Yahoo Tuning Groups Ultimate Backup tuning Re: More on 29-tet -- a beautiful tuning

Hello, there, and thanks to Paul Erlich and Joseph Pehrson for
spurring me on to offer another view on one of my favorite neo-Gothic
tunings: 29-tone equal temperament (29-tet).

This is a really beautiful tuning for neo-Gothic music, and I find
myself using it eagerly and often both for 13th-14th century European
pieces and for improvisations in related styles. Recently I seem to be
using it more than any other tuning -- with interludes of medieval
Pythagorean or Renaissance 1/4-comma meantone -- so maybe this
discussion of 29-tet comes when "the iron is hot."

Here I would like to look first at the usual (and very beautiful)
aspects of 29-tet as a regular neo-Gothic temperament, and then some
more esoteric and quasi-historical aspects which may draw interesting
replies, especially from the Monz.

At the beginning, I should emphasize that how one views a tuning such
as 29-tet may depend on one's musical viewpoint. Someone like Paul
Erlich using 29-tet as a kind of 5-limit or higher meantone tuning
would likely have some quite different observations to offer than what
follows. Even such a basic statement as "29-tet has nice major thirds"
could refer to two or possibly three different sizes of intervals
depending on one's stylistic viewpoint.

The following discussion approaches 29-tet as a hypermeantone -- that
is, a regular temperament with fifths wider than Pythagorean or pure.
The spectrum of hypermeantones ranges from Pythagorean to 5-tet, with
the portion of this spectrum most relevant for Gothic or neo-Gothic
music going out at least as far as 22-tet -- or 27-tet in certain
timbres.

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1. Some musical observations: usual styles
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From my musical perspective, 29-tet is a subtly accentuated version of
a standard medieval Pythagorean tuning. The fifths are tempered by
around 1.49 cents in the wide direction (and fourths in the narrow
direction), leaving these intervals quite close to pure.

If asked for an overall impression of the tuning, I might say, "rather
mild, gentle, and charming." As a close neighbor of Pythagorean,
29-tet offers some intriguing shades of difference; at the same time,
it's on the mild side of 17-tet, which is a bit more of a clearly
_neo_-Gothic tuning.

Major thirds in 29-tet at ~413.38 cents are not too far from
Pythagorean, and about midway between a Pythagorean 81:64 (~407.82
cents) or a 19:15 (~409.24 cents) and a 14:11 (~417.51 cents). I like
them very much in certain voicelike timbres on a Yamaha TX-802
synthesizer, and also for example in a mixture of two harpischord-like
timbres.

A slight timbral nuance: while I tend in Pythagorean to play the lower
voices with the richer harpsichord-like timbre, in 29-tet I tend to
use this timbre for the upper parts, and a more "harp-like" or
"lute-like" registration for the lower parts. Possibly this preference
reflects the somewhat greater acoustical tension involving the fifth
partial of the richer timbre in 29-tet. The goal in either case is a
"semi-concordant" major third, at once pleasant and intriguing in
itself, and moving toward some expected resolution.

Interestingly, the 29-tet minor third at ~289.66 cents is an excellent
approximation of 13:11 (~289.21 cents), and this interval very nicely
contracts to a unison -- or expands to a fifth also in 13th-century
styles.

Major seconds at ~206.90 cents and minor sevenths at ~993.10 cents are
around 2.99 cents from their pure Pythagorean counterparts at 9:8 and
16:9. These intervals and the near-pure fifths and fourths produce
very nice Gothic or neo-Gothic sonorities such as 4:6:9, 9:12:16,
6:8:9, and 8:9:12.

A pleasant feature of this tuning, both in its mathematical simplicity
and in its aural effect, is the division of the whole-tone into five
equal parts. The usual diatonic semitone or limma is 2/5-tone or
~82.76 cents, a very nice size for usual cadences, and narrower than
the already compact Pythagorean interval of 256:243 (~90.22 cents) or
about 4/9-tone. The chromatic semitone or apotome is 3/5-tone or
~124.14 cents, an interval lending its flavor to a neat cadential
variation.

This variation brings us to the topic of alternative thirds (augmented
seconds and diminished fourths) in 29-tet, another neat nuance of this
tuning. While Pythagorean versions of these intervals ("schisma
thirds") very closely approach 5-limit ratios, 29-tet gives them their
own distinct character. Our alternative major third or diminished
fourth is ~372.41 cents, and our alternative minor third or augmented
second is ~331.03 cents.

These intervals differ from 5:4 and 6:5 by amounts roughly comparable
to regular major and minor thirds in 12-tet -- but in the opposite
directions, of course, maybe approaching the "zones of influence" of
intervals such as 21:17 (~365.83 cents) or 17:14 (~336.13 cents). A
tuning like my "e-based hypermeantone" (with the ratio between the
whole-tone and diatonic semitone equal to Euler's e, ~2.71828) takes
us more clearly into such zones of influence, but the intermediate
realm of 29-tet can have its own special attractions.

Thus in 29-tet both the regular and alternative thirds have an active
quality, and the latter intervals invite resolutions like the
following involving the wide chromatic semitone or apotome, with C4 as
middle C and higher note numbers showing higher octaves. Vertical
intervals in cents are shown in parentheses, while signed numbers show
melodic intervals in the ascending (positive) or descending (negative)
directions:

Bb3 -- ~+124.14 -- C4
(~331.03) (~703.45)
G3 -- ~-206.90 -- F3
(~372.41) (0.00)
Eb3 -- ~+124.14 -- F3

(m3-1 + M3-5)

The difference between the diatonic and chromatic semitones is
1/5-tone, or one scale step -- about 41.38 cents. Interestingly this
is very close to the 128:125 diesis of 1/4-comma meantone (~41.06
cents), and maybe invites some experiments in neo-Gothic chromaticism
with melodic figures alternating between these contrasting semitones.

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2. Quasi-historical aspects: 29-tet and "ultra-Gothic" cadences
---------------------------------------------------------------

As a tuning system which rather closely approximates Pythagorean while
dividing the whole-tone into five equal parts, 29-tet has another and
more "radical" side. It permits variants on the usual cadences
featuring the use of a melodic "diesis" of only 1/5-tone or ~41.38
cents, and "superwide" major thirds near 13:10 and major sixths near
26:15.

Before getting into the possible "quasi-historical" implications of
these progressions, let's look at a characteristic example, using the
ASCII asterisk (*) to show a note raised by a diesis or 1/5-tone:

E*4 -- ~+41.38 -- F4
(~951.72) (1200.00)
B*3 -- ~+41.38 -- C4
(~455.17) (~703.45)
G3 -- ~-206.90 -- F3

(Mx3-5 + Mx6-8)

Here the cadential major third expanding to a fifth is a diesis wider
than usual, or ~455.17 cents, very close to 13:10 (~454.21 cents); and
likewise the major sixth (~951.72 cents) expanding to an octave, very
close to 26:15 (~952.26 cents). To show that these intervals differ
from usual major thirds and sixths, I call them "maximal" thirds and
sixths, or Mx3 and Mx6, thus notating this cadence as (Mx3-5 + Mx6-8).

As a feature of neo-medieval style, this kind of "ultra-Gothic"
cadence can be very striking, and I like the effect especially with
certain voice-like timbres on synthesizer.

Now for the quasi-historical part, which the Monz (and maybe others)
may have anticipated. Marchettus of Padua, in his _Lucidarium_ of
1318, advocates that singers follow an intonation based on the usual
2:1 octaves, 3:2 fifths, 4:3 fourths, and 9:8 whole-tones of
Pythagorean tuning -- but deriving other intervals, especially certain
cadential ones, from a division of the whole-tone into "five parts."

Each of these parts is known as a "diesis," and here the
interpretive debate swiftly gets under way. Are these dieses equal or
unequal? Is the "fivefold division" of the tone a geometric one
(calling for the element of a temperament) or possibly an arithmetic
one (as in a monochord division)? Are the "five parts" of the tone a
mathematical formulation, or more of a guide for singers -- who are
not bound to any fixed tuning?

Of special interest is the cadential diesis of Marchettus, which he
defines as equal to "one of the five parts of a tone." If this is an
equal or roughly equal division, than a cadence like the above
"ultra-Gothic" example would result.

While scholars including our own Monz have come up with many ingenious
interpretations of Marchettus, there is one passage which to me
suggests that 29-tet may in fact provide one intriguing realization
of his cadential aesthetics.

In his discussion of the resolution of unstable intervals or
"dissonances" -- Marchettus refers to thirds and sixths as "tolerable
dissonances," while 14th-century writers in the French tradition, for
example, tend to call them "imperfect concords" or the like -- he
compares two resolutions of the cadential major sixth, one standard
and the other a "feigned color" (loosely translated, a "deceptive
cadential inflection"):

D4 C#4 D4 D4 C#4 C4
D3 E3 D3 D3 E3 F3

8 M6 8 8 M6 5

In the notation of Marchettus, the C# shows that the note C is raised
by an extra-wide "chromatic semitone" equal to "four parts of a tone,"
leaving only "one part of a tone" for the remaining diesis C#-D. The
problem which Marchettus considers is why the first resolution is
standard, while the second is less expected and favored.

Having earlier explained that a cadential dissonance should approach
the consonance which it seeks (e.g. M3-5, M6-8) as closely as possible
-- thus calling for the cadential diesis of only "one of the five
parts" of a tone -- he remarks that the major sixth of these two
progressions is equally far from either the octave or the fifth. Thus
the question arises: why does the standard resolution to the octave
seem better to fit the norm of "closest approach"?

One answer, the first offered by Marchettus, is that the octave at 2:1
is a purer consonance than the fifth at 3:2, and thus a more
satisfying destination for the major sixth, located as it is "six
dieses" from either stable goal. However, both the framing of the
question, and the second answer of Marchettus, are of special
interest.

In saying that a cadential major sixth is equally "six dieses" from
either the octave or the fifth, Marchettus may be implying that his
dieses are at least roughly equal, so that one can validly compare
distances between intervals using them.

In 29-tet, an extra-wide or "maximal" major sixth at 23 steps is
indeed precisely 6/5-tone (or 6/29 octave) from either the usual goal
of the octave (29 steps) or the less usual goal by contrary motion of
the fifth (17 steps). This doesn't mean that Marchettus necessarily
intended vocal intonation precisely in 29-tet, or using this scale as
the basis for an "adaptive JI" scheme to make fifths, fourths, and
major seconds pure -- only that 29-tet seems to realize the idea of a
cadential major sixth (or "maximal sixth") about equally far from the
octave or the fifth.

As his second answer to the problem of why the resolution to the
octave is more favored, Marchettus analyzes the melodic motion of each
voice in the cadence. Generally, he presents ideal cadential action as
involving movement by _both_ voices in contrary motion: both voices
share in the "dissonance," and both participate in its resolution.

Comparing the two resolutions by contrary motion of the major sixth,
he notes that in the usual resolution to the octave, the lower voice
descends by a regular whole-tone or five dieses, while the upper voice
ascends by one diesis, the narrowest possible interval of the system.

In contrast, in the unusual resolution to the fifth, the lower voice
ascends by the usual diatonic semitone or limma (to Marchettus, an
"enharmonic semitone") of two dieses, while the upper voice descends
by its wide "chromatic semitone" of four dieses.

If we look at these progressions in 29-tet, using the maximal sixth in
either case, these descriptions exactly and intuitively fit. Here
numbers in parentheses show vertical intervals in scale steps, and
signed numbers show melodic motion in scale steps or "dieses":

C#*4 -- +1 -- D4 C#*4 -- -4 -- C4
(23) (29) (23) (17)
E3 -- -5 -- D3 E3 -- +2 -- F3

Mx6 8 Mx6 5

Marchettus explains that the usual resolution to the octave is more
satisfying in part because one of the voices moves by the narrowest
interval possible, the cadential diesis equal to "one of the five
parts" of a tone -- here C#*-D in the upper voice. In contrast, the
smallest interval involved in the unusual resolution to the fifth is
the usual semitone of "two dieses," here E3-F3 in the lower voice.

At the least, this comparison of the cadential M6-8 and M6-5
progressions suggests that Marchettus indeed regards the cadential
diesis as substantially smaller than the usual semitone or limma --
with the 29-tet distinction between these melodic intervals at
2/5-tone and 1/5-tone nicely fitting the terms of his comparison.

In a slightly different realization, one could tune the cadential
major thirds and sixths (our "maximal" thirds and sixths) at precisely
13:10 and 26:15 in a JI system with a 3-odd-limit of stability but a
13-prime-limit or higher for generating unstable intervals. Many
neo-Gothic temperaments including 29-tet tend to approximate such a
state of affairs.

Finally, I should note that the 29-tet interval of 23 steps can be
used either as a Marchettus-like(?) "maximal sixth" or as a "minimal
seventh" regularly resolving to the fifth by a progression featuring a
cadential diesis, e.g.:

D4 -- -5 -- C4
(23) (17)
E*3 -- +1 -- F3

Although Marchettus focuses on thirds and sixths as unstable cadential
intervals, the resolution from minor seventh to fifth occurs in
various 13th-14th century styles. Like the progression from maximal
sixth to octave, this intonational variation using 23/29 octave as a
"minimal seventh" contracting to a fifth features motion by a
whole-tone (five dieses) in one voice, and by a single cadential
diesis in the other.

In short, 29-tet is a temperament rich with many neo-medieval
possibilities, ranging from the rather gently neo-Gothic to the
adventurous ultra-Gothic. This isn't to exclude all the other uses of
this tuning, only to offer a sketch of one very attractive side of its
musical flavor.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: More on 29-tet -- a beautiful tuning

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>