Re: Neo-Gothic Well-Temperament -- 17 notes

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A Neo-Gothic Well-Temperament in 17 notes:
Just and tempered intervals interwoven
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Hello, there, everyone, and it's a great pleasure to introduce what
may be a new tuning genre: a "neo-Gothic well-temperament" in 17 notes
per octave.

Please let me begin by acknowledging that some people may have very
reasonable objections to the use of the term "well-temperament" for a
tuning which not only has 17 notes per octave rather than the usual
12, but is designed to fit a Gothic or neo-Gothic musical framework
radically different than the world of 5-limit major/minor tonality
from Werckmeister to Beethoven.

As to the number of notes, I can say that regardless of any verdict
ultimately reached on this question of terminology, I am indebted to a
remark of Paul Erlich some months ago that a well-temperament could
have more than 12 notes, for example 19. This remark was part of a
dialogue on modality, tonality, and tuning in 17th-century music; and
it has ultimately proved catalytic in moving me to explore unequal
circulating neo-Gothic temperaments, whether appropriately called
"well-temperaments" or something else.

This article is indeed warmly dedicated to Paul as a scholar,
musician, mathematician, theorist, and educator.

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1. Introduction: a tuning takes shape
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Upon seeing Paul Erlich's recent articles on well-temperaments of the
conventional kind, I was moved to ask, "What about neo-Gothic
well-temperaments? Isn't time that I explored this aspect of
neo-medieval intonation?"

Quickly I decided that while conventional 5-limit well-temperaments
have 12 notes, a 17-note format would seem much friendlier for a
neo-Gothic well-temperament. I may write an article explaining this
decision in more detail, but for now let's consider the attractiveness
of 17 as a tuning size.

In a circulating 17-note tuning, we must disperse the quantity by
which 17 pure 3:2 fifths would fall short of ten pure 2:1 octaves --
about 66.76 cents.[1] Thus we must temper our fifths by an average of
about 3.93 cents in the _wide_ direction.

If we temper each fifth by this identical amount, then we have the
special case of 17-tone equal temperament or (17-tet). Since 17-tet is
squarely in "the middle of the road" within the spectrum of neo-Gothic
regular tunings or hypermeantones ranging from Pythagorean JI (or
53-tet) to around 22-tet or 27-tet, it offers a very friendly point of
departure for unequal well-temperaments.

Very happily, I quickly came upon a pleasant way to allocate that
66.76 cents: why not temper the 11 "regular" fifths (Eb-G#) so as to
produce pure 14:11 major thirds, and the other six more remote or
"supplementary" fifths (Gb-Eb, G#-A#) so as to obtain pure 9:7 major
thirds at the far side of the tuning circle?

In other words, my ideal was to tune 11 fifths in what I call a pure
14:11 hypermeantone (fifths ~2.421 cents wider than pure), and the
other six fifths in a pure 9:7 hypermeantone (fifths ~6.816 cents
wider than pure). As it turned out, this scheme would exceed the total
tuning "budget" of 66.76 cents by only around 0.773 cents.

An obvious and pleasant solution is to temper five of the six more
remote fifths as in a pure 9:7 hypermeantone, as intended, and to
temper the sixth "odd" fifth by this amount less the adjustment of
~0.773 cents, or ~6.043 cents, thus "balancing the budget" and
making the tuning cycle come out to precisely 10 octaves.

My goal was to maximize the number of just intervals -- just intervals
with a neo-Gothic flavor, of course. We get eight pure 14:11 major
thirds in the nearer portion of the circle -- actually a complete
12-note hypermeantone tuning of Eb-G# in its own right, then ranging
through intermediate sizes until we reach two pure 9:7 major thirds at
the far side of the circle. Another major third in this remote region
is only ~0.773 cents from a pure 9:7, being formed from a chain of
three 9:7 hypermeantone fifths plus the slightly narrower "odd" fifth
balancing out the circle.

Now just one decision remained: where to put that 17th odd fifth? At
first blush, I considered placing it at the most remote fifth of the
circle A#-Gb, where sharp and flat extremes meet in a kind of
intonational International Date Line. However, this would break the
chain of 9:7 hypermeantone fifths needed to get some pure 9:7 major
thirds. The following diagram, with "S" showing a "supplementary" 9:7
hypermeantone fifth and "O" the odd fifth, may help show my dilemma:

... G# - D# - A# - Gb - Db - Ab - Eb ...
S S O S S S

To get a pure 9:7, we need a chain of four consecutive "S" fifths.
Fortunately, a ready solution was at hand: why not place the odd fifth
at Ab-Eb -- a traditional location in a 12-note meantone tuning for
the "odd" or "Wolf" fifth (i.e. G#-Eb)? Also, since this odd 17th
fifth is actually intermediate in size between the 14:11 and 9:7
hypermeantone fifths, wouldn't placing it at an intermediate point in
the circle be in line with "the conventions of well-temperament"?

At any rate, the Ab-Eb solution nicely restores an unbroken chain of
five "S" fifths -- producing two pure 9:7 thirds (G#-Db, D#-Ab) as
intended:

... G# - D# - A# - Gb - Db - Ab - Eb ...
S S S S S O

Thus our full 17-note tuning circle looks like this, with approximate
sizes in cents shown for each fifth, and the "International Date Line"
at A#-Gb appearing at the bottom of the diagram:

D
704.377 704.377
G A
704.377 704.377
C E
704.377 704.377
F B
704.377 704.377
Bb F#
704.377 704.377
Eb C#
707.998 704.377
Ab G#
708.771 708.771
Db D#
708.771 708.771
Gb A#
708.771

Now the tuning itself was complete, and it was time to explore the
ramifications.

Charting out the various categories and sizes of intervals, I came
upon the serendipitous discovery that the tuning includes another just
interval: a pure 11:9 third at the augmented second Db-E. Also, I was
delighted to find a relationship involving the ratio -- shall I call
it a schisma? -- of 2058:2057.

Having considered the theory, it was time for some musical practice:
tuning up the temperament and trying it out with some 13th-14th
century compositions, and some neo-Gothic improvisations in similar
styles.

Using a mild "positive organ" timbre, I found that the pure 14:11
major thirds were "sunny" and beautiful, resolving splendidly to
stable sonorities with fifths. At the keyboard, as on paper, there
were striking contrasts as one journeyed about the territory of the
more remote accidentals and cadences. The neutral or "neutral-like"
thirds and sixths of this region -- actually diminished fourths or
augmented seconds -- can have a "mysterious" flavor, like similar
intervals in 17-tet.

Yesterday a visiting friend heard the tuning, and offered some
fascinating reactions which I hope to share in another article.
For the moment, I'll just note that she when she heard the "neutral"
17-tet-like intervals, she described them as "mysterious" -- and said
that the tuning generally had for her a "dark" quality like "a forest
in Northern Europe." (Actually, as I reread this early on Monday, the
visit took place "the day before yesterday," Saturday.)

In the remainder of this first presentation, I'll seek to survey a few
features of the tuning and document some of its "vital stastics."
Section 2 offers an overview including a keyboard chart and a sample
of a few sonorities and cadences.

Within the next day or two I hope to post a list of interval
categories and sizes in the tuning.

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2. When tunings coalesce: a just and tempered fabric
----------------------------------------------------

The following chart shows the tuning with the five more common or
"regular" accidentals (Eb-G#) above the seven diatonic notes, and the
five less common or "supplementary" accidentals below. The octave C-C
is arbitrarily taken as a reference:

196:121
130.639 286.869 626.262 835.016 991.246
c#' eb' f#' g#' bb'
_130.6|78.1_78.1|130.6_ _130.6|78.1_130.6|78.1_78.1|130.6_
c' d' e' f' g' a' b' c''
1:1 14:11
0 208.754 417.508 495.623 704.377 913.131 1121.885 1200
208.754 208.754 78.115 208.754 208.754 208.754 78.115
db' d#' gb' ab' a#'
_70.1|138.7_135.03|73.7_ _65.7|143.0_74.5|134.3_139.4|69.3_
70.100 343.786 561.329 778.871 1052.558
126:121

One way of mapping this tuning to two standard 12-note keyboards is to
place the five regular accidentals on one keyboard and the
supplementary ones on the other, with the seven diatonic notes shared
by both keyboards. Another approach to place the five flats on one
manual and the five sharps on the other.[2]

If we take the first approach, then our manual with the regular
accidentals will offer a full 12-note chromatic scale (Eb-G#) in our
hypermeantone with pure 14:11 major thirds.[3] Note that we can
incorporate this beautiful 12-note tuning, complete and unaltered,
into our 17-note tuning circle. This situation may suggest the
creative possibilities of 17-note neo-Gothic well-temperaments, and
also their forgiving nature when compared with the more earnest
constraints typically imposed by historical 12-note well-temperaments.

On our second manual, the five supplementary accidentals also define a
tuning in themselves: a pentatonic scale in a hypermeantone with pure
9:7 major thirds. Trying this scale, e.g. Db-D#-Gb-Ab-A#, we find that
the two "minor thirds" actually are spelled as doubly augmented primes
(Db-D#, Ab-A#), while "major seconds" occur at the doubly diminished
fourths D#-Gb and A#-Db as well as the expected Gb-Ab.[4]

As we move from the usual Eb-G# range to more remote cadential centers
and modal transpositions, intervals and progressions take on
intriguingly different shadings and colors, something I would like to
describe more fully in another article. For now, let us consider a few
examples of regular, alternative, and "remote" sonorities and cadences.

In the "regular" 12-note hypermeantone range, the mildly unstable
Gothic _quinta fissa_ or split fifth sonority consists of an outer
fifth divided into a pure 14:11 major third (~417.51 cents) and a
minor third at about 286.87 cents, or ~2.34 cents narrower than
13:11.

This kind of sonority could be interpreted either as a tempered
version of 22:28:33 or 28:33:42, with 14:11 and 33:28 thirds making up
a pure 3:2 fifth; or as a tempering of 121:143:181 or 121:154:181,
with a pure 14:11 major third and 13:11 minor third joined to form a
fifth at 196:121, or about 706.72 cents (~4.76 cents wider than pure).

In the second interpretation of a sonority combining 14:11 and 13:11,
where the fifth would be "naturally tempered" as it were, we are in
effect slightly tempering the 13:11 but actually making the fifth
closer to a pure 3:2 (~2.42 cents wide) than in the "just" tuning.

Within our regular Eb-G# hypermeantone range, we also have four
diminished fourths or alternate major thirds at 121:98 or ~364.98
cents, smaller than a pure 21:17 by a schisma of 2058:2057, or about
0.84 cents. These thirds have a pleasing "submajor" or "subditonal"
quality, while augmented seconds or alternate minor thirds in this
regular 12-note range at ~339.39 cents, or ~3.26 cents wider than
17:14, may have a "superminor" or "supersemiditonal" quality. An
alternative split fifth sonority combining these two thirds might be
taken as an approximation of 14:17:21 or 34:42:51.

Comparing standard cadential resolutions of regular and alternate
split fifth sonorities within the 12-note Eb-G# range may also show
the contrast between diatonic and chromatic semitones. Here C4 is
middle C, with higher note numbers for higher octaves; numbers in
parentheses show vertical intervals between adjacent voices, and
signed numbers show melodic intervals either ascending (positive) or
descending (negative):

B3 -- +78.1 -- C4 Bb3 -- +130.6 -- B3
(417.5) (704.4) (365.0) (704.4)
G3 -- -208.8 -- F3 F#3 -- -208.8 -- E3
(286.9) (0.0) (339.4) (0.0)
E3 -- +78.1 -- F3 Eb3 -- +130.6 -- E3

The difference in color between these two versions of the common
resolution in which a minor third contracts to a unison while a major
third expands to a fifth (m3-1 + M3-5) results not only from the
varying sizes of the unstable thirds but also from the contrast
between diatonic semitones (~78.12 cents) in the usual version and
chromatic semitones (~130.64 cents) in the alternate one. The
contrast becomes yet greater as we venture into more remote
accidentals and cadential transpositions.

The octave complement of the diminished fourth is an augmented fifth
(e.g. F3-C#4) or alternative minor sixth at 196:121 or ~835.02 cents,
rather close to the Golden Ratio or Phi at ~1.61803398875 (~833.09
cents).

Here are two possible resolution of a sonority involving this interval
in combination with the augmented second, both variants on a common
13th-century cadence in which minor sixth expands to octave and minor
third to fifth (m6-8 + m3-5), with the numbers in parenthesis this
time showing vertical intervals between the lower and outer pairs of
voices. The second example introduces some more remote accidentals:

C#4 -- +156.2 -- Eb4 C#4 -- +213.1 -- D#4
(835.0) (1200.0) (835.0) (1200.0)
G#3 -- +156.2 -- Bb3 G#3 -- +217.5 -- A#3
(339.4) (704.4) (339.4) (708.8)
F3 -- -208.8 -- Eb3 F3 -- -151.8 -- D#3

In the first progression, within the usual 12-note range, the lower
voice descends by a major second while the upper voices ascend in
parallel fourths by the melodic interval of a diminished third,
something like 3/4-tone. In the second, the lower voice descends by a
similar diminished third while the upper voices ascend together in
fourths by major seconds. In the latter progression, leading us into
the more remote portion of the tuning circle, note how major seconds
and fifths grow larger, and diminished thirds smaller.

A final example may demonstrate the dramatic contrasts offered by a
17-tone well-temperament; here the basic progression is the favorite
14th-century cadence of major sixth to octave and major third to fifth
(M6-8 + M3-5):

C#4 -- +78.1 -- D4 Db4 -- +138.7 -- D4
(913.1) (1200.0) (852.6) (1200.0)
G#3 -- +78.1 -- A3 Ab3 -- +134.3 -- A3
(417.5) (704.4) (361.4) (704.4)
E3 -- -208.8 -- D3 E3 -- -208.8 -- D3

In the second example, the usual major sixth E-C# is replaced by the
remote diminished seventh E3-Db4 at a pure 18:11, and the usual major
third E-G# by the diminished fourth E3-Ab3 in a transitional region of
the tuning circle where this interval shades from the usual ~21:17
toward 16:13. The very compact 78-cent diatonic semitones of the first
progression are replaced by chromatic semitones further enlarged by
the well-temperament process: Db-D exceeds C#-D by a full 60 cents.

This kind of 17-note circulating tuning might be compared to a fabric
or quilt of many threads: it combines a generous set of just intervals
with the color contrasts of extended (hyper)meantone and of more
traditional well-temperaments. We might add that the first 12 notes in
14:11 hypermeantone are almost identical to 46-tet (see also Note 3
below), while the five remote accidentals in 9:7 hypermeantone are
very close to 22-tet.

-----
Notes
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1. This quantity of ~66.76 cents, or 134217728:129140163, is the
"supercompact" semitone or "sublimma" of Xeno-Gothic music based on a
24-note Pythagorean tuning (17 fifths down, e.g. G#-Bbb), equal to a
usual diatonic semitone or limma of 256:243 (~90.22 cents) less a
Pythagorean comma of 531441:524288 (~23.46 cents).

2. The arrangement of the usual accidentals from Eb to G# in one
"rank" or tier, and of the less common accidentals in another, follows
a 16th-century practice illustrated by Nicola Vicentino's archicembalo
(1555). The arrangement with five flats on one keyboard and five
sharps on the other might at least conceptually have a certain
affinity to the 17-note Pythagoreah monochord divisions of Prosdocimus
de Beldemandis (1413) and Ugolino of Orvieto (c. 1430-1440?). These
theorists first present separate chromatic monochords with five flats
or five sharps, showing how the two monochords must be combined into a
17-note monochord in order to avoid certain cadential inconsistencies.
While specifically describes how the "intelligent organist" could take
advantage of such a tuning, Ugolino does not actually address the
question of a keyboard arrangement. My first impression is that the
Eb-G#/other scheme may be more practical if one only occasionally uses
the "other" accidentals, while the flats/sharps scheme may facilitate
neo-Gothic styles where these other accidentals are used with some
frequency.

3. if extended to 46 notes, this tuning would provide a circulating or
"virtually closed" temperament almost identical to 46-tet. The last
fifth would be made about 1.34 cents narrower than the others (~703.04
cents, ~1.08 cents wider than pure) to achieve a cycle of 46 fifths
precisely equal to 27 octaves.

4. Another pentatonic scale in 9:7 hypermeantone is defined by the
chain of fifths G#-D#-A#-Gb-Db, e.g. Db-D#-Gb-G#-A# (equivalent to
m3-M2-m3-M2-M2). In this specific mode, incidentally, there is no
fifth above the chosen center Db, but some world polyphonies actually
prefer fourths as the main concords, a preference exhibited in some
Western European polyphony of the 9th-11th centuries (e.g. Guido
d'Arezzo). Thus the fourth Db-Gb, or a three-voice sonority with the
fourth below and fifth above such as Db3-Gb-Db4, could serve as a
concord of repose in this mode.

Most respectfully,

Margo Schulter
mschulter@value.net