Yahoo Tuning Groups Ultimate Backup tuning For Julia Werntz -- Comparing notes (1)

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For Julia Werntz -- Comparing Notes (1)
Adding Pitches: A game of "least steps"
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Hello, there, Julia and everyone.

Please let me begin by thanking you warmly for sharing in the process
of our tuning forum, and by apologizing for taking so long to write my
first promised "Comparing Notes" article.

In approaching such a pleasant and rewarding dialogue, I am especially
concerned about two points, of which I would invite you or others to
remind me at any opportune point in this conversation.

First, I hope that I may focus especially on a composerly and
musicianly view, since the enlivening realities of practice can at
once make theory more comprehensible and promote an understand of what
may be human commonalities of diverse musical worlds and experiences.

Secondly, I hope that this may be a free and open colloquy, not unduly
impeded by my penchant at times for involved technical presentations.
Each step of the way, I welcome your feedback and guidance.

Please let me also send my apologies both to you and to your renowned
colleague Joe Maneri for the variant spelling of his last name in my
previous post. Much in doubt on this point when I wrote, I attempted
quickly to canvass a few possible sources -- not then having your
article at hand. Unfortunately, I followed a source giving what would
prove to be the less orthodox orthography, a mistake promptly revealed
the next day when I revisited the library and again checked your
article. Neither did my foible go unnoticed in this forum, where it
was not unprecedented.

To draw some redeeming value from my misadventure, I might wonder
whether this potential ambiguity of spelling could resemble that I
noted in Florio's Italian-English dictionary of 1611, with either
_manera_ or _maniera_ as a form of the word meaning "manner" or
"style."

However spelled, this happy word may express the greatness both of a
composer such as Vicentino or Gesualdo, and of Joe Maneri as one
boldly exploring a new musical universe and manner of expression.

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1. "Adding Pitches" -- one overview
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Repeating my concern that any first impressions here should facilitate
but not constrain the free give and take of our conversation, I might
speak of three main themes that your PNM article "Adding Pitches"
sounds for me.

The first is an enriching counterpoint to my own musical practice,
which I will discuss in this article.

The second is an invitation for me, as someone who makes music with an
assortment of just intonation systems and tempered systems, to
distinguish this practice and theory from "rules" of "pure tuning"
which might exclude most of my favorite JI tunings as yet more
"dissonant" or "out of tune" than 12-tone equal temperament. Your
eloquent praise for certain JI compositions celebrates prime musical
values toward which I also strive: complex ratios, and novel types of
intervals both vertical and melodic.

The third is an opportunity for an open exploration of what I might
call "stylized tuning" -- the way that 72-tone equal temperament, as
you discuss and celebrate, so satisfyingly fits your musical
worldview, while I am enchanted by systems with fifths impure by about
the same amount, but in the wide direction. It feels almost like an
alternative history novel, each road having its own compelling logic
and grace.

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2. Adding pitches -- a generalized approach?
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In your article, you show how the basic tuning of 12-note equal
temperament (12-tET) can serve as the starting point for a process of
"adding pitches" to build complex microtonal systems such as 24-tET,
36-tET, 48-tET, and 72-tET.

Here I am tempted to suggest that my own approach to music seems to
involve a parallel process in which a 12-note tuning based on a
regular chain of fifths -- typically either pure (Pythagorean JI) or
gently wider than pure -- serves as a basis for "adding pitches" to
arrive at a larger system of 17 or 24 notes per octave.

Your article, of course, focuses on this process as applied to a
"microtonal equal temperament," and more specifically to what I might
term in shorthand as "6n/12n-tET" -- that is, equally tempered systems
with a number of notes evenly divisible by 6 or 12.

However, many elements of this paradigm -- with curious variations --
could also apply to many of my most characteristic tuning systems for
what I term "neo-Gothic" or "neo-medieval" music largely based on the
13th-14th century styles of Gothic Europe, with some influences also
from the medieval Near East.

To illustrate some of these parallels and variations, I have chosen a
regular but unequal as well as impure 24-note tuning based on a chain
of fifths at about 704.61 cents, or about 2.65 cents wide. As
mentioned above, these fifths are about as impure as the 700-cent
fifths of 12-tET and its extensions such as 72-tET (about 1.95 cents
narrow) -- actually a bit more impure -- but in the opposite
direction.

My choice to begin with an _impure_ 24-note system might be taken as
an exercise of a bit of prudent diplomacy: a 24-note system based on
Pythagorean tuning would take us into the realm of just intonation,
the not unexciting topic for my next post.

For now, it might suffice to add that these 24-note JI systems
(Pythagorean-based, pure fifths) and tempered systems (wide fifths)
typically share many of the same types of intervals, and support many
of the same Gothic or neo-medieval patterns and progressions. In this
musical environment, the "just/tempered" distinction can seem more of
a nuance than a gaping dichotomy.

In a Pythagorean-based JI system, as we shall see in my next article,
the "user interface" at the keyboard would be much like that we are
about to encounter: two 12-note manuals, each arranged in a regular
chain of fifths. As a medievalist, on such a keyboard I feel cozily at
home -- much as another musician might feel in 12-tET, also based on
such a regular chain, albeit with some important differences.

Before "adding pitches" to build a 24-note system, let us take a look
at our basic 12-note temperament, a familiar musical space in a
neo-medieval setting, just as 12-tET is in many other settings. Here
I show intervals in rounded cents:

132 286 628 837 991
C# Eb F# G# Bb
C D E F G A B C
0 209 418 495 705 914 1123 1200
209 209 77 209 209 209 77

From a melodic point of view, this temperament has whole-tones
(e.g. C-D, Eb-F) at about 209.21 cents, a bit larger than 12-tET, and
diatonic semitones (e.g. E-F, A-Bb, C#-D) at about 76.97 cents --
considerably narrower and more compact than the 100 cents of 12-tET.

Additionally, there is a dramatic contrast between these narrow
diatonic semitones and the wide chromatic semitones (e.g. C-C#, Eb-E)
at around 132.25 cents -- a difference of no less than 55 cents,
greater than the 50-cent quartertone of 24-tET and its subdivisions.

This disparity of semitone sizes invites striking forms of
chromaticism, for example those illustrated and advocated by
Marchettus of Padua in his _Lucidarium_ of 1318, while opening the way
in a 24-note extension of this tuning for captivating patterns of
metachromaticism or "ultrachromaticism" -- the latter term gratefully
borrowed from your "Adding Pitches" in PNM, where you discuss its use
by Ivan Wyschnegradsky

Vertically, as is characteristic of Gothic and neo-medieval tunings,
there is a contrast between pure or near-pure fifths and fourths and
rather complex thirds and sixths. Regular major and minor thirds at
about 418.43 cents and 286.18 cents are quite close to the complex
JI ratios of 14:11 (~417.51 cents) and 33:28 (~284.45 cents). These
are favorite sizes for neo-medieval thirds, either just or, as here,
slightly tempered.

Without getting into fine points of harmony and style, I should
explain that regular major thirds typically expand by stepwise
contrary motion to stable fifths, while minor thirds contract to
unisons: these intervals are themselves deemed _relatively_ concordant
or blending, but unstable. The compact 77-cent diatonic semitones lend
an incisive air to such directed progressions, as in these cadences
with an octave notation using C4 for middle C:

E4 F4 G#4 A4
B3 C4 E4 D4
G3 F3 C#4 D4

The first three-voice cadence also shows the characteristic expansion
of major sixth to octave, arriving at a complete stable sonority of
outer octave, lower fifth, and upper fourth (F3-C4-F4). The second
cadence arrives at a simple fifth (D4-A4). These 14th-century European
progressions represent one side of what Bill Alves terms "quintal
counterpoint" in the same issue of PNM as your article, a category
embracing other world traditions including gamelan and its
cross-cultural offshoots, the topic of his article.

Before from moving from 12 notes to 24, and to metachromatic or
ultrachromatic refinements explored in a musical game I call "Least
Steps," let us consider one more feature of this basic tuning: the
family of "alternative thirds and sixths" formed by intervals such as
augmented seconds and diminished fourths.

In 12-tET, regular minor thirds and augmented seconds have the
identical size of 300 cents, and likewise major thirds and diminished
fourths at 400 cents -- although, as you point out, their perceived
effects in musical context can be quite different.

In our temperament with wide fifths, however, their is an intonational
distinction between a regular 286-cent minor third (e.g. E3-G3) and an
augmented second (e.g. Eb3-F#3) at about 341.46 cents, the latter a
"supraminor third" a bit wider than the complex just ratio of 17:14
(~336.13 cents) or the 36-tET/72-tET interval of 333.33 cents.

Likewise, in contrast to the 418-cent major third (e.g. C-E), a
diminished fourth (e.g. F#-Bb) or "submajor third" has a size of about
363.14 cents, a bit narrower than a just 21:17 (~365.83 cents) or the
36-tET/72-tET interval of 366.67 cents.

These alternative thirds, while differing in size and color from
their regular counterparts, share a certain complex and active
quality, while inviting resolutions involving 132-cent chromatic
semitone steps, for example:

Bb3 B3
F#3 E3
Eb3 E3

Here the lower supraminor third contracts to a unison, while the upper
submajor third expands to a fifth, with chromatic steps of Eb3-E3 and
Bb3-B3.

Having introduced some themes of variety, contrast, and drama, I would
like to expand the stage to 24 notes, and to the smaller intervals
which are the special focus of my game of "Least Steps."

In the case of 12n-tET, as discussed in your article, the process of
"adding pitches" involves dividing each equal semitone into a smaller
number of equal parts -- quartertones (24-tET), sixth-tones (36-tET),
eighth-tones (48-tET), or twelfth-tones (72-tET), etc.

For our regular but unequal temperament, the simplest way to add
pitches is merely to continue the chain of fifths, let us say in the
sharp direction, so generating 12 additional notes which we map to a
second or upper manual. This new manual has an arrangement of steps
and intervals identical to that of the first or lower manual -- a
"carbon copy," as it were, but with each note higher by 55.28 cents,
the difference in size between the 77-cent and 132-cent semitones.
Here an asterisk (*) shows a note raised by this 55-cent interval:

188 341 683 892 1046
C#* Eb*/D# F#* G#* Bb*/A#
C* D* E* F* G* A* B* C*
55 264 474 551 760 969 1178 1255
209 209 77 209 209 209 77
------------------------------------------------------------------
132 286 628 837 991
C# Eb F# G# Bb
C D E F G A B C
0 209 418 495 705 914 1123 1200
209 209 77 209 209 209 77

By mixing notes from the two manuals, either successively or
simultaneously, we have an expanded universe of pitches, melodic
steps, and sonorities to explore, with the identical patterns on the
two keyboards providing, at least for me, a "user-friendly interface."

Melodically, we have two types of smaller steps: the 55-cent interval
I term a _diesis_, somewhere between the 66.67-cent thirdtone and
50-cent quartertone of 72-tET (e.g. E4-E*4); and the 21.68-cent
microinterval I term a "subdiesis" or comma (e.g. F#*3-G3). The latter
is equal to the difference between the regular 77-cent semitone and
the 55-cent diesis -- its 22-cent size is between that of the
1/12-tone (16.67 cents) and 1/6-tone (33.33 cents) of 72-tET.

Having adopted the term "metachromatic" for progressions involving the
55-cent diesis, I now find myself delightedly borrowing the term
"ultrachromatic" to describe this 22-cent step. Thanks to you, and to
Wyschnegradsky, whose work you celebrate in a new century.

Harmonically, our 24-note tuning introduces some intervals of a kind
you mentioned enjoying in your article: 264.50-cent minor thirds close
to 7:6 (~266.67 cents). We also find, for example, 440.11-cent major
thirds near 9:7 (~435.08 cents), and 550.76-cent "superfourths" very
close to 11:8 (~551.32 cents). As it happens, there are some small
minor sevenths at 969.10 cents, a virtually pure approximation of 7:4
(~968.83 cents).

The overall effect, at least in a neo-medieval setting, is one of a
subtle continuum of simplicity/complexity or "concord/discord" -- an
ethos which one could also, of course, realize in a larger system such
as 72-tET, where these diverse types of intervals have close
counterparts.

Here, our special interest is in melody, and in a challenge which your
article raised for me: it's nice to have all those small intervals
available on the keyboard, but how I can be sure that in my music they
actually get used, and in a telling rather than tangential manner?

My own approach to this problem is contrapuntal, bringing together the
dimensions of part-writing and distinctions of vertical sonority. To
start the game of "Least Steps," let us begin with this progression,
with numbers in parentheses showing vertical intervals in cents above
the lowest voice, and signed numbers show melodic motions, either
ascending (positive) or descending (negative). Here the upper voices
move in fourths, as they often do in 13th-14th century Gothic
progressions; the tempo should be quite leisurely, to permit
appreciation of the fine nuances:

E4 -- +55 -- E*4 -- +22 -- F4 -- +132 -- F#4 -- +77 -- G4
(705) (760) (782) (914)
B3 -- +55 -- B*4 -- +22 -- C4 -- +132 -- C#4 -- +77 -- D4
(209) (264) (286) (418)
A3 ------------------------------------------ -- -209 -- G3

The voices start at the relatively concordant sonority of A3-B3-E4,
common in some styles of 13th-century music, and in 20th-century terms
a form of fourth chord or fifth chord. As in 12n-tET, this type of
sonority has ratios quite close to 8:9:12.

The upper voices ascend by 55-cent steps to a small minor third and
sixth -- about 264 cents and 760 cents -- above the lowest voice. On
the continuum, these intervals might be described as close to 7:6
and 14:9. For me, as also for an analyst of LaMonte Young's music in
an earlier PNM, the 7:6 is rather "cozy" -- while, as you discuss,
minor sixths at various ratios may tend toward complexity with their
potential tension between partials.

We move next by 22-cent ultrachromatic steps to a regular minor third
and sixth at 286 cents and 782 cents, the latter a near-just 11:7,
a favorite size for a regular minor sixth in various tunings I use.
Here the small melodic steps are associated with a subtle shift of
vertical color or "mood" which may help to bring out this nuance.

The upper voices now move by comparatively large 132-cent steps --
chromatic semitones -- from minor to major third and sixth at 418
cents and 914 cents. This step size reminded me of the 133.33-cent
motions you mentioned in your article.

There follows a usual medieval or neo-medieval cadence, with these
intervals of regular major third and sixth expanding to fifth and
octave, the upper voices ascending by 77-cent diatonic semitones, and
the lowest voice, stationary to this point, descending by a regular
209-cent whole-tone.

After coming up with this first exercise at the keyboard, I realized
that I could subdivide the motion further, adding more steps and
sonorities:

E4 - +55 - E*4 - +22 - F4 - +55 - F*4 - +77 - F#4 - +77 -- G4
(705) (760) (782) (837) (914)
B3 - +55 - B*4 - +22 - C4 - +55 - C*4 - +77 - C#4 - +77 -- D4
(209) (264) (286) (341) (418)
A3 ------------------------------------------------- -209 -- G3

Here the opening sonorities and motions are the same; the regular
minor third and sixth (A3-C4-F4), however, is followed by a 55-cent
motion of the two upper voices to an intermediate sonority A3-C*4-F*4
with supraminor third and sixth at 341 and 837 cents, and then a
77-cent motion leading to the cadential major third and sixth
(A3-C#4-F#4) which resolve as before.

This exercise reminded me of your observation as to how the same
interval can have different musical meanings depending on context --
as with the minor third and augmented second in 12n-tET at 300 cents.
Here the first 77-cent motion in the upper voices, taking them from
supraminor to major third and sixth, is clearly a kind of "chromatic"
or maybe better "enharmonic" inflection; the following 77-cent motion
is a routine cadential semitone. The pleasant parallel seems like a
bridge between our worlds of intricate microtonality.

Most appreciatively,

Margo Schulter
mschulter@value.net

> Please forgive the delay in my response. Your post was full of
> information, and I wanted to read it carefully and digest it, but was
> held back by a busy schedule during the past week.

Please let me thank you not only for having the patience to read a
quite long post, but for taking the time to reply so thoughtfully. I
hope that my response to your article shows such reflection and
consideration.

The give and take of this kind of dialogue is a special treat and
honor for me, and I deeply appreciate any time you can give to this
process along with all of your responsibilities and commitments.

This is a precious learning process, mutual, of course, but something
which I find especially in keeping with your role as a teacher as well
as a composer. Already, both your article and your responses here have
taught me new ways of looking at music, and I hope that, to borrow an
old saying, a major part of the syllabus will prove to be a deeper
empathy and understanding regarding the different ways we approach this
wonderful world of music.

> It was very interesting to read about your own process of adding
> pitches; it is truly a fascinating, parallel world, with some common
> points as well as many foreign aspects, as you pointed out. It
> occurs to me that one of the primary differences in approach is that
> mine (and that of others who use microtonal equal temperaments with
> no JI, Pythagorean, etc. intentions) doesn't really have much to do
> with "tuning," per se. I realize that this also is certainly what
> Carl Lumma meant in his criticism of my use of the term
> "temperament" in this context. He had a point. It's a murky,
> semantic issue, but one which illustrates the difference in musical
> point of view, or tradition.

The shared perception we have of "parallel worlds" seems like a very
fertile direction to take, and at this point I might just offer a
quick aside. While some people here have adopted the term "equal
division of the octave" (EDO), for example "72-EDO," to avoid some of
the implications of "tuning" or "temperament" that you and Carl are
raising, I do want to say that I consider "72-tone equal temperament"
(72-tET or 72-ET) standard usage, and often go either way.

> "Tuning" would seem most often to imply the adjusting (whether in
> the pure or impure directions) of the pitch members of a preexisting
> hierarchical system, such as a "functionally" harmonic idiom, or a
> mode or scale. With music from a "pan-tonal" tradition (...I just
> don't want to use that nasty other "a-word" anymore), like mine, the
> most "tuning" can mean is "playing the intervals accurately," since
> the "system," or underlying functional logic, is idiosyncratic to
> each piece. The intervals are there, the musician's ears are
> trained, but the chromatic in this case isn't something that's
> "tuned"; it's more of a grid that's been made, like 12-note equal
> temperament (and which similarly allows for some coloristic or
> expressive inflection). Ezra Sims' scale, derived from the same
> 72-note chromatic, *can* be more properly called a "tuning," for
> example. (Though I believe Ezra also allows room for some minute
> pitch inflection.)

Thank you for shedding some light on your parallel universe: the idea
of a "given grid" here is very expressive. In response, I should begin
by acknowledging that indeed my "tuning" process _is_ colored by a
certain "preexisting hierarchical system" or "harmonic idiom" -- in my
case, that of 13th-14th century Europe, with some new elements mixed
in.

In your approach -- a tuning system as a grid that's simply "given"
rather than fine-tuned, so to speak -- there's a special kind of
discipline and devotion that I deeply respect: taking the everyday and
familiar, and doing something new and creative with it. This seems to
me a part of the creative impulse of 12-note pantonalism, and your
carrying of this kind of approach to a 72-note grid, your concept of
"Atwelve-tonality," is a very impressive musical as well as
intellectual development of this kind of current of musicmaking.

At one level, our musical worlds may have a certain mirrored contrast
and symmetry. You seem to be taking a more or less traditional grid --
with the important element of the added pitches, of course -- and
doing new and exciting things, including making sure those microsteps
really stand out in their own right. In contrast, I often seem to be
devising a range of new tunings, and doing more or less traditional
things with them. As it happens, my tradition is a few centuries older
than some people are used to, so it can sound "new," but that makes it
no less traditional or routine to me.

At another level, however, I wonder whether even a carefully
calculated and nuanced tuning system isn't also to a very considerable
degree a "found object" with surprises and unintended consequences --
or, one might say, random acts of musical kindness waiting to be
discovered.

Those of us who do play the game of "designer tunings," and then try
them out at the keyboard or wherever -- and people like George Secor
here have been at this for far more years than I have -- find that
there are all kinds of surprises that didn't necessarily enter to the
design.

Often serendipity is where it's at: pick some numbers that look
intuitively interesting, tune it up, play a bit, and _then_ say: "Hey,
this is a new and wonderful type of interval or progression or
whatever."

Maybe I come up with something in 12 notes, like it, and then a few
months later suddenly ask, "What if I carry this to 24 notes?" -- and
find some more surprises waiting for me.

Having said this, I'd like to express my admiration for you in taking
what seems to me the more radical and exacting path: seeking new
melodic patterns and properties from the more or less "familiar," and
adopting a discipline to make it more likely that new patterns emerge,
as distinct from either historical styles or the academic kind of
composition with a few quartertone inflections that's addressed in
your article.

To express some of this discipline in your "Atwelve-tonality," maybe
I'm tempted to propose the term "panmelody" or "panmelodiousness" --
an emphasis on the smaller steps as basic ingredients rather than mere
"modifications" of the larger and more familiar ones.

Like "pantonal," "panmelodic" could be a more positive assertion of
this "emancipation of microsteps." I wonder how well these terms seem
to fit together.

> I do realize that there are people like you who come to JI from this
> angle - seeking "novel types of intervals" - and that this is
> more-or-less where I was coming from in choosing to add pitches in
> the perhaps cruder way that I do, just grabbing them directly from
> in-between the semitones. I also realize that there certainly is a
> good amount of disagreement about the term "just intonation" itself,
> as there is about most general musical terms of this kind. I noticed
> you include complex, "out of tune" ratios in your definition, while
> others don't (some even insist that beats must not be present for
> music to be truly justly tuned). (Some would call Ezra's music just,
> but he wouldn't.)

Please let me apologize for the length of what follows, an attempt to
bring a bit of historical and cross-cultural perspective to the
question of complex JI and related issues.

First, with due musicianly humility, I would like to say our methods
of adding pitches might often seem to share a certain "automatic" or
"mechanical" aspect: whether it's dividing semitones equally, or
taking a chain of identically sized fifths and adding some more to
generate more notes.

Also, from my perspective, "just grabbing" notes "directly from
in-between the semitones" can also be a fine art. I reflect on Busoni,
whose "tripartite" division of the tone into thirdtones and then
sixthtones (36-tET) was based on an exquisitely discerning
appreciation for the melodic inflections and nuances of string
players. He also very carefully considered the musical consequences of
his division, its compatibility with existing music, and so forth.

To me, 36-tET is a thing of beauty melodically and harmonically, and
maybe I should compose something in it in your honor, and Busoni's

As far as JI, what it says to me mainly is that one chooses to
approach pitches or intervals in terms of integer ratios -- large or
small, simple or complex, with whatever "concord/discord" conventions
or character might attach in a given cultural or stylistic concept.

Maybe I'd compare JI -- tuning by integer ratios -- to a technique of
counterpoint like retrograde inversion. Someone familiar only with a
15th-century European use of this technique, for example, might assume
that the specific stylistic conventions regarding things like
"concord" and "discord" are inherent in the technique itself.

However, a 20th-century pantonalist might study 15th-century examples
in order to apply retrograde inversion in a serialist context with a
radically different approach to intervals -- as I understand has
actually happened. It's not "retrograde inversion" or "JI" (tuning by
ratios) itself that imposes the conventions, but a specific cultural,
stylistic, or personal preference.

Personally, I find the larger or more complex ratios a beautiful and
invigorating aspect of JI, musically and intellectually, and often
part of the idea of a tuning as at least in part "found object."

It's the nature of small integer ratios to generate large and more
complex ones -- a kind of branching or ramification, and something
that often produces extra and unplanned kinds of intervals, musical
opportunities waiting to be discovered and appreciated.

It can also be a kind of game of free association. For example, on
this list, the ratio 74:64 came up, which reduces to 37:32. Some
people would call that "the 37th harmonic": I just consider it a neat
ratio, maybe like one of those mathematical parameters you mention for
a piece by Xenakis.

It turns out that 37:32 has a size of about 251.34 cents, a very
interesting size, in conventional terms about midway between a major
second and a minor third. In 24/48/72-tET, there's a quartertone step
at 250 cents -- not quite identical, but very close. As it happens,
one of my JI systems based on two 12-note Pythagorean sets has an
interval of 14336:12393, about 252.14 cents -- also very close.

I very much like this part of the spectrum, whether approached in a
just or tempered tuning: it can have pleasantly "cool" and
sophisticated color for me, a new kind of "concord" or character. By
the way, I find 24-tET a tuning with some stunning musical qualities;
it's also a way for me to get in touch with certain aspects of the
20th century, another side of the realm of world musics in many times
and places.

At the end of this article, I'm including a link to a short
composition I did in this tuning, using the 14336:12393 as one of the
intervals.

Quickly, I should say that approaches to JI that recognize larger
ratios aren't new: this is, for example, an important aspect of
13th-14th century European practice and theory, where the simple
device of tuning a chain of pure 3:2 fifths or 4:3 fourths generates
some very large and complex ratios that play an important role in the
music.

For example, a theorist named Jacobus of Liege recognizes a degree of
"concord" or "compatibility" in ratios such as 81:64, 32:27, 27:16,
9:8, and 9:4 -- as in some 20th-century theory, there is a subtle
continuum or gradation. Of course, _any_ scale of "concord/discord" is
a cultural or stylistic construct, but my point is that JI theory can
involve diversity and contrast.

From a cross-cultural perspective, I might add that I've discussed the
European tradition because that's an important source for my own
outlook; but the medieval Near Eastern tradition recognizes an even
wider range of JI ratios like 14:13 and 13:12 and 7:6 and 27:22. It's
interesting that Ibn Sina (a name Latinized as Avicenna), a Persian
philosopher who exerted an immense influence on Gothic Europe, also
wrote about some beautiful scales with ratios like these.

To all these numbers, I'd add your vital reminder about "coloristic or
expressive inflections" -- integer ratios or cents are a kind of map
which singers or even players of fretted instruments dance _around_.
What I'm visualizing is a kind of decorative grid pattern on a dance
floor: the grid is there, now how do we dance around it, pantonally or
otherwise?

With the music of Ezra Sims, I might guess that his point is that
certain 72-tET intervals are very close to JI ratios, but that he
regards this as an approximation rather than a precise equivalence.
What I'd say is that _some_ JI ratios are heard as beatless when tuned
with sufficient precision -- a 3:2, for example, in contrast to
something considerably more complex. Thus a fifth tuned about 2 cents
narrow in 72-tET, or 2 cents wide in one of my "near-just" tunings,
could have some slow but audible beating making the texture subtly
different than a pure 3:2. Timbre plays a role here also.

From another point of view, Sims might be recognizing that 72-tET has
a somewhat different "mapping" or architecture than a scheme with
integer ratios for very similar types of intervals. The conceptual
structure is a bit different.

For example, as mentioned above, a 250-cent interval in 72-tET is very
close to 37:32, but a system featuring a just 37:32 might have a
certain association or "branching out" of ratios making the "grid" at
least a bit more asymmetrical and uneven. For example, we might find
37:32 associated with 36:32 or 9:8 (a usual JI major second) plus a
37:36 (about 47 cents, a kind of "quartertone" close to 24-tET).

The maps are a bit different, although they often can lead to similar
destinations. One might say either, "This tempered interval sounds to
me identical to an integer-based ratio, so I'll call it just"; or "The
maps are a bit different, and maybe the sounds a bit different, also,
so I'd rather keep the two categories distinct."

Again, the footsteps of the dancers in a performance, with their
"minute pitch inflection," might be larger than the theoretical
distance between the ideal points on these "maps" -- like art, and
life generally, it's an open question.

> Thanks for such an interesting post. I would be very interested to
> hear the music! How is this possible?

One first approach, however imperfect, is to share a few items through
the MIDI format -- that brief piece I mentioned, _Salutation for Mary
Beth Ackerley, envision'd as Lysistrata_, and some odd snippets in the
temperament I mentioned in my previous post.

Coding a score on computer and having it translated automatically into
a MIDI file isn't the most expressive performance, and I realize that
not everyone has the means to listen to these files (I can generate
them and proofread the output codes, but not hear them). However, I
offer these items as a possible start.

Here's the piece, a kind of keyboard prelude, including that sonority
with 14436:13392, a tad larger than the 72-tET interval of 250 cents:

http://value.net/~mschulter/mary002.mid

For quick MIDI snippets, some involving "ultrachromaticism," I'm
giving a URL to the third article in a series on the temperament I've
been mentioning -- cautioning that the musical examples, the main
point of interest here, are interspersed among lots of theory. For
anyone curious, there are links at the start of the article to the
previous two portions (without musical examples), but I'd emphasize
that my main purpose here is to share the musical examples themselves:

/tuning/topicId_24448.html#24448

To give a bit of balance, here's a piece I composed some years ago
without any special consideration as to "tuning," although here it's
in Pythagorean -- please enjoy:

http://value.net/~mschulter/library1.mid

Here's a link to a score of the piece in PostScript for printing or
viewing either with a dedicated PostScript printer, or with PostScript
interpreter utilities such as the free Ghostscript available on the
Internet:

http://value.net/~mschulter/library1.ps

Most appreciatively,

Margo Schulter
mschulter@value.net

For Julia Werntz -- Comparing notes (1)

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