/5-tone), and new intervals such as the extra-wide
cadential major third of 11/29-octave (with a ratio of ~13:10)
inviting resolution to a fifth.

In contrast to a model of circulation based on a closed (or virtually
closed) tuning cycle, this "parallel universes" model may lend itself
to forms of musical motion, and possibly systematic "perambulation"
involving leaps between the two 12-note gamuts.

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4. Concluding comments
----------------------

Note that all of the above discussion assumes a compositional or
improvisational "paradigm" that one brings to a new tuning. However,
it is also quite possible to take the tuning itself as a starting
point, and ask, "What kind of patterns can I discover in these
intervals that might lead to some interesting compositions or
improvisations?"

There may be a kind of middle ground: "I'm choosing this tuning
because it permits me to use 'recognizable' patterns of a style with
which I'm familiar, but introduces complications or ambiguities."

Various ways of analyzing tunings, including mine, are based on
musical categories or assumptions which may apply to some styles but
not to others.

For Paul Erlich, for example, one important consideration may be
"consistency" for an n-tET at various limits.

In a neo-Gothic setting, I might consider the "symmetry" or
"asymmetry" of a tuning for a few important intervals: for example,
does the best approximation of a 4:3 fourth plus the best
approximation of a 9:8 major second add up to the best approximation
of a 3:2 fifth. If not, the tuning is "asymmetrical" -- an intriguing
feature, as with 23-tET.

One point which Ivor Darreg and Brian McLaren have made bears special
emphasis: each n-tET has its own "flavor" -- an observation applying
to other kinds of tunings also -- and, I might add, different
"flavors" to different people depending on their backgrounds and
expectations.

Most respectfully,

Margo Schulter
mschulter@value.net