Listen to Merciful/Just cadences - a gem from Margo Schulter

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Dave Keenan
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Listen to Merciful/Just cadences - a gem from Margo Schulter

Post by Dave Keenan »

Thanks to Douglas Blumeyer's recent resurrection of the Yahoo tuning group archive, I found the following gem of a post, by @mschulter. The complete thread containing this post can be found here: https://yahootuninggroupsultimatebackup ... 73833.html

Margo Schulter <>
11/12/2007 10:29:13 PM

Noble Intonation Approximations in Zest-24
An mp3 sampler of metastable interval zones

Hello, everyone.

Recently David Keenan and others have taken part in discussions about
the "metastable" or "Nobly Intoned" (NI) intervals which the two of us
described in a paper of 2000:


Here I would like to share some mp3 files of cadential progressions
involving reasonably close approximations of many of these NI sizes as
found in the 24-note Zarlino Encompassing Spectrum Tuning, or Zest-24.
In short, an interval of around NI size may represent the zone of
maximum harmonic complexity between two simpler ratios. For example,
the 5:4 major third at 386 cents and 9:7 major third at 435 cents may
have an intervening region of "gravitational equipoise," or maximum
complexity, at around 422 cents.

The paper explains David Keenan's ingenious process for estimating
this NI value by using a Phi-weighted mediant of the two simpler
ratios. Here, however, the purpose is not so much to analyze this
process as to hear the audible results.

Zest-24 is called an "Encompassing Spectrum Tuning" because it surveys
many different regions and subregions of interval space. In my notes
for the mp3 examples which follow, I describe these regions in terms
which hopefully will be reasonably self-explanatory, but are more
fully developed in my newly completed paper exploring one possible
approach to the interval spectrum.

< ... egions.txt>

As mentioned at the beginning of this paper, I am much indebted to the
very thoughtful and musically sensitive writings of David Keenan on
interval names and regions and much else. Our counterpoint of ideas,
like a musical counterpoint, seems to proceed now by similar and now
by contrary motion.


The Zest-24 system uses two 12-note circles of a modified meantone
tuning based on Gioseffo Zarlino's 2/7-comma meantone. Each circle has
eight regular meantone fifths (F-C#), with the other four equally
wide; the two circles are at the distance of Zarlino's meantone
diesis, 50.276 cents. A Scala file is included at the end of this

The name for each mp3 file starts with "NIA" for "Noble Intonation
Approximation," and includes the sizes in rounded cents for the
relevant NI approximations used in a given example. Here I have
specified the sizes which theoretically should occur in my specific
synthesizer realization of Zest-24 in 1024-EDO, which may sometimes
vary by something on the order of a cent from the values appearing in
the Scala file below.



This example includes two simple and very characteristic neomedieval
cadences using a 422-cent major third and 926-cent major sixth, the
first virtually coinciding with the estimated NI zone at 422 cents or
so, and the second slightly wide of this zone at around 923 cents.
The sonority Ab-C-F (or G#-C-F in a typical meantone spelling) found
in either 12-note circle includes these intervals. The resolution has
the major third expand to a fifth and the major sixth to an octave.



This example features the small neutral third and sixth at 337 and 829
cents, not too far from NI values of 339 and 833 cents -- these
intervals, in either shading, may also be called supraminor thirds and
sixths. Here the third expands to the fifth, and the sixth to the
octave. A helpful spelling might be C-Eb*-Ab*, with the asterisk (*)
showing a note on the upper keyboard raised by a 50-cent diesis.



This example shows the use of the 337-cent small neutral third in a
progression where it contracts to a unison; the unstable sonority is



An interesting feature of the theory of metastable or Nobly Intoned
intervals is that sometimes these zones can involve quite familiar
sizes, here a 791-cent minor sixth almost identical with the
Pythagorean 128:81 or rounded NI value (both 792 cents), and a minor
tenth at an even 1500 cents which approximates the NI value of 1503
cents. These intervals (G-Eb-Bb in either circle) expand to the octave
and 12th.



This standard neomedieval cadence uses a minor third at 287 cents,
slightly larger than the NI value of around 284 cents, with the NI
major third at around 422 cents between the middle and upper voices
(0-422-708 cents), a tuning found at Eb-Gb-Bb in either circle.
Here minor third contracts to unison and major third expands to


< ... longer.mp3>

This example has the same final cadence as the last, but approached
from the diminished seventh sonority D-F-Ab-Cb (0-313-587-888 cents)
available in either circle (compare 15:18:21:25, 0-316-583-884 cents).
The diminished seventh contracts to the outer fifth of a sonority with
a 4-3 suspension, the fourth resolving to a _minor_ third that then
resolves to a unison, as in the previous example.



This cadence uses the sonority G-Eb-B with its two almost identical
intervals of 791 cents (G-Eb) and 792 cents (Eb-B) in a kind of
Manneristic meantone idiom treating Eb-B as an augmented fifth.
The very slight inequality of these two intervals is a quirk of the
synthesizer temperament.



This neomedieval cadence features what I term in my new paper an
_interseptimal_ interval of around 946 cents, located in the
fascinated region between the septimal ratios of 12:7 (933 cents) for
a large major sixth and 7:4 (969 cents) for a small minor seventh.
Here it is coupled with another dynamic interseptimal interval at
around 454 cents, virtually identical to a just 13:10, in the region
between the septimal 9:7 major third at 435 cents and the narrow 21:16
fourth at 471 cents. In this context, these intervals (C*-F-Bb) serve
as an "ultramajor" third and sixth expanding to fifth and octave with
the upper voices rising by 50-cent steps.

The 454-cent interval showcased in this example, by the way, is almost
identical to Gene Ward Smith's "ratwolf" equal to precisely to 13:10,
also his name for a regular meantone temperament differing minutely
from 2/7-comma which produces a "RAT(ional) WOLF" at this just ratio
from a chain of 11 fifths up. In Zest-24, the 2/7-comma "near-ratwolf"
is tempered out or "domesticated" within each 12-note circle, but
happily "reintroduced into the wild" by the 50-cent diesis between the
two circles.



This Manneristic cadence in a meantone style uses a 926-cent major
sixth (or here diminished seventh) in the sonority C#-E-G-Bb at around
0-313-626-926 cents, followed by a 4-3 suspension and resolution.



An interesting feature of metastable or NI intervals underscored in a
recent list contributed by David Keenan is that octave extensions can
make a difference. Here the metastable or NI major third at 422 cents
is coupled with a near-NI major 13th at 2113 cents, close to the
estimated NI zone at around 2109 cents. These intervals (Gb3-Bb3-Eb5
with C4 as middle C), available within either circle, resolve to fifth
and 15th (or double octave) respectively.



Another example of an NI interval larger than octave, this cadence has
a 2055-cent neutral 13th, here used together with a 2752-cent neutral
third plus two octaves or neutral 17th (Ab2-E*4-B*4). These intervals
expand to the 15th or double octave and 19th or fifth plus double
octave. This wide a spacing of voices is rather rare in a neomedieval
style; but this example and the last show how certain NI intervals
well beyond the octave may provide an incentive.



This concluding example shows how the 422-cent NI major third may be
used as an idiomatic meantone diminished fourth in a Manneristic
style. Here we start at D-G-Bb-D, with the fourth and minor sixth
above the bass suggesting a cadential approach. The fourth G then
proceeds to the major third F# -- forming the diminished fourth F#-Bb
at 422 cents, which leads to a 4-3 suspension and resolution.


As promised, here is the Zest-24 tuning:

! zest24.scl
Zarlino Encompassing Spectrum Temperament (two circles at ~50.28c apart)

Most appreciatively,

Margo Schulter

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Re: Listen to Merciful/Just cadences - a gem from Margo Schulter

Post by cmloegmcluin »

I was actually about to share this exact same link! For some reason I didn't get notified by email that you'd posted it, but I found it when searching for the thread about noble notation. Indeed it is quite the metallic gem.

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