Yahoo Tuning Groups Ultimate Backup tuning Re: Wilsonian studies, optimized complexity, etc.

Hello, there, Dave, Kraig, and everyone.

Please let me begin by saying that in approaching any document,
whether from a 14th-century theorist in medieval Europe, or from Ervin
Wilson, or from a Balinese or Javanese musician regarding gamelan, my
first concern should be seek understanding with due prudence and
caution.

In this dialogue, we have seen one possibly fertile area for
misunderstandings: the "Noble Mediant" concept can refer either to
fractions of an octave used in scale construction, or to interval
ratios.

Certainly I agree both with you, Dave, and with Paul Erlich that this
distinction cannot be emphasized too early or often; and also with
you, Kraig, that these two different applications are distinct but not
necessarily uncongenial to each other.

Kraig, thank you again for responding to my article on Keenan Pepper's
"Noble Mediant" tuning, which included a diagram of the portion of the
spectrum of intervals between 5:4 and 9:7, by posting a link to Erv
Wilson's beautiful "scale tree" diagram showing the same region of
ratios between 5:4 and 9:7, including the Classic and Noble Mediants
and other intermediate ratios of my diagram plus many more.

Thanks to both of you, Kraig and Dave, for calling my attention to the
paper by Dr. Temes dating back to 1970 on the topic of Phi as an
interval ratio of "maximal dissonance" -- or, I would say, "maximal
complexity" (a musical distinction maybe inviting a lot more
dialogue).

[Dave, on ratios between "(L)arge" and "(s)mall" steps in scales]

> One should note that the good melodic properties of noble generators
> are very "broadly tuned". i.e. It may be quite acceptable for ratios
> between steps of a scale (in cents) L/s may be anywhere from about
> 1.3 to 2.5 (centered around phi) and still give recognisable
> scales. [...]

Yes, and I have also found L/s=3 (17-tone equal temperament or
17-tET), L/s=4 (22-tET in a Pythagorean arrangement), and even L/s=5
(27-tET for neo-Gothic with a metallophone-like timbre) to be fine.

In fact I consider 29-tET (L/s=2.5) to be on the very delectably
"gentle" side of the neo-Gothic spectrum, and something around 46-tET
(L/s=2.67) to be especially "characteristic."

Thus high L/s ratios, like contrasts between pure or near-pure 3-limit
concords and various types of complex unstable intervals at other
points on the concord/discord spectrum, may be a feature of neo-Gothic
music; and this melodic side of the question should receive due
attention in my presentations. At times I have discussed compact
diatonic semitones and large whole-tones, being much indebted to Mark
Lindley (and also Easley Blackwood's diatonic "R"), but maybe this
point should get more attention from a scale-generation perspective;
it's the defining feature of Keenan Pepper's tuning, of course.

[Dave on Noble Mediants between interval ratios]

> For example Noble_Mediant(4:5, 7:9) = (4+7phi):(5+9phi) gives a
> local maximum of dissonance but Noble_Mediant(4:5, 11:14) does
> not. Somewhat misleadingly, Noble_Mediant(7:9, 11:14) _does_ give a
> local dissonance maximum. But that is only because it is exactly the
> same number as Noble_Mediant(4:5, 7:9).

Other interesting cases producing identical Noble_Mediants are either
(4:5, 7:9) or (4:5, 3:4); and either (7:11, 5:8) or (2:3, 5:8).

> Margo, In your excellent articles on Keenan Pepper's neo-Gothic
> tuning you make use of noble mediants for both purposes (melodic and
> harmonic). I think the reader would benefit from a clear distinction
> between the two.

Yes, this may be a distinction difficult to overstate or overclarify.

> Also, part 2 may give the reader the impression that I would
> consider the neo-Gothic near 17-tET to be somehow ideal because it
> maximises the complexity/dissonance of the pythagorean major
> third. I think your excellent spectrum chart could benefit from also
> showing the points that correspond to maximally complex minor thirds
> and major and minor sixths. This would show that they cannot all be
> maximised together and would, I think, suggest a compromise
> somewhere around 29-tET.

Thank you for making a very fair and indeed very important point,
while offering feedback not only clarifying your viewpoint but
leading to an engaging question regarding a bit of an optimization
dilemma.

Interestingly, if I take your ideal to be one of maximizing
"complexity/dissonance," then the best solution may depend on how much
weight we choose to give the minor sixth, in contrast to the major and
minor thirds and major sixth.

A table of Noble Mediants and regular fifth sizes for the three latter
intervals, and then for the minor sixth, may make this problem
clearer:

---------------------------------------------------------------------
Interval Valley Ratios NobleMediant Fifth Tempering
---------------------------------------------------------------------
M3 (4:5, 7:9) or (4:5, 3:4) ~422.487 ~705.622 ~+3.667
m3 (5:6, 6:7) ~283.605 ~705.465 ~+3.510
M6 (3:5, 7:12) ~923.029 ~707.676 ~+5.721
.....................................................................
m6 (5:8, 7:11) or (2:3, 5:8) ~792.105 ~701.974 ~+0.019
---------------------------------------------------------------------

Here the regular temperaments giving Noble Mediants of maximal
complexity for major and minor thirds are quite close to 17-tET (fifth
~705.882 cents, ~3.927 cents wide) with fifths just a bit smaller,
while the major sixth would actually call for _more_ tempering in the
wide direction, 39-tET (fifths ~707.692 cents) being almost precisely
optimal under this criterion (major sixths at ~923.077 cents).

Thus for these three intervals, the mathematics would seem consistent
with my musical intuition that a tuning in the neighborhood between
around 29-tET and 46-tET, including Keenan Pepper's temperament, is
"milder" than 17-tET.

However, if we consider the minor sixth and especially if we give it
equal weight, this can throw our calculations askew: this interval is
at almost precisely maximal complexity in Pythagorean tuning with pure
3:2 fifths (~701.955 cents), where it has a size of 128:81 or ~792.180
cents.

From an historical point of view, one might ask: "Could this feature
of Pythagorean tuning have influenced the 13th-century classification
of the minor sixth as a 'perfect discord' along with m2 and M7 and the
tritonic intervals Aug4 or dim5?"

In any event, whether or goal is to maximize complexity or to seek
some other "optimal" level of complexity, for me maybe exemplified by
the Classic Mediants for major and minor thirds (5+9):(7+4) or 14:11
and (6+7):(5+6) or 13:11, the question of the minor sixth is an
interesting musical dilemma.

My own inclination not to give this interval very much weight in such
optimizations might have three possible musical rationales:

(1) In 13th-century music, where the minor sixth is indeed a very
important cadential interval expanding to the octave, it is also
viewed as a rather strong or even acute "discord," while the
relatively blending major and minor thirds and the relatively tense
major sixth are regarded as more "compatible." Maximizing the tension
of an interval already considered quite "discordant" may be a less
important priority.

(2) More generally, as I recall Paul Erlich has pointed out, a minor
sixth even at a simple ratio such as 5:8 is not especially
"concordant," so fine-tuning its degree of tension in a style where it
is an unstable interval may be less important. My crude explanation of
this acoustical quality might focus on the tension between the third
partial of the lower note and the second partial of the upper note,
which is there whether our tuning is 128:81 or 7:11, etc.

(3) In 14th-century style, which emphasizes "closest approach"
progressions by contrary motion where one voice moves by a whole-tone
and the other by a diatonic semitone (e.g. m3-1, M3-5, M6-8), the
minor sixth is less important as a cadential interval, since in this
scheme it would resolve to the fourth, a less conclusive interval than
unison, fifth, or octave.

However one decides to define the musical "ideal," and to weigh these
intervals, it is a notable observation that a tuning with the major
third at or near 14:11 will have minor sixths at or close to a local
_minimum_ of tension, 7:11. Note that I write 14:11 with terms in
descending order to show that it is a complex integer ratio, but 7:11
in ascending order to suggest that it may be simple and aurally
recognizable by locking-in of partials.

Maybe this kind of minor sixth might be described as a bit "languid"
in comparison with the Pythagorean flavor. In 13th-century style, in
addition to m6-8 by contrary motion, it has an oblique resolution to
the fifth by a descending semitone, around 80 cents in regular
neo-Gothic tunings with a minor sixth of ~7:11, which can be very
expressive.

To sum up on 17-tET as a nice tuning for "maximal complexity" -- apart
from the question of the minor sixth -- here's a quick table:

---------------------------------------------------------------------
Interval Valley Ratios NobleMediant 17-tET Variance
---------------------------------------------------------------------
M3 (4:5, 7:9) or (4:5, 3:4) ~422.487 ~423.529 ~+1.042
m3 (5:6, 6:7) ~283.605 ~282.353 ~-1.252
M6 (3:5, 7:12) ~923.029 ~917.647 ~-5.382
.....................................................................
m6 (5:8, 7:11) or (2:3, 5:8) ~792.105 ~776.471 ~-15.635
---------------------------------------------------------------------

Incidentally, if one seeks "maximal complexity" for all four
intervals, then 17-tET might have another argument in its favor. If we
consider 7:11 (~782.492 cents) as a rather "shallow" valley (a*b=77),
then staying a few cents to _either_ side of it may be enough. With
17-tET, we're about 6.021 cents on the low side, in contrast to
tunings with major thirds at or close to 14:11 where we're squarely in
the 7:11 valley.

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: Wilsonian studies, optimized complexity, etc.

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>